eISSN: 2577-8242 FMRIJ

Fluid Mechanics Research International Journal
Research Article
Volume 2 Issue 3

Exact solution of plane Poiseuille flow in a channel filled with a porous medium

Amey Joshi
Director of Fidelity Investments, Bangalore, India
Received: May 03, 2018 | Published: June 14, 2018

Correspondence: Amey Joshi, Director of Fidelity Investments, Pinehurst, Embassy Golf Links business park, Bangalore, 56007, Tel +91-8691-2046, Email

Citation: Joshi A. Exact solution of plane Poiseuille flow in a channel filled with a porous medium. Fluid Mech Res Int. 2018;2(3):131‒136. DOI: 10.15406/fmrij.2018.02.00029

Abstract

In this paper, I have presented an analytical solution to the problem of plane Poiseuille flow in a channel filled with a porous medium. Nield et al.,1 have expressed it in terms of elliptic integrals. I express it in an equivalent, but a more conventional form, in terms of Weierstrass elliptic function. I also present the R code to plot the velocity profile and compute the flux.

Introduction

The flow of an incompressible, Newtonian fluid through a porous medium of porosity ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew7aLbaa@37ED@  and permeability K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUeaaaa@3716@  is described by the equation2
u t + u .( u ε )=( pε ρ )+ ν e Δ u + F , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaey OaIyRabmyDayaalaaabaGaeyOaIyRaamiDaaaacqGHRaWkceWG1bGb aSaacaaIUaGaey4bIe9aaeWaaeaadaWcaaqaaiqadwhagaWcaaqaai abew7aLbaaaiaawIcacaGLPaaacaaI9aGaeyOeI0Iaey4bIe9aaeWa aeaadaWcaaqaaiaadchacqaH1oqzaeaacqaHbpGCaaaacaGLOaGaay zkaaGaey4kaSIaeqyVd42aaSbaaSqaaiaadwgaaeqaaOGaeuiLdqKa bmyDayaalaGaey4kaSIabmOrayaalaGaaGilaaaa@556F@                                           (1)
where ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg8aYbaa@3806@ is the density of the fluid, u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadwhagaWcaa aa@3752@  is the velocity, p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchaaaa@373B@  the pressure, ν e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabe27aUnaaBa aaleaacaWGLbaabeaaaaa@3914@  the effective viscosity, ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabe27aUbaa@37FE@  the kinematic viscosity, Δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfs5aebaa@37AC@ denotes the Laplace operator and the total force is
F = F b εν K u ε F 0 K u u . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadAeagaWcai aai2daceWGgbGbaSaadaWgaaWcbaGaamOyaaqabaGccqGHsisldaWc aaqaaiabew7aLjabe27aUbqaaiaadUeaaaGabmyDayaalaGaeyOeI0 YaaSaaaeaacqaH1oqzcaWGgbWaaSbaaSqaaiaaicdaaeqaaaGcbaWa aOaaaeaacaWGlbaaleqaaaaakiaadwhaceWG1bGbaSaacaaIUaaaaa@482E@                                                                 (2)
It is a sum of the body force F b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadAeagaWcam aaBaaaleaacaWGIbaabeaaaaa@3836@  and the effect of the porosity of the medium. The geometric function F 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaa WcbaGaaGimaaqabaaaaa@37F7@ in equation (2) depends only on the porosity and the nature of the solid particle matrix of the porous medium. For a fully developed and steady flow, the left hand side of equation (1) vanishes. Further, if the flow is driven only by a constant pressure gradient and not an external body force, then equation (1) becomes
ν e Δ u εν K u ε F 0 K u u +ε G =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabe27aUnaaBa aaleaacaWGLbaabeaakiabfs5aejqadwhagaWcaiabgkHiTmaalaaa baGaeqyTduMaeqyVd4gabaGaam4saaaaceWG1bGbaSaacqGHsislda Wcaaqaaiabew7aLjaadAeadaWgaaWcbaGaaGimaaqabaaakeaadaGc aaqaaiaadUeaaSqabaaaaOGaamyDaiqadwhagaWcaiabgUcaRiabew 7aLjqadEeagaWcaiaai2dacaaIWaGaaGilaaaa@4EC0@                                                     (3)
where ρ G =p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg8aYjqadE eagaWcaiaai2dacqGHsislcqGHhis0caWGWbaaaa@3D13@ . I will solve equation (3) for a flow between parallel solid plates with a porous medium between them.

The physical setup

Consider a steady, fully-developed flow of a fluid along the x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhaaaa@3743@  axis between two infinite parallel plates, a distance H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeaaaa@3713@ apart. Let the plates concide with planes y=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhacaaI9a GaaGimaaaa@38C5@  and y=H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhacaaI9a Gaamisaaaa@38D8@ . If the channel is long enough that the end effects are negligible along most of the channel's length then we can express the fluid's velocity along the x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhaaaa@3743@  axis as u(y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacaaIOa GaamyEaiaaiMcaaaa@39A3@ . Equation (3), in this case, becomes
ν e ε u '' ν K u F 0 K u 2 +εG=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaeq yVd42aaSbaaSqaaiaadwgaaeqaaaGcbaGaeqyTdugaaiaadwhadaah aaWcbeqaaiaadEcacaWGNaaaaOGaeyOeI0YaaSaaaeaacqaH9oGBae aacaWGlbaaaiaadwhacqGHsisldaWcaaqaaiaadAeadaWgaaWcbaGa aGimaaqabaaakeaadaGcaaqaaiaadUeaaSqabaaaaOGaamyDamaaCa aaleqabaGaaGOmaaaakiabgUcaRiabew7aLjaadEeacaaI9aGaaGim aiaaiYcaaaa@4D03@                                                      (4)
where u '' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaahaa WcbeqaaiaadEcacaWGNaaaaaaa@38C5@  denotes the second derivative with respect to y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhaaaa@3744@ . Since the channel is bounded by solid walls, the usual no-slip boundary conditions apply. I will solve (4) subject to the conditions u(y=0)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacaaIOa GaamyEaiaai2dacaaIWaGaaGykaiaai2dacaaIWaaaaa@3CA5@  and u ' (H/2)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaahaa WcbeqaaiaadEcaaaGccaaIOaGaamisaiaai+cacaaIYaGaaGykaiaa i2dacaaIWaaaaa@3D4B@ , that is, the velocity has an extremum (actually, a maximum) at the center of the channel. Nield et al.,1 have given an analytical solution which expresses the coordinate y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhaaaa@3744@  as an elliptic integral in the velocity u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhaaaa@3740@ . I will develop an equivalent solution in which u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhaaaa@3740@  is expressed as a Weierstrass elliptic function of y. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhacaaMc8 UaaiOlaaaa@3981@

The solution

 We convert equation (4) in a non-dimensional form using the relations,
u = u U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWF gls5aeqaaOGaaGypamaalaaabaGaamyDaaqaaiaadwfaaaaaaa@45FD@                                                                                            (5)
y = y H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhadaWgaa WcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWF gls5aeqaaOGaaGypamaalaaabaGaamyEaaqaaiaadIeaaaaaaa@45F8@                                                                                           (6)
G = HG U 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEeadaWgaa WcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWF gls5aeqaaOGaaGypamaalaaabaGaamisaiaadEeaaeaacaWGvbWaaW baaSqabeaacaaIYaaaaaaakiaaiYcaaaa@4817@                                                                                       (7)
where U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwfaaaa@3720@  is the velocity at the center of the channel, and the dimensionless quantities
Re= HU ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkfacaWGLb GaaGypamaalaaabaGaamisaiaadwfaaeaacqaH9oGBaaaaaa@3C3D@                                                                                        (8)
Da= K H 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadseacaWGHb GaaGypamaalaaabaGaam4saaqaaiaadIeadaahaaWcbeqaaiaaikda aaaaaaaa@3B52@                                                                                        (9)
J= ν e ν . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadQeacaaI9a WaaSaaaeaacqaH9oGBdaWgaaWcbaGaamyzaaqabaaakeaacqaH9oGB aaGaaGOlaaaa@3D34@                                                                                           (10)
Re MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkfacaWGLb aaaa@3807@ , Da MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadseacaWGHb aaaa@37F5@  and J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadQeaaaa@3715@  are the Reynold's number, the Darcy number and the viscosity ratio of the flow. Dropping the MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae8NXIuoaaa@4222@  subscript for sake of notation clarity, we get,
u '' A 2 u 2 B 2 u+ C 2 =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaahaa WcbeqaaiaadEcacaWGNaaaaOGaeyOeI0IaamyqamaaCaaaleqabaGa aGOmaaaakiaadwhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGcb WaaWbaaSqabeaacaaIYaaaaOGaamyDaiabgUcaRiaadoeadaahaaWc beqaaiaaikdaaaGccaaI9aGaaGimaiaaiYcaaaa@45D7@                                                             (11)
where
A 2 = F 0 εRe J Da MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaahaa WcbeqaaiaaikdaaaGccaaI9aWaaSaaaeaacaWGgbWaaSbaaSqaaiaa icdaaeqaaOGaeqyTduMaamOuaiaadwgaaeaacaWGkbWaaOaaaeaaca WGebGaamyyaaWcbeaaaaaaaa@4092@                                                                                  (12)
B 2 = ε JDa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkeadaahaa WcbeqaaiaaikdaaaGccaaI9aWaaSaaaeaacqaH1oqzaeaacaWGkbGa amiraiaadggaaaaaaa@3CFC@                                                                                      (13)
C 2 = ε 2 ReG J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaahaa WcbeqaaiaaikdaaaGccaaI9aWaaSaaaeaacqaH1oqzdaahaaWcbeqa aiaaikdaaaGccaWGsbGaamyzaiaadEeaaeaacaWGkbaaaaaa@3ECE@                                                                                  (14)
are all positive numbers. Introduce a new variable u 1 (y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaaGymaaqabaGccaaIOaGaamyEaiaaiMcaaaa@3A94@ ,
u 1 (y)=Au(y)+ 1 2 B 2 A , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaaGymaaqabaGccaaIOaGaamyEaiaaiMcacaaI9aGaamyqaiaa dwhacaaIOaGaamyEaiaaiMcacqGHRaWkdaWcaaqaaiaaigdaaeaaca aIYaaaamaalaaabaGaamOqamaaCaaaleqabaGaaGOmaaaaaOqaaiaa dgeaaaGaaGilaaaa@452D@                                                                   (15)
so that equation (11) becomes,
u 1 '' A u 1 2 +( A C 2 + B 4 4A )=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaqhaa WcbaGaaGymaaqaaiaadEcacaWGNaaaaOGaeyOeI0Iaamyqaiaadwha daqhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHRaWkdaqadaqaaiaadg eacaWGdbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaSaaaeaacaWG cbWaaWbaaSqabeaacaaI0aaaaaGcbaGaaGinaiaadgeaaaaacaGLOa GaayzkaaGaaGypaiaaicdacaaISaaaaa@493A@                                                         (16)
Multiplying equation (16) by u 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaqhaa WcbaGaaGymaaqaaiaadEcaaaaaaa@38D4@ , the first derivative of u 1 (y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaaGymaaqabaGccaaIOaGaamyEaiaaiMcaaaa@3A94@ with respect to y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhaaaa@3744@ , and integrating we get,
( u 1 ' ) 2 2 3 A u 1 3 +2( A C 2 + B 4 4A ) u 1 + 36 g 3 A 2 =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam yDamaaDaaaleaacaaIXaaabaGaam4jaaaaaOGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaakiabgkHiTmaalaaabaGaaGOmaaqaaiaaio daaaGaamyqaiaadwhadaqhaaWcbaGaaGymaaqaaiaaiodaaaGccqGH RaWkcaaIYaWaaeWaaeaacaWGbbGaam4qamaaCaaaleqabaGaaGOmaa aakiabgUcaRmaalaaabaGaamOqamaaCaaaleqabaGaaGinaaaaaOqa aiaaisdacaWGbbaaaaGaayjkaiaawMcaaiaadwhadaWgaaWcbaGaaG ymaaqabaGccqGHRaWkdaWcaaqaaiaaiodacaaI2aGaam4zamaaBaaa leaacaaIZaaabeaaaOqaaiaadgeadaahaaWcbeqaaiaaikdaaaaaaO GaaGypaiaaicdacaaISaaaaa@5542@                               (17)
where 36 g 3 / A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiodacaaI2a Gaam4zamaaBaaaleaacaaIZaaabeaakiaai+cacaWGbbWaaWbaaSqa beaacaaIYaaaaaaa@3C0A@  is a constant of integration. Introduce yet another variable, u 2 (y)=(A/6) u 1 (y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaaGOmaaqabaGccaaIOaGaamyEaiaaiMcacaaI9aGaaGikaiaa dgeacaaIVaGaaGOnaiaaiMcacaWG1bWaaSbaaSqaaiaaigdaaeqaaO GaaGikaiaadMhacaaIPaaaaa@434E@ , to transform equation (17) to
( u 2 ' ) 2 =4 u 2 3 1 3 ( A 2 C 2 + B 4 4 ) u 2 g 3 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam yDamaaDaaaleaacaaIYaaabaGaam4jaaaaaOGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaakiaai2dacaaI0aGaamyDamaaDaaaleaaca aIYaaabaGaaG4maaaakiabgkHiTmaalaaabaGaaGymaaqaaiaaioda aaWaaeWaaeaacaWGbbWaaWbaaSqabeaacaaIYaaaaOGaam4qamaaCa aaleqabaGaaGOmaaaakiabgUcaRmaalaaabaGaamOqamaaCaaaleqa baGaaGinaaaaaOqaaiaaisdaaaaacaGLOaGaayzkaaGaamyDamaaBa aaleaacaaIYaaabeaakiabgkHiTiaadEgadaWgaaWcbaGaaG4maaqa baGccaaIUaaaaa@4FD8@                                           (18)
This equation is of the form,
( u 2 ' ) 2 =4 u 2 3 g 2 u 2 g 3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiIcacaWG1b Waa0baaSqaaiaaikdaaeaacaWGNaaaaOGaaGykamaaCaaaleqabaGa aGOmaaaakiaai2dacaaI0aGaamyDamaaDaaaleaacaaIYaaabaGaaG 4maaaakiabgkHiTiaadEgadaWgaaWcbaGaaGOmaaqabaGccaWG1bWa aSbaaSqaaiaaikdaaeqaaOGaeyOeI0Iaam4zamaaBaaaleaacaaIZa aabeaakiaaiYcaaaa@479F@                                                                (19)
where
g 2 = 1 3 ( A 2 C 2 + B 4 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaWgaa WcbaGaaGOmaaqabaGccaaI9aWaaSaaaeaacaaIXaaabaGaaG4maaaa daqadaqaaiaadgeadaahaaWcbeqaaiaaikdaaaGccaWGdbWaaWbaaS qabeaacaaIYaaaaOGaey4kaSYaaSaaaeaacaWGcbWaaWbaaSqabeaa caaI0aaaaaGcbaGaaGinaaaaaiaawIcacaGLPaaaaaa@42DC@                                                                     (20)
Its solution is u 2 (y)=(y+ k 1 ; g 2 , g 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaaGOmaaqabaGccaaIOaGaamyEaiaaiMcacaaI9aGaeyigHmSa aGikaiaadMhacqGHRaWkcaWGRbWaaSbaaSqaaiaaigdaaeqaaOGaaG 4oaiaadEgadaWgaaWcbaGaaGOmaaqabaGccaaISaGaam4zamaaBaaa leaacaaIZaaabeaakiaaiMcaaaa@4742@ , where k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaaGymaaqabaaaaa@381D@  is another constant of integration. The function MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIriddaa@37CE@  is the Weierstrass elliptic function with invariants g 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaWgaa WcbaGaaGOmaaqabaaaaa@381A@  and g 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaWgaa WcbaGaaG4maaqabaaaaa@381B@ .3 Therefore, the solution of (11) is
u(y)= 6 A 2 [ ( y+ k 1 | 1 3 ( A 2 C 2 + B 4 4 ), g 3 ) B 2 12 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacaaIOa GaamyEaiaaiMcacaaI9aWaaSaaaeaacaaI2aaabaGaamyqamaaCaaa leqabaGaaGOmaaaaaaGcdaWadaqaaiabgIridpaabmaabaGaamyEai abgUcaRiaadUgadaWgaaWcbaGaaGymaaqabaGccaaI8bWaaSaaaeaa caaIXaaabaGaaG4maaaadaqadaqaaiaadgeadaahaaWcbeqaaiaaik daaaGccaWGdbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaSaaaeaa caWGcbWaaWbaaSqabeaacaaI0aaaaaGcbaGaaGinaaaaaiaawIcaca GLPaaacaaISaGaam4zamaaBaaaleaacaaIZaaabeaaaOGaayjkaiaa wMcaaiabgkHiTmaalaaabaGaamOqamaaCaaaleqabaGaaGOmaaaaaO qaaiaaigdacaaIYaaaaaGaay5waiaaw2faaaaa@5771@                        (21)
The constants g 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaWgaa WcbaGaaG4maaqabaaaaa@381B@ and k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaaGymaaqabaaaaa@381D@  are found using the boundary conditions.

Some properties of Weierstrass elliptic function

The Weierstrass elliptic function is defined as the solution of the differential equation,
[ ' (u) ] 2 =4 3 (u) g 2 (u) g 3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaey igHm8aaWbaaSqabeaacaWGNaaaaOGaaGikaiaadwhacaaIPaaacaGL BbGaayzxaaWaaWbaaSqabeaacaaIYaaaaOGaaGypaiaaisdacqGHye YWdaahaaWcbeqaaiaaiodaaaGccaaIOaGaamyDaiaaiMcacqGHsisl caWGNbWaaSbaaSqaaiaaikdaaeqaaOGaeyigHmSaaGikaiaadwhaca aIPaGaeyOeI0Iaam4zamaaBaaaleaacaaIZaaabeaakiaaiYcaaaa@4E89@  (22)
where u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhaaaa@3740@  is a complex variable and g 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaWgaa WcbaGaaGOmaaqabaaaaa@381A@  and g 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaWgaa WcbaGaaG4maaqabaaaaa@381B@ are complex constants, called invariants of MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIriddaa@37CE@ . Weierstrass elliptic function is a doubly periodic function. We follow Lawden4 and call the periods 2 ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaikdacqaHjp WDdaWgaaWcbaGaaGymaaqabaaaaa@39B6@  and 2 ω 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaikdacqaHjp WDdaWgaaWcbaGaaG4maaqabaaaaa@39B8@ . There is another half period 2 ω 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaikdacqaHjp WDdaWgaaWcbaGaaGOmaaqabaaaaa@39B7@ , where ω 2 = ω 1 ω 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIYaaabeaakiaai2dacqGHsislcqaHjpWDdaWgaaWcbaGa aGymaaqabaGccqGHsislcqaHjpWDdaWgaaWcbaGaaG4maaqabaaaaa@411A@ . Whittaker et al.,3 denote the fundamental periods by ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIXaaabeaaaaa@38FA@  and ω 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIYaaabeaaaaa@38FB@  and define ω 3 = ω 1 ω 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIZaaabeaakiaai2dacqGHsislcqaHjpWDdaWgaaWcbaGa aGymaaqabaGccqGHsislcqaHjpWDdaWgaaWcbaGaaGOmaaqabaaaaa@411A@ .) If the constants e j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaamOAaaqabaaaaa@384B@ are defined as
e j =( ω j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaamOAaaqabaGccaaI9aGaeyigHmSaaGikaiabeM8a3naaBaaa leaacaWGQbaabeaakiaaiMcaaaa@3EFB@                                                                                     (23)
then it can be shown that [by Lawden4 p.160] they are solutions of the cubic equation 4 s 3 g 2 s g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaWGZb WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam4zamaaBaaaleaacaaI YaaabeaakiaadohacqGHsislcaWGNbWaaSbaaSqaaiaaiodaaeqaaO GaaGypaiaaicdaaaa@4100@ . Further, they are related to the invariants as
g 2 =4( e 1 e 2 + e 2 e 3 + e 3 e 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaWgaa WcbaGaaGOmaaqabaGccaaI9aGaeyOeI0IaaGinaiaaiIcacaWGLbWa aSbaaSqaaiaaigdaaeqaaOGaamyzamaaBaaaleaacaaIYaaabeaaki abgUcaRiaadwgadaWgaaWcbaGaaGOmaaqabaGccaWGLbWaaSbaaSqa aiaaiodaaeqaaOGaey4kaSIaamyzamaaBaaaleaacaaIZaaabeaaki aadwgadaWgaaWcbaGaaGymaaqabaGccaaIPaaaaa@48E7@                                                             (24)
g 3 =4 e 1 e 2 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaWgaa WcbaGaaG4maaqabaGccaaI9aGaaGinaiaadwgadaWgaaWcbaGaaGym aaqabaGccaWGLbWaaSbaaSqaaiaaikdaaeqaaOGaamyzamaaBaaale aacaaIZaaabeaaaaa@3F34@                                                                                   (25)
Additionally [Lawden4 p.159],
e 1 + e 2 + e 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGymaaqabaGccqGHRaWkcaWGLbWaaSbaaSqaaiaaikdaaeqa aOGaey4kaSIaamyzamaaBaaaleaacaaIZaaabeaakiaai2dacaaIWa aaaa@3F1F@                                                                               (26)

The discriminant of the cubic equation 4 t 3 g 2 t g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaWG0b WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam4zamaaBaaaleaacaaI YaaabeaakiaadshacqGHsislcaWGNbWaaSbaaSqaaiaaiodaaeqaaO GaaGypaiaaicdaaaa@4102@  is D=16( g 2 3 27 g 3 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadseacaaI9a GaaGymaiaaiAdacaaIOaGaam4zamaaDaaaleaacaaIYaaabaGaaG4m aaaakiabgkHiTiaaikdacaaI3aGaam4zamaaDaaaleaacaaIZaaaba GaaGOmaaaakiaaiMcaaaa@4258@ .
If

  1. D>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadseacaaI+a GaaGimaaaa@3891@ , then all roots of 4 t 3 g 2 t g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaWG0b WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam4zamaaBaaaleaacaaI YaaabeaakiaadshacqGHsislcaWGNbWaaSbaaSqaaiaaiodaaeqaaO GaaGypaiaaicdaaaa@4102@  are real and distinct.
  2. D=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadseacaaI9a GaaGimaaaa@3890@ , then the roots of 4 t 3 g 2 t g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaWG0b WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam4zamaaBaaaleaacaaI YaaabeaakiaadshacqGHsislcaWGNbWaaSbaaSqaaiaaiodaaeqaaO GaaGypaiaaicdaaaa@4102@  are real and not distinct.
  3. D<0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadseacaaI8a GaaGimaaaa@388F@ , then 4 t 3 g 2 t g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaWG0b WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam4zamaaBaaaleaacaaI YaaabeaakiaadshacqGHsislcaWGNbWaaSbaaSqaaiaaiodaaeqaaO GaaGypaiaaicdaaaa@4102@  has one real root and two complex roots. The complex roots are conjugates of each other.

Theorem 1: If the invariants of Weierstrass elliptic function MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIriddaa@37CE@  are real and if the roots of the cubic 4 s 3 g 2 s g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaWGZb WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam4zamaaBaaaleaacaaI YaaabeaakiaadohacqGHsislcaWGNbWaaSbaaSqaaiaaiodaaeqaaO GaaGypaiaaicdaaaa@4100@  are real then ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIXaaabeaaaaa@38FA@  is real and ω 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIZaaabeaaaaa@38FC@  is imaginary.

Proof: The constants e 1 , e 2 , e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGymaaqabaGccaaISaGaamyzamaaBaaaleaacaaIYaaabeaa kiaaiYcacaWGLbWaaSbaaSqaaiaaiodaaeqaaaaa@3D3C@  are solutions of the cubic 4 s 3 g 2 s g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaWGZb WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam4zamaaBaaaleaacaaI YaaabeaakiaadohacqGHsislcaWGNbWaaSbaaSqaaiaaiodaaeqaaO GaaGypaiaaicdaaaa@4100@ . If they are all real, we can choose them to be such that e 1 > e 2 > e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGymaaqabaGccaaI+aGaamyzamaaBaaaleaacaaIYaaabeaa kiaai6dacaWGLbWaaSbaaSqaaiaaiodaaeqaaaaa@3D60@ . Then (22) can be written as
{ ((u) e 1 )((u) e 2 )((u) e 3 ) } 1/2 = 1 2 ' (u) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaG ikaiabgIridlaaiIcacaWG1bGaaGykaiabgkHiTiaadwgadaWgaaWc baGaaGymaaqabaGccaaIPaGaaGikaiabgIridlaaiIcacaWG1bGaaG ykaiabgkHiTiaadwgadaWgaaWcbaGaaGOmaaqabaGccaaIPaGaaGik aiabgIridlaaiIcacaWG1bGaaGykaiabgkHiTiaadwgadaWgaaWcba GaaG4maaqabaGccaaIPaaacaGL7bGaayzFaaWaaWbaaSqabeaacaaI XaGaaG4laiaaikdaaaGccaaI9aGaeyOeI0YaaSaaaeaacaaIXaaaba GaaGOmaaaacqGHyeYWdaahaaWcbeqaaiaadEcaaaGccaaIOaGaamyD aiaaiMcaaaa@5B22@
that is,
d(u) 4((u) e 1 )((u) e 2 )((u) e 3 ) =du MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izaiabgIridlaaiIcacaWG1bGaaGykaaqaamaakaaabaGaaGinaiaa iIcacqGHyeYWcaaIOaGaamyDaiaaiMcacqGHsislcaWGLbWaaSbaaS qaaiaaigdaaeqaaOGaaGykaiaaiIcacqGHyeYWcaaIOaGaamyDaiaa iMcacqGHsislcaWGLbWaaSbaaSqaaiaaikdaaeqaaOGaaGykaiaaiI cacqGHyeYWcaaIOaGaamyDaiaaiMcacqGHsislcaWGLbWaaSbaaSqa aiaaiodaaeqaaOGaaGykaaWcbeaaaaGccaaI9aGaeyOeI0Iaamizai aadwhaaaa@57DF@
equivalently,
dt 4(t e 1 )(t e 2 )(t e 3 ) =du, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izaiaadshaaeaadaGcaaqaaiaaisdacaaIOaGaamiDaiabgkHiTiaa dwgadaWgaaWcbaGaaGymaaqabaGccaaIPaGaaGikaiaadshacqGHsi slcaWGLbWaaSbaaSqaaiaaikdaaeqaaOGaaGykaiaaiIcacaWG0bGa eyOeI0IaamyzamaaBaaaleaacaaIZaaabeaakiaaiMcaaSqabaaaaO GaaGypaiabgkHiTiaadsgacaWG1bGaaGilaaaa@4CDD@
where t=(u) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshacaaI9a GaeyigHmSaaGikaiaadwhacaaIPaaaaa@3BED@ . Integrating this equation so that u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhaaaa@3740@  goes from 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaicdaaaa@3700@ to U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwfaaaa@3720@ , and hence t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshaaaa@373F@  goes from (0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIridlaaiI cacaaIWaGaaGykaaaa@39ED@  to (U) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIridlaaiI cacaWGvbGaaGykaaaa@3A0D@ , we have two possibilities because of the double pole of MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIriddaa@37CE@  at the origin,
(U) dt 4(t e 1 )(t e 2 )(t e 3 ) =U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaapedabeWcba GaeyOhIukabaGaeyigHmSaaGikaiaadwfacaaIPaaaniabgUIiYdGc daWcaaqaaiaadsgacaWG0baabaWaaOaaaeaacaaI0aGaaGikaiaads hacqGHsislcaWGLbWaaSbaaSqaaiaaigdaaeqaaOGaaGykaiaaiIca caWG0bGaeyOeI0IaamyzamaaBaaaleaacaaIYaaabeaakiaaiMcaca aIOaGaamiDaiabgkHiTiaadwgadaWgaaWcbaGaaG4maaqabaGccaaI Paaaleqaaaaakiaai2dacqGHsislcaWGvbaaaa@528A@                                              (27)
or
(U) dt 4(t e 1 )(t e 2 )(t e 3 ) =U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaapedabeWcba GaeyOeI0IaeyOhIukabaGaeyigHmSaaGikaiaadwfacaaIPaaaniab gUIiYdGcdaWcaaqaaiaadsgacaWG0baabaWaaOaaaeaacaaI0aGaaG ikaiaadshacqGHsislcaWGLbWaaSbaaSqaaiaaigdaaeqaaOGaaGyk aiaaiIcacaWG0bGaeyOeI0IaamyzamaaBaaaleaacaaIYaaabeaaki aaiMcacaaIOaGaamiDaiabgkHiTiaadwgadaWgaaWcbaGaaG4maaqa baGccaaIPaaaleqaaaaakiaai2dacqGHsislcaWGvbaaaa@5377@                                              (28)

Choose U= ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwfacaaI9a GaeqyYdC3aaSbaaSqaaiaaigdaaeqaaaaa@3A9B@  in equation (27), so that
ω 1 = ( ω 1 ) dt 4(t e 1 )(t e 2 )(t e 3 ) = e 1 dt 4(t e 1 )(t e 2 )(t e 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIXaaabeaakiaai2dacqGHsisldaWdXaqabSqaaiabg6Hi LcqaaiabgIridlaaiIcacqaHjpWDdaWgaaqaaiaaigdaaeqaaiaaiM caa0Gaey4kIipakmaalaaabaGaamizaiaadshaaeaadaGcaaqaaiaa isdacaaIOaGaamiDaiabgkHiTiaadwgadaWgaaWcbaGaaGymaaqaba GccaaIPaGaaGikaiaadshacqGHsislcaWGLbWaaSbaaSqaaiaaikda aeqaaOGaaGykaiaaiIcacaWG0bGaeyOeI0IaamyzamaaBaaaleaaca aIZaaabeaakiaaiMcaaSqabaaaaOGaaGypamaapedabeWcbaGaamyz amaaBaaabaGaaGymaaqabaaabaGaeyOhIukaniabgUIiYdGcdaWcaa qaaiaadsgacaWG0baabaWaaOaaaeaacaaI0aGaaGikaiaadshacqGH sislcaWGLbWaaSbaaSqaaiaaigdaaeqaaOGaaGykaiaaiIcacaWG0b GaeyOeI0IaamyzamaaBaaaleaacaaIYaaabeaakiaaiMcacaaIOaGa amiDaiabgkHiTiaadwgadaWgaaWcbaGaaG4maaqabaGccaaIPaaale qaaaaaaaa@6EAF@
Since e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGymaaqabaaaaa@3817@  is the largest root of 4(t e 1 )(t e 2 )(t e 3 )=4 t 3 g 2 t g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaaIOa GaamiDaiabgkHiTiaadwgadaWgaaWcbaGaaGymaaqabaGccaaIPaGa aGikaiaadshacqGHsislcaWGLbWaaSbaaSqaaiaaikdaaeqaaOGaaG ykaiaaiIcacaWG0bGaeyOeI0IaamyzamaaBaaaleaacaaIZaaabeaa kiaaiMcacaaI9aGaaGinaiaadshadaahaaWcbeqaaiaaiodaaaGccq GHsislcaWGNbWaaSbaaSqaaiaaikdaaeqaaOGaamiDaiabgkHiTiaa dEgadaWgaaWcbaGaaG4maaqabaGccaaI9aGaaGimaaaa@51FC@ , throughout the path of integration, the integrand stays finite. Further, the integrand is always real and hence ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIXaaabeaaaaa@38FA@  is real.

Choose U= ω 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwfacaaI9a GaeyOeI0IaeqyYdC3aaSbaaSqaaiaaiodaaeqaaaaa@3B8A@  in equation (28), so that
ω 3 = ( ω 3 ) dt 4(t e 1 )(t e 2 )(t e 3 ) = e 3 dt 4(t e 1 )(t e 2 )(t e 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIZaaabeaakiaai2dacqGHsisldaWdXaqabSqaaiabgkHi Tiabg6HiLcqaaiabgIridlaaiIcacqaHjpWDdaWgaaqaaiaaiodaae qaaiaaiMcaa0Gaey4kIipakmaalaaabaGaamizaiaadshaaeaadaGc aaqaaiaaisdacaaIOaGaamiDaiabgkHiTiaadwgadaWgaaWcbaGaaG ymaaqabaGccaaIPaGaaGikaiaadshacqGHsislcaWGLbWaaSbaaSqa aiaaikdaaeqaaOGaaGykaiaaiIcacaWG0bGaeyOeI0IaamyzamaaBa aaleaacaaIZaaabeaakiaaiMcaaSqabaaaaOGaaGypaiabgkHiTmaa pedabeWcbaGaeyOeI0IaeyOhIukabaGaamyzamaaBaaabaGaaG4maa qabaaaniabgUIiYdGcdaWcaaqaaiaadsgacaWG0baabaWaaOaaaeaa caaI0aGaaGikaiaadshacqGHsislcaWGLbWaaSbaaSqaaiaaigdaae qaaOGaaGykaiaaiIcacaWG0bGaeyOeI0IaamyzamaaBaaaleaacaaI YaaabeaakiaaiMcacaaIOaGaamiDaiabgkHiTiaadwgadaWgaaWcba GaaG4maaqabaGccaaIPaaaleqaaaaaaaa@717C@
or,
ω 3 =i e 3 dt 4( e 1 t)( e 2 t)( e 3 t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIZaaabeaakiaai2dacaWGPbWaa8qmaeqaleaacqGHsisl cqGHEisPaeaacaWGLbWaaSbaaeaacaaIZaaabeaaa0Gaey4kIipakm aalaaabaGaamizaiaadshaaeaadaGcaaqaaiaaisdacaaIOaGaamyz amaaBaaaleaacaaIXaaabeaakiabgkHiTiaadshacaaIPaGaaGikai aadwgadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWG0bGaaGykaiaa iIcacaWGLbWaaSbaaSqaaiaaiodaaeqaaOGaeyOeI0IaamiDaiaaiM caaSqabaaaaaaa@5355@
Since e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaG4maaqabaaaaa@3819@  is the smallest root of 4(t e 1 )(t e 2 )(t e 3 )=4 t 3 g 2 t g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaaIOa GaamiDaiabgkHiTiaadwgadaWgaaWcbaGaaGymaaqabaGccaaIPaGa aGikaiaadshacqGHsislcaWGLbWaaSbaaSqaaiaaikdaaeqaaOGaaG ykaiaaiIcacaWG0bGaeyOeI0IaamyzamaaBaaaleaacaaIZaaabeaa kiaaiMcacaaI9aGaaGinaiaadshadaahaaWcbeqaaiaaiodaaaGccq GHsislcaWGNbWaaSbaaSqaaiaaikdaaeqaaOGaamiDaiabgkHiTiaa dEgadaWgaaWcbaGaaG4maaqabaGccaaI9aGaaGimaaaa@51FC@ , throughout the path of integration, the integrand stays finite. Further, the integrand is always real and hence ω 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIZaaabeaaaaa@38FC@  is imaginary.

Remark: The converse of theorem 1 is also true [Lawden4 p.163]. That is, if ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIXaaabeaaaaa@38FA@  is real and ω 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIZaaabeaaaaa@38FC@  is imaginary then e j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaamOAaaqabaaaaa@384B@ , the roots of 4 s 3 g 2 s g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaWGZb WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam4zamaaBaaaleaacaaI YaaabeaakiaadohacqGHsislcaWGNbWaaSbaaSqaaiaaiodaaeqaaO GaaGypaiaaicdaaaa@4100@  are all real.

If ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIXaaabeaaaaa@38FA@  is real and ω 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIZaaabeaaaaa@38FC@  is imaginary then section 6.11 shows that4
Theorem 2: MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIriddaa@37CE@  takes real values on the the rectangle OXAY MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad+eacaWGyb GaamyqaiaadMfaaaa@399B@  in Figure 1, where O(0,0),X( ω 1 ,0),A( ω 1 , ω 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad+eacqGHHj IUcaaIOaGaaGimaiaaiYcacaaIWaGaaGykaiaaiYcacaWGybGaeyyy IORaaGikaiabeM8a3naaBaaaleaacaaIXaaabeaakiaaiYcacaaIWa GaaGykaiaaiYcacaWGbbGaeyyyIORaaGikaiabeM8a3naaBaaaleaa caaIXaaabeaakiaaiYcacqaHjpWDdaWgaaWcbaGaaG4maaqabaGcca aIPaaaaa@503F@  and Y(0, ω 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMfacqGHHj IUcaaIOaGaaGimaiaaiYcacqaHjpWDdaWgaaWcbaGaaG4maaqabaGc caaIPaaaaa@3E82@ . Further, if u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhaaaa@3740@  is taken round OXAY MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad+eacaWGyb GaamyqaiaadMfaaaa@399B@ , (u) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIridlaaiI cacaWG1bGaaGykaaaa@3A2D@  decreases from + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgUcaRiabg6 HiLcaa@3899@  at O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad+eaaaa@371A@  to e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGymaaqabaaaaa@3817@  at X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfaaaa@3723@ , further decreases to e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGOmaaqabaaaaa@3818@  along XA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaWGbb aaaa@37E9@ , to e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaG4maaqabaaaaa@3819@  along AY MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeacaWGzb aaaa@37EA@  and finally to MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabg6 HiLcaa@38A4@  as it reaches the origin along YA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMfacaWGbb aaaa@37EA@ .

Figure 1 Positive discrimimant.

If the invariants g 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaWgaa WcbaGaaGOmaaqabaaaaa@381A@ and g 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaWgaa WcbaGaaG4maaqabaaaaa@381B@  of the Weierstrass elliptic function are real but the discriminant of the cubic equation 4 t 3 g 2 t g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaWG0b WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam4zamaaBaaaleaacaaI YaaabeaakiaadshacqGHsislcaWGNbWaaSbaaSqaaiaaiodaaeqaaO GaaGypaiaaicdaaaa@4102@  is negative, then referring to Figure 2 or Handbook of Mathematical Functions with Formulas5, the function MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIriddaa@37CE@  takes value + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgUcaRiabg6 HiLcaa@3899@  at O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad+eaaaa@371A@ . Its value decreases monotonically upto ω 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabeM 8a3naaBaaaleaacaaIYaaabeaaaaa@39E8@ , where it is e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGOmaaqabaaaaa@3818@  and then rises once again to + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgUcaRiabg6 HiLcaa@3899@  at 2 ω 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiaaik dacqaHjpWDdaWgaaWcbaGaaGOmaaqabaaaaa@3AA4@ . If the function has a singularity at the origin, it will have it at points 2 ω 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiaaik dacqaHjpWDdaWgaaWcbaGaaGOmaaqabaaaaa@3AA4@ , 2 ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaikdacqaHjp WDdaWgaaWcbaGaaGymaaqabaaaaa@39B6@  and 2 ω 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaikdacqaHjp WDdaWgaaWcbaGaaG4maaqabaaaaa@39B8@ because of its periodicity. The function also takes real values along the segment BD MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkeacaWGeb aaaa@37D6@ .

Figure 2 Negative discrimimant.

Finding g 3 and k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaWgaa WcbaGaaG4maaqabaGccaaMc8UaaCyyaiaah6gacaWHKbGaaGPaVlaa dUgadaWgaaWcbaGaaGymaaqabaaaaa@3FE0@

Since the velocity has an extremum at the center of the channel, its derivative with respect to y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhaaaa@3744@  vanishes there. Therefore, derivatives of u 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaaGymaaqabaaaaa@3827@ and u 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaaGOmaaqabaaaaa@3828@  also vanish at the center of the channel. If the magnitude of the non-dimensional velocity u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhaaaa@3740@  at the center is u c =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaahaa WcbeqaaiaadogaaaGccaaI9aGaaGymaaaa@39E1@  then we immediately have u 1 c =A+ B 2 /(2A) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaqhaa WcbaGaaGymaaqaaiaadogaaaGccaaI9aGaamyqaiabgUcaRiaadkea daahaaWcbeqaaiaaikdaaaGccaaIVaGaaGikaiaaikdacaWGbbGaaG ykaaaa@40E3@  and
u 2 c = A 2 6 + B 2 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaqhaa WcbaGaaGOmaaqaaiaadogaaaGccaaI9aWaaSaaaeaacaWGbbWaaWba aSqabeaacaaIYaaaaaGcbaGaaGOnaaaacqGHRaWkdaWcaaqaaiaadk eadaahaaWcbeqaaiaaikdaaaaakeaacaaIXaGaaGOmaaaaaaa@408E@                                                                                (29)
Since at the center,
4[ u 2 c ] 3 g 2 u 2 c g 3 =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaaIBb GaamyDamaaDaaaleaacaaIYaaabaGaam4yaaaakiaai2fadaahaaWc beqaaiaaiodaaaGccqGHsislcaWGNbWaaSbaaSqaaiaaikdaaeqaaO GaamyDamaaDaaaleaacaaIYaaabaGaam4yaaaakiabgkHiTiaadEga daWgaaWcbaGaaG4maaqabaGccaaI9aGaaGimaiaaiYcaaaa@473C@
u 2 c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaqhaa WcbaGaaGOmaaqaaiaadogaaaaaaa@3911@  is one of the roots of the cubic equation 4 s 3 g 2 s g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaWGZb WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam4zamaaBaaaleaacaaI YaaabeaakiaadohacqGHsislcaWGNbWaaSbaaSqaaiaaiodaaeqaaO GaaGypaiaaicdaaaa@4100@ . If e 1 , e 2 , e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGymaaqabaGccaaISaGaamyzamaaBaaaleaacaaIYaaabeaa kiaaiYcacaWGLbWaaSbaaSqaaiaaiodaaeqaaaaa@3D3C@  are the roots, without loss of generality, let
u 2 c = e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaqhaa WcbaGaaGOmaaqaaiaadogaaaGccaaI9aGaamyzamaaBaaaleaacaaI Yaaabeaaaaa@3BB4@                                                                                           (30)
From these equations (24) to (26) we readily get,
e 1 = e 2 + g 2 3 e 2 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGymaaqabaGccaaI9aWaaSaaaeaacqGHsislcaWGLbWaaSba aSqaaiaaikdaaeqaaOGaey4kaSYaaOaaaeaacaWGNbWaaSbaaSqaai aaikdaaeqaaOGaeyOeI0IaaG4maiaadwgadaqhaaWcbaGaaGOmaaqa aiaaikdaaaaabeaaaOqaaiaaikdaaaaaaa@4390@                                                                     (31)
e 3 = e 2 g 2 3 e 2 2 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaG4maaqabaGccaaI9aWaaSaaaeaacqGHsislcaWGLbWaaSba aSqaaiaaikdaaeqaaOGaeyOeI0YaaOaaaeaacaWGNbWaaSbaaSqaai aaikdaaeqaaOGaeyOeI0IaaG4maiaadwgadaqhaaWcbaGaaGOmaaqa aiaaikdaaaaabeaaaOqaaiaaikdaaaGaaGilaaaa@4453@                                                                   (32)
Knowing e 1 , e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGymaaqabaGccaaISaGaamyzamaaBaaaleaacaaIYaaabeaa aaa@3AA9@ and e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaG4maaqabaaaaa@3819@ equation (25) gives,
g 3 =4 e 1 e 2 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaWgaa WcbaGaaG4maaqabaGccaaI9aGaaGinaiaadwgadaWgaaWcbaGaaGym aaqabaGccaWGLbWaaSbaaSqaaiaaikdaaeqaaOGaamyzamaaBaaale aacaaIZaaabeaaaaa@3F34@                                                                                   (33)
We will now use the second boundary condition u(0)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacaaIOa GaaGimaiaaiMcacaaI9aGaaGimaaaa@3AE0@  to find k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaaGymaaqabaaaaa@381D@ . The equation u(0)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacaaIOa GaaGimaiaaiMcacaaI9aGaaGimaaaa@3AE0@  is equivalent to, (0+ k 1 ; g 2 , g 3 )= B 2 /12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIridlaaiI cacaaIWaGaey4kaSIaam4AamaaBaaaleaacaaIXaaabeaakiaaiUda caWGNbWaaSbaaSqaaiaaikdaaeqaaOGaaGilaiaadEgadaWgaaWcba GaaG4maaqabaGccaaIPaGaaGypaiaadkeadaahaaWcbeqaaiaaikda aaGccaaIVaGaaGymaiaaikdaaaa@4699@  or
k 1 = 1 ( B 2 12 | g 2 , g 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaaGymaaqabaGccaaI9aGaeyigHm8aaWbaaSqabeaacqGHsisl caaIXaaaaOWaaeWaaeaadaWcaaqaaiaadkeadaahaaWcbeqaaiaaik daaaaakeaacaaIXaGaaGOmaaaacaaI8bGaam4zamaaBaaaleaacaaI YaaabeaakiaaiYcacaWGNbWaaSbaaSqaaiaaiodaaeqaaaGccaGLOa Gaayzkaaaaaa@4698@                                                                    (34)

Computation of the velocity profile

The inverse of Weiestrass elliptic function can be readily evaluated on the wolframalpha website using the Mathematica ® MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGGaaiab=5kasd aa@3855@ function.

Listing 1: Evaluation of 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIridpaaCa aaleqabaGaeyOeI0IaaGymaaaaaaa@39A3@ ,

InverseWeierstrassP[B^2/12, g_2, g_3] 

Let us assume that the equation 4 s 3 g 2 s g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaWGZb WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam4zamaaBaaaleaacaaI YaaabeaakiaadohacqGHsislcaWGNbWaaSbaaSqaaiaaiodaaeqaaO GaaGypaiaaicdaaaa@4100@  has a positive discriminant. Then we saw in section 3.1 that MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIriddaa@37CE@  will take real values on the the rectangle OXAY MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad+eacaWGyb GaamyqaiaadMfaaaa@399B@  where O(0,0),X( ω 1 ,0),A( ω 1 , ω 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad+eacqGHHj IUcaaIOaGaaGimaiaaiYcacaaIWaGaaGykaiaaiYcacaWGybGaeyyy IORaaGikaiabeM8a3naaBaaaleaacaaIXaaabeaakiaaiYcacaaIWa GaaGykaiaaiYcacaWGbbGaeyyyIORaaGikaiabeM8a3naaBaaaleaa caaIXaaabeaakiaaiYcacqaHjpWDdaWgaaWcbaGaaG4maaqabaGcca aIPaaaaa@503F@  and Y(0, ω 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMfacqGHHj IUcaaIOaGaaGimaiaaiYcacqaHjpWDdaWgaaWcbaGaaG4maaqabaGc caaIPaaaaa@3E82@ . Further, if u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhaaaa@3740@  is taken round OXAY MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad+eacaWGyb GaamyqaiaadMfaaaa@399B@ , (u) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIridlaaiI cacaWG1bGaaGykaaaa@3A2D@  decreases from + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgUcaRiabg6 HiLcaa@3899@  at O MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad+eaaaa@371A@  to e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGymaaqabaaaaa@3817@  at X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfaaaa@3723@ , further decreases to e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGOmaaqabaaaaa@3818@  along XA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaWGbb aaaa@37E9@ , to e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaG4maaqabaaaaa@3819@  along AY MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeacaWGzb aaaa@37EA@  and finally to MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabg6 HiLcaa@38A4@  as it reaches the origin along YA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMfacaWGbb aaaa@37EA@ . Thus, k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaaGymaaqabaaaaa@381D@  will lie on OX MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad+eacaWGyb aaaa@37F7@ is it is greater than e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGymaaqabaaaaa@3817@  or on XA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaWGbb aaaa@37E9@  is it is between e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGymaaqabaaaaa@3817@  and e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGOmaaqabaaaaa@3818@ or on AB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeacaWGcb aaaa@37D3@  if it is less than e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGOmaaqabaaaaa@3818@ . It cannot be less than e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaG4maaqabaaaaa@3819@  because e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaG4maaqabaaaaa@3819@  is always less than zero while B 2 /12>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkeadaahaa WcbeqaaiaaikdaaaGccaaIVaGaaGymaiaaikdacaaI+aGaaGimaaaa @3BB2@ .

Supposing e 1 B 2 /12 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaWgaa WcbaGaaGymaaqabaGccqGHLjYScaWGcbWaaWbaaSqabeaacaaIYaaa aOGaaG4laiaaigdacaaIYaGaeyyzImRaamyzamaaBaaaleaacaaIYa aabeaaaaa@4169@ , that is, k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaaGymaaqabaaaaa@381D@  lies on XA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaWGbb aaaa@37E9@ . Then we have to align the imaginary axis along the y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhaaaa@3744@  axis and plot values of
6 A 2 ( z| 1 3 ( A 2 C 2 + B 4 4 ), k 0 A 2 36 ) 1 2 B 2 A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaG OnaaqaaiaadgeadaahaaWcbeqaaiaaikdaaaaaaOGaeyigHm8aaeWa aeaacaWG6bGaaGiFamaalaaabaGaaGymaaqaaiaaiodaaaWaaeWaae aacaWGbbWaaWbaaSqabeaacaaIYaaaaOGaam4qamaaCaaaleqabaGa aGOmaaaakiabgUcaRmaalaaabaGaamOqamaaCaaaleqabaGaaGinaa aaaOqaaiaaisdaaaaacaGLOaGaayzkaaGaaGilamaalaaabaGaam4A amaaBaaaleaacaaIWaaabeaakiaadgeadaahaaWcbeqaaiaaikdaaa aakeaacaaIZaGaaGOnaaaaaiaawIcacaGLPaaacqGHsisldaWcaaqa aiaaigdaaeaacaaIYaaaamaalaaabaGaamOqamaaCaaaleqabaGaaG OmaaaaaOqaaiaadgeadaahaaWcbeqaaiaaikdaaaaaaaaa@539F@
where the complex variable z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadQhaaaa@3745@  is z= ω 1 + z 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadQhacaaI9a GaeqyYdC3aaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamOEamaaBaaa leaacaaIWaaabeaaaaa@3D91@ , where z 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadQhadaWgaa WcbaGaaGimaaqabaaaaa@382B@  ranges from 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaicdaaaa@3700@  to ±2Im( k 1 )i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgglaXkaaik dacaWGjbGaamyBaiaaiIcacaWGRbWaaSbaaSqaaiaaigdaaeqaaOGa aGykaiaadMgaaaa@3EE4@ . Since k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaaGymaaqabaaaaa@381D@  lies on XA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaWGbb aaaa@37E9@ , ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBa aaleaacaaIXaaabeaaaaa@38FA@  is identical to Re k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkfacaWGLb Gaam4AamaaBaaaleaacaaIXaaabeaaaaa@39DE@  and hence z=Re( k 1 )+ z 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadQhacaaI9a GaamOuaiaadwgacaaIOaGaam4AamaaBaaaleaacaaIXaaabeaakiaa iMcacqGHRaWkcaWG6bWaaSbaaSqaaiaaicdaaeqaaaaa@3FDA@ . The following R code, with the library 'elliptic' was used to generate the velocity profile of Figure 3.6

Listing 2: Velocity profile of flow in porous medium

Figure 3 Velocity profile for flow in a porous channel bounded by flat plates.

library(elliptic)
epsilon <- 0.1   # Porosity
Re <- 0.1          # Reynold's number
Da <- 5e-1       # Darcy number
J <- 1               # Viscosity ratio
G <- 1e-4        # Pressure gradient/density

F0 <- 1.75/sqrt(150 * epsilon^3)  # Geometric function

# Calculation of A, B, C
A <- sqrt((F0 * epsilon * Re)/(J * sqrt(Da)))
B <- sqrt(epsilon/(J * Da))
C <- sqrt(epsilon^2 * Re * G/J)

g2 <- (A^2*C^2 + B^4/4)/3
u2_center <- A^2/6 + B^2/12

# Choose e2 as u2_center
e2 <- u2_center

e1 <- (-e2 + sqrt(g2 - 3*1i*(-1i)*e2^2))/2
e3 <- (-e2 - sqrt(g2 - 3*1i*(-1i)*e2^2))/2

g3 <- 4*e1*e2*e3

# Get half periods
Omegas <- half.periods(g = c(g2, g3))

# Recreate e's so that they are consistent with R
es <- e1e2e3(g = c(g2, g3))

# Computed using Mathematica
k1 <- 8.020971603238442 + 3.116271691054093i

# Observe that k1 is equal to Omegas[2].

# Print Omegas. Let omega1 be the real period and omega3 be the imaginary period.
#
# We want to plot the Weierstrass function along the imaginary # axis. We know that P(omega1) = e1 while e1 >= P0. The function # P falls below e1 both above and below omega1 if we move along # the imaginary axis. If it takes a value B^2/12 a distance Im(k1) # above omega1, it will take the same value a distance Im(k1) below # omega1. Therefore, we plot the function from omega1 - 1i*Im(k1) # omega1 + 1i * Im(k1).

z <- Omegas[2] + seq(from = 0, to = -2*Im(k1), len = 100)*1i y <- -Im(z)

u <- 6/A^2*( P(z + Im(k1)*1i, g = c(g2, g3)) - B^2/12)

# We choose real part alone. Check that the imaginary part # is very small. Use the command max(abs(Im(u))) to check # it.

# If the imaginary part is indeed quite small xl <- 'Dimensionless velocity' yl <- 'y' ml <- 'Velocity profile with porous media' plot(Re(u), y, type = 'l', xlab = xl, ylab = yl, main = ml)

The same code can be used if the equation 4 s 3 g 2 s g 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaisdacaWGZb WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaam4zamaaBaaaleaacaaI YaaabeaakiaadohacqGHsislcaWGNbWaaSbaaSqaaiaaiodaaeqaaO GaaGypaiaaicdaaaa@4100@  has a negative discriminant and k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaaGymaaqabaaaaa@381D@  is complex. In the event k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaaGymaaqabaaaaa@381D@  is real, the code should be modified so that the real axis is along the y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhaaaa@3744@  axis.

When we introduced the non-dimensional variables, we defined y =y/H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhadaWgaa WcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWF gls5aeqaaOGaaGypaiaadMhacaaIVaGaamisaaaa@46A1@ . Therefore, y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhadaWgaa WcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWF gls5aeqaaaaa@434C@  should range from 0 to 1. However, we notice in Figure 3 that it does not. We will introduce another transformation that brings the non-dimensional displacement variable between 0 and 1 after proving its validity.

Theorem 3: If we introduce a new non-dimensional displacement variable y ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadMhagaacam aaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb aiab=zSiLdqabaaaaa@435B@  defined as
y ˜ = y α , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadMhagaacam aaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb aiab=zSiLdqabaGccaaI9aWaaSaaaeaacaWG5bWaaSbaaSqaaiab=z SiLdqabaaakeaacqaHXoqyaaGaaGilaaaa@4A17@
where α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHbaa@37E5@ is a positive real number then this transformation keeps the form of equation (11) unchanged.

Proof: For sake of clarity, let us write the derivatives in full and re-introduce the MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGGaaiab=jdiIo rr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae4NXIuUa e8NmGikaaa@451D@  subscript for non-dimensional variables. Thus, equation (11) is written as
d 2 u d y 2 A 2 u 2 B 2 u + C 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izamaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaWefv3ySLgz nfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFgls5aeqaaaGcba GaamizaiaadMhadaWgaaWcbaGae8NXIuoabeaakmaaCaaaleqabaGa aGOmaaaaaaGccqGHsislcaWGbbWaaWbaaSqabeaacaaIYaaaaOGaam yDamaaDaaaleaacqWFgls5aeaacaaIYaaaaOGaeyOeI0IaamOqamaa CaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGae8NXIuoabeaaki abgUcaRiaadoeadaahaaWcbeqaaiaaikdaaaGccaaI9aGaaGimaaaa @5BCC@  (35)
If the non-dimensionalization of y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhaaaa@3744@  is done with αH MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHjaadI eaaaa@38B2@ , where α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHbaa@37E5@  is a positive real number then
y ˜ = y αH = y α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadMhagaacam aaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb aiab=zSiLdqabaGccaaI9aWaaSaaaeaacaWG5baabaGaeqySdeMaam isaaaacaaI9aWaaSaaaeaacaWG5bWaaSbaaSqaaiab=zSiLdqabaaa keaacqaHXoqyaaaaaa@4DA2@
If f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgaaaa@3731@  is a function of y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhaaaa@3744@  then
df d y = df d y ˜ d y ˜ d y = 1 α df d y ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izaiaadAgaaeaacaWGKbGaamyEamaaBaaaleaatuuDJXwAK1uy0Hwm aeHbfv3ySLgzG0uy0Hgip5wzaGqbaiab=zSiLdqabaaaaOGaaGypam aalaaabaGaamizaiaadAgaaeaacaWGKbGabmyEayaaiaWaaSbaaSqa aiab=zSiLdqabaaaaOWaaSaaaeaacaWGKbGabmyEayaaiaWaaSbaaS qaaiab=zSiLdqabaaakeaacaWGKbGaamyEamaaBaaaleaacqWFgls5 aeqaaaaakiaai2dadaWcaaqaaiaaigdaaeaacqaHXoqyaaWaaSaaae aacaWGKbGaamOzaaqaaiaadsgaceWG5bGbaGaadaWgaaWcbaGae8NX Iuoabeaaaaaaaa@5FD2@
and hence
d 2 u d y 2 = 1 α 2 d 2 u d y ˜ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izamaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaWefv3ySLgz nfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFgls5aeqaaaGcba GaamizaiaadMhadaWgaaWcbaGae8NXIuoabeaakmaaCaaaleqabaGa aGOmaaaaaaGccaaI9aWaaSaaaeaacaaIXaaabaGaeqySde2aaWbaaS qabeaacaaIYaaaaaaakmaalaaabaGaamizamaaCaaaleqabaGaaGOm aaaakiaadwhadaWgaaWcbaGae8NXIuoabeaaaOqaaiaadsgaceWG5b GbaGaadaWgaaWcbaGae8NXIuoabeaakmaaCaaaleqabaGaaGOmaaaa aaaaaa@5999@
If we denote the new dimensionless numbers with a tilde on top then from equations (8) to (10),
R ˜ e=αRe MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadkfagaacai aadwgacaaI9aGaeqySdeMaamOuaiaadwgaaaa@3C3D@                                                                                       (36)
D ˜ a= Da α 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadseagaacai aadggacaaI9aWaaSaaaeaacaWGebGaamyyaaqaaiabeg7aHnaaCaaa leqabaGaaGOmaaaaaaaaaa@3D12@                                                                                       (37)
J ˜ =J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadQeagaacai aai2dacaWGkbaaaa@38BA@                                                                                             (38)

The new pressure gradient is G ˜ =α G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadEeagaacam aaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb aiab=zSiLdqabaGccaaI9aGaeqySdeMaam4ramaaBaaaleaacqWFgl s5aeqaaaaa@48E3@  while the new constants A ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadgeagaacaa aa@371B@ , B ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadkeagaacaa aa@371C@  and C ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadoeagaacaa aa@371D@  become
A ˜ 2 = F 0 ε R ˜ e J ˜ D ˜ a = α 2 A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadgeagaacam aaCaaaleqabaGaaGOmaaaakiaai2dadaWcaaqaaiaadAeadaWgaaWc baGaaGimaaqabaGccqaH1oqzceWGsbGbaGaacaWGLbaabaGabmOsay aaiaWaaOaaaeaaceWGebGbaGaacaWGHbaaleqaaaaakiaai2dacqaH XoqydaahaaWcbeqaaiaaikdaaaGccaWGbbWaaWbaaSqabeaacaaIYa aaaaaa@45E0@                                                                      (39)
B ˜ 2 = ε J ˜ D ˜ a = α 2 B 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadkeagaacam aaCaaaleqabaGaaGOmaaaakiaai2dadaWcaaqaaiabew7aLbqaaiqa dQeagaacaiqadseagaacaiaadggaaaGaaGypaiabeg7aHnaaCaaale qabaGaaGOmaaaakiaadkeadaahaaWcbeqaaiaaikdaaaaaaa@4232@                                                                         (40)
C ˜ 2 = ε 2 R ˜ e G ˜ J ˜ = α 2 C 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadoeagaacam aaCaaaleqabaGaaGOmaaaakiaai2dadaWcaaqaaiabew7aLnaaCaaa leqabaGaaGOmaaaakiqadkfagaacaiaadwgaceWGhbGbaGaaaeaace WGkbGbaGaaaaGaaGypaiabeg7aHnaaCaaaleqabaGaaGOmaaaakiaa doeadaahaaWcbeqaaiaaikdaaaaaaa@4414@                                                                     (41)

Therefore, equation (35) becomes,
1 α 2 d 2 u d y ˜ 2 A ˜ 2 α 2 u 2 B ˜ 2 α 2 u + C ˜ 2 α 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaG ymaaqaaiabeg7aHnaaCaaaleqabaGaaGOmaaaaaaGcdaWcaaqaaiaa dsgadaahaaWcbeqaaiaaikdaaaGccaWG1bWaaSbaaSqaamrr1ngBPr wtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae8NXIuoabeaaaOqa aiaadsgaceWG5bGbaGaadaWgaaWcbaGae8NXIuoabeaakmaaCaaale qabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiqadgeagaacamaaCaaa leqabaGaaGOmaaaaaOqaaiabeg7aHnaaCaaaleqabaGaaGOmaaaaaa GccaWG1bWaa0baaSqaaiab=zSiLdqaaiaaikdaaaGccqGHsisldaWc aaqaaiqadkeagaacamaaCaaaleqabaGaaGOmaaaaaOqaaiabeg7aHn aaCaaaleqabaGaaGOmaaaaaaGccaWG1bWaaSbaaSqaaiab=zSiLdqa baGccqGHRaWkdaWcaaqaaiqadoeagaacamaaCaaaleqabaGaaGOmaa aaaOqaaiabeg7aHnaaCaaaleqabaGaaGOmaaaaaaGccaaI9aGaaGim aaaa@674B@
or
d 2 u d y ˜ 2 A ˜ 2 u 2 B ˜ 2 u + C ˜ 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izamaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaWefv3ySLgz nfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFgls5aeqaaaGcba GaamizaiqadMhagaacamaaBaaaleaacqWFgls5aeqaaOWaaWbaaSqa beaacaaIYaaaaaaakiabgkHiTiqadgeagaacamaaCaaaleqabaGaaG OmaaaakiaadwhadaqhaaWcbaGae8NXIuoabaGaaGOmaaaakiabgkHi TiqadkeagaacamaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcba Gae8NXIuoabeaakiabgUcaRiqadoeagaacamaaCaaaleqabaGaaGOm aaaakiaai2dacaaIWaaaaa@5C08@
The vertical axis of the graph in Figure 3 extends from 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaicdaaaa@3700@  to 2Im( k 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaikdacaWGjb GaamyBaiaaiIcacaWGRbWaaSbaaSqaaiaaigdaaeqaaOGaaGykaaaa @3C08@ . To shrink it to 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaicdaaaa@3700@  to 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaigdaaaa@3701@ , we choose α=2*Im( k 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHjaai2 dacaaIYaGaaGOkaiaadMeacaWGTbGaaGikaiaadUgadaWgaaWcbaGa aGymaaqabaGccaaIPaaaaa@3F22@ , in this case 6.232543 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaiAdacaaIUa GaaGOmaiaaiodacaaIYaGaaGynaiaaisdacaaIZaaaaa@3C2D@ . After changing the constants A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeaaaa@370C@ , B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkeaaaa@370D@  and C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeaaaa@370E@  we proceed in similar fashion as before to get the plot in Figure 4.

Figure 4 Rescaled velocity profile for flow in a channel bounded by flat plates.

The code to generate the plot is

Listing 3: Velocity profile of flow in porous medium

library(elliptic)

epsilon <- 0.1     # Porosity
Re <- 0.1            # Reynold's number
Da <- 5e-1          # Darcy number
J <- 1                 # Viscosity ratio
G <- 1e-4           # Pressure gradient
F0 <- 1.75/sqrt(150 * epsilon^3)              # Geometric function

# Re-normalize y using this factor
alpha <- 6.232543

# Calculation of A, B, C
# However, for sake of simplicity, we call the variables A, B and C.
A <- sqrt((F0 * epsilon * Re)/(J * sqrt(Da))) * alpha
B <- sqrt(epsilon/(J * Da)) * alpha
C <- sqrt(epsilon^2 * Re * G/J) * alpha

g2 <- (A^2*C^2 + B^4/4)/3
u2_center <- A^2/6 + B^2/12
# Choose e2 as u2_center
e2 <- u2_center
e1 <- (-e2 + sqrt(g2 - 3*1i*(-1i)*e2^2))/2
e3 <- (-e2 - sqrt(g2 - 3*1i*(-1i)*e2^2))/2

g3 <- 4*e1*e2*e3

Omegas <- half.periods(g = c(g2, g3)) print(Omegas)

# Computed using Mathematica
k1 <- 1.286949924601859 + 0.5000000786725393i
P(k1, g = c(g2, g3), give.all.3 = T)

z <- Re(k1) + seq(from = 0, to = -2*Im(k1), len = 100)*1i
y <- -Im(z)
u <- 6/A^2*( P(z + Im(k1)*1i, g = c(g2, g3)) - B^2/12 )

# We choose real part alone. Check that the imaginary part # is very small. Use the command max(abs(Im(u))) to check # it.
xl <- 'Dimensionless velocity'
yl <- 'Dimensionless y'
ml <- 'Velocity profile with porous media' plot(Re(u), y, type = 'l', xlab = xl, ylab = yl, main = ml)

To compare the flow profiles with and without porous medium we choose the pressure gradient in the latter case so that the flow has the same peak velocity as in the former case. If we run the code in listing 4 after the one in listing 3, we can have the two profiles in the same plot as shown in Figure 5.

Listing 4: Comparison of velocity profiles

# To compare flows with and without porous media,
# adjusting the pressure gradient in the second
# case so that they have the same maximum velocity.
umax <- max(abs(Re(u)))
G.eq <- 2*mu*umax/(Im(k1)^2)
w.eq <- G.eq/mu * y * (Im(k1) - y/2)
xl <- 'Dimensionless velocity'
yl <- 'Dimensionless y'
ml <- 'Comparing flows with and without porous medium.'
plot(Re(u), y, type = 'l', col = 'blue', xlab = xl, ylab = yl, main = ml)

lines(w.eq, y, col = 'red')
legend.list <- c('With porous medium', 'Without porous medium'
legend(x = 'topright', legend.list, lty = c(1, 1), lwd=c(2.5,2.5),col=c('blue','red'))

Calculation of flux

The flux of volume of fluid past a section of the channel is probably more important in practice than the velocity profile. In this section, I will calculate the flux in the channel. If s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadohaaaa@373E@  denotes the transverse dimension of the channel then the flux Q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfaaaa@371C@  is Q= 0 h uds, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfacaaI9a Waa8qmaeqaleaacaaIWaaabaGaamiAaaqdcqGHRiI8aOGaamyDaiaa dsgacaWGZbGaaGilaaaa@3F4F@

where h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgaaaa@3733@  is the width of the channel and u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhaaaa@3740@  is velocity of the fluid. If the channel does not have porous medium then
u= G 2μ s(hs) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacaaI9a WaaSaaaeaacaWGhbaabaGaaGOmaiabeY7aTbaacaWGZbGaaGikaiaa dIgacqGHsislcaWGZbGaaGykaaaa@4084@
and hence the flux Q 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaaGimaaqabaaaaa@3802@ is
Q 0 = G h 3 12μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaaGimaaqabaGccaaI9aWaaSaaaeaacaWGhbGaamiAamaaCaaa leqabaGaaG4maaaaaOqaaiaaigdacaaIYaGaeqiVd0gaaaaa@3EBD@                                                                                       (42)
On the other hand if the channel is stuffed with porous medium then, the flux Q 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaaGymaaqabaaaaa@3803@ is
Q 1 = 0 h 6 A 2 [ ( s+ k 1 | 1 3 ( A 2 C 2 + B 4 4 ), g 3 ) B 2 12 ]ds MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaaGymaaqabaGccaaI9aWaa8qmaeqaleaacaaIWaaabaGaamiA aaqdcqGHRiI8aOWaaSaaaeaacaaI2aaabaGaamyqamaaCaaaleqaba GaaGOmaaaaaaGcdaWadaqaaiabgIridpaabmaabaGaam4CaiabgUca RiaadUgadaWgaaWcbaGaaGymaaqabaGccaaI8bWaaSaaaeaacaaIXa aabaGaaG4maaaadaqadaqaaiaadgeadaahaaWcbeqaaiaaikdaaaGc caWGdbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaSaaaeaacaWGcb WaaWbaaSqabeaacaaI0aaaaaGcbaGaaGinaaaaaiaawIcacaGLPaaa caaISaGaam4zamaaBaaaleaacaaIZaaabeaaaOGaayjkaiaawMcaai abgkHiTmaalaaabaGaamOqamaaCaaaleqabaGaaGOmaaaaaOqaaiaa igdacaaIYaaaaaGaay5waiaaw2faaiaadsgacaWGZbaaaa@5B91@                    (43)
In terms of non-dimensional parameters h=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgacaaI9a GaaGymaaaa@38B5@  and hence Q 0 =G/(12μ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaaGimaaqabaGccaaI9aGaam4raiaai+cacaaIOaGaaGymaiaa ikdacqaH8oqBcaaIPaaaaa@3EEA@ . If we adjust pressure gradient so that peak velocity matches with that when porous medium is present, then code listing 5 gives Q 0 =0.6665793 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaaGimaaqabaGccaaI9aGaaGimaiaai6cacaaI2aGaaGOnaiaa iAdacaaI1aGaaG4naiaaiMdacaaIZaaaaa@3F85@ . Carrying out the integration in equation (43) and noting that s=iy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadohacaaI9a GaamyAaiaadMhaaaa@39F1@ , we get Q 1 =0.6321661 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaaGymaaqabaGccaaI9aGaaGimaiaai6cacaaI2aGaaG4maiaa ikdacaaIXaGaaGOnaiaaiAdacaaIXaaaaa@3F75@ . As expected from Figure 5, Q 1 < Q 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaaGymaaqabaGccaaI8aGaamyuamaaBaaaleaacaaIWaaabeaa aaa@3A8F@ . Note that listing 5 is meant to be run after running listing (4).

Figure 5 Comparison of velocity profiles.

Listing 5: Calculation of flux,

# Flux in presence of porous medium
# up denotes the 'velocity function' when porous medium is present
up <- function(x) {6/A^2*( P(x + Im(k1)*1i, g = c(g2, g3)) - B^2/12)}

# We multiply the integral by i because the integral was carried out along the imaginary axis.
Q1 <- integrate.segments(up, c(z[1], z[100]), close = F)*1i

# Flux in absence of porous medium
Q0 <- G.eq/(12*mu)

There is another way to compute the flux of flow in a porous medium. It uses the relation between Weierstrass MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIriddaa@37CE@  function and Weierstrass (not Riemann) ζ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeA7a6baa@3803@ function.4
(y)= ζ ' (y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIridlaaiI cacaWG5bGaaGykaiaai2dacqGHsislcqaH2oGEdaahaaWcbeqaaiaa dEcaaaGccaaIOaGaamyEaiaaiMcaaaa@40E8@

(44)

Therefore, we get an analytical expression for Q 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaaGymaaqabaaaaa@3803@ ,
Q 1 = 6 A 2 [ ζ( k 1 | g 2 , g 3 )ζ(1+ k 1 | g 2 , g 3 ) ] B 2 2 A 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaaGymaaqabaGccaaI9aWaaSaaaeaacaaI2aaabaGaamyqamaa CaaaleqabaGaaGOmaaaaaaGcdaWadaqaaiabeA7a6jaaiIcacaWGRb WaaSbaaSqaaiaaigdaaeqaaOGaaGiFaiaadEgadaWgaaWcbaGaaGOm aaqabaGccaaISaGaam4zamaaBaaaleaacaaIZaaabeaakiaaiMcacq GHsislcqaH2oGEcaaIOaGaaGymaiabgUcaRiaadUgadaWgaaWcbaGa aGymaaqabaGccaaI8bGaam4zamaaBaaaleaacaaIYaaabeaakiaaiY cacaWGNbWaaSbaaSqaaiaaiodaaeqaaOGaaGykaaGaay5waiaaw2fa aiabgkHiTmaalaaabaGaamOqamaaCaaaleqabaGaaGOmaaaaaOqaai aaikdacaWGbbWaaWbaaSqabeaacaaIYaaaaaaaaaa@59F3@                          (45)

The library 'elliptic' allows us to evaluate Weierstrass ζ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeA7a6baa@3803@  function. The listing 5, if run after listing 5 computes
Q 1 =0.6321664 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaaGymaaqabaGccaaI9aGaaGimaiaai6cacaaI2aGaaG4maiaa ikdacaaIXaGaaGOnaiaaiAdacaaI0aaaaa@3F78@ .6

Listing 6: Another way to compute flux

# We calculate the two terms in equation (45) separately.
Q11 <- 6/A^2*(zeta(z[1] + Im(k1)*1i, g = c(g2, g3)) - zeta(z[100] + Im(k1)*1i, g = c(g2, g3)))
Q12 <- B^2/(2*A^2)*(z[100] - z[1])

# We multiply the result by i because the integral was carried out along the imaginary axis.
Q1.using.zeta <- (Q11 - Q12)*1i

Acknowledgements

None.

Conflict of interest

Author declares there is no conflict of interest in publishing the article.

References

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