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

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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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