278 P. J. Oliveira <strong>and</strong> 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. Pr<strong>of</strong>iles <strong>of</strong> the normalized shear <strong>and</strong> normal stress components: (a) <strong>for</strong> varying ɛ, atconstant De = 2 (solid lines: UCM fluid; dashed lines: ɛ =0.01, 0.1, 0.25 <strong>and</strong> 1.0); (b) <strong>for</strong> varyingDe, at constant ɛ =0.25 (solid lines: Newtonian fluid; dashed lines: De =0.1, 1, 2, 5 <strong>and</strong> 10).a straight<strong>for</strong>ward but robust bisection method <strong>and</strong> the <strong>solution</strong> is shown in figure 1;<strong>for</strong> the sake <strong>of</strong> 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, <strong>for</strong> identical pressure gradients, the PTT fluid cancarry a larger <strong>flow</strong> rate than the Newtonian or UCM fluids, especially <strong>for</strong> ɛ 1/2 Delarger than 2. This effect is due to an increased shear-thinning behaviour with theparameter ɛ 1/2 De <strong>and</strong> is more intense with the exponential <strong>for</strong>m <strong>of</strong> the PTT model.The shear-thinning behaviour is also observed in the flatter velocity pr<strong>of</strong>iles pertaining
Fully <strong>developed</strong> <strong>flow</strong> <strong>of</strong> <strong>Phan</strong>-Thien–Tanner fluids 279to the plane <strong>flow</strong> <strong>of</strong> the linear PTT fluid in figure 2. As ɛ 1/2 De increases the velocitypr<strong>of</strong>iles flatten in the centre in a similar way to those <strong>of</strong> shear-thinning power-lawfluids (Skell<strong>and</strong> 1967). The exponential <strong>for</strong>m <strong>of</strong> the PTT model leads to velocitydistributions (equations (20), (24) <strong>and</strong> (25)) that are similar to those in figure 2 except<strong>for</strong> the increased shear-thinning behaviour as a consequence <strong>of</strong> the correspondinghigher values <strong>of</strong> 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; <strong>for</strong> example, <strong>for</strong> De = 2 <strong>and</strong> ɛ =0.1,˙γ w increases by 34% relative to the Newtonian case, but µ w is reduced by 63.4% <strong>and</strong>the net effect is a reduction <strong>of</strong> 51% <strong>of</strong> the wall shear stress.The distributions <strong>of</strong> the normalized τ xy <strong>and</strong> τ xx across the <strong>channel</strong> width areshown together <strong>for</strong> the linear PTT model in figure 4(a), <strong>for</strong> varying ɛ at constantDe, <strong>and</strong> in figure 4(b) <strong>for</strong> varying De at constant ɛ. The trends in figure 4(a) areexpected since ɛ → 0 brings the PTT model close to the UCM model <strong>and</strong> so thestresses should increase in magnitude. In the latter graph, however, the trend is notmonotonic <strong>and</strong> <strong>for</strong> high elasticity (high De) the normal stresses are seen to decrease,an unexpected outcome. Inspection <strong>of</strong> 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 <strong>of</strong> τ xx with elasticity seen in figure 4(b). This peculiareffect can be removed if the stresses are made non-dimensional with their own value<strong>of</strong> shear stress at the wall. Then, the variation <strong>of</strong> τ xy /(τ xy ) w will coincide with that <strong>for</strong>the UCM or the Newtonian models, <strong>and</strong> the normal stress will be given byτ xx=4κDe(ū N /ū)(y/H) 2 ,(τ xy ) wwhich, <strong>for</strong> 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 <strong>of</strong> viscoelastic <strong>flow</strong>s througha planar contraction. J. Non-Newtonian Fluid Mech. 62, 253–277.Baaijens,F.P.T.1993 Numerical analysis <strong>of</strong> start-up planar <strong>and</strong> axisymmetric contraction <strong>flow</strong>susing 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 expansion<strong>and</strong> contraction <strong>flow</strong>s. J. Non-Newtonian Fluid Mech. 65, 133–149.Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics <strong>of</strong> Polymeric Liquids. Volume 1:Fluid Mechanics, 2nd ed. John Wiley <strong>and</strong> Sons.Carew, E. O. A., Townsend, P. & Webster, M. F. 1993 A Taylor-Petrov-Galerkin algorithm <strong>for</strong>viscoelastic <strong>flow</strong>. J. Non-Newtonian Fluid Mech. 50, 253–287.<strong>Phan</strong>-Thien, N. 1978 A nonlinear network viscoelastic model. J. Rheol. 22, 259–283.<strong>Phan</strong>-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 <strong>of</strong> coupled birefringence <strong>and</strong> LDV