Analytical solution for fully developed channel and pipe flow of Phan ...

J. Fluid Mech. (1999), vol. 387, pp. 271–280. Printed in the United Kingdomc○ 1999 Cambridge University Press271Analytical solution for fullydeveloped channel and pipe flow ofPhan-Thien–Tanner fluidsBy PAULO J. OLIVEIRA 1 AND FERNANDO T. PINHO 21 Universidade da Beira Interior, Departamento Eng a Electromecânica,6200 Covilhã, Portugal2 Centro de Estudo de Fenómenos de Transporte, DEMEGI,Faculdade de Engenharia da Universidade do Porto, Rua dos Bragas,4099 Porto Codex, Portugal(Received 8 July 1998 and in revised form 12 December 1998)Analytical expressions are derived for the velocity vector, the stress componentsand the viscosity function in fully developed channel and pipe flow of Phan-Thien–Tanner (PTT) fluids; both the linearized and the exponential forms of the PTTequation are considered. The solution shows that the wall shear stress of a PTTfluid is substantially smaller than the corresponding value for a Newtonian or upperconvectedMaxwell fluid, with implications for comparing predicted and measuredvalues in a non-dimensional form.1. IntroductionThe constitutive equation for Phan-Thien–Tanner (PTT) fluid (Phan-Thien & Tanner1977; Phan-Thien 1998) can be written in a general form asf(tr(τ))τ + λ τ =2ηD, (1)where τ and D are the extra-stress and deformation-rate tensors, λ is the relaxationtime, η is a constant viscosity coefficient and τ denotes Oldroyd’s upper-convectedderivative,τ =Du/Dt − τ ·∇u − ∇u t · τ.Two forms of the PTT model are in common use, namely the linearized form givenin the original paper (Phan-Thien & Tanner 1997), where the function f isf(tr(τ))=1+ ɛλ tr(τ) (2)ηand the exponential form (Phan-Thien 1978) with( ) ɛλf(tr(τ)) = expη tr(τ) . (3)In both forms ɛ is a parameter related to the elongational behaviour of the model.Note that the linearized form results from (3) if the trace of the stress tensor is smalland that both forms reduce to the well-known upper-convected Maxwell (UCM)model when ɛ vanishes.

Fully developed flow of Phan-Thien–Tanner fluids 273the equations of motion that the pressure gradient p ,x is constant and integration ofthe longitudinal momentum equation, subjected to the boundary condition τ xy =0atthe symmetry line y = 0, yieldsyτ xy = p ,x2 . (7)jWith the above simplifications and upon division of the expressions for the two nonvanishingstresses (equations (4) and (6)) the specific function f cancels out resulting inτ xx = 2λ y 2η p2 ,x . (8)22j Note that equation (8) is consistent with the required boundary condition at thecenterline, i.e. τ xx =0aty = 0. After combining (6), (7) and (8) we come to theimportant conclusion that the velocity gradient is given by˙γ ≡ du ( ) 2λdy = f y 2 p,x yη p2 ,x2 2j η 2 , (9)jan explicit differential equation in y which can be directly integrated to give thevelocity profile once the function f is specified.2.1. Linear PTT modelWith f given by equation (2), equation (9) can be integrated subject to the no-slipboundary condition at the wall (y = H) and the resulting velocity profile is thefollowing cubic equation in the pressure gradient:()u(y) = −p ,x2 j+1 η (H2 − y 2 )1+ ɛλ2 p 2 ,x2 2j η 2 (H2 + y 2 ). (10a)It is often more convenient to work with the non-dimensional form of this equation,obtained after scaling distances with H, velocities with the cross-sectional averagevelocity ū and stresses or pressure with ηū/H. In this way, it will be easier to derivethe relevant equations for the inverse problem of determining the pressure gradientwhen the flow rate is known, the common situation in an experiment. After somemanipulation, the non-dimensional velocity profile can be written asu(y)ū= κūN ū(1 −( yH) 2 )(1+4κ 2 ɛDe 2 (ūNū) 2 (1+( yH) 2 ))(10b)where κ takes the values 1.5 and 2 for plane or axisymmetric flows, respectively. Inequation (10b) the parameter ū N is defined asū N ≡ −p ,xH 2(11)2 j+1 kηand represents the cross-sectional average velocity for the Newtonian or the upperconvectedMaxwell fluid cases. Note also that ū N /ū is nothing other than a nondimensionalpressure gradient. The dimensionless group De = λū/H is the Deborahnumber, a measure of the level of elasticity in the fluid, and is based on the averagevelocity ū. ForDe =0orɛ = 0, the above equations reduce to the well knownparabolic profile with a maximum velocity at the centreline given by (u 0 ) N = κū N .Thus the second term in the brackets of equation (10) represents a corrective (oradditional) term, relatively to the parabolic profile, and is connected to the PTTmodel.

274 P. J. Oliveira and F. T. PinhoThe pressure gradient −p ,x is usually unknown but can be related to the crosssectionalaverage velocity through the definition of the flow rate following Bird,Armstrong & Hassager (1987)ū ≡ 1 ∫ H2 j y j u(y)dyH j+1 0and the integration yields the solution of the direct problem, the flux for a givenpressure gradientū =()−p ,x H 21+ ɛλ2 p 2 ,xH 2 (j +3). (12)η(j + 1)(j +3) 2 (2j−1) η 2 (j +5)For the solution of the inverse problem, the determination of the pressure gradientfor a given flux, it is advantageous to work with the normalized velocity profile (10b)which needs to be integrated to yield the following non-dimensional cubic equationfor ū N /ū:1=ūN ū( ) 2 ) (ūN1+būwith8(3 + j)κ2b ≡ ɛDe 2 .(5 + j)This equation shows that ū N < ū, i.e. for an identical longitudinal pressure gradient ahigher flow rate results for the PTT fluid than for the Newtonian fluid on account ofthe shear-thinning behaviour. From the Cardan–Tartaglia formula for the solution ofalgebraic cubic equations it can be readily shown that the real solution of (13) isū Nū = (432)1/6 (δ 2/3 − 2 2/3 )(14)6b 1/2 δ 1/3with the following definitions used to simplify the notation:α =3 3 b +4; β =3 3/2 b 1/2 ; δ = α 1/2 + β.Equation (14) gives the explicit relation for the pressure gradient as a function of thecross-sectional average velocity ū, once ū N is substituted by its definition (equation(11)). The main results of the analysis for the linearized PTT fluid are therefore thevelocity profile (10), the flux equation (12) and the unknown driving pressure gradientat given flow rate obtained from (14). The maximum velocity at the centreline (y =0)is also useful and is given byu 0ū = κ1+4κ2 ɛDe 2 (ū N /ū) 21+b(ū N /ū) 2showing that it is smaller than in the Newtonian case. Expressions for the normalizedstress components are readily obtained after scaling with the wall shear stress forthe Newtonian (or UCM) fluid. From equation (8), the non-dimensional normalstress is) 2 ( ) 2τ xx(ūN y2κη ū/H =4κDe (15)ū H(13)

Fully developed flow of Phan-Thien–Tanner fluids 275and the normalized shear stress component is calculated from (7):)( )τ xy(ūN y2κη ū/H = − . (16)ū HThe shear strain rate, from (9), becomes˙γ(y)2κ ū/H = −ūN ūand the viscosity profile isµ(˙γ) ≡ τ xy˙γ⇒ µ(˙γ)η( yH)(1+8κ 2 ɛDe 2 (ūNū=) 2 ( yH) 2 )(17)() 2 ( ) 2 ) −1 (ūN y1+8κ 2 ɛDe 2 . (18)ū HWall values for these quantities are useful to define non-dimensional quantities andare obtained after setting y = H:) 2 ) −1µ w(ūN(1+8κη = 2 ɛDe 2 (τ xy ) w,ū 2κη ū/H = ūN andū) 2(τ xx ) w(ūN2κη ū/H =4κDe ,ū(19)where (τ xy ) w is defined positive. These values are smaller than the corresponding valuesfor the UCM fluid, a point to be taken into account when comparing non-dimensionalvalues.2.2. Exponential PTT modelThe normal and shear stress profiles are independent of the function f(trτ), thereforethey are still given by equations (15) and (16), respectively. However, the ratio ū N /ūin those expressions is different since it depends on the new velocity distribution. Thisdistribution is obtained in a similar way by inserting the new f function (equation(3)) into equation (9) followed by integration, to yield the dimensional and thecorresponding non-dimensional forms of the velocity profile, respectively:andu(y) = exp(ɛλ2 H 2 p 2 ,x/(2 (2j−1) η 2 ))−p ,x ɛλ 2 /(2 j−2 η)u(y)ū= κūN ū(1 − exp(− ɛλ2 p 2 ,xη 2 2 2j−1 (H2 − y 2 )))(20a)exp (b(ū N /ū) 2 )b(ū N /ū) 2 (1 − exp(−b(ū N /ū) 2 (1 − (y/H) 2 ))), (20b)where now b ≡ 8κ 2 ɛDe 2 . This is also valid for the axisymmetric case, after effectingthe appropriate changes. Appropriately, in the limit of small b(ū N /ū) 2 , equation (20b)tends to the parabolic profile, as it should.The equation relating the flow rate and the pressure gradient, required for thesolution of the direct problem, is no longer common to both geometries. For thechannel, we obtain( 2ɛλ 2 p 2 ,xH 2ū =−η (exp4ɛλ 2 p ,xη 2 )−η2λp ,x Hπ 1/2 erf (i(λp ,x H/η) √ )2ɛ)i √ 2ɛ(21a)and for the pipeū =−η ( ɛλ 2 p 2 ,xH 2 )(exp1+ exp(−ɛλ2 p 2 ,xH 2 /(2η 2 )))−1. (21b)2ɛλ 2 p ,x 2η 2 ɛλ 2 p 2 ,xH 2 /(2η 2 )

276 P. J. Oliveira and F. T. Pinho1.00.80.6u Nu0.40.20 2 4 6 8 10ε 1/2 DeFigure 1. Variation of the average velocity ratio ū N /ū with ɛ 1/2 De (solid line: plane flow; dashedline: axisymmetric flow; no symbols: linear PTT; symbols: exponential PTT).ɛ 1/2 De Channel Pipe ɛ 1/2 De Channel Pipe0.05 0.9746 0.9526 1.0 0.3134 0.24710.1 0.9132 0.8545 2.0 0.1858 0.14390.2 0.7695 0.6732 3.0 0.1342 0.10330.3 0.6515 0.5492 4.0 0.1059 0.081140.4 0.5628 0.4642 5.0 0.08778 0.067130.5 0.4953 0.4028 6.0 0.07519 0.057400.6 0.4427 0.3565 7.0 0.06589 0.050240.7 0.4007 0.3203 8.0 0.05873 0.044740.8 0.3663 0.2911 9.0 0.05303 0.040360.9 0.3376 0.2672 10.0 0.04839 0.03680Table 1. Numerical solution of equations (24) and (25). Values of ū N /ū.The normalized shear-rate and viscosity profiles are readily obtained from (20b) and(16):( ) () 2 ( ) 2 )˙γ(y)y(ūN y2κ ū/H = −ūN exp 8κ 2 ɛDe 2 (22)ū H ū Hand() 2 ( ) 2 )µ(˙γ)(ūN y= exp − 8κ 2 ɛDe 2 . (23)ηū HAgain, for the inverse problem of determining the pressure gradient for a given flux,the non-dimensional velocity profile is integrated across the channel or pipe sectionsto give the parameter ū N /ū. Here, also, different equations are obtained for the planar,1= 3 ()ū N exp (b(ū N /ū) 2 )1+ iπ1/2 exp (−b(ū N /ū) 2 ) erf (ib 1/2 ū N /ū)(24)2 ū b(ū N /ū) 2 2b 1/2 ū N /ū

1.0Fully developed flow of Phan-Thien–Tanner fluids 2770.8yH0.60.40.2UCM0.10.51.05.00 0.5 1.0 1.5u/uFigure 2. Velocity profile of the linear PTT fluid in channel flow as a function of the dimensionlessgroup ɛ 1/2 De (solid line: parabolic profile; dashed lines: ɛ 1/2 De =0.1, 0.5, 1.0 and 5.0).1.00.8yH0.60.4ChannelPipe0.20 0.5 1.0 1.5u/u2.0Figure 3. Comparison of the velocity profiles for the linear and exponential PTT model in channeland pipe flow (De = 2 and ɛ = 0.1) (solid lines: parabolic profile; dashed lines: linear andexponential PTT).and the axisymmetric cases1=2ūN ū(exp (b(ū N /ū) 2 )1 − 1 − exp (−b(ū )N/ū) 2 ). (25)b(ū N /ū) 2 b(ū N /ū) 2This time, these nonlinear equations are not amenable to an analytical solution andtherefore numerical methods are required. We have solved the above equations with

278 P. J. Oliveira and F. T. Pinho1.00.8s xysxx(a)yH0.60.40.20–10 1 2De =2UCMε = 0.01ε = 0.1ε = 0.25ε =1.031.0s xys xxDe =1(b)0.8yH0.60.40.2ε = 0.25De = 0.1De =2De =5De =100–10 1 2s ij/(3gu/H )3Figure 4. Profiles of the normalized shear and normal stress components: (a) for varying ɛ, atconstant De = 2 (solid lines: UCM fluid; dashed lines: ɛ =0.01, 0.1, 0.25 and 1.0); (b) for varyingDe, at constant ɛ =0.25 (solid lines: Newtonian fluid; dashed lines: De =0.1, 1, 2, 5 and 10).a straightforward but robust bisection method and the solution is shown in figure 1;for the sake of completeness, a few such values are also given in table 1 from whereother values can be extracted by interpolation.3. DiscussionIt is clear from figure 1 that, for identical pressure gradients, the PTT fluid cancarry a larger flow rate than the Newtonian or UCM fluids, especially for ɛ 1/2 Delarger than 2. This effect is due to an increased shear-thinning behaviour with theparameter ɛ 1/2 De and is more intense with the exponential form of the PTT model.The shear-thinning behaviour is also observed in the flatter velocity profiles pertaining

Fully developed flow of Phan-Thien–Tanner fluids 279to the plane flow of the linear PTT fluid in figure 2. As ɛ 1/2 De increases the velocityprofiles flatten in the centre in a similar way to those of shear-thinning power-lawfluids (Skelland 1967). The exponential form of the PTT model leads to velocitydistributions (equations (20), (24) and (25)) that are similar to those in figure 2 exceptfor the increased shear-thinning behaviour as a consequence of the correspondinghigher values of the function f, as seen in figure 3. Shear thinning reduces the wallshear stress since the increased shear rate at the wall (cf. du/dy at y = H in figure 2)is outweighted by the reduction in the viscosity; for example, for De = 2 and ɛ =0.1,˙γ w increases by 34% relative to the Newtonian case, but µ w is reduced by 63.4% andthe net effect is a reduction of 51% of the wall shear stress.The distributions of the normalized τ xy and τ xx across the channel width areshown together for the linear PTT model in figure 4(a), for varying ɛ at constantDe, and in figure 4(b) for varying De at constant ɛ. The trends in figure 4(a) areexpected since ɛ → 0 brings the PTT model close to the UCM model and so thestresses should increase in magnitude. In the latter graph, however, the trend is notmonotonic and for high elasticity (high De) the normal stresses are seen to decrease,an unexpected outcome. Inspection of the relevant equations shows that both stresscomponents depend only on the dimensionless group ɛ 1/2 De, but the normal stressalso depends separately on De alone. For high De, equation (15) shows τ xx ∝ De −1/3hence justifying the decrease of τ xx with elasticity seen in figure 4(b). This peculiareffect can be removed if the stresses are made non-dimensional with their own valueof shear stress at the wall. Then, the variation of τ xy /(τ xy ) w will coincide with that forthe UCM or the Newtonian models, and the normal stress will be given byτ xx=4κDe(ū N /ū)(y/H) 2 ,(τ xy ) wwhich, for high De, tends to ≈ (De/ɛ) 1/3 at the wall, because ū N /ū ≈ 1/b 1/3 ≈1/(ɛDe 2 ) 1/3 (see equation (13)). Hence, the above non-dimensional normal stress nowincreases monotonically with De.The authors would like to acknowledge the financial support provided by JuntaNacional de Investigação Científica e Tecnológica (JNICT, Portugal) under projectPBIC/P/QUI/1980/95. Useful comments by the referees are also acknowledged. Theauthors are alphabetically listed.REFERENCESAzaiez, J., Guénette, R. & Aït-Kadi, A. 1996 Numerical simulation of viscoelastic flows througha planar contraction. J. Non-Newtonian Fluid Mech. 62, 253–277.Baaijens,F.P.T.1993 Numerical analysis of start-up planar and axisymmetric contraction flowsusing multi-mode differential constitutive models. J. Non-Newtonian Fluid Mech. 48, 147–180.Baloch, A., Townsend, P. & Webster, M. F. 1996 On vortex development in viscoelastic expansionand contraction flows. J. Non-Newtonian Fluid Mech. 65, 133–149.Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids. Volume 1:Fluid Mechanics, 2nd ed. John Wiley and Sons.Carew, E. O. A., Townsend, P. & Webster, M. F. 1993 A Taylor-Petrov-Galerkin algorithm forviscoelastic flow. J. Non-Newtonian Fluid Mech. 50, 253–287.Phan-Thien, N. 1978 A nonlinear network viscoelastic model. J. Rheol. 22, 259–283.Phan-Thien, N. & Tanner, R. I. 1977 A new constitutive equation derived from network theory.J. Non-Newtonian Fluid Mech. 2, 353–365.Quinzani, L., Armstrong, R. C. & Brown, R. A. 1995 Use of coupled birefringence and LDV

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