eISSN: 2576-4500 AAOAJ

Aeronautics and Aerospace Open Access Journal
Research Article
Volume 2 Issue 6

Slip analysis of magnetohydrodynamics flow of an upper-convected Maxwell viscoelastic nanofluid in a permeable channel embedded in a porous medium
MG Sobamowo, AA Yinusa, AA Oluwo, SI Alozie
Department of Mechanical Engineering, University of Lagos, Nigeria
Received: September 12, 2018 | Published: October 11, 2018

Correspondence: MG Sobamowo, Department of Mechanical Engineering, University of Lagos, Akoka, Lagos, Nigeria, Tel 2347034717417, Email

Citation: Sobamowo MG, Yinusa AA, Oluwo AA, et al. Slip analysis of magnetohydrodynamics flow of an upper-convected Maxwell viscoelastic nanofluid in a permeable channel embedded in a porous medium. Aeron Aero Open Access J. 2018;2(5):310‒318. DOI: 10.15406/aaoaj.2018.02.00065

Abstrat

The continuous applications of viscoelastic fluids in biomedical engineering and industrial processes require some studies that provide better physical insights into the flow phenomena of the fluids. In this work, homotopy perturbation method is applied to investigate the simultaneous effects of slip and magnetic field on the flow of an upper convected Maxwell nanofluid through a permeable microchannel embedded in a porous medium. The results of the approximate analytical solution depict very good agreements with the results of the fourth order Runge-Kutta Fehlberg numerical method for the verification of the mathematical method used in analyzing the flow. Thereafter, the obtained analytical solutions are used to investigate the effects of pertinent rheological parameters on the flow process. It is observed from the results that increase in slip parameter, nanoparticle concentration and Darcy number lead to increase in the velocity of the upper-convected Maxwell fluid. However, when the Deborah’s number increases, the Hartmann, and Reynold numbers decrease the fluid flow velocity towards the lower plate but as the upper plate is approached, a reverse trend is observed. The study can be used to advance the application of upper convected Maxwell flow in the areas of in biomedical, geophysical and astrophysics.

Keywords: slip analysis, upper-convected Maxwell flow, viscoelastic nanofluid, magnetic field, porous medium

Introduction

There have been increasing research works in the past few years on analysis of flow of viscoelastic fluid. This is due to its various applications in gaseous diffusion, blood flow through oxygenators, flow in blood capillaries have continue to aroused the research interests. However, complex rheological fluids such as blood, paints, synovial fluid, saliva, jam which cannot be adequately described by Navier Stokes. Consequently, complex constitutive relations that capture the flow behaviour of the complex fluids have been developed.1 Among the newly developed integral and differential-type fluid flow models, Upper convected Maxwell fluid model has showed to be an effective fluid model that captures the complex flow phenomena of fluids especially of the fluid with high elastic behaviours such as polymer melts. Such highly elastic fluids have high Deborah number.2‒3 In the analysis of Maxwell flow, Fetecau4 presented a new exact solution for the flow of fluid through infinite microchannel while Hunt5 studied convective fluid flow through rectangular duct. Sheikholeslami et al.6 investigated magneto hydrodynamic field effect on flow through semi-porous channel utilizing analytical methods. Shortly after, Sheikholeslami7‒9 adopted numerical solutions in the investigations of nanofluid in semi-annulus enclosure.

Flow of upper convected Maxwell fluid through porous stretch sheet was investigated by Raftari & Yildirim.10 Entophy generation in fluid in the presence of magnetic field was analyzed by Sheikholeslami & Ganji11 using lattice Boltzmann method while Ganji et al.12 explored analytical and numerical methods to analyze the fluid flow problems under the influence of magnetic field. The flow of viscoelastic fluid through a moving plate was analyzed by Sadeghy & Sharifi13 using local similarity solutions. Vajrevulu et al.14 investigated the mass transfer and flow of chemically reactive upper convected Maxwell fluid under induced magnetic field. Not long after Raftari & Vajrevulu15 adopted the homotopy analysis method in the study of flow and heat transfer in stretching wall channels considering MHD. Hatami et al.16 presented forced convective MHD nanofluid flow conveyed through horizontal parallel plates. Laminar thermal boundary flow layer over flat plate considering convective fluid surface was analyzed by Aziz17 using similarity solution. Beg & Makinde18 examined the flow of viscoelastic fluid through Darcian microchannel with high permeability.

Most of the above reviews studies focused on the analysis of fluid flow under no slip condition. However, such an assumption of no slip condition does not hold in a flow system with small size characteristics size or low flow pressure. The pioneer work of flow with slip boundary condition was first initiated by Navier.19 The slip conditions occur in various flows such as nanofluids, polymeric liquids, fluids containing concentrated suspensions, flow on multiple interfaces, thin film problems and rarefied fluid problems.19‒31 Due to the practical implications of the condition on the flow processes, several studies on the effects of slip boundary conditions on fluids flow behaviours have been presented by many researchers.19‒35 Abbasi et al.36 investigated the MHD flow characteristics of upper-convected Maxwell viscoelastic flow in a permeable channel under slip conditions. However, an analytical study on simultaneous effects of slip, magnetic field, nanoparticle and porous medium on the flow characteristics of an upper-convected Maxwell viscoelastic nanofluid has not been carried out in literature. Therefore, in this work, homotopy perturbation method is used to analyze the slip flow of an upper-convected Maxwell viscoelastic nanofluid through a permeable microchannel embedded in porous medium under the influence of magnetic field is analyzed. Also, the effects of other pertinent parameters of the flow process are investigated and discussed.

Model development and analytical solution

Consider a laminar slip flow of an electrically conducting fluid in a microchannel is considered. Along the y axis, magnetic fields are imposed uniformly, as described in the physical model diagram Figure 1. It is assumed external electric field is zero and constant of electrical conductivity is constant. Therefore, magnetic Reynolds number is small and magnetic field induced by fluid motion is negligible.

Figure 1 Flow of upper-convected Maxwell fluid between in permeable channel embedded in porous medium.

Based on the assumptions, the governing equation for the Maxwell fluid is presented as8

T=-PI+S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWHub Gaeyypa0JaaeylaiaadcfacaWHjbGaey4kaSIaaC4uaaaa@3C7D@ (1)

where the Cauchy stress tensor is T and S is the extra-stress Tensor which satisfies

S+λ( S t +(V)S-LSS L T )=μ A L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWHtb Gaey4kaSIaeq4UdWMaaiikaKqbaoaalaaakeaajugibiabgkGi2kaa hofaaOqaaKqzGeGaeyOaIyRaamiDaaaacqGHRaWkcaGGOaGaaCOvai abgwSixlabgEGirlaacMcacaWHtbGaaeylaiaahYeacaWHtbGaeyOe I0IaaC4uaiaahYeajuaGdaahaaWcbeqaaKqzadGaamivaaaajugibi aacMcacqGH9aqpcqaH8oqBcaWGbbqcfa4aaSbaaSqaaKqzadGaamit aaWcbeaaaaa@581D@ (2)

The Rivlin-Ericksen tensor is defined by

A L =V+ ( V ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb qcfa4aaSbaaSqaaKqzadGaamitaaWcbeaajugibiabg2da9iabgEGi rlaadAfacqGHRaWkjuaGdaqadaGcbaqcLbsacqGHhis0caWGwbaaki aawIcacaGLPaaajuaGdaahaaWcbeqaaKqzadGaamivaaaaaaa@46C4@ (3)

The continuity and momentum equations for steady, incompressible two dimensional flows are expressed in Eqs. (4) -(6) as

u ¯ x + v ¯ y =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyRabmyDayaaraaakeaajugibiabgkGi2kaadIha aaGaey4kaSscfa4aaSaaaOqaaKqzGeGaeyOaIyRabmODayaaraaake aajugibiabgkGi2kaadMhaaaGaeyypa0JaaGimaaaa@45F0@ (4)

ρ nf ( u u x +v u y )= P x + S xx x + S xy y σ nf B 2 (t)u μ nf u K p , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbp GCjuaGdaWgaaWcbaqcLbmacaWGUbGaamOzaaWcbeaajuaGdaqadaGc baqcLbsacaWG1bqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaamyDaaGcba qcLbsacqGHciITcaWG4baaaiabgUcaRiaadAhajuaGdaWcaaGcbaqc LbsacqGHciITcaWG1baakeaajugibiabgkGi2kaadMhaaaaakiaawI cacaGLPaaajugibiabg2da9iabgkHiTKqbaoaalaaakeaajugibiab gkGi2kaadcfaaOqaaKqzGeGaeyOaIyRaamiEaaaacqGHRaWkjuaGda WcaaGcbaqcLbsacqGHciITcaWGtbqcfa4aaSbaaSqaaKqzGeGaamiE aiaadIhaaSqabaaakeaajugibiabgkGi2kaadIhaaaGaey4kaSscfa 4aaSaaaOqaaKqzGeGaeyOaIyRaam4uaKqbaoaaBaaaleaajugibiaa dIhacaWG5baaleqaaaGcbaqcLbsacqGHciITcaWG5baaaiabgkHiTi abeo8aZLqbaoaaBaaaleaajugWaiaad6gacaWGMbaaleqaaKqzGeGa amOqaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaaiikaiaads hacaGGPaGaamyDaiabgkHiTKqbaoaalaaakeaajugibiabeY7aTLqb aoaaBaaaleaajugWaiaad6gacaWGMbaaleqaaKqzGeGaamyDaaGcba qcLbsacaWGlbqcfa4aaSbaaSqaaKqzadGaamiCaaWcbeaaaaqcLbsa caGGSaaaaa@8A1D@ (5)                             

ρ nf ( u ¯ v ¯ x + v ¯ v ¯ y )= P x + S yx x + S yy y μ nf v ¯ K p , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbp GCjuaGdaWgaaWcbaqcLbmacaWGUbGaamOzaaWcbeaajuaGdaqadaGc baqcLbsaceWG1bGbaebajuaGdaWcaaGcbaqcLbsacqGHciITceWG2b GbaebaaOqaaKqzGeGaeyOaIyRaamiEaaaacqGHRaWkceWG2bGbaeba juaGdaWcaaGcbaqcLbsacqGHciITceWG2bGbaebaaOqaaKqzGeGaey OaIyRaamyEaaaaaOGaayjkaiaawMcaaKqzGeGaeyypa0JaeyOeI0sc fa4aaSaaaOqaaKqzGeGaeyOaIyRaamiuaaGcbaqcLbsacqGHciITca WG4baaaiabgUcaRKqbaoaalaaakeaajugibiabgkGi2kaadofajuaG daWgaaWcbaqcLbmacaWG5bGaamiEaaWcbeaaaOqaaKqzGeGaeyOaIy RaamiEaaaacqGHRaWkjuaGdaWcaaGcbaqcLbsacqGHciITcaWGtbqc fa4aaSbaaSqaaKqzadGaamyEaiaadMhaaSqabaaakeaajugibiabgk Gi2kaadMhaaaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaeqiVd0wcfa4a aSbaaSqaaKqzadGaamOBaiaadAgaaSqabaqcLbsaceWG2bGbaebaaO qaaKqzGeGaam4saKqbaoaaBaaaleaajugWaiaadchaaSqabaaaaKqz GeGaaiilaaaa@7D81@ (6)

where the effective density  and effective dynamic viscosity of the nanofluid are defined as follows:

ρ nf =( 1ϕ ) ρ f +ϕ ρ s , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqyWdi xcfa4aaSbaaSqaaKqzadGaamOBaiaadAgaaSqabaqcLbsacqGH9aqp juaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0Iaeqy1dygakiaawIcaca GLPaaajugibiabeg8aYLqbaoaaBaaaleaajugWaiaadAgaaSqabaqc LbsacqGHRaWkcqaHvpGzcqaHbpGCjuaGdaWgaaWcbaqcLbmacaWGZb aaleqaaKqzGeGaaiilaaaa@521A@

μ nf = μ f ( 1ϕ ) 2.5 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 wcfa4aaSbaaSqaaKqzadGaamOBaiaadAgaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacqaH8oqBjuaGdaWgaaWcbaqcLbmacaWGMb aaleqaaaGcbaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiabew9a MbGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaajugWaiaaikdacaGGUa GaaGynaaaaaaqcLbsacaGGSaaaaa@4EEA@

σ nf = σ f [ 1+ 3{ σ s σ f 1 }ϕ { σ s σ f +2 }ϕ{ σ s σ f 1 }ϕ ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Wdm xcfa4aaSbaaSqaaKqzadGaamOBaiaadAgaaSqabaqcLbsacqGH9aqp cqaHdpWCjuaGdaWgaaWcbaqcLbmacaWGMbaaleqaaKqbaoaadmaake aajugibiaaigdacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaIZaqcfa4a aiWaaOqaaKqbaoaalaaakeaajugibiabeo8aZLqbaoaaBaaaleaaju gibiaadohaaSqabaaakeaajugibiabeo8aZLqbaoaaBaaaleaajugW aiaadAgaaSqabaaaaKqzGeGaeyOeI0IaaGymaaGccaGL7bGaayzFaa qcLbsacqaHvpGzaOqaaKqbaoaacmaakeaajuaGdaWcaaGcbaqcLbsa cqaHdpWCjuaGdaWgaaWcbaqcLbmacaWGZbaaleqaaaGcbaqcLbsacq aHdpWCjuaGdaWgaaWcbaqcLbmacaWGMbaaleqaaaaajugibiabgUca RiaaikdaaOGaay5Eaiaaw2haaKqzGeGaeqy1dyMaeyOeI0scfa4aai WaaOqaaKqbaoaalaaakeaajugibiabeo8aZLqbaoaaBaaaleaajugW aiaadohaaSqabaaakeaajugibiabeo8aZLqbaoaaBaaaleaajugWai aadAgaaSqabaaaaKqzGeGaeyOeI0IaaGymaaGccaGL7bGaayzFaaqc LbsacqaHvpGzaaaakiaawUfacaGLDbaajugibiaacYcaaaa@8205@

and

Sxx,Sxy,Syx and Syy are extra stress tensors and ρ is the density of the fluid. Using the shear-stress strain for a upper-convected liquid, The governing equations of fluid motion is easily expressed as16

u x + v y =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyRaamyDaaGcbaqcLbsacqGHciITcaWG4baaaiab gUcaRKqbaoaalaaakeaajugibiabgkGi2kaadAhaaOqaaKqzGeGaey OaIyRaamyEaaaacqGH9aqpcaaIWaaaaa@45C0@ (7)

u u x +v u y +λ( u 2 2 u x 2 +v 2 u y 2 +2uv 2 u xy )= v nf 2 u y 2 σ nf B 2 (t)u ρ nf ν nf u K p , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG1b qcfa4aaSaaaOqaaKqzGeGaeyOaIyRaamyDaaGcbaqcLbsacqGHciIT caWG4baaaiabgUcaRiaadAhajuaGdaWcaaGcbaqcLbsacqGHciITca WG1baakeaajugibiabgkGi2kaadMhaaaGaey4kaSIaeq4UdWwcfa4a aeWaaOqaaKqzGeGaamyDaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaK qbaoaalaaakeaajugibiabgkGi2MqbaoaaCaaaleqabaqcLbmacaaI YaaaaKqzGeGaamyDaaGcbaqcLbsacqGHciITcaWG4bqcfa4aaWbaaS qabeaajugWaiaaikdaaaaaaKqzGeGaey4kaSIaamODaKqbaoaalaaa keaajugibiabgkGi2MqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGe GaamyDaaGcbaqcLbsacqGHciITcaWG5bqcfa4aaWbaaSqabeaajugW aiaaikdaaaaaaKqzGeGaey4kaSIaaGOmaiaadwhacaWG2bqcfa4aaS aaaOqaaKqzGeGaeyOaIyBcfa4aaWbaaSqabeaajugWaiaaikdaaaqc LbsacaWG1baakeaajugibiabgkGi2kaadIhacqGHciITcaWG5baaaa GccaGLOaGaayzkaaqcLbsacqGH9aqpcaWG2bqcfa4aaSbaaSqaaKqz adGaamOBaiaadAgaaSqabaqcfa4aaSaaaOqaaKqzGeGaeyOaIyBcfa 4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacaWG1baakeaajugibiab gkGi2kaadMhajuaGdaahaaWcbeqaaKqzadGaaGOmaaaaaaqcLbsacq GHsisljuaGdaWcaaGcbaqcLbsacqaHdpWCjuaGdaWgaaWcbaqcLbma caWGUbGaamOzaaWcbeaajugibiaadkeajuaGdaahaaWcbeqaaKqzad GaaGOmaaaajugibiaacIcacaWG0bGaaiykaiaadwhaaOqaaGGaaKqz GeGae8xWdixcfa4aaSbaaSqaaKqzadGaamOBaiaadAgaaSqabaaaaK qzGeGaeyOeI0scfa4aaSaaaOqaaKqzGeGaeqyVd4wcfa4aaSbaaSqa aKqzadGaamOBaiaadAgaaSqabaqcLbsacaWG1baakeaajugibiaadU eajuaGdaWgaaWcbaqcLbmacaWGWbaaleqaaaaajugibiaacYcaaaa@B373@ (8)

where flow velocity component (u,v) are velocity component along the x and y directions respectively. Since flow is symmetric about channel center line, attention is given to the flow region 0<y<H. Appropriate boundary conditions are given as

y=0: u x =0,v=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5b Gaeyypa0JaaGimaiaacQdajuaGdaWcaaGcbaqcLbsacqGHciITcaWG 1baakeaajugibiabgkGi2kaadIhaaaGaeyypa0JaaGimaiaacYcaca WG2bGaeyypa0JaaGimaaaa@45BF@ (9)

y=H: u y =βu,v= V w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5b Gaeyypa0JaamisaiaacQdajuaGdaWcaaGcbaqcLbsacqGHciITcaWG 1baakeaajugibiabgkGi2kaadMhaaaGaeyypa0JaeyOeI0IaeqOSdi MaamyDaiaacYcacaaMf8UaamODaiabg2da9iaadAfajuaGdaWgaaWc baqcLbmacaWG3baaleqaaaaa@4D3F@ (10)

where Vw and β are the wall characteristic suction velocity and sliding friction respectively.

The physical and thermal properties of the base fluid and nanoparticles are given in Table 1 and Table 2, respectively.

Base fluid                          

ρ(kg/m3)            

Cp (J/kgK)            

k(W/mK)            

σ(Ω-1m-1)

Pure water                           

997.1

4179

0.613

5.5

Ethylene Glycol                  

1115

2430

0.253

1.07

Engine oil                           

884

1910

0.144

4.02

Kerosene   

783

2010

0.145

4.01

Table 1 Physical and thermal properties of the base fluid

Nanoparticles 

ρ(kg/m3)            

Cp(J/kgK)            

k(W/mK)            

σ(Ω-1m-1)

Copper (Cu)                                         

8933

385

401

59.6

Aluminum oxide (Al2O3)                    

3970

765

40

16.7

SWCNTs

2600

42.5

6600

1.26

Silver (Ag)                                          

10500

235

429

Titanium dioxide (TiO2)                     

4250

686.2

8.9538

Copper (II) Oxide (CuO)                    

783

540

18

 

Table 2 Physical and thermal properties of nanoparticles

The similarity variables are introduced as:

η= y H ,u= V w x f ' (y);v= V w f(y);k= μ Hβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH3o aAcqGH9aqpjuaGdaWcaaGcbaqcLbsacaWG5baakeaajugibiaadIea aaGaaiilaiaaywW7caWG1bGaeyypa0JaeyOeI0IaamOvaKqbaoaaBa aaleaajugWaiaadEhaaSqabaqcLbsacaWG4bGaamOzaKqbaoaaCaaa leqabaqcLbsacaGGNaaaaiaacIcacaWG5bGaaiykaiaacUdacaaMf8 UaamODaiabg2da9iaadAfajuaGdaWgaaWcbaqcLbmacaWG3baaleqa aKqzGeGaamOzaiaacIcacaWG5bGaaiykaiaacUdacaaMf8Uaam4Aai abg2da9KqbaoaalaaakeaajugibiabeY7aTbGcbaqcLbsacaWGibGa eqOSdigaaaaa@62AA@ (11)

With the aid of the dimensionless parameters in Eq. (11), the constitutive relation is satisfied. Equation (2-4) can be expressed as:

f ''' ( M 2 + 1 Da ) f ' + ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w ( f '2 f f '' )+De ( 1ϕ ) 2.5 ( 2f f ' f '' f 2 f ''' )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaWbaaSqabeaajugibiaacEcacaGGNaGaai4jaaaacqGHsisl juaGdaqadaGcbaqcLbsacaWGnbqcfa4aaWbaaSqabeaajugWaiaaik daaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugi biaadseacaWGHbaaaaGccaGLOaGaayzkaaqcLbsacaWGMbqcfa4aaW baaSqabeaajugibiaacEcaaaGaey4kaSscfa4aaeWaaOqaaKqzGeGa aGymaiabgkHiTiabew9aMbGccaGLOaGaayzkaaqcfa4aaWbaaSqabe aajugWaiaaikdacaGGUaGaaGynaaaajuaGdaqadaGcbaqcfa4aaeWa aOqaaKqzGeGaaGymaiabgkHiTiabew9aMbGccaGLOaGaayzkaaqcLb sacqGHRaWkcqaHvpGzjuaGdaWcaaGcbaqcLbsacqaHbpGCjuaGdaWg aaWcbaqcLbmacaWGZbaaleqaaaGcbaqcLbsacqaHbpGCjuaGdaWgaa WcbaqcLbmacaWGMbaaleqaaaaaaOGaayjkaiaawMcaaKqzGeGaamOu aiaadwgajuaGdaWgaaWcbaqcLbmacaWG3baaleqaaKqbaoaabmaake aajugibiaadAgajuaGdaahaaWcbeqaaKqzadGaai4jaiaaikdaaaqc LbsacqGHsislcaWGMbGaamOzaKqbaoaaCaaaleqabaqcLbsacaGGNa Gaai4jaaaaaOGaayjkaiaawMcaaKqzGeGaey4kaSIaamiraiaadwga juaGdaqadaGcbaqcLbsacaaIXaGaeyOeI0Iaeqy1dygakiaawIcaca GLPaaajuaGdaahaaWcbeqaaKqzadGaaGOmaiaac6cacaaI1aaaaKqb aoaabmaakeaajugibiaaikdacaWGMbGaamOzaKqbaoaaCaaaleqaba qcLbsacaGGNaaaaiaadAgajuaGdaahaaWcbeqaaKqzGeGaai4jaiaa cEcaaaGaeyOeI0IaamOzaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaK qzGeGaamOzaKqbaoaaCaaaleqabaqcLbsacaGGNaGaai4jaiaacEca aaaakiaawIcacaGLPaaajugibiabg2da9iaaicdaaaa@A2A3@ (12)

Taking boundary condition as

η=0: f '' =0;f=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH3o aAcqGH9aqpcaaIWaGaaiOoaiaadAgajuaGdaahaaWcbeqaaKqzGeGa ai4jaiaacEcaaaGaeyypa0JaaGimaiaacUdacaWGMbGaeyypa0JaaG imaaaa@4364@ (13)

where Re w = V w H υ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGsb GaaiyzaKqbaoaaBaaaleaajugWaiaadEhaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacaWGwbqcfa4aaSbaaSqaaKqzadGaam4Daa WcbeaajugibiaadIeaaOqaaKqzGeGaeqyXduhaaaaa@4587@ is the Reynolds number, De= λ V w 2 υ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb Gaamyzaiabg2da9KqbaoaalaaakeaajugibiabeU7aSjaadAfajuaG daqhaaWcbaqcLbmacaWG3baaleaajugWaiaaikdaaaaakeaajugibi abew8a1baaaaa@443E@ is the Deborah’s number, M 2 = σ B 0 2 H μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacqGH9aqpjuaGdaWc aaGcbaqcLbsacqaHdpWCcaWGcbqcfa4aa0baaSqaaKqzadGaaGimaa WcbaqcLbmacaaIYaaaaKqzGeGaamisaaGcbaqcLbsacqaH8oqBaaaa aa@4795@  is the Hartman parameter, Da= K p H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamirai aadggacqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGlbqcfa4aaSbaaSqa aKqzadGaamiCaaWcbeaaaOqaaKqzGeGaamisaaaaaaa@3F84@  is the Darcy’s number. For Rew>0 corresponds to suction flow while Rew<0 correspond to injection flow respectively.

Equ. (13) is a third-order differential equation with four boundary conditions. Through a creative differentiation of Eq. (12). Hence introducing fourth order equation as:

f iv ( M 2 + 1 Da ) f + ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w ( f f f f )+De ( 1ϕ ) 2.5 ( 2 f 2 f '' 2f f 2 + f 2 f iv )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaWbaaSqabeaajugibiaadMgacaWG2baaaiabgkHiTKqbaoaa bmaakeaajugibiaad2eajuaGdaahaaWcbeqaaKqzadGaaGOmaaaaju gibiabgUcaRKqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaamir aiaadggaaaaakiaawIcacaGLPaaajugibiqadAgagaGbaiabgUcaRK qbaoaabmaakeaajugibiaaigdacqGHsislcqaHvpGzaOGaayjkaiaa wMcaaKqbaoaaCaaaleqabaqcLbmacaaIYaGaaiOlaiaaiwdaaaqcfa 4aaeWaaOqaaKqbaoaabmaakeaajugibiaaigdacqGHsislcqaHvpGz aOGaayjkaiaawMcaaKqzGeGaey4kaSIaeqy1dywcfa4aaSaaaOqaaK qzGeGaeqyWdixcfa4aaSbaaSqaaKqzadGaam4CaaWcbeaaaOqaaKqz GeGaeqyWdixcfa4aaSbaaSqaaKqzadGaamOzaaWcbeaaaaaakiaawI cacaGLPaaajugibiaadkfacaWGLbqcfa4aaSbaaSqaaKqzadGaam4D aaWcbeaajuaGdaqadaGcbaqcLbsaceWGMbGbauaaceWGMbGbayaacq GHsislcaWGMbGabmOzayaasaaakiaawIcacaGLPaaajugibiabgUca RiaadseacaWGLbqcfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiabew 9aMbGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaajugWaiaaikdacaGG UaGaaGynaaaajuaGdaqadaGcbaqcLbsacaaIYaGabmOzayaafaqcfa 4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacaWGMbqcfa4aaWbaaSqa beaajugibiaacEcacaGGNaaaaiabgkHiTiaaikdacaWGMbGabmOzay aagaqcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacqGHRaWkcaWG Mbqcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacaWGMbqcfa4aaW baaSqabeaajugWaiaadMgacaWG2baaaaGccaGLOaGaayzkaaqcLbsa cqGH9aqpcaaIWaaaaa@A2DD@ (14)

The above Eq. (14) study satisfies all the four boundary conditions

Principles of homotopy perturbation method

The following equation is considered in explaining the fundamentals of the homotopy perturbation method [10]

 

A(u)f(r)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyqai aacIcacaWG1bGaaiykaiabgkHiTiaadAgacaGGOaGaamOCaiaacMca cqGH9aqpcaaIWaaaaa@3F7B@ rΩ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOCai abgIGiolabfM6axbaa@3A83@ (15)

Utilizing the boundary condition

B(u, u η )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOqai aacIcacaWG1bGaaiilaKqbaoaalaaakeaajugibiabgkGi2kaadwha aOqaaKqzGeGaeyOaIyRaeq4TdGgaaiaacMcacqGH9aqpcaaIWaaaaa@4346@ rΓ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOCai abgIGiolabfo5ahbaa@3A5D@ (16)

A is the general differential operator, B is the boundary operator, f(r) is the analytical function and Γ is the boundary domain of Ω. Separating A into two components of linear and nonlinear terms L and N respectively. The Eq. (21) is reconstructed as

                                                  

L(u)+N(u)f(r)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitai aacIcacaWG1bGaaiykaiabgUcaRiaad6eacaGGOaGaamyDaiaacMca cqGHsislcaWGMbGaaiikaiaadkhacaGGPaGaeyypa0JaaGimaaaa@438E@ rΩ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOCai abgIGiolabfM6axbaa@3A83@ (17)

Homotopy perturbation structure takes the form

                                                      

    H(v,p)=(1p)[L(v)L( u 0 )]+P[A(v)f(r)]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamisai aacIcacaWG2bGaaiilaiaadchacaGGPaGaeyypa0Jaaiikaiaaigda cqGHsislcaWGWbGaaiykaiaacUfacaGGmbGaaiikaiaacAhacaGGPa GaeyOeI0IaaiitaiaacIcacaGG1bqcfa4aaSbaaSqaaKqzadGaaGim aaWcbeaajugibiaacMcacaGGDbGaey4kaSIaamiuaiaacUfacaWGbb GaaiikaiaadAhacaGGPaGaeyOeI0IaamOzaiaacIcacaWGYbGaaiyk aiaac2facqGH9aqpcaaIWaaaaa@58E3@ (18)

Where

v(r,p):Ωx[0,1]R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamODai aacIcacaWGYbGaaiilaiaadchacaGGPaGaaiOoaiabfM6axjaadIha caGGBbGaaGimaiaacYcacaaIXaGaaiyxaiabgkHiTiaadkfaaaa@445C@ (19)

p ϵ(0, 1) is the embedding parameter and U0 is taken as the initial term that satisfies boundary condition. The power series of Eq. (24) can be expressed as:

                                                     

v= v 0 +p v 1 + p 2 v 2 + p 3 v 3 +... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamODai abg2da9iaadAhajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqzGeGa ey4kaSIaamiCaiaadAhajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaK qzGeGaey4kaSIaamiCaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqz GeGaamODaKqbaoaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqGHRa WkcaWGWbqcfa4aaWbaaSqabeaajugWaiaaiodaaaqcLbsacaWG2bqc fa4aaSbaaSqaaKqzadGaaG4maaWcbeaajugibiabgUcaRiaac6caca GGUaGaaiOlaaaa@5843@ (20)

Most appropriate solution for the problem takes the form

 

u= lim p1 ( v 0 +p v 1 + p 2 v 2 + p 3 v 3 +...)= v 0 + v 1 + v 2 + v 3 +... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyDai abg2da9KqbaoaaxabakeaajugibiGacYgacaGGPbGaaiyBaaWcbaqc LbsacaWGWbGaeyOKH4QaaGymaaWcbeaajugibiaacIcacaWG2bqcfa 4aaSbaaSqaaKqzadGaaGimaaWcbeaajugibiabgUcaRiaadchacaWG 2bqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabgUcaRiaadc hajuaGdaahaaWcbeqaaKqzadGaaGOmaaaajugibiaadAhajuaGdaWg aaWcbaqcLbmacaaIYaaaleqaaKqzGeGaey4kaSIaamiCaKqbaoaaCa aaleqabaqcLbmacaaIZaaaaKqzGeGaamODaKqbaoaaBaaaleaajugW aiaaiodaaSqabaqcLbsacqGHRaWkcaGGUaGaaiOlaiaac6cacaGGPa Gaeyypa0JaamODaKqbaoaaBaaaleaajugWaiaaicdaaSqabaqcLbsa cqGHRaWkcaWG2bqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibi abgUcaRiaadAhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGa ey4kaSIaamODaKqbaoaaBaaaleaajugWaiaaiodaaSqabaqcLbsacq GHRaWkcaGGUaGaaiOlaiaac6caaaa@7A17@ (21)

Application of the homotopy perturbation method to the flow problem

The homotopy pertubation method which is an analytical scheme for providing approximate solutions to the ordinary differential equations, is adopted in generating solutions to the coupled ordinary nonlinear differential e quation .Upon constructing the homotopy, the Eqs. (11)- (12) can be expressed as

(1p)( f iv )+p( f iv ( M 2 + 1 Da ) f + ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w ( f f f f ) +De ( 1ϕ ) 2.5 ( 2 f 2 f '' 2f f 2 + f 2 f iv )=0 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaiikai aaigdacqGHsislcaWGWbGaaiykaKqbaoaabmaakeaajugibiaadAga juaGdaahaaWcbeqaaKqzadGaamyAaiaadAhaaaaakiaawIcacaGLPa aajugibiabgUcaRiaadchajuaGdaqadaqcLbsaeaqabOqaaKqzGeGa amOzaKqbaoaaCaaaleqabaqcLbmacaWGPbGaamODaaaajugibiabgk HiTKqbaoaabmaakeaajugibiaad2eajuaGdaahaaWcbeqaaKqzadGa aGOmaaaajugibiabgUcaRKqbaoaalaaakeaajugibiaaigdaaOqaaK qzGeGaamiraiaadggaaaaakiaawIcacaGLPaaajugibiqadAgagaGb aiabgUcaRKqbaoaabmaakeaajugibiaaigdacqGHsislcqaHvpGzaO GaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLbmacaaIYaGaaiOlaiaa iwdaaaqcfa4aaeWaaOqaaKqbaoaabmaakeaajugibiaaigdacqGHsi slcqaHvpGzaOGaayjkaiaawMcaaKqzGeGaey4kaSIaeqy1dywcfa4a aSaaaOqaaKqzGeGaeqyWdixcfa4aaSbaaSqaaKqzadGaam4CaaWcbe aaaOqaaKqzGeGaeqyWdixcfa4aaSbaaSqaaKqzadGaamOzaaWcbeaa aaaakiaawIcacaGLPaaajugibiaadkfacaWGLbqcfa4aaSbaaSqaaK qzadGaam4DaaWcbeaajuaGdaqadaGcbaqcLbsaceWGMbGbauaaceWG MbGbayaacqGHsislcaWGMbGabmOzayaasaaakiaawIcacaGLPaaaae aajugibiabgUcaRiaadseacaWGLbqcfa4aaeWaaOqaaKqzGeGaaGym aiabgkHiTiabew9aMbGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaaju gWaiaaikdacaGGUaGaaGynaaaajuaGdaqadaGcbaqcLbsacaaIYaGa bmOzayaafaqcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacaWGMb qcfa4aaWbaaSqabeaajugibiaacEcacaGGNaaaaiabgkHiTiaaikda caWGMbGabmOzayaagaqcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLb sacqGHRaWkcaWGMbqcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsa caWGMbqcfa4aaWbaaSqabeaajugWaiaadMgacaWG2baaaaGccaGLOa GaayzkaaqcLbsacqGH9aqpcaaIWaaaaOGaayjkaiaawMcaaKqzGeGa eyypa0JaaGimaaaa@B772@ (22)

Taking power series of velocity and rotation fields yields

f= f 0 + p 1 f 1 + p 2 f 2 + p 3 f 3 .+.. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzai abg2da9iaadAgajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqzGeGa ey4kaSIaamiCaKqbaoaaCaaaleqabaqcLbmacaaIXaaaaKqzGeGaam OzaKqbaoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGHRaWkcaWG Wbqcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacaWGMbqcfa4aaS baaSqaaKqzadGaaGOmaaWcbeaajugibiabgUcaRiaadchajuaGdaah aaWcbeqaaKqzadGaaG4maaaajugibiaadAgajuaGdaWgaaWcbaqcLb macaaIZaaaleqaaKqzGeGaaiOlaiabgUcaRiaac6cacaGGUaaaaa@5B26@ (23)       

Substituting Eq. (23) into (22) and collecting the like terms of the various order yields

p 0 : f 0 iv =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiCaK qbaoaaCaaaleqabaqcLbmacaaIWaaaaKqzGeGaaiOoaiaadAgajuaG daqhaaWcbaqcLbmacaaIWaaaleaajugWaiaadMgacaWG2baaaKqzGe Gaeyypa0JaaGimaaaa@445E@ (24)

p 1 : f 1 iv ( M 2 + 1 Da ) f 0 + ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w ( f 0 f 0 f 0 f 0 )      +De ( 1ϕ ) 2.5 ( 2 f 2 0 f 0 2 f 0 f 0 2 + f 0 2 f 0 iv )=0=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsaca WGWbqcfa4aaWbaaSqabeaajugWaiaaigdaaaqcLbsacaGG6aGaamOz aKqbaoaaDaaaleaajugWaiaaigdaaSqaaKqzadGaamyAaiaadAhaaa qcLbsacqGHsisljuaGdaqadaGcbaqcLbsacaWGnbqcfa4aaWbaaSqa beaajugWaiaaikdaaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsaca aIXaaakeaajugibiaadseacaWGHbaaaaGccaGLOaGaayzkaaqcLbsa ceWGMbGbayaajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqzGeGaey 4kaSscfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiabew9aMbGccaGL OaGaayzkaaqcfa4aaWbaaSqabeaajugWaiaaikdacaGGUaGaaGynaa aajuaGdaqadaGcbaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiab ew9aMbGccaGLOaGaayzkaaqcLbsacqGHRaWkcqaHvpGzjuaGdaWcaa GcbaqcLbsacqaHbpGCjuaGdaWgaaWcbaqcLbmacaWGZbaaleqaaaGc baqcLbsacqaHbpGCjuaGdaWgaaWcbaqcLbmacaWGMbaaleqaaaaaaO GaayjkaiaawMcaaKqzGeGaamOuaiaadwgajuaGdaWgaaWcbaqcLbma caWG3baaleqaaKqbaoaabmaakeaajugibiqadAgagaqbaKqbaoaaBa aaleaajugWaiaaicdaaSqabaqcLbsaceWGMbGbayaajuaGdaWgaaWc baqcLbmacaaIWaaaleqaaKqzGeGaeyOeI0IaamOzaKqbaoaaBaaale aajugWaiaaicdaaSqabaqcLbsaceWGMbGbaibajuaGdaWgaaWcbaqc LbmacaaIWaaaleqaaaGccaGLOaGaayzkaaaabaqcLbsacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiabgUcaRiaadseacaWGLbqcfa4aaeWa aOqaaKqzGeGaaGymaiabgkHiTiabew9aMbGccaGLOaGaayzkaaqcfa 4aaWbaaSqabeaajugWaiaaikdacaGGUaGaaGynaaaajuaGdaqadaGc baqcLbsacaaIYaGabmOzayaafaqcfa4aaWbaaSqabeaajugWaiaaik daaaqcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaajugibiqadAgagaGb aKqbaoaaBaaaleaajugWaiaaicdaaSqabaqcLbsacqGHsislcaaIYa GaamOzaKqbaoaaBaaaleaajugWaiaaicdaaSqabaqcLbsaceWGMbGb ayaajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqbaoaaCaaaleqaba qcLbmacaaIYaaaaKqzGeGaey4kaSIaamOzaKqbaoaaDaaaleaajugW aiaaicdaaSqaaKqzadGaaGOmaaaajugibiaadAgajuaGdaqhaaWcba qcLbmacaaIWaaaleaajugWaiaadMgacaWG2baaaaGccaGLOaGaayzk aaqcLbsacqGH9aqpcaaIWaGaeyypa0JaaGimaaaaaa@CC92@ (25)                                                     

p 2 : f 2 iv ( M 2 + 1 Da ) f 1 + ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w ( f 1 f 0 + f 0 f 1 f 1 f 0 f 0 f 1 )      +De ( 1ϕ ) 2.5 ( 2 f 2 0 f 1 2 f 1 f 0 2 + f 0 2 f 1 iv )=0=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsaca WGWbqcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacaGG6aGaamOz aKqbaoaaDaaaleaajugWaiaaikdaaSqaaKqzadGaamyAaiaadAhaaa qcLbsacqGHsisljuaGdaqadaGcbaqcLbsacaWGnbqcfa4aaWbaaSqa beaajugWaiaaikdaaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsaca aIXaaakeaajugibiaadseacaWGHbaaaaGccaGLOaGaayzkaaqcLbsa ceWGMbGbayaajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaey 4kaSscfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiabew9aMbGccaGL OaGaayzkaaqcfa4aaWbaaSqabeaajugWaiaaikdacaGGUaGaaGynaa aajuaGdaqadaGcbaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTiab ew9aMbGccaGLOaGaayzkaaqcLbsacqGHRaWkcqaHvpGzjuaGdaWcaa GcbaqcLbsacqaHbpGCjuaGdaWgaaWcbaqcLbmacaWGZbaaleqaaaGc baqcLbsacqaHbpGCjuaGdaWgaaWcbaqcLbmacaWGMbaaleqaaaaaaO GaayjkaiaawMcaaKqzGeGaamOuaiaadwgajuaGdaWgaaWcbaqcLbma caWG3baaleqaaKqbaoaabmaakeaajugibiqadAgagaqbaKqbaoaaBa aaleaajugWaiaaigdaaSqabaqcLbsaceWGMbGbayaajuaGdaWgaaWc baqcLbmacaaIWaaaleqaaKqzGeGaey4kaSIabmOzayaafaqcfa4aaS baaSqaaKqzadGaaGimaaWcbeaajugibiqadAgagaGbaKqbaoaaBaaa leaajugWaiaaigdaaSqabaqcLbsacqGHsislcaWGMbqcfa4aaSbaaS qaaKqzadGaaGymaaWcbeaajugibiaadAgajuaGdaWgaaWcbaqcLbma caaIWaaaleqaaKqbaoaaCaaaleqabaqcLbsacWaGGBOmGiQaaGzaVl adacUHYaIOcaaMb8Uamai4gkdiIcaacqGHsislcaWGMbqcfa4aaSba aSqaaKqzadGaaGimaaWcbeaajugibiqadAgagaGeaKqbaoaaBaaale aajugWaiaaigdaaSqabaaakiaawIcacaGLPaaaaeaajugibiaabcca caqGGaGaaeiiaiaabccacaqGGaGaey4kaSIaamiraiaadwgajuaGda qadaGcbaqcLbsacaaIXaGaeyOeI0Iaeqy1dygakiaawIcacaGLPaaa juaGdaahaaWcbeqaaKqzadGaaGOmaiaac6cacaaI1aaaaKqbaoaabm aakeaajugibiaaikdaceWGMbGbauaajuaGdaahaaWcbeqaaKqzadGa aGOmaaaajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqzGeGabmOzay aagaqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabgkHiTiaa ikdacaWGMbqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiqadA gagaGbaKqbaoaaBaaaleaajugWaiaaicdaaSqabaqcfa4aaWbaaSqa beaajugWaiaaikdaaaqcLbsacqGHRaWkcaWGMbqcfa4aa0baaSqaaK qzadGaaGimaaWcbaqcLbmacaaIYaaaaKqzGeGaamOzaKqbaoaaDaaa leaajugWaiaaigdaaSqaaKqzadGaamyAaiaadAhaaaaakiaawIcaca GLPaaajugibiabg2da9iaaicdacqGH9aqpcaaIWaaaaaa@EBA7@ (26)

The boundary conditions are

η=0: f 0 = f 1 = f 2 =...=0, f 0 = f 1 = f 2 =...=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH3o aAcqGH9aqpcaaIWaGaaiOoaiaadAgajuaGdaWgaaWcbaqcLbmacaaI WaaaleqaaKqzGeGaeyypa0JaamOzaKqbaoaaBaaaleaajugWaiaaig daaSqabaqcLbsacqGH9aqpcaWGMbqcfa4aaSbaaSqaaKqzadGaaGOm aaWcbeaajugibiabg2da9iaac6cacaGGUaGaaiOlaiabg2da9iaaic dacaGGSaGaaGzbVlqadAgagaGbaKqbaoaaBaaaleaajugWaiaaicda aSqabaqcLbsacqGH9aqpceWGMbGbayaajuaGdaWgaaWcbaqcLbmaca aIXaaaleqaaKqzGeGaeyypa0JabmOzayaagaqcfa4aaSbaaSqaaKqz adGaaGOmaaWcbeaajugibiabg2da9iaac6cacaGGUaGaaiOlaiabg2 da9iaaicdaaaa@63D4@ (27)
η=1: f 0 = f 1 = f 2 =...=1, f 0 =k f 0 , f 1 =k f 1 , f 2 =k f 2 ,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH3o aAcqGH9aqpcaaIXaGaaiOoaiaadAgajuaGdaWgaaWcbaqcLbmacaaI WaaaleqaaKqzGeGaeyypa0JaamOzaKqbaoaaBaaaleaajugWaiaaig daaSqabaqcLbsacqGH9aqpcaWGMbqcfa4aaSbaaSqaaKqzadGaaGOm aaWcbeaajugibiabg2da9iaac6cacaGGUaGaaiOlaiabg2da9iaaig dacaGGSaGaaGzbVlqadAgagaqbaKqbaoaaBaaaleaajugWaiaaicda aSqabaqcLbsacqGH9aqpcqGHsislcaWGRbGabmOzayaagaqcfa4aaS baaSqaaKqzadGaaGimaaWcbeaajugibiaacYcacaaMe8UaaGjbVlqa dAgagaqbaKqbaoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqGH9a qpcqGHsislcaWGRbGaamOzaKqbaoaaBaaaleaajugWaiaaigdaaSqa baqcfa4aaWbaaSqabeaajugibiadacUHYaIOcaaMb8Uamai4gkdiIc aacaGGSaGaaGjbVlaaysW7ceWGMbGbauaajuaGdaWgaaWcbaqcLbma caaIYaaaleqaaKqzGeGaeyypa0JaeyOeI0Iaam4AaiaadAgajuaGda WgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaaCaaaleqabaqcLbsacWaG GBOmGiQaaGzaVladacUHYaIOaaGaaiilaiaac6cacaGGUaGaaiOlaa aa@8C9D@

On solving Eq. (24) applying the boundary conditions yields

f 0 (η)= 1 2(3k+1) η 3 + 3(2k+1) 2(3k+1) η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaajugibiaacIcacqaH3oaA caGGPaGaeyypa0JaeyOeI0scfa4aaSaaaOqaaKqzGeGaaGymaaGcba qcLbsacaaIYaGaaiikaiaaiodacaWGRbGaey4kaSIaaGymaiaacMca aaGaeq4TdGwcfa4aaWbaaSqabeaajugWaiaaiodaaaqcLbsacqGHRa WkjuaGdaWcaaGcbaqcLbsacaaIZaGaaiikaiaaikdacaWGRbGaey4k aSIaaGymaiaacMcaaOqaaKqzGeGaaGOmaiaacIcacaaIZaGaam4Aai abgUcaRiaaigdacaGGPaaaaiabeE7aObaa@5B8B@ (26)

Also, solving Eq. (25) applying the corresponding boundary conditions yields

f 1 (η)= 1 2 { 5 336 De ( 1ϕ ) 2.5 η 9 (3k+1) 3 2 ( 0.0071649 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w 0.042857De 0.0023810 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w 0.085714Dek ) η 7 27 k 3 +27 k 2 +9k+1 +3 ( 0.15( M 2 + 1 Da ) k 2 0.1( M 2 + 1 Da )k0.016667( M 2 + 1 Da )+0.3De k 2 +0.3De ( 1ϕ ) 2.5 k+0.075De ( 1ϕ ) 2.5 ) η 5 27 k 3 +27 k 2 +9k+1 1 420 [ ( 378De ( 1ϕ ) 2.5 k 3 1890( M 2 + 1 Da ) k 3 189 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w k 2 +2268DDe ( 1ϕ ) 2.5 2 1638( M 2 + 1 Da ) k 2 90 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w k462( M 2 + 1 Da )k+468De ( 1ϕ ) 2.5 k ) η 3 1+54 k 2 +12k+108 k 3 +81 k 4 + ( 9 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w +52De ( 1ϕ ) 2.5 42( M 2 + 1 Da ) ) 1+54 k 2 +12k+108 k 3 +81 k 4 ] + 1 560 [ ( 8 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w 96 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w k1764( M 2 + 1 Da ) k 3 +1440De ( 1ϕ ) 2.5 k 2 +77De ( 1ϕ ) 2.5 k364( M 2 + 1 Da )k )η 1+54 k 2 +12k+108 k 3 +81 k 4 + ( 1428( M 2 + 1 Da ) k 2 +3528De ( 1ϕ ) 2.5 k 3 216 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w k 2 28( M 2 + 1 Da )+7De ( 1ϕ ) 2.5 )η 1+54 k 2 +12k+108 k 3 +81 k 4 ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaaeaaju gibiaadAgajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaaiik aiabeE7aOjaacMcacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaake aajugibiaaikdaaaqcfa4aaiWaaKqzGeabaeqakeaajuaGdaWcaaGc baqcLbsacaaI1aaakeaajugibiaaiodacaaIZaGaaGOnaaaajuaGda WcaaGcbaqcLbsacaWGebGaamyzaKqbaoaabmaakeaajugibiaaigda cqGHsislcqaHvpGzaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLb macaaIYaGaaiOlaiaaiwdaaaqcLbsacqaH3oaAjuaGdaahaaWcbeqa aKqzadGaaGyoaaaaaOqaaKqzGeGaaiikaiaaiodacaWGRbGaey4kaS IaaGymaiaacMcaaaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaaG4maaGc baqcLbsacaaIYaaaaKqbaoaalaaakeaajuaGdaqadaqcLbsaeaqabO qaaKqzGeGaeyOeI0IaaGimaiaac6cacaaIWaGaaGimaiaaiEdacaaI 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By the definition of homotopy perturbation method, we have

f(η)= 1 2(3k+1) η 3 + 3(2k+1) 2(3k+1) η+ 1 2 { 5 336 De ( 1ϕ ) 2.5 η 9 (3k+1) 3 2 ( 0.0071649 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w 0.042857De 0.0023810 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w 0.085714Dek ) η 7 27 k 3 +27 k 2 +9k+1 +3 ( 0.15( M 2 + 1 Da ) k 2 0.1( M 2 + 1 Da )k0.016667( M 2 + 1 Da )+0.3De k 2 +0.3De ( 1ϕ ) 2.5 k+0.075De ( 1ϕ ) 2.5 ) η 5 27 k 3 +27 k 2 +9k+1 1 420 [ ( 378De ( 1ϕ ) 2.5 k 3 1890( M 2 + 1 Da ) k 3 189 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w k 2 +2268DDe ( 1ϕ ) 2.5 2 1638( M 2 + 1 Da ) k 2 90 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w k462( M 2 + 1 Da )k+468De ( 1ϕ ) 2.5 k ) η 3 1+54 k 2 +12k+108 k 3 +81 k 4 + ( 9 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w +52De ( 1ϕ ) 2.5 42( M 2 + 1 Da ) ) 1+54 k 2 +12k+108 k 3 +81 k 4 ] + 1 560 [ ( 8 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w 96 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w k1764( M 2 + 1 Da ) k 3 +1440De ( 1ϕ ) 2.5 k 2 +77De ( 1ϕ ) 2.5 k364( M 2 + 1 Da )k )η 1+54 k 2 +12k+108 k 3 +81 k 4 + ( 1428( M 2 + 1 Da ) k 2 +3528De ( 1ϕ ) 2.5 k 3 216 ( 1ϕ ) 2.5 ( ( 1ϕ )+ϕ ρ s ρ f )R e w k 2 28( M 2 + 1 Da )+7De ( 1ϕ ) 2.5 )η 1+54 k 2 +12k+108 k 3 +81 k 4 ] } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaaeaaju gibiaadAgacaGGOaGaeq4TdGMaaiykaiabg2da9iabgkHiTKqbaoaa laaakeaajugibiaaigdaaOqaaKqzGeGaaGOmaiaacIcacaaIZaGaam 4AaiabgUcaRiaaigdacaGGPaaaaiabeE7aOLqbaoaaCaaaleqabaqc LbmacaaIZaaaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGaaG4mai aacIcacaaIYaGaam4AaiabgUcaRiaaigdacaGGPaaakeaajugibiaa ikdacaGGOaGaaG4maiaadUgacqGHRaWkcaaIXaGaaiykaaaacqaH3o aAcqGHRaWkjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaikda aaqcfa4aaiWaaKqzGeabaeqakeaajuaGdaWcaaGcbaqcLbsacaaI1a aakeaajugibiaaiodacaaIZaGaaGOnaaaajuaGdaWcaaGcbaqcLbsa caWGebGaamyzaKqbaoaabmaakeaajugibiaaigdacqGHsislcqaHvp 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GccaGLOaGaayzkaaqcfa4aaWbaaSqabeaajugWaiaaikdacaGGUaGa aGynaaaaaaGccaGLOaGaayzkaaqcLbsacqaH3oaAaOqaaKqzGeGaaG ymaiabgUcaRiaaiwdacaaI0aGaam4AaKqbaoaaCaaaleqabaqcLbma caaIYaaaaKqzGeGaey4kaSIaaGymaiaaikdacaWGRbGaey4kaSIaaG ymaiaaicdacaaI4aGaam4AaKqbaoaaCaaaleqabaqcLbmacaaIZaaa aKqzGeGaey4kaSIaaGioaiaaigdacaWGRbqcfa4aaWbaaSqabeaaju gWaiaaisdaaaaaaaaakiaawUfacaGLDbaaaaGaay5Eaiaaw2haaaaa aa@338D@ (28)

Results and discussion

The results of the homotopy perturbation method for the problem investigated have been with the results of the fourth-order Runge-Kutta Fehlberg numerical method (for the simplified case) as shown in Table 3. As observed from the Table, good agreements are established between the results of the numerical and homotopy analysis methods. Using copper nanoparticle and water, the results obtained from the analytical solution are shown graphically in Figures 2-9, when Rew = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabg2da9aaa@37AB@ 8, De = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabg2da9aaa@37AB@ 0.1, M = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabg2da9aaa@37AB@ 2, =0.1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabl+qiOjabg2da9iaaicdacaGGUaGaaGymaaaa@3AFB@ ,Da = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabg2da9aaa@37AB@ 2 and ϕ=0.01 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabew9aMjabg2da9iaaicdacaGGUaGaaGimaiaaigdaaaa@3C54@ , unless otherwise stated. Figure 2 depicts the influence of nanoparticle concentration ( ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaGcbaqcLbsacqaHvpGzaOGaayjkaiaawMcaaaaa@3A98@ on the flow process. As shown from the Figure, as the nanoparticle concentration increases, there is an increase in the velocity distribution. It is very important to indicate viscoelastic nature of the fluid. Therefore, the effects of Deborah’s number on the flow process are depicted in Figure 3. In the Figure, it illustrated in that increase in Deborah’s number (De) which illustrates the UCM as highly elastic fluid (such as polymeric melts) depicts decrease in fluid flow velocity. The influence of magnetic field parameter on flow of the UCM fluid under is depicted in Figure 4. As observed in the figure, the numerical increase of the magnetic or Hartmann parameter (M) shows decreasing velocity profile. This is because the applied magnetic field produces a damping effect (Lorentz force) on the flow process. This damping affects increases as the quantitative or numerical value of the Hartmann number increases. It should be noted that the effects magnetic field parameter is maximum towards the upper flow channel. In order to shown the effect of the permeability of the porous medium on the flow, effect of Darcy parameter (Da) on fluid transport is illustrated in Figure 5. Increasing Darcy number demonstrates increasing velocity profile as shown in the figure.

  η

     RKFNM

 HPM

 |RKFNM-HPM|

0

0

0

0

0.05

0.070154

0.0701

5.46E-05

0.1

0.139997

0.139899

9.76E-05

0.15

0.209217

0.209199

1.77E-05

0.2

0.259219

0.259201

1.76E-05

0.25

0.344546

0.344499

4.76E-05

0.3

0.410038

0.409999

3.96E-05

0.35

0.473672

0.473598

7.37E-05

0.4

0.535148

0.535102

4.59E-05

0.45

0.594153

0.594098

5.51E-05

0.5

0.650402

0.650348

5.37E-05

0.55

0.7036

0.703084

5.16E-05

0.6

0.75345

0.753398

5.19E-05

0.65

0.79968

0.799599

8.18E-05

0.7

0.842013

0.841977

3.56E-05

0.75

0.880181

0.88013

5.06E-05

0.8

0.913929

0.913899

2.94E-05

0.85

0.94301

0.942976

3.41E-05

0.9

0.967193

0.967159

3.45E-05

0.95

0.986257

0.9862

5.74E-05

1

1

1

0

Table 3 Comparison of results of numerical and homotopy analysis method for f(η), when De=0.1, Da-1= ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp Gzaaa@384D@ =M=0, K=0.1, Rew=4

Figure 2 Effecct of nanoparticle concentration number ( ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp Gzaaa@384D@ ) on the axial velocity of the flow process.

Figure 3 Effect of Deborah’s number (De) on the axial velocity of the flow process.

Figure 4 Effect of Hartmann parameter (M) on the axial velocity of the flow process.

Figure 5 Effect of Darcy’s number (Da) on the axial velocity of the flow process.

Figure 6 Effect of slip parameter (k) on the axial velocity of the flow process.

Figure 7 Effect of Reynold’s number (Rew) on the axial velocity of the flow process.

Figure 8 Effect of Reynold’s number (Rew) on the radial velocity of the flow process.

Figure 9 Effect of Hartman parameter (M) on the radial velocity of the flow process.

Figure 6 shows the effect of fluid slip parameter (k) on the velocity of the fluid flow. It should be noted that the slip parameter depicts that the fluid velocity at the boundary is not at equal velocity with fluid particles closest to flow boundary due to large variance in macro and micro fluid flow. As observed from the Figure 6, increasing the slip parameter leads to decreasing velocity distributions of the process. In order to show the relative significance of the inertia effect as compared to the viscous effect, the effect of Reynolds number on the flow phenomena is illustrated in the Figure 7. It is established form the graphical display that increasing Reynolds number (Rew) causes decrease in flow profile which effect is maximum towards the upper plate. Figure 8 shows the effect of Reynolds number on the radial velocity component of the flow. It is shown that increasing the Reynolds number causes decrease in velocity distribution but as flow reaches the mid plate around η=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeE7aOjabg2da9iaaicdacaGGUaGaaGynaaaa@3B82@ (not determined accurately) an increasing velocity distribution is seen. However, effect is minimal towards the upper plate. Also, influence of magnetic field on radial velocity is depicted in Figure 9, as shown significant increase in velocity is seen due to quantitative increase of Hartmann parameter (M) towards the lower plate while as upper plate is approached a reverse trend is observed.

Conclusion

In this work, homotopy perturbation method is used to analyze the flow of an upper convective Maxwell (UCM) nanofluid through a permeable microchannel embedded in a porous medium and under the influence of slip condition has been presented. Important fluid parameter effect such as Deborah’s number, Darcy parameter and Hartman parameter on the fluid flow was investigated. that increase in slip parameter, nanoparticle concentration and Darcy number lead to increase in the velocity of the upper-convected Maxwell fluid while increase in Deborah’s, Hartmann and Reynold numbers decrease the fluid flow velocity towards the lower plate but as the upper plate is approached, a reverse trend is observed. The results obtained in this work can be used to further the applications of UCM fluid in biomedical, astrophysics, geosciences etc.

Acknowledgements

None.

Conflict of interest

The authors declare that there is no conflict of interest.

Nomenclature

Re w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGsb GaaiyzaKqbaoaaBaaaleaajugWaiaadEhaaSqabaaaaa@3B35@                Reynolds number

M                    Hartman parameter

K                     Slip parameter

De                   Deborah’s number

HAM                 Homotopy analysis method

MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqWIpe cAaaa@37AE@                       Auxilliary parameter

v * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaWbaaSqabeaajugWaiaacQcaaaaaaa@3A17@                       y axis velocity component

u * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG1b qcfa4aaWbaaSqabeaajugWaiaacQcaaaaaaa@3A16@                       x axis velocity component

x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b aaaa@3782@                       Dimensionless horizontal coordinate

y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5b aaaa@3783@                       Dimensionless vertical coordinate

x * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b qcfa4aaWbaaSqabeaajugWaiaacQcaaaaaaa@3A19@                       Distance in x axis parallel to plate

y * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG5b qcfa4aaWbaaSqabeaajugWaiaacQcaaaaaaa@3A1A@                       Distance in y axis parallel to plate

Da                     Darcy number

Symbols

ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbp GCaaa@3845@                       Fluid density

λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBaaa@3839@                       Relaxation time

υ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDaaa@384C@                       Kinematic viscosity

β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo Gyaaa@3826@                       Sliding friction

ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp Gzaaa@384D@                       Nanoparticle concentration

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