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Open Access Journal of
eISSN: 2641-9335

Mathematical and Theoretical Physics

Research Article Volume 1 Issue 5

Laminar flow of Newtonian liquids in ducts of rectangular cross-section a model for both physics and mathematics

Franck Delplace

ESI Group Scientific Committee, France

Correspondence: Franck Delplace, ESI Group Scientific Committee, 100 Av. De Suffren, Paris, France

Received: August 16, 2018 | Published: September 24, 2018

Citation: Delplace F. Laminar flow of Newtonian liquids in ducts of rectangular cross-section a model for both physics and mathematics. Open Acc J Math Theor Phy. 2018;1(5):198-201 DOI: 10.15406/oajmtp.2018.01.00034

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Abstract

In this paper, we considered the laminar fully developed flow, of a Newtonian fluid, inducts of rectangular cross-section. Poisson’s partial differential equation Saint-Venant solution was used, to calculate Poiseuille number values whatever is rectangles aspect ratio. From these results, we considered limit cases of square duct and plane Poiseuille flow (infinite parallel plates). We showed there exists a rectangle equivalent to a circular cross-section for energy dissipation through viscous friction. Finally, we gave some mathematical consequences of this approach for odd integers zeta function calculations and Catalan’s constant.

Keywords: rectangular ducts, poisson’s equation, saint-venant solution, viscous friction, zeta function, catalan’s constant.

Introduction

Pipes used in most applications always have a circular cross-section. That is why Poiseuille law/equation is used to calculate the pressure drop produced by a liquid flowing in a pipe in the laminar flow regime. Poiseuille famous equation tells us that pressure drop is proportional to liquid flow-rate Q( m 3 . s 1 ). MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyuamaabmaapaqaa8qacaWGTbWdamaaCaaabeqcfasaa8qa caaIZaaaaKqbakaac6cacaWGZbWdamaaCaaabeqcfasaa8qacqGHsi slcaaIXaaaaaqcfaOaayjkaiaawMcaaiaac6caaaa@3FEC@ In Engineering, this relationship is expressed using dimensionless numbers: The Fanning friction factor (f/2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaiikaiaadAgacaGGVaGaaGOmaiaacMcaaaa@3973@ and the Reynolds number (Re): MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaiikaiaadkfacaWGLbGaaiykaiaacQdaaaa@3998@
f 2 = 8 Re MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiaadAgaa8aabaWdbiaaikdaaaGaeyypa0Za aSaaa8aabaWdbiaaiIdaa8aabaWdbiaadkfacaWGLbaaaaaa@3B8C@  (1)

With

f 2 = τ w ρ v ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiaadAgaa8aabaWdbiaaikdaaaGaeyypa0Za aSaaa8aabaWdbiabes8a09aadaWgaaqcfasaa8qacaWG3baapaqaba aajuaGbaWdbiabeg8aYjqadAhagaqea8aadaahaaqcfasabeaapeGa aGOmaaaaaaaaaa@40C4@  (2)

And

Re= ρ v ¯ D η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOuaiaadwgacqGH9aqpdaWcaaWdaeaapeGaeqyWdiNabmOD ayaaraGaamiraaWdaeaapeGaeq4TdGgaaaaa@3E1D@  (3)

In equation (2), D( m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiramaabmaapaqaa8qacaWGTbaacaGLOaGaayzkaaaaaa@3923@ is the pipe diameter, τ w ( Pa ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiXdq3damaaBaaajuaibaWdbiaadEhaaKqba+aabeaapeWa aeWaa8aabaWdbiaadcfacaWGHbaacaGLOaGaayzkaaaaaa@3CFF@ is the wall shear stress due to liquid friction on pipe wall. In the case of the perfectly symmetric circular cross-section, its value is identical whatever is the position along the perimeter giving the local value τ w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiXdq3damaaBaaajuaibaWdbiaadEhaaKqba+aabeaaaaa@398C@ equal to the mean value τ ¯ w .ρ(kg. m 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGafqiXdqNbaebapaWaaSbaaKqbGeaapeGaam4Daaqcfa4daeqa aiaac6cacqaHbpGCcaGGOaGaam4AaiaadEgacaGGUaGaamyBamaaCa aajuaibeqaaiabgkHiTiaaiodaaaqcfaOaaiykaaaa@4377@ is the liquid density and v ¯ (m. s 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmODayaaraGaaiikaiaad2gacaGGUaGaam4CamaaCaaajuai beqaaiabgkHiTiaaigdaaaqcfaOaaiykaaaa@3D4E@ its mean velocity calculated from the flow-rate Q( m 3 . s 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyuamaabmaapaqaa8qacaWGTbWdamaaCaaajuaibeqaa8qa caaIZaaaaKqbakaac6cacaWGZbWdamaaCaaajuaibeqaa8qacqGHsi slcaaIXaaaaaqcfaOaayjkaiaawMcaaaaa@3F3A@ measurement using v ¯ =Q/S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmODayaaraGaeyypa0Jaamyuaiaac+cacaWGtbaaaa@3A3A@ where S( m 2 )=π R 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4uamaabmaapaqaa8qacaWGTbWdamaaCaaajuaibeqaa8qa caaIYaaaaaqcfaOaayjkaiaawMcaaiabg2da9iabec8aWjaadkfapa WaaWbaaeqajuaibaWdbiaaikdaaaaaaa@3FB0@ is the cross-section area. From a balance between pressure drop ΔP( Pa ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaeiLdiaadcfadaqadaWdaeaapeGaamiuaiaadggaaiaawIca caGLPaaaaaa@3B12@ and viscous friction on pipe wall, it is possible to obtain a simple relationship between τ w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiXdq3damaaBaaajuaibaWdbiaadEhaaKqba+aabeaaaaa@398C@  and ΔP: MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaeiLdiaadcfacaGG6aaaaa@386D@

τ w = ΔPD 4L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiXdq3damaaBaaajuaibaWdbiaadEhaaKqba+aabeaapeGa eyypa0ZaaSaaa8aabaWdbiaabs5acaWGqbGaamiraaWdaeaapeGaaG inaiaadYeaaaaaaa@3F37@  (4)

In this equation, D( m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiramaabmaapaqaa8qacaWGTbaacaGLOaGaayzkaaaaaa@3923@ is the pipe diameter and L(m) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamitaiaacIcacaWGTbGaaiykaaaa@38DC@ is the pipe length where pressure drop ΔP MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaeiLdiaadcfaaaa@37AF@ is measured by use of a pressure sensor. The last parameter involved in equation (3) is well-known Newtonian liquid dynamic viscosity η( Pa.s ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4TdG2aaeWaa8aabaWdbiaadcfacaWGHbGaaiOlaiaadoha aiaawIcacaGLPaaacaGGUaaaaa@3D2B@

Finally, in equation (2), terms τ w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiXdq3damaaBaaajuaibaWdbiaadEhaaKqba+aabeaaaaa@398C@ and ρ v ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqyWdiNabmODayaaraWdamaaCaaajuaibeqaa8qacaaIYaaa aaaa@39BE@ are energy concentrations, respectively energy dissipated by viscous friction and kinetic energy introduced in the liquid by the pumping system. From these considerations, dimensionless number ΔP MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaeiLdiaadcfaaaa@37AF@ represents the percentage of energy concentration dissipated by the liquid at pipe wall. Of course, this mechanical energy is converted into heat.

From equation (1), we can form the product:

f/2 Re=Po MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOzaiaac+cacaaIYaGaaiiOaiaadkfacaWGLbGaeyypa0Ja amiuaiaad+gaaaa@3DCE@  (5)

The new dimensionless quantity Po MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gaaaa@3789@ is called Poiseuille number in honour of important Poiseuille work on liquids laminar flow. In the simple case of pipe flow, we have Po=8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpcaaI4aaaaa@3951@ (you sometimes find Po=16 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpcaaIXaGaaGOnaaaa@3A0A@ or Po=64 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpcaaI2aGaaGinaaaa@3A0D@ depending on how you define Fanning friction factor).

Now, the question is what is the situation when a duct has a non-circular cross-section?

As reported in famous Shah and London1 source book entitled “Laminar forced convection in ducts”, and experimentally or numerically verified by numerous authors,2 we generally have Po8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGHGjsUcaaI4aaaaa@3A12@ (we will explain why we say generally in the following of this paper).

An interesting and important geometry to investigate is rectangular ducts, from square cross-section exhibiting high symmetry properties (regular compact convex shape) to all rectangles of aspect ratio we called b/a. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbGaaiOlaaaa@38F2@ In fluid mechanics, we consider a limit case for rectangular geometries: the often called “Plane Poiseuille flow” corresponds to a rectangle such as ba MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiablUMi=iaadggaaaa@38EA@  giving b/a+. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbGaeyOKH4Qaey4kaSIaeyOhIuQaaiOl aaaa@3D32@  This ideal type of flow is highly symmetric like the flow in a pipe because small side length has no influence on the velocity field which remains the same along large side length. The following Figure 1 illustrates rectangular geometries considered in fluid mechanics.

Figure 1 Rectangular cross-section considered in fluid mechanics.

It is well-known that ideal plane Poiseuille flow gives a theoretical value Po=12. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpcaaIXaGaaGOmaiaac6caaaa@3AB8@

Moreover, as recently showed by Delplace,2 Po MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gaaaa@3789@ values could explain critical Reynolds number values for the change in the flow regime from laminar to transition and turbulent.

The objective of this paper is then to recall how Po MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gaaaa@3789@ values are obtained from Poisson partial differential equation (PDE) and also to try to explain why these results could be very important in both Physics and Mathematics.

Theory of laminar flow in rectangular ducts

Considering cartesian coordinates ( x,y,z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGPaVpaabmaapaqaa8qacaWG4bGaaiilaiaadMhacaGGSaGa amOEaaGaayjkaiaawMcaaaaa@3D4D@ with origin at the centre of the duct of rectangular cross-section, the fully established laminar flow of a Newtonian liquid is described by the following well-known Poisson equation:

2 v z x 2 + 2 v z y 2 = ΔP ηL    MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiabgkGi2+aadaahaaqcfasabeaapeGaaGOm aaaajuaGcaWG2bWdamaaBaaajuaibaWdbiaadQhaaKqba+aabeaaae aapeGaeyOaIyRaamiEamaaCaaabeqcfasaaiaaikdaaaaaaKqbakab gUcaRmaalaaapaqaa8qacqGHciITdaahaaqcfasabeaacaaIYaaaaK qbakaadAhapaWaaSbaaKqbGeaapeGaamOEaaWdaeqaaaqcfayaa8qa cqGHciITcaWG5bWaaWbaaKqbGeqabaGaaGOmaaaaaaqcfaOaeyypa0 JaaGPaVlabgkHiTiaaykW7caaMc8+aaSaaa8aabaWdbiaabs5acaWG qbaapaqaa8qacqaH3oaAcaWGmbaaaiaacckacaGGGcaaaa@58DB@  (6)

Solutions of this PDE depend on the boundary conditions (Dirichlet problem) and the general case of rectangles with aspect ratio b/a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbaaaa@3840@ can be solved by use of Saint-Venant method3 giving the velocity field: v z ( x,y ):  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamODa8aadaWgaaqcfasaa8qacaWG6baajuaGpaqabaWdbmaa bmaapaqaa8qacaWG4bGaaiilaiaadMhaaiaawIcacaGLPaaacaGG6a GaaiiOaaaa@3F0A@

v z ( x,y )= 16ΔP a 2 ηL π 3 n=1,3,5, + ( 1 ) ( n1 )/2 n 3 ( 1 ch( nπ 2a y ) ch( nπ 2a b )   )cos( nπ 2a x )   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamODa8aadaWgaaqaa8qacaWG6baapaqabaWdbmaabmaapaqa a8qacaWG4bGaaiilaiaadMhaaiaawIcacaGLPaaacqGH9aqpdaWcaa WdaeaapeGaaGymaiaaiAdacaqGuoGaamiuaiaadggapaWaaWbaaeqa baWdbiaaikdaaaaapaqaa8qacqaH3oaAcaWGmbGaeqiWda3damaaCa aabeqaa8qacaaIZaaaaaaadaGfWbqab8aabaWdbiaad6gacqGH9aqp caaIXaGaaiilaiaaiodacaGGSaGaaGynaiaacYcacqGHMacVa8aaba WdbiabgUcaRiabg6HiLcWdaeaapeGaeyyeIuoaamaalaaapaqaa8qa daqadaWdaeaapeGaeyOeI0IaaGymaaGaayjkaiaawMcaa8aadaahaa qabeaapeWaaeWaa8aabaWdbiaad6gacqGHsislcaaIXaaacaGLOaGa ayzkaaGaai4laiaaikdaaaaapaqaa8qacaWGUbWdamaaCaaabeqaa8 qacaaIZaaaaaaadaqadaWdaeaapeGaaGymaiabgkHiTmaalaaapaqa a8qacaWGJbGaamiAamaabmaapaqaa8qadaWcaaWdaeaapeGaamOBai abec8aWbWdaeaapeGaaGOmaiaadggaaaGaamyEaaGaayjkaiaawMca aaWdaeaapeGaam4yaiaadIgadaqadaWdaeaapeWaaSaaa8aabaWdbi aad6gacqaHapaCa8aabaWdbiaaikdacaWGHbaaaiaadkgaaiaawIca caGLPaaaaaGaaiiOaaGaayjkaiaawMcaaiaadogacaWGVbGaam4Cam aabmaapaqaa8qadaWcaaWdaeaapeGaamOBaiabec8aWbWdaeaapeGa aGOmaiaadggaaaGaamiEaaGaayjkaiaawMcaaiaacckacaGGGcaaaa@83BA@  (7)

This equation allows components of wall shear-rate:

( v z ( x,y ) x ) w and  ( v z ( x,y ) y ) w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaeWaa8aabaWdbmaalaaapaqaa8qacqGHciITcaWG2bWdamaa BaaajuaibaWdbiaadQhaaKqba+aabeaapeWaaeWaa8aabaWdbiaadI hacaGGSaGaamyEaaGaayjkaiaawMcaaaWdaeaapeGaeyOaIyRaamiE aaaaaiaawIcacaGLPaaapaWaaSbaaKqbGeaapeGaam4Daaqcfa4dae qaa8qacaWGHbGaamOBaiaadsgacaGGGcWaaeWaa8aabaWdbmaalaaa paqaa8qacqGHciITcaWG2bWdamaaBaaajuaibaWdbiaadQhaaKqba+ aabeaapeWaaeWaa8aabaWdbiaadIhacaGGSaGaamyEaaGaayjkaiaa wMcaaaWdaeaapeGaeyOaIyRaamyEaaaaaiaawIcacaGLPaaapaWaaS baaKqbGeaapeGaam4Daaqcfa4daeqaaaaa@5814@  (8)

To be calculated and then components of wall shear stress τ w ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiXdq3damaaBaaajuaibaWdbiaadEhaaKqba+aabeaapeWa aeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3C42@ and τ w ( y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiXdq3damaaBaaajuaibaWdbiaadEhaaKqba+aabeaapeWa aeWaa8aabaWdbiaadMhaaiaawIcacaGLPaaaaaa@3C43@ by use of the rheological equation of state:

τ w ( )=η  ( v z ( x,y ) ( ) ) w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiXdq3damaaBaaajuaibaWdbiaadEhaaKqba+aabeaapeWa aeWaa8aabaWdbiablgAjxbGaayjkaiaawMcaaiabg2da9iabeE7aOj aacckadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2kaadAhapaWa aSbaaKqbGeaapeGaamOEaaqcfa4daeqaa8qadaqadaWdaeaapeGaam iEaiaacYcacaWG5baacaGLOaGaayzkaaaapaqaa8qacqGHciITdaqa daWdaeaapeGaeSyOLCfacaGLOaGaayzkaaaaaaGaayjkaiaawMcaa8 aadaWgaaqcfasaa8qacaWG3baajuaGpaqabaaaaa@5250@  (9)

The average wall shear stress can then be calculated using classical integral mean value:

τ ¯ w = 1 a+b ( 0 a τ w ( x )dx+ 0 b τ w ( y )dy ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGafqiXdqNbaebapaWaaSbaaKqbGeaapeGaam4Daaqcfa4daeqa a8qacqGH9aqpdaWcaaWdaeaapeGaaGymaaWdaeaapeGaamyyaiabgU caRiaadkgaaaWaaeWaa8aabaWdbmaawahabeqcfaYdaeaapeGaaGim aaWdaeaapeGaamyyaaqcfa4daeaapeGaey4kIipaaiabes8a09aada Wgaaqcfasaa8qacaWG3baajuaGpaqabaWdbmaabmaapaqaa8qacaWG 4baacaGLOaGaayzkaaGaamizaiaadIhacqGHRaWkdaGfWbqabKqbG8 aabaWdbiaaicdaa8aabaWdbiaadkgaaKqba+aabaWdbiabgUIiYdaa cqaHepaDpaWaaSbaaKqbGeaapeGaam4Daaqcfa4daeqaa8qadaqada WdaeaapeGaamyEaaGaayjkaiaawMcaaiaadsgacaWG5baacaGLOaGa ayzkaaaaaa@5BF2@  (10)

We can also calculate from equation (7) the velocity mean value:

v ¯ = 1 a b   0 a 0 b v z ( x,y )dxdy MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmODayaaraGaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaWd biaadggacaGGGcGaamOyaaaacaGGGcWaaybCaeqajuaipaqaa8qaca aIWaaapaqaa8qacaWGHbaajuaGpaqaa8qacqGHRiI8aaWaaybCaeqa juaipaqaa8qacaaIWaaapaqaa8qacaWGIbaajuaGpaqaa8qacqGHRi I8aaGaamODa8aadaWgaaqaa8qacaWG6baapaqabaWdbmaabmaapaqa a8qacaWG4bGaaiilaiaadMhaaiaawIcacaGLPaaacaWGKbGaamiEai aadsgacaWG5baaaa@5196@  (11)

To finally obtain:

f 2 = τ ¯ w ρ v ¯ 2 = Po Re MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiaadAgaa8aabaWdbiaaikdaaaGaeyypa0Za aSaaa8aabaWdbiqbes8a0zaaraWdamaaBaaajuaibaWdbiaadEhaaK qba+aabeaaaeaapeGaeqyWdiNabmODayaaraWdamaaCaaabeqcfasa a8qacaaIYaaaaaaajuaGcqGH9aqpdaWcaaWdaeaapeGaamiuaiaad+ gaa8aabaWdbiaadkfacaWGLbaaaaaa@4649@  (12)

Which is the analogous of relation (1) for the case of rectangular ducts. From knowledge of τ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGafqiXdqNbaebaaaa@379E@ and v ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmODayaaraGaaiilaaaa@3784@ we obtain:

Po= π 4 b 2 8 ( a+b ) 2 n=1,3,5, + 1 n 4 ( 1 2a nπb th( nπb 2a ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpdaWcaaWdaeaapeGaeqiWda3damaa Caaajuaibeqaa8qacaaI0aaaaKqbakaadkgapaWaaWbaaKqbGeqaba WdbiaaikdaaaaajuaGpaqaa8qacaaI4aWaaeWaa8aabaWdbiaadgga cqGHRaWkcaWGIbaacaGLOaGaayzkaaWdamaaCaaajuaibeqaa8qaca aIYaaaaKqbaoaavadabeqcfaYdaeaapeGaamOBaiabg2da9iaaigda caGGSaGaaG4maiaacYcacaaI1aGaaiilaiabgAci8cWdaeaapeGaey 4kaSIaeyOhIukajuaGpaqaa8qacqGHris5aaWaaSaaa8aabaWdbiaa igdaa8aabaWdbiaad6gapaWaaWbaaKqbGeqabaWdbiaaisdaaaaaaK qbaoaabmaapaqaa8qacaaIXaGaeyOeI0YaaSaaa8aabaWdbiaaikda caWGHbaapaqaa8qacaWGUbGaeqiWdaNaamOyaaaacaWG0bGaamiAam aabmaapaqaa8qadaWcaaWdaeaapeGaamOBaiabec8aWjaadkgaa8aa baWdbiaaikdacaWGHbaaaaGaayjkaiaawMcaaaGaayjkaiaawMcaaa aaaaa@6896@  (13)

It is now interesting to evaluate this last result for different aspect ratios b/a. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbGaaiOlaaaa@38F3@  The first elementary case is of course the square cross-section giving b=a. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiabg2da9iaadggacaGGUaaaaa@3946@

Equation (13) reduces to:

Po= π 4 32 n=1,3,5, + 1 n 4 ( 1 2 nπ th( nπ 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpdaWcaaWdaeaapeGaeqiWda3damaa Caaajuaibeqaa8qacaaI0aaaaaqcfa4daeaapeGaaG4maiaaikdada qfWaqabKqbG8aabaWdbiaad6gacqGH9aqpcaaIXaGaaiilaiaaioda caGGSaGaaGynaiaacYcacqGHMacVa8aabaWdbiabgUcaRiabg6HiLc qcfa4daeaapeGaeyyeIuoaamaalaaapaqaa8qacaaIXaaapaqaa8qa caWGUbWdamaaCaaajuaibeqaa8qacaaI0aaaaaaajuaGdaqadaWdae aapeGaaGymaiabgkHiTmaalaaapaqaa8qacaaIYaaapaqaa8qacaWG UbGaeqiWdahaaiaadshacaWGObWaaeWaa8aabaWdbmaalaaapaqaa8 qacaWGUbGaeqiWdahapaqaa8qacaaIYaaaaaGaayjkaiaawMcaaaGa ayjkaiaawMcaaaaaaaa@5D03@  (14)

Considering now, the well-known mathematical result coming from Euler-Riemann zeta function knowledge:

n=1,3,5, + 1 n 4 = π 4 96   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaybCaeqajuaipaqaa8qacaWGUbGaeyypa0JaaGymaiaacYca caaIZaGaaiilaiaaiwdacaGGSaGaeyOjGWlapaqaa8qacqGHRaWkcq GHEisPaKqba+aabaWdbiabggHiLdaadaWcaaWdaeaapeGaaGymaaWd aeaapeGaamOBa8aadaahaaqabKqbGeaapeGaaGinaaaaaaqcfaOaey ypa0ZaaSaaa8aabaWdbiabec8aW9aadaahaaqcfasabeaapeGaaGin aaaaaKqba+aabaWdbiaaiMdacaaI2aaaaiaacckaaaa@4E57@  (15)

We obtain:

Po= π 4 π 4 3 64 π n=1,3,5, + 1 n 5 th( n π 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpdaWcaaWdaeaapeGaeqiWda3damaa Caaabeqcfasaa8qacaaI0aaaaaqcfa4daeaapeWaaSaaa8aabaWdbi abec8aW9aadaahaaqabKqbGeaapeGaaGinaaaaaKqba+aabaWdbiaa iodaaaGaeyOeI0YaaSaaa8aabaWdbiaaiAdacaaI0aaapaqaa8qacq aHapaCaaWaaubmaeqajuaipaqaa8qacaWGUbGaeyypa0JaaGymaiaa cYcacaaIZaGaaiilaiaaiwdacaGGSaGaeyOjGWlapaqaa8qacqGHRa WkcqGHEisPaKqba+aabaWdbiabggHiLdaadaWcaaWdaeaapeGaaGym aaWdaeaapeGaamOBa8aadaahaaqabKqbGeaapeGaaGynaaaaaaqcfa OaamiDaiaadIgadaqadaWdaeaapeGaamOBamaalaaapaqaa8qacqaH apaCa8aabaWdbiaaikdaaaaacaGLOaGaayzkaaaaaaaa@5D7A@  (16)

The series in equation (16) can easily be evaluated numerically giving:

n=1,3,5, + 1 n 5 th( n π 2 )0.92167516 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaybCaeqajuaipaqaa8qacaWGUbGaeyypa0JaaGymaiaacYca caaIZaGaaiilaiaaiwdacaGGSaGaeyOjGWlapaqaa8qacqGHRaWkcq GHEisPaKqba+aabaWdbiabggHiLdaadaWcaaWdaeaapeGaaGymaaWd aeaapeGaamOBa8aadaahaaqabKqbGeaapeGaaGynaaaaaaqcfaOaam iDaiaadIgadaqadaWdaeaapeGaamOBamaalaaapaqaa8qacqaHapaC a8aabaWdbiaaikdaaaaacaGLOaGaayzkaaGaeyyrIaKaaGimaiaac6 cacaaI5aGaaGOmaiaaigdacaaI2aGaaG4naiaaiwdacaaIXaGaaGOn aiabgAci8caa@584F@  (17)

Finally, we obtain the Po MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gaaaa@378A@ value for a duct of square cross-section shape:

Po= π 4 π 4 3 64 π  0.92167516 =7.11353554 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpdaWcaaWdaeaapeGaeqiWda3damaa Caaajuaibeqaa8qacaaI0aaaaaqcfa4daeaapeWaaSaaa8aabaWdbi abec8aW9aadaahaaqabKqbGeaapeGaaGinaaaaaKqba+aabaWdbiaa iodaaaGaeyOeI0YaaSaaa8aabaWdbiaaiAdacaaI0aaapaqaa8qacq aHapaCaaGaaiiOaiaaicdacaGGUaGaaGyoaiaaikdacaaIXaGaaGOn aiaaiEdacaaI1aGaaGymaiaaiAdaaaGaeyypa0JaaG4naiaac6caca aIXaGaaGymaiaaiodacaaI1aGaaG4maiaaiwdacaaI1aGaaGinaiab gAci8caa@57C6@  (18)

This purely theoretical result is in perfect agreement with experimental results obtained by many authors2 and of course with the value reported in Shah and London1 source book.

Let us now consider the other limit case described above i.e. the plane Poiseuille flow obtained for infinite parallel plates. As previously reported, this highly symmetric case gives a well-known value of Poiseuille number: Po=12. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpcaaIXaGaaGOmaiaac6caaaa@3AB9@

If we consider ba MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiablUMi=iaadggaaaa@38EB@ in equation (13), we obtain:

Po= π 4 8 n=1,3,5, + 1 n 4 = π 4 8  π 4 96 =12  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpdaWcaaWdaeaapeGaeqiWda3damaa Caaabeqcfasaa8qacaaI0aaaaaqcfa4daeaapeGaaGioamaavadabe qcfaYdaeaapeGaamOBaiabg2da9iaaigdacaGGSaGaaG4maiaacYca caaI1aGaaiilaiabgAci8cWdaeaapeGaey4kaSIaeyOhIukajuaGpa qaa8qacqGHris5aaWaaSaaa8aabaWdbiaaigdaa8aabaWdbiaad6ga paWaaWbaaeqajuaibaWdbiaaisdaaaaaaaaajuaGcqGH9aqpdaWcaa WdaeaapeGaeqiWda3damaaCaaajuaibeqaa8qacaaI0aaaaaqcfa4d aeaapeGaaGioaiaacckadaWcaaWdaeaapeGaeqiWda3damaaCaaabe qcfasaa8qacaaI0aaaaaqcfa4daeaapeGaaGyoaiaaiAdaaaaaaiab g2da9iaaigdacaaIYaGaaiiOaaaa@5D96@  (19)

Remarkably, this result is in perfect agreement with both experimental and theoretical results reported above. Moreover, it shows that Saint-Venant solution of Poisson PDE, established for elasticity theory,3 is of great importance for the study of laminar flow in rectangular ducts. Equation (13) established for aspect ratios b/a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbaaaa@3841@  varying in the range 1 (square) to + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaey4kaSIaeyOhIukaaa@3814@ (infinite parallel plates) is then of major interest for all these geometrical shapes.

We know from experiments that for these shapes, we have 7.1135Po12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaG4naiaac6cacaaIXaGaaGymaiaaiodacaaI1aGaeyOjGWRa eyizImQaamiuaiaad+gacqGHKjYOcaaIXaGaaGOmaaaa@425E@ and this result is in perfect agreement with equation (13). We can then write the following theorem:

Theorem 1: For b/a [ 1;+ [ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbGaeyicI48aaKGia8aabaWdbiaaigda caGG7aGaey4kaSIaeyOhIukacaGLBbGaay5waaaaaa@3FC6@ we have, according to equation (12): 7.1135Po12. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaG4naiaac6cacaaIXaGaaGymaiaaiodacaaI1aGaeyOjGWRa eyizImQaamiuaiaad+gacqGHKjYOcaaIXaGaaGOmaiaac6caaaa@4310@

This fundamental result clearly demonstrates that every value of Poiseuille number are possible between 7.1135… and 12. Particularly, it exists an aspect ratio b/a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbaaaa@3841@ which gives Po=8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpcaaI4aaaaa@3952@  like in pipes i.e. for a circular cross-section shape. Numerical calculations performed with equation (13) gave:

b a =2.26930413446618  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiaadkgaa8aabaWdbiaadggaaaGaeyypa0Ja aGOmaiaac6cacaaIYaGaaGOnaiaaiMdacaaIZaGaaGimaiaaisdaca aIXaGaaG4maiaaisdacaaI0aGaaGOnaiaaiAdacaaIXaGaaGioaiab gAci8kaacckaaaa@4767@  (20)

This result signifies that, in fluid mechanics, it exists a rectangle having the same property than a circle for mechanical energy dissipation through viscous friction and this rectangular duct has an aspect ratio b/a=2.2693 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbGaeyypa0JaaGOmaiaac6cacaaIYaGa aGOnaiaaiMdacaaIZaGaeyOjGWlaaa@3F3F@  

Discussion on mathematical consequences

Of course, these results give equivalence between rectangular and circular geometries in terms of energy dissipation and we can write the following theorem:

Theorem 2: Considering energy dissipation by viscous friction during the fully established laminar flow of a Newtonian fluid, the equivalent geometry for a pipe of circular cross-section is a rectangular duct having an aspect ratio b/a=2.2693 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbGaeyypa0JaaGOmaiaac6cacaaIYaGa aGOnaiaaiMdacaaIZaGaeyOjGWlaaa@3F3F@

This result could be extended to others geometries like triangles. We know that for an equilateral triangle, Po=20/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpcaaIYaGaaGimaiaac+cacaaIZaaa aa@3B76@  and stretching of this triangle giving isosceles triangles increases Po MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gaaaa@378A@ values until it also reaches Po=12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpcaaIXaGaaGOmaaaa@3A07@ for an infinite triangle comparable to infinite parallel plates.2 In that sense, there also exists a triangle for which Po=8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpcaaI4aaaaa@3952@ meaning a triangle equivalent to a circle. We can then propose the following conjecture:

Conjecture 1: For any compact convex shape, there exists a non-regular geometry giving Po=8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpcaaI4aaaaa@3952@ and then giving equivalence with circular geometry in terms of mechanical energy degradation by viscous friction.

If this conjecture was true, signification of Poiseuille number values could be very important in Physics and Mathematics. Considering well-known membrane deformation problem giving Poisson’s PDE, equation (6) is clearly its analogous for the laminar flow of a Newtonian liquid in a duct of arbitrary cross-section shape. The Saint-Venant solution given by equation (7) gives the velocity field shape which depends on the boundary conditions i.e. the shape of the duct cross-section perimeter. In the case of a pipe with circular cross-section, the high symmetry allows simple calculations and velocity field has a parabolic shape according to Poiseuille law. But for polygonal geometries like rectangles or triangles, shape is much more complicated. But, at the end, calculation of τ ¯ w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGafqiXdqNbaebapaWaaSbaaKqbGeaapeGaam4Daaqcfa4daeqa aaaa@39A5@  and v ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmODayaaraaaaa@36D4@ allows a simple dimensionless equation of the same form (equation (12)) to be obtained and this equation involves Po MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gaaaa@378A@ numbers varying in the range 20/3 to 12.

Another consequence of the rectangular approach is the close relation between Po MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gaaaa@378A@ numbers and the Euler-Riemann zeta function. The problem of ζ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOTdO3aaeWaa8aabaWdbiaadohaaiaawIcacaGLPaaaaaa@3A1E@ values for odd integer remains unsolved because at this time, we have no idea of a closed form for4 s=3, 5, 7,  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Caiabg2da9iaaiodacaGGSaGaaiiOaiaaiwdacaGGSaGa aiiOaiaaiEdacaGGSaGaaiiOaiabgAci8caa@4106@ Equation (13) gives interesting properties which could help approaching a closed form for ζ( 5 ). MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOTdO3aaeWaa8aabaWdbiaaiwdaaiaawIcacaGLPaaacaGG Uaaaaa@3A97@

By considering well-known properties of hyperbolic tangent function th( x ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiDaiaadIgadaqadaWdaeaapeGaamiEaaGaayjkaiaawMca aiaacYcaaaa@3AFC@ this function reaches very rapidly asymptotic value of 1 whenreach sufficiently large values (greater than 10). We can then consider that for sufficiently high values of ratio b/a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbaaaa@3841@ in equation (13), the quantity th( nπb/2a )=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiDaiaadIgadaqadaWdaeaapeGaamOBaiabec8aWjaadkga caGGVaGaaGOmaiaadggaaiaawIcacaGLPaaacqGH9aqpcaaIXaaaaa@40FC@ giving the following relationship for the sum over odd integers of 1/ n 5 : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGymaiaac+cacaWGUbWdamaaCaaajuaibeqaa8qacaaI1aaa aKqba+aacaGG6aaaaa@3AAB@  

n=1, 3, 5, + 1 n 5 = b 192 a   π 5   b 3 16 a  ( a+b ) 2  Po   π 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaybCaeqajuaipaqaa8qacaWGUbGaeyypa0JaaGymaiaacYca caGGGcGaaG4maiaacYcacaGGGcGaaGynaiaacYcacqGHMacVa8aaba WdbiabgUcaRiabg6HiLcqcfa4daeaapeGaeyyeIuoaamaalaaapaqa a8qacaaIXaaapaqaa8qacaWGUbWdamaaCaaabeqcfasaa8qacaaI1a aaaaaajuaGcqGH9aqpdaWcaaWdaeaapeGaamOyaaWdaeaapeGaaGym aiaaiMdacaaIYaGaaiiOaiaadggaaaGaaiiOaiabec8aW9aadaahaa qcfasabeaapeGaaGynaaaajuaGcqGHsislcaGGGcGaaGPaVpaalaaa paqaa8qacaWGIbWdamaaCaaabeqcfasaa8qacaaIZaaaaaqcfa4dae aapeGaaGymaiaaiAdacaGGGcGaamyyaiaacckadaqadaWdaeaapeGa amyyaiabgUcaRiaadkgaaiaawIcacaGLPaaapaWaaWbaaKqbGeqaba WdbiaaikdaaaqcfaOaaiiOaiaadcfacaWGVbaaaiaacckacqaHapaC paWaaWbaaKqbGeqabaWdbiaaiwdaaaaaaa@6C89@  (21)

Of course, b, a and Po MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaacYcacaGGGcGaamyyaiaacckacaWGHbGaamOBaiaa dsgacaGGGcGaamiuaiaad+gaaaa@4035@ values are linked together (for example you have Po=11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpcaaIXaGaaGymaaaa@3A06@ for b/a=14.84241923166) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbGaeyypa0JaaGymaiaaisdacaGGUaGa aGioaiaaisdacaaIYaGaaGinaiaaigdacaaI5aGaaGOmaiaaiodaca aIXaGaaGOnaiaaiAdacaGGPaaaaa@444B@ but equation (21) is surely an interesting result for understanding of ζ( 5 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOTdO3aaeWaa8aabaWdbiaaiwdaaiaawIcacaGLPaaaaaa@39E5@ behaviour even if the sum only concerns odd values of n. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOBaiaac6caaaa@3766@

Complex calculations in rectangular ducts also give others surprising and interesting results in numbers theory. For example, it is possible to calculate the wall shear stress along the side length a: MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyyaiaacQdaaaa@3765@

τ w ( x )=η  ( v z y ) y=b = 8ΔPa π 2 L   n=1,3,5, + 1 n 2 ( 1 ) n+1/2 th( nπb 2a )cos( nπx 2a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiXdq3damaaBaaajuaibaWdbiaadEhaaKqba+aabeaapeWa aeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacqGH9aqpcqaH3oaAca GGGcWaaeWaa8aabaWdbmaalaaapaqaa8qacqGHciITcaWG2bWdamaa BaaajuaibaWdbiaadQhaa8aabeaaaKqbagaapeGaeyOaIyRaamyEaa aaaiaawIcacaGLPaaapaWaaSbaaKqbGeaapeGaamyEaiabg2da9iaa dkgaaKqba+aabeaapeGaeyypa0ZaaSaaa8aabaWdbiaaiIdacqGHuo arcaWGqbGaamyyaaWdaeaapeGaeqiWda3damaaCaaabeqcfasaa8qa caaIYaaaaKqbakaadYeaaaGaaiiOamaawahabeqcfaYdaeaapeGaam OBaiabg2da9iaaigdacaGGSaGaaG4maiaacYcacaaI1aGaaiilaiab gAci8cWdaeaapeGaey4kaSIaeyOhIukajuaGpaqaa8qacqGHris5aa WaaSaaa8aabaWdbiaaigdaa8aabaWdbiaad6gapaWaaWbaaeqajuai baWdbiaaikdaaaaaaKqbaoaabmaapaqaa8qacqGHsislcaaIXaaaca GLOaGaayzkaaWdamaaCaaajuaibeqaa8qacaWGUbGaey4kaSIaaGym aiaac+cacaaIYaaaaKqbakaadshacaWGObWaaeWaa8aabaWdbmaala aapaqaa8qacaWGUbGaeqiWdaNaamOyaaWdaeaapeGaaGOmaiaadgga aaaacaGLOaGaayzkaaGaam4yaiaad+gacaWGZbWaaeWaa8aabaWdbm aalaaapaqaa8qacaWGUbGaeqiWdaNaamiEaaWdaeaapeGaaGOmaiaa dggaaaaacaGLOaGaayzkaaaaaa@8412@  (22)

The maximum value of τ w ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiXdq3damaaBaaajuaibaWdbiaadEhaaKqba+aabeaapeWa aeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3C42@ is obtained for x=0, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiEaiabg2da9iaaicdacaGGSaaaaa@392E@ giving:

τ w max = 8ΔPa π 2 L   n=1,3,5, + 1 n 2 ( 1 ) n+1/2 th( nπb 2a )   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiXdq3damaaDaaajuaibaWdbiaadEhaa8aabaWdbiaad2ga caWGHbGaamiEaaaajuaGcqGH9aqpdaWcaaWdaeaapeGaaGioaiaabs 5acaWGqbGaamyyaaWdaeaapeGaeqiWda3damaaCaaajuaibeqaa8qa caaIYaaaaKqbakaadYeaaaGaaiiOamaawahabeqcfaYdaeaapeGaam OBaiabg2da9iaaigdacaGGSaGaaG4maiaacYcacaaI1aGaaiilaiab gAci8cWdaeaapeGaey4kaSIaeyOhIukajuaGpaqaa8qacqGHris5aa WaaSaaa8aabaWdbiaaigdaa8aabaWdbiaad6gapaWaaWbaaeqajuai baWdbiaaikdaaaaaaKqbaoaabmaapaqaa8qacqGHsislcaaIXaaaca GLOaGaayzkaaWdamaaCaaajuaibeqaa8qacaWGUbGaey4kaSIaaGym aiaac+cacaaIYaaaaKqbakaadshacaWGObWaaeWaa8aabaWdbmaala aapaqaa8qacaWGUbGaeqiWdaNaamOyaaWdaeaapeGaaGOmaiaadgga aaaacaGLOaGaayzkaaGaaiiOaiaacckaaaa@6B93@ s (23)

If we consider the limit case of infinite parallel plates giving ba: MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiablUMi=iaadggacaGG6aaaaa@39A9@ we obtain:

τ w max = 8ΔPa π 2 L   n=1,3,5, + 1 n 2 ( 1 ) n+1/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiXdq3damaaDaaajuaibaWdbiaadEhaa8aabaWdbiaad2ga caWGHbGaamiEaaaajuaGcqGH9aqpdaWcaaWdaeaapeGaaGioaiaabs 5acaWGqbGaamyyaaWdaeaapeGaeqiWda3damaaCaaajuaibeqaa8qa caaIYaaaaKqbakaadYeaaaGaaiiOamaawahabeqcfaYdaeaapeGaam OBaiabg2da9iaaigdacaGGSaGaaG4maiaacYcacaaI1aGaaiilaiab gAci8cWdaeaapeGaey4kaSIaeyOhIukajuaGpaqaa8qacqGHris5aa WaaSaaa8aabaWdbiaaigdaa8aabaWdbiaad6gapaWaaWbaaKqbGeqa baWdbiaaikdaaaaaaKqbaoaabmaapaqaa8qacqGHsislcaaIXaaaca GLOaGaayzkaaWdamaaCaaajuaibeqaa8qacaWGUbGaey4kaSIaaGym aiaac+cacaaIYaaaaaaa@5FA8@  (24)

The series can be written as followed:

n=1,3,5, + 1 n 2 ( 1 ) n+1/2 = n=0 + ( 1 ) n ( 2n+1 ) 2 =G  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaybCaeqajuaipaqaa8qacaWGUbGaeyypa0JaaGymaiaacYca caaIZaGaaiilaiaaiwdacaGGSaGaeyOjGWlapaqaa8qacqGHRaWkcq GHEisPaKqba+aabaWdbiabggHiLdaadaWcaaWdaeaapeGaaGymaaWd aeaapeGaamOBa8aadaahaaqcfasabeaapeGaaGOmaaaaaaqcfa4aae Waa8aabaWdbiabgkHiTiaaigdaaiaawIcacaGLPaaapaWaaWbaaKqb GeqabaWdbiaad6gacqGHRaWkcaaIXaGaai4laiaaikdaaaqcfaOaey ypa0JaeyOeI0YaaybCaeqajuaipaqaa8qacaWGUbGaeyypa0JaaGim aaWdaeaapeGaey4kaSIaeyOhIukajuaGpaqaa8qacqGHris5aaWaaS aaa8aabaWdbmaabmaapaqaa8qacqGHsislcaaIXaaacaGLOaGaayzk aaWdamaaCaaajuaibeqaa8qacaWGUbaaaaqcfa4daeaapeWaaeWaa8 aabaWdbiaaikdacaWGUbGaey4kaSIaaGymaaGaayjkaiaawMcaa8aa daahaaqcfasabeaapeGaaGOmaaaaaaqcfaOaeyypa0Jaam4raiaacc kaaaa@68A3@  (25)

Where G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4raaaa@368D@ is the well-known Catalan’s constant. Until now, we ignore if this number is irrational even if it is conjectured. What we obtained from equation (24) gives interesting information about this number. Moreover, if we calculate the mean value of wall shear-stress along the same side, we obtain:

τ ¯ w = 1 a   0 a τ w ( x )dx= 16ΔPa π 3 L   n=1,3,5, + 1 n 3 ( 1 ) n th( nπb 2a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGafqiXdqNbaebapaWaaSbaaKqbGeaapeGaam4Daaqcfa4daeqa a8qacqGH9aqpdaWcaaWdaeaapeGaaGymaaWdaeaapeGaamyyaaaaca GGGcWaaybCaeqajuaipaqaa8qacaaIWaaapaqaa8qacaWGHbaajuaG paqaa8qacqGHRiI8aaGaeqiXdq3damaaBaaajuaibaWdbiaadEhaa8 aabeaajuaGpeWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacaWG KbGaamiEaiabg2da9maalaaapaqaa8qacaaIXaGaaGOnaiaabs5aca WGqbGaamyyaaWdaeaapeGaeqiWda3damaaCaaajuaibeqaa8qacaaI ZaaaaKqbakaadYeaaaGaaiiOamaawahabeqcfaYdaeaapeGaamOBai abg2da9iaaigdacaGGSaGaaG4maiaacYcacaaI1aGaaiilaiabgAci 8cWdaeaapeGaey4kaSIaeyOhIukajuaGpaqaa8qacqGHris5aaWaaS aaa8aabaWdbiaaigdaa8aabaWdbiaad6gapaWaaWbaaeqabaWdbiaa iodaaaaaamaabmaapaqaa8qacqGHsislcaaIXaaacaGLOaGaayzkaa WdamaaCaaajuaibeqaa8qacaWGUbaaaKqbakaadshacaWGObWaaeWa a8aabaWdbmaalaaapaqaa8qacaWGUbGaeqiWdaNaamOyaaWdaeaape GaaGOmaiaadggaaaaacaGLOaGaayzkaaaaaa@7512@  (26)

Considering now ba MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiablUMi=iaadggaaaa@38EB@  gives:

τ ¯ w = 16ΔPa π 3 L   n=1,3,5, + 1 n 3 ( 1 ) n = 16ΔPa π 3 L   π 3 32 = ΔPa 2L    MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGafqiXdqNbaebapaWaaSbaaKqbGeaapeGaam4Daaqcfa4daeqa a8qacqGH9aqpdaWcaaWdaeaapeGaaGymaiaaiAdacaqGuoGaamiuai aadggaa8aabaWdbiabec8aW9aadaahaaqcfasabeaapeGaaG4maaaa juaGcaWGmbaaaiaacckadaGfWbqabKqbG8aabaWdbiaad6gacqGH9a qpcaaIXaGaaiilaiaaiodacaGGSaGaaGynaiaacYcacqGHMacVa8aa baWdbiabgUcaRiabg6HiLcqcfa4daeaapeGaeyyeIuoaamaalaaapa qaa8qacaaIXaaapaqaa8qacaWGUbWdamaaCaaabeqcfasaa8qacaaI ZaaaaaaajuaGdaqadaWdaeaapeGaeyOeI0IaaGymaaGaayjkaiaawM caa8aadaahaaqcfasabeaapeGaamOBaaaajuaGcqGH9aqpdaWcaaWd aeaapeGaaGymaiaaiAdacaqGuoGaamiuaiaadggaa8aabaWdbiabec 8aW9aadaahaaqcfasabeaapeGaaG4maaaajuaGcaWGmbaaaiaaccka daWcaaWdaeaapeGaeqiWda3damaaCaaajuaibeqaa8qacaaIZaaaaa qcfa4daeaapeGaaG4maiaaikdaaaGaeyypa0ZaaSaaa8aabaWdbiaa bs5acaWGqbGaamyyaaWdaeaapeGaaGOmaiaadYeaaaGaaiiOaiaacc kaaaa@7373@  (27)

Reporting this result in equation (24) gives:

τ w max τ ¯ = 16 G π 2 1.484907491  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiabes8a09aadaqhaaqcfasaa8qacaWG3baa paqaa8qacaWGTbGaamyyaiaadIhaaaaajuaGpaqaa8qacuaHepaDga qeaaaacqGH9aqpdaWcaaWdaeaapeGaaGymaiaaiAdacaGGGcGaam4r aaWdaeaapeGaeqiWda3damaaCaaajuaibeqaa8qacaaIYaaaaaaaju aGcqGHfjcqcaaIXaGaaiOlaiaaisdacaaI4aGaaGinaiaaiMdacaaI WaGaaG4naiaaisdacaaI5aGaaGymaiabgAci8kaacckaaaa@52DD@  (28)

This result clearly shows that Catalan’s constant is proportional to π 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiWda3damaaCaaabeqcfasaa8qacaaIYaaaaaaa@38A9@ which is an irrational number and then it could be considered as a proof of Catalan’s constant irrationality.
We can then write the following theorem:

Theorem 3: Catalan’s constant is proportional to π 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiWda3damaaCaaabeqcfasaa8qacaaIYaaaaaaa@38A9@ and then is an irrational number.
Finally, it is also possible to calculate the ratio   v max / v ¯   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaiiOaiaadAhapaWaaSbaaKqbGeaapeGaamyBaiaadggacaWG 4baajuaGpaqabaWdbiaac+capaGabmODayaaraWdbiaacckaaaa@3ED9@ of the maximum velocity at the centre of the rectangular duct and the average velocity. For maximum velocity, we obtain:

v max = ΔP a 2 2ηL 16ΔP a 2 π 3 ηL   n=1,3,5, + 1 n 3 ( 1 ) n1/2   1 ch( nπb 2a )   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamODa8aadaWgaaqcfasaa8qacaWGTbGaamyyaiaadIhaa8aa beaajuaGpeGaeyypa0ZaaSaaa8aabaWdbiaabs5acaWGqbGaamyya8 aadaahaaqcfasabeaapeGaaGOmaaaaaKqba+aabaWdbiaaikdacqaH 3oaAcaWGmbaaaiabgkHiTmaalaaapaqaa8qacaaIXaGaaGOnaiaabs 5acaWGqbGaamyya8aadaahaaqcfasabeaapeGaaGOmaaaaaKqba+aa baWdbiabec8aW9aadaahaaqcfasabeaapeGaaG4maaaajuaGcqaH3o aAcaWGmbaaaiaacckadaGfWbqabKqbG8aabaWdbiaad6gacqGH9aqp caaIXaGaaiilaiaaiodacaGGSaGaaGynaiaacYcacqGHMacVa8aaba WdbiabgUcaRiabg6HiLcqcfa4daeaapeGaeyyeIuoaamaalaaapaqa a8qacaaIXaaapaqaa8qacaWGUbWdamaaCaaabeqcfasaa8qacaaIZa aaaaaajuaGdaqadaWdaeaapeGaeyOeI0IaaGymaaGaayjkaiaawMca a8aadaahaaqcfasabeaapeGaamOBaiabgkHiTiaaigdacaGGVaGaaG OmaaaajuaGcaGGGcWaaSaaa8aabaWdbiaaigdaa8aabaWdbiaadoga caWGObWaaeWaa8aabaWdbmaalaaapaqaa8qacaWGUbGaeqiWdaNaam OyaaWdaeaapeGaaGOmaiaadggaaaaacaGLOaGaayzkaaaaaiaaccka aaa@77F1@  (29)

For mean velocity, we obtain from equation (11) and Fubini theorem:

v ¯ = 32ΔP a 2 π 4 ηL   n=1,3,5, + 1 n 4  ( 1 2a nπb  th( nπb 2a ) )   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadAhaga qeaabaaaaaaaaapeGaeyypa0ZaaSaaa8aabaWdbiaaiodacaaIYaGa aeiLdiaadcfacaWGHbWdamaaCaaajuaibeqaa8qacaaIYaaaaaqcfa 4daeaapeGaeqiWda3damaaCaaajuaibeqaa8qacaaI0aaaaKqbakab eE7aOjaadYeaaaGaaiiOamaawahabeqcfaYdaeaapeGaamOBaiabg2 da9iaaigdacaGGSaGaaG4maiaacYcacaaI1aGaaiilaiabgAci8cWd aeaapeGaey4kaSIaeyOhIukajuaGpaqaa8qacqGHris5aaWaaSaaa8 aabaWdbiaaigdaa8aabaWdbiaad6gapaWaaWbaaKqbGeqabaWdbiaa isdaaaaaaKqbakaacckadaqadaWdaeaapeGaaGymaiaaykW7cqGHsi slcaaMc8UaaGPaVpaalaaapaqaa8qacaaIYaGaamyyaaWdaeaapeGa amOBaiabec8aWjaadkgaaaGaaiiOaiaadshacaWGObWaaeWaa8aaba Wdbmaalaaapaqaa8qacaWGUbGaeqiWdaNaamOyaaWdaeaapeGaaGOm aiaadggaaaaacaGLOaGaayzkaaaacaGLOaGaayzkaaGaaiiOaiaacc kaaaa@7188@  (30)

Giving,

v max v ¯ = π 4 64 n=1,3,5, + 1 n 4 ( 1 2a nπb  th( nπb 2a ) ) π  n=1,3,5, + 1 n 3 ( 1 ) n1/2 1 ch( nπb 2a ) 2  n=1,3,5, + 1 n 4  ( 1 2a nπb  th( nπb 2a ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiaadAhapaWaaSbaaKqbGeaapeGaamyBaiaa dggacaWG4baajuaGpaqabaaabaGabmODayaaraaaa8qacqGH9aqpca aMc8+aaSaaa8aabaWdbiabec8aW9aadaahaaqabKqbGeaapeGaaGin aaaaaKqba+aabaWdbiaaiAdacaaI0aWaaubmaeqajuaipaqaa8qaca WGUbGaeyypa0JaaGymaiaacYcacaaIZaGaaiilaiaaiwdacaGGSaGa eyOjGWlapaqaa8qacqGHRaWkcqGHEisPaKqba+aabaWdbiabggHiLd aadaWcaaWdaeaapeGaaGymaaWdaeaapeGaamOBa8aadaahaaqabKqb GeaapeGaaGinaaaaaaqcfa4aaeWaa8aabaWdbiaaigdacqGHsislda WcaaWdaeaapeGaaGOmaiaadggaa8aabaWdbiaad6gacqaHapaCcaWG IbaaaiaacckacaWG0bGaamiAamaabmaapaqaa8qadaWcaaWdaeaape GaamOBaiabec8aWjaadkgaa8aabaWdbiaaikdacaWGHbaaaaGaayjk aiaawMcaaaGaayjkaiaawMcaaaaacaaMc8UaaGPaVlabgkHiTiaayk W7caaMc8+aaSaaa8aabaWdbiabec8aWjaacckadaqfWaqabKqbG8aa baWdbiaad6gacqGH9aqpcaaIXaGaaiilaiaaiodacaGGSaGaaGynai aacYcacqGHMacVa8aabaWdbiabgUcaRiabg6HiLcqcfa4daeaapeGa eyyeIuoaamaalaaapaqaa8qacaaIXaaapaqaa8qacaWGUbWdamaaCa aajuaibeqaa8qacaaIZaaaaaaajuaGdaqadaWdaeaapeGaeyOeI0Ia aGymaaGaayjkaiaawMcaa8aadaahaaqcfasabeaapeGaamOBaiabgk HiTiaaigdacaGGVaGaaGOmaaaajuaGdaWcaaWdaeaapeGaaGymaaWd aeaapeGaam4yaiaadIgadaqadaWdaeaapeWaaSaaa8aabaWdbiaad6 gacqaHapaCcaWGIbaapaqaa8qacaaIYaGaamyyaaaaaiaawIcacaGL Paaaaaaapaqaa8qacaaIYaGaaiiOamaavadabeqcfaYdaeaapeGaam OBaiabg2da9iaaigdacaGGSaGaaG4maiaacYcacaaI1aGaaiilaiab gAci8cWdaeaapeGaey4kaSIaeyOhIukajuaGpaqaa8qacqGHris5aa WaaSaaa8aabaWdbiaaigdaa8aabaWdbiaad6gapaWaaWbaaKqbGeqa baWdbiaaisdaaaaaaKqbakaacckadaqadaWdaeaapeGaaGymaiabgk HiTmaalaaapaqaa8qacaaIYaGaamyyaaWdaeaapeGaamOBaiabec8a WjaadkgaaaGaaiiOaiaadshacaWGObWaaeWaa8aabaWdbmaalaaapa qaa8qacaWGUbGaeqiWdaNaamOyaaWdaeaapeGaaGOmaiaadggaaaaa caGLOaGaayzkaaaacaGLOaGaayzkaaaaaaaa@BC38@  (31)

It is then easy to consider the limit case of infinite parallel plates by taking ba: MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiablUMi=iaadggacaGG6aaaaa@39A9@

v max v ¯ = π 4 64  π 4 96 = 96 64 = 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiaadAhapaWaaSbaaKqbGeaapeGaamyBaiaa dggacaWG4baajuaGpaqabaaabaGabmODayaaraaaa8qacqGH9aqpda WcaaWdaeaapeGaeqiWda3damaaCaaajuaibeqaa8qacaaI0aaaaaqc fa4daeaapeGaaGOnaiaaisdacaGGGcWaaSaaa8aabaWdbiabec8aW9 aadaahaaqabKqbGeaapeGaaGinaaaaaKqba+aabaWdbiaaiMdacaaI 2aaaaaaacqGH9aqpdaWcaaWdaeaapeGaaGyoaiaaiAdaa8aabaWdbi aaiAdacaaI0aaaaiabg2da9maalaaapaqaa8qacaaIZaaapaqaa8qa caaIYaaaaaaa@4FC7@  (32)

Remarkably, this last result is well-known in fluid mechanics for the case of infinite parallel plates. For pipes of circular cross-section, we have: v max / v ¯ =2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamODa8aadaWgaaqcfasaa8qacaWGTbGaamyyaiaadIhaa8aa beaajuaGpeGaai4la8aaceWG2bGbaebapeGaeyypa0JaaGOmaiaac6 caaaa@3F05@

Equation (31) allows this ratio to be calculated whatever is the rectangle aspect ratio b/a. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbGaaiOlaaaa@38F3@

Conclusion

In this paper, we investigated the Poisson’s PDE describing the fully established laminar flow of a Newtonian fluid in a duct of rectangular cross-section. We used the Saint-Venant solution, established for torsion of prismatical bars to obtain the velocity field whatever is the rectangle aspect ratio b/a. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbGaaiOlaaaa@38F3@

From this equation, we showed how Poiseuille number values can be calculated giving a simple theorem for evolution of  7.1135Po12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaiiOaiaaiEdacaGGUaGaaGymaiaaigdacaaIZaGaaGynaiab gAci8kabgsMiJkaadcfacaWGVbGaeyizImQaaGymaiaaikdaaaa@4382@ when b/a [ 1;+ [ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbGaeyicI48aaKGia8aabaWdbiaaigda caGG7aGaey4kaSIaeyOhIukacaGLBbGaay5waaGaaGPaVlaac6caaa a@4203@  This result allowed the rectangle, giving the same value than the circular cross-section, to be defined with an aspect ratio b/a=2.2693 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOyaiaac+cacaWGHbGaeyypa0JaaGOmaiaac6cacaaIYaGa aGOnaiaaiMdacaaIZaGaeyOjGWlaaa@3F3F@  

We tried to give some mathematical consequences of this approach. Among them, we conjecture that for any convex shape of non-circular cross-section, there always exists one having a Poiseuille number value equal to the circle value i.e. Po=8. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiuaiaad+gacqGH9aqpcaaI4aGaaiOlaaaa@3A04@

We also showed, from Poiseuille number equation for rectangular ducts, that Euler-Riemann zeta function ζ( s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOTdO3aaeWaa8aabaWdbiaadohaaiaawIcacaGLPaaaaaa@3A1E@ for odd integer s=5, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4Caiabg2da9iaaiwdacaGGSaaaaa@392E@ for summation over odd integers n=1,3,5,... MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOBaiabg2da9iaaigdacaGGSaGaaG4maiaacYcacaaI1aGa aiilaiaac6cacaGGUaGaaiOlaaaa@3E17@ can be calculated as proportional to π 5 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiWda3aaWbaaKqbGeqabaGaaGynaaaajuaGcaGGUaaaaa@39CD@

From calculation of both, maximum wall shear stress, and average wall shear stress, we showed that famous Catalan’s constant G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4raaaa@368D@ is proportional to π 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqiWda3aaWbaaeqajuaibaGaaGOmaaaaaaa@388A@ which could be a proof of its irrationality.

Finally, by integrating the velocity field, we found an expression for the ratio v max / v ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamODa8aadaWgaaqcfasaa8qacaWGTbGaamyyaiaadIhaaKqb a+aabeaapeGaai4la8aaceWG2bGbaebaaaa@3C81@ which gave, for the limit case of plane Poiseuille flow, a value of 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVuYJK8sipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaiodaca GGVaGaaGOmaaaa@37CD@ in perfect agreement with fluid mechanics results.

Acknowledgements

None.

Conflict of interest

The author declares that there is no conflict of interest.

References

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  2. Delplace F. Fluids flow stability in ducts of arbitrary cross-section. J Mod Appl Physics. 2018;2(2):10‒15.
  3. Timoshenko SP, Goodier JN. Theory of elasticity. 3rd Ed. New York: Mc Graw-Hill; 1970.
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