Korea-Australia Rheology Journal, Vol.24, No.4, pp.277-286 (2012) www.springer.com/13367 DOI: 10.1007/s13367-012-0034-x

Isothermal and non-isothermal viscoelastic flow of PTT fluid in lid-driven polar cavity

Hatice Mercan and Kunt Atal k* o Mechanical Engineering Department, Bog aziçi University, 34342 Bebek, Istanbul, Turkey (Received January 24, 2012; final revision received October 16, 2012; accepted October 31, 2012)

The isothermal and non-isothermal viscoelastic flow of Phan-Thien-Tanner (PTT) fluids is considered in lid- driven polar cavity geometry, using a numerical solution method with parameter continuation technique. Thermoelastic effects, in terms of elastic/elongational effects and viscous dissipation, are demonstrated by the changes in vortical structure, temperature/stress distributions and heat transfer characteristics in the curved cavity. Central vortex/maximum temperature location shifts are observed under elastic and elon- gational (strain hardening and strain softening/shear thinning) effects for isothermal and non-isothermal conditions. The growth in size and strength of a secondary vortex is denoted in the downstream stationary corner of the cavity for the viscoelastic fluid under strain hardening, which also introduces an increase in stress gradients. Viscous heating is observed with elongational effects near the central vortex in the cavity. Stress components and their gradients decrease under viscous dissipation. The changes in temperature field and heat transfer properties in the cavity are revealed. Keywords: Lid-driven polar cavity, viscoelastic, thermoelastic, non-isothermal, viscous dissipation, Phan- Thien-Tanner fluid

1. Introduction tally and numerically the increase of temperature due to viscous dissipation in Newtonian and non-Newtonian In this study we investigate non-isothermal effects on power-law (shear thinning) fluids for the flow in a scraped two-dimensional viscoelastic flow in a polar cavity, under surface heat exchanger. They noticed an increase in vis- low inertia. The polar cavity geometry can also be con- cous heating as the rotor speed increases. However, they sidered as a typical cross section of a single screw reported that the effect of viscous dissipation was not extruder, and is suitable to illustrate the curved geometry observed for the power-law fluid. They attributed this effect on an internal flow field. This geometry (with cavity behavior to the shear thinning character of the fluid which angle of 1 radian and gap ratio of 1/2) has been described considerably reduces the apparent viscosity and therefore and used in a study by Fuchs and Tillmark (1985), where the viscous dissipation in the high shear rate zones near experimental and numerical simulation results have been the blade. obtained for Newtonian flow at moderate Reynolds num- Peters and Baaijens (1997) considered the modeling of bers. Wu and co-workers (2007) solved numerically the non-isothermal viscoelastic flows using different consti- polar cavity flow of a Newtonian fluid up to Reynolds tutive equations and analyzed the partition between dis- number Re = 1000 to validate their Lagrange interpolation sipated and elastically stored energy in terms of entropy method. Also, Kim (1989), Lei and co-workers (2000), and energy elasticity. Yeçs ilata (2002) formulated the prob- and Darbandi and Vikilipour (2008) used this geometry as lem for the non-isothermal flow of an Oldroyd-B fluid a test and example case for their numerical schemes to between two rotating parallel plates and investigated how develop solvers for Navier-Stokes equations. to predict the material properties of viscoelastic fluids The effects of viscous dissipation on Newtonian and under viscous dissipation effects. non-Newtonian flow fields have been studied by many Purely elastic and thermoelastic instabilities have been researchers, for curved geometries. Al-Mubaiyedh and co- revealed in flows with curved streamlines like Taylor- workers (2002) investigated the influence of viscous heat- Couette, parallel plates, cone-and-plate and Dean flow ing on the stability of Taylor-Couette flow for Newtonian geometries (Muller, 2008). In addition to purely elastic fluids with thermally sensitive viscosity. Based on the lin- instabilities, viscous heating leads to temperature gradi- ear stability analysis, they showed that viscous heating ents in the flow field which in turn produce gradients in leads to significant destabilization in Taylor-Couette flow. fluid viscosity and elasticity. These gradients are observed Yataghane and co-workers (2009) analyzed experimen- to lead to new thermoelastic instability modes. Thomas and co-workers (2004) investigated the effect of base flow temperature and inertia on the stability of Taylor-Couette *Corresponding author: [email protected] flow and noted that inertia effects may become important

© 2012 The Korean Society of Rheology and Springer 277 Hatice Mercan and Kunt Atali k for sufficiently large temperature ranges. Coelho and co-workers (2003) presented a theoretical study of viscoelastic thermal entry flow (Graetz problem) in channels and tubes for the PTT model. For the imposed temperature and fluid heating case, they observed that vis- cous heating due to dissipation tends to increase in regions where the velocity gradients are steeper, especially near the wall. They also mentioned that as elongational/elastic effects increase the overall heat transfer tends to increase in fully developed flow regime. Pinho and Coelho (2006) investigated theoretically a simplified viscoelastic PTT flow in annuli under viscous dissipation and found that for imposed wall temperature case, shear thinning and elas- ticity lead to lower heat transfer when viscous dissipation is weak and to higher heat transfer when viscous dissi- pation is strong. In this study we investigate numerically the isothermal and non-isothermal viscoelastic flow of a PTT fluid in a lid-driven polar cavity. Elastic, elongational (strain hard- ening and strain softening/shear thinning) and viscous dis- Fig. 1. Cavity geometry. sipation effects are considered in the analysis. As a general model the Williams-Lendel-Ferry equation is used application, (Karwe and Jaluria, 1990). to describe the temperature dependence of the material The mass and momentum conservation equations in parameters such as the viscosity and the relaxation time. dimensionless vector form for unsteady, non-isothermal, Thermoelastic effects on the flow and thermal fields are incompressible viscoelastic flow can be written as, shown and compared with respect to the Newtonian case. ∇⋅ V = 0 (1) The results are presented and discussed in terms of the flow structure, the temperature and stress distributions, as DV well as the heat transfer properties in the cavity. Re------= –∇p+∇⋅ []∇21()–ωr D + ⋅ T (2) The isothermal and non-isothermal modeling of vis- Dt coelastic flow for PTT fluids is given in Section 2 in terms where V is the velocity vector, D is the rate of deformation of non-dimensional parameters and equations. The details tensor, T is the viscoelastic extra-stress tensor, p is the of the numerical solution method together with numerical pressure, T is the temperature, D/Dt denotes the material tests and comparisons to available data in the literature are derivative. The is defined as presented in Section 3. The results are shown and dis- Re = ρV0ri ⁄µ0 , where ρ is the fluid density, V0 is the char- cussed in Section 4, before concluding with final remarks acteristic lid velocity and µ0 is the total viscosity calcu- in Section 5. lated at reference temperature. The pressure and viscoelastic stress are scaled with µ0V0 ⁄ri . The non-dimen- ˜ 2. Mathematical Modeling sional temperature is defined as TT= ()–Ti ⁄()To –Ti where T˜ is the dimensional temperature. The retardation param- The two-dimensional flow of a polymer solution is con- eter ωr is defined as the ratio of zero shear rate polymer sidered in a polar cavity with cavity angle θp = π ⁄2 and viscosity µp at reference temperature, to µ0 . The vis- inner radius ri = 1 , outer radius ro = 1.75 (Fig. 1). The coelastic stress (T) is defined using linear PTT constitutive outer and side walls are fixed, while the inner lid moves relation. In dimensionless form the non-isothermal PTT in the transverse θ direction with a predefined velocity model reads, distribution. The fluid is initially at rest and no-slip bound- ∇ ε ()⎛⎞() ω () ary conditions are specified at the walls. T+2We T ⎝⎠T+ ------tr T T = r T D (3) ωr()T The outer wall is kept at a constant high temperature To , while the moving inner lid is kept at a constant low tem- where tr denotes the trace. In the above constitutive rela- perature Ti (To >Ti ). The stationary side walls are insu- tion, the temperature dependency of the connector force lated. Since this geometry is a better approximation in related to polymeric fluid structure in kinetic theory, is two-dimensional space for single screw extruders com- neglected following the final formulation in Peters and pared to the rectangular cavity geometry, the thermal Baaijens (1997). The Weissenberg number is defined as boundary conditions are determined according to this We = λVo ⁄ri , where λ is the relaxation time of the fluid

278 Korea-Australia Rheology J., Vol. 24, No. 4 (2012) Isothermal and non-isothermal viscoelastic flow of PTT fluid in lid-driven polar cavity and ε is the elongational parameter of the PTT model. This lowing the assumption that the temperatures are close to parameter characterizes the strain hardening and strain glass transition temperature, for strong thermorheological softening/shear ∇thinning feature of the polymer additive. coupling (Peters and Baaijens, 1997). In the above for- The operator () denotes the upper convected derivative mulation (Eq.6), the temperature dependence of the steady as follows, state recoverable compliance (the ratio of polymer relax-

∇ ation time to polymer viscosity) which is found to be weak DT T T = ------–L ⋅TTL– ⋅ (4) in temperature ranges close to glass transition temperature Dt and often neglected in the literature (Ngai and Plazek, where L is the velocity gradient tensor. 1996; Peters and Baaijens, 1997), is also ignored. In the formulation by Peters and Baaijens (1997), the energy equation includes both the entropy and energy 3. Method of Solution elasticity effects, since for a viscoelastic fluid, the mechanical energy can be assumed to be partially dissi- The streamfunction-vorticity formulation in cylindrical pated and partially stored elastically. They define a par- coordinates is adopted. In cylindrical coordinates the rela- tition parameter α (related to temperature dependence of tions between streamfunction, vorticity and velocity field stress) for these effects. The pure energy elasticity case are as follows: corresponds to α = 0 and the pure entropy elasticity case 1∂ψ α = 1 ur = ------corresponds to , respectively. The latter case is gen- ∂θ erally considered in numerical applications for non-iso- r thermal flows in the literature. The non-dimensional ∂ψ formulation of the energy equation for a PTT fluid with a uθ = –------∂r Newtonian solvent is then given by: ∂ DT 1 2 Br uθ uθ 1∂ur 2 ------= ------∇ T+ ------21()()()–ωr D :D + T :D (5) ω ==----- + ------– ------–∇ ψ (8) Dt Pe Pe r ∂r r ∂θ where Péclet number is defined as Pe = ρCpV0ri ⁄k with Cp , where ψ is the streamfunction, ω is the vorticity, ur and uθ the heat capacity and k, the thermal conductivity of the are the radial and transverse velocity components, respec- fluid. represents the ratio of mechanical tively. With this formulation the mass conservation (Eq.1) energy dissipation to heat conduction resulting from is automatically satisfied and the vorticity transport equa- imposed temperature difference, and is defined as tion is obtained by taking the curl of the momentum equa- 2 Br = µ0V0 ⁄k()To –Ti . tion (Eq.2) and eliminating the pressure term. We also consider the temperature difference (To –Ti ) to The governing equations are solved numerically using a be small such that the fluid can be assumed incompress- second order centered finite difference scheme. The ible. The density and thermal conductivity are assumed to explicit Runge-Kutta-Fehlberg method with time step be independent of the temperature field. We also neglect adjustment is used for time integration. Elliptic stream- the temperature dependency of the solvent viscosity. How- function-vorticity equation is solved by successive over ever polymer viscosity and relaxation time of the polymer relaxation (SOR) method with Chebychev acceleration. A additive are assumed to be temperature dependent. The parameter continuation technique is applied to accelerate temperature dependence of the non-dimensional Weissen- convergence to steady state values for a given parameter berg number and retardation parameter are as follows, set, by making use of the solutions for a nearby parameter set as initial conditions. We() T = Wef() T The vorticity boundary conditions for the boundaries shown in Fig. 1 can be written as follows: ωr()T = ωrfT() (6)

ψw–1 where We is the Weissenberg number defined with respect ωwall, AB– CD = 2------2∆θ2 to relaxation time at reference temperature, and ωr is the r retardation parameter defined with respect to viscosity val- ()ψw–1 +Ui∆r ues at reference temperature. The temperature dependency ωlid, AD = 2------∆ 2 function, f()T , is defined according to Williams-Lendel- r Ferry (WLF) model which reads, ψ ω w–1 wall, CB = 2------2- (9) c1T ∆ f()T = exp –------(7) r c2 ⁄()To –Ti +T where the subscript w–1 represents the value of the vari- where the constants are set as c1 = 15 and c2 = 50 , fol- able next to the wall, ∆θ and ∆r are space increments for

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π 2 π 2 (),θ 8 ()()⎛⎞--- θ ⎛⎞--- θ Ui t = ------14 + tanh 8t–4 ⎝⎠+ ⎝⎠– (10) ()π ⁄2 4 4 where t denotes the non-dimensional time. The velocities are zero at the corners and symmetric bell like distribution reaches steady state conditions in short times. To investigate whether this new form affects the flow structure in the cavity with respect to constant lid veloc- ity case, we compared the results for these different boundary condition formulations for Newtonian flow in polar cavity geometry at Reynolds number Re = 60 with θp = 1 , ri = 1 , ro = 2 (Fig. 2 (a) and 2 (b)). It can be observed that the vortex structure is qualitatively similar and the streamfunction values at the center of the pri- mary vortex are also close. This simple analysis suggests that the introduction of distributed lid velocity does not qualitatively modify the flow behavior in the cavity, and therefore it is acceptable as a boundary condition for our case. The system of equations is integrated in time and steady state solutions are presented. The time step adjustment Fig. 2. Newtonian polar cavity flow, velocity plots along cen- technique is combined with Runge-Kutta-Fehlberg scheme θ terline, Re = 60 and Re= 350 ( p =1, ri = 1 and ro = 2) (a) trans- (Matthews and Fink, 1999). The smallest adjusted time θ –6 verse velocity component distribution along =0 and step is in the order of 10 for the non-isothermal vis- comparison with Kim (1989) (b) radial velocity component dis- coelastic case Re = 0.5 , We = 3.0 , ωr = 0.3 , ε = 0.5 , tribution along r= 1.5 and comparison with Kim (1989). Pe = 900 and Br = 100 . It is accepted that the steady state is reached according to a relative difference norm E of the th discretization in transverse and radial directions, respec- vorticity at each ()k+1 Runge Kutta Fehlberg adjusted th tively and Ui is the predefined lid velocity. time step with respect to the previous k time step as To validate the numerical scheme, solutions for New- Nr,Nθ tonian case are obtained at Reynolds number Re = 60 and k +1 k 2 ∑ ()ω ()ij, –ω ()ij, Re = 350 with θp = 1 radian, ri = 1 , ro = 2 . The centerline E = ------ij, = 1 (11) Nr,Nθ velocity component distributions are compared with the k +1 2 literature in Fig. 2, where it can be observed that the ∑ ()ω ()ij, ij, = 1 results are in good agreement. For viscoelastic fluid flows, it is well known that corner where Nr and Nθ denote the number of grid points in the singularities cause elastic instabilities and numerical con- r and θ directions respectively. In the simulations for pre- vergence problems. At high Weissenberg numbers, the sented steady state results this E value is of the order of –9 singularity near the lid corner in cavity flow produces high 510× and below. stresses and stress gradients, leading to numerical con- The parameter continuation technique is used to solve vergence problems and even to blow-up of solutions. the unsteady system of differential equations such that an Some remedies have been reported in the literature to enhancement for the initial guess is obtained. The stability overcome this problem. Grillet and co-workers (2000) of the numerical scheme is observed to be mostly sensitive introduced a controlled amount of leakage near the cor- to the ratio of elongational to retardation parameters, ners and achieved a convergent numerical result for vis- εω⁄ r , and the ratio of elastic effects to inertial effects, coelastic cavity flow. Fattal and Kupferman (2005) We⁄ Re . In this study the values of these ratios are in the pointed out that the velocity gradient near a singularity, in intervals 1≤≤εω⁄ r 3 and 1≤We⁄ Re≤3 for isothermal case this case near the moving lid corner, should be eliminated and in the intervals 1≤≤εω⁄ r 2 and 2≤We⁄ Re≤6 for the and impulsive start should be smoothed out. In this study non-isothermal case. we used a symmetric bell like distributed velocity function In Table 1, minimum and maximum streamfunction val- for the lid velocity (Ui ) to smooth out corner singularities, ues as well as minimum streamfunction locations are tab- inspired from the work by Fattal and Kupferman (2005) ulated for three different grid densities. Based on these for square cavity. The moving inner lid velocity in trans- tests, a structured grid with (65× 49 ) grid density is cho- verse direction is then given by, sen for simulations.

280 Korea-Australia Rheology J., Vol. 24, No. 4 (2012) Isothermal and non-isothermal viscoelastic flow of PTT fluid in lid-driven polar cavity

Table 1. Minimum streamfunction values/locations and maxi- mum streamfunction values for three different grid densities at Re = 0.5 , We = 1.0 , ωr = 0.3 , ε = 0.3 , Pe = 900 and Br = 10

Grid ψmin ()xmin,ymin ψmax 51×39 -0.080529 (1.2187,0.000121) 3.3661 e-4 65×49 -0.079479 (1.2158,0.000119) 2.9354 e-4 81×61 -0.079828 (1.2126,0.000102) 2.6253 e-4

Fig. 3. Newtonian polar cavity flow streamlines at Re =60 (θp =1, ri =1 and ro = 2) (a) constant lid velocity case, ψmin = –0.116336 (b) distributed lid velocity case, ψmin = –0.106369 . Contour lev- els are: -0.1, -0.09, -0.08, -0.07, -0.06, -0.05, -0.04, -0.03, -0.02, -0.01, 0.

Fig. 5. ε effect on streamlines for isothermal viscoelastic PTT flow (Re = 1.0 , We = 1 , ωr = 0.35 ) (a) Newtonian, (b) ε = 0.3 , (c) ε = 0.5 , (d) ε = 0.9 . The stream function contour levels shown are -0.09, -0.08, -0.07, -0.06, -0.05, -0.04, -0.03, -0.02, -0.01, -0.002, 0.0., 0.00006.

eter and viscous dissipation for the non-isothermal vis- coelastic flow. Finally, the changes in heat transfer rate through the moving lid and stationary curved wall under strain hardening/softening and elastic effects are examined at low and high viscous dissipation. Fig. 4. Central vortex shift (a) Center location of primary vortex as a function of Reynolds number in Newtonian case (b) Center 4.1. The effect of elasticity and elongational param- location of primary vortex as a function of PTT elongational eter for isothermal flow: parameter (ε) in isothermal viscoelastic case, (Re = 1.0, We = 1 , When we compare the isothermal Newtonian and vis- ωr = 0.35 ) (c) Center location of primary vortex as a function of coelastic PTT fluid flow (We = 1 , Re = 1.0 ) cases, we Weissenberg number in isothermal viscoelastic case, (Re = 1.0 , observe a shift in central vortex location compared to ε ω = 0.9 , r = 0.35 ). Newtonian case, in the direction opposite to lid motion and towards the lid under elastic and strain hardening effects, as the elongational parameter is decreased from 4. Results and Discussion ε = 0.9 (most strain softening and shear thinning case) to ε = 0.3 (strain hardening case) (Fig. 4(b)). When the elas- We first discuss the elastic and elongational effects for ticity is increased from We = 1 to We = 3 at a fixed elon- the isothermal flow in lid-driven polar cavity by com- gational parameter value (ε = 0.9 ) the central vortex shifts paring the isothermal Newtonian and PTT fluid flow again towards the lid and in the opposite direction to the cases. Then we present the effects of elongational param- lid motion (Fig. 4(c)). A similar observation can be found

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Fig. 7. ε effect on stress distributions along radial direction for isothermal viscoelastic PTT flow (We = 1.0 , Re = 1.0 , ωr = 0.35 ) (a) shear stress ()Trθ along θ = 0.2209 (b) normal stress ()Tθθ along θ = 0.0491 .

numbers, the vortical structure is opposite in the sense that Fig. 6. Velocity component profiles for isothermal viscoelastic the upstream stationary corner vortex (corner B in Fig. 1) PTT flow (Re = 1.0 , We = 1.0 , ωr = 0.35 ) at different ε values increases in size/strength and becomes much larger com- (a) radial velocity component comparison along r = 1.6406 (b) pared to the size of the downstream stationary corner vortex transverse velocity component comparison along θ = 0.5154 . as inertia effects are increased (Fuchs and Tillmark, 1985). The velocity component distributions near this corner are shown along transverse and radial directions in Figs. 6 in the literature for elastic but constant shear/finite exten- (a)-(b) respectively. As the elongational parameter is sional viscosity fluid (FENE-CR dumbbell model) flow in increased from ε = 0.3 to ε = 0.9 and the fluid shear thins, lid-driven square cavity as the fluid elasticity is increased the value of transverse velocity component is increased as Grillet and co-workers (1999), which has been attributed expected and the value of radial component decreases. by the authors to the localization of large polymer stresses The stress gradients near the moving lid are increased with near the corners and high polymer stretching downstream increasing strain hardening effect. Stress distributions in of the corners. For Newtonian flow, the shift is in the radial directions are shown in Fig. 7 for different ε values direction of the lid motion and away from the moving lid at a section where the steepest gradients are observed. at high inertia as the Reynolds number is increased, which High normal and shear stress values are localized in the can be observed in Fig. 4 (a). Thus elasticity (increase in region close to the moving lid and these values decrease Weissenberg number) and strain softening/shear thinning with strain softening/shear thinning. (increase in elongational parameter) have opposite effects on the shift of the primary vortex location in the cavity. 4.2. The effect of elasticity and elongational parameter In Fig. 5 the vortex structures are compared at Re = 1.0 , for non-isothermal flow with viscous dissipation We = 1.0 , ωr = 0.3 for different elongational parameters. We compare the non-isothermal Newtonian case and non- We observe a growth in downstream stationary corner vor- isothermal viscoelastic PTT case under pure viscous dis- tex (corner C in Fig. 1) which then decreases in size and sipation. It is observed from Table 2 that the downstream strength as strain softening/shear thinning is increased with corner vortex is considerably increased in size and strength the increase in elongational parameter up to ε = 0.9 . It compared to Newtonian case at the value of the elonga- should be noted that for Newtonian flow at high Reynolds tional parameter ε = 0.3 . We observe that the downstream

282 Korea-Australia Rheology J., Vol. 24, No. 4 (2012) Isothermal and non-isothermal viscoelastic flow of PTT fluid in lid-driven polar cavity

Table 2. Center (C) and downstream stationary corner (DSC ) vortex strength values for Re = 0.5 , We = 1 , ωr = 0.3 , Pe = 900 under changing viscous heating and elongational effects. Br = 1 Br = 10 Br = 50 Br = 100 ε ψC ψDSC ψC ψDSC ψC ψDSC ψC ψDSC 0.3 -0.0763080 1.1891e-04 -0.0794790 2.9354e-04 -0.0863519 7.7310e-05 -0.0898415 2.820e-05 0.4 -0.0794638 3.3650e-04 -0.0817558 2.3703e-04 -0.0871260 6.5930e-05 -0.0900200 2.823e-05 0.5 -0.0813174 2.7902-e04 -0.0832344 1.7744e-04 -0.0876205 5.9180e-05 -0.0901387 2.873e-05 0.6 -0.0826833 2.1729e-04 -0.0842437 1.4228e-04 -0.08799407 5.3930e-05 -0.0902234 3.039e-05

Fig. 8. Br effect on stress distributions along radial direction for non-isothermal viscoelastic PTT flow (We = 1.0 , Re = 0.5 , ωr = 0.3 , ε = 0.5 , Pe = 900 ) (a) shear stress ()Trθ for θ = 0.2945 (b) normal stress ()Tθθ for θ = 0.0982 . Fig. 9. Temperature profiles for non-isothermal viscoelastic PTT flow (We = 1.0 , Re = 0.5 , ωr = 0.3 , Pe = 900 , ε = 0.5 ) at dif- θ stationary corner vortex decreases in size/strength as the ferent Br values (a) along radius for = 0.2945 (b) along radius for θ = 0.0982 . value of the elongational parameter is increased to ε = 0.6 , similar to isothermal case as it is shown in Table 2 in terms of the strength ψDSC , (which is also an indication for the and stress gradients, in the region close to the moving lid size) of the corner vortex. When viscous dissipation is for increasing viscous dissipation effects. Temperature increased to Br = 50 the same effect is observed. At high profiles at the same section are given in Fig. 9, where the viscous dissipation (Br = 100 ), elongational effects seem to heating and increase in temperature gradients are clearly have an opposite effect on the corner vortex size and observed for increasing Br values. Thus stresses and strength. When viscous dissipation effects are increased their gradients decrease due to increase in temperature from Br = 1 to Br = 100 , we observe from the same table and temperature gradients under viscous dissipation, as it that the downstream stationary corner vortex is decreased can be expected. for all considered values of the elongational parameter, i.e. If we increase elasticity increasing the Weissenberg for both strain hardening and strain softening cases. number for ε = 0.5 , we observe a decrease in normal The elongational parameter effect on stress distribu- stress values near the moving lid, however they increase tions is similar to isothermal case. However stress values near the stationary curved wall (Fig. 10). These results are are affected by viscous heating. Shear stress (Trθ ) and similar to the observation of reduction in normal stresses normal stress (Tθθ ) distributions along radial direction in near the moving wall as We is increased in the vis- central region are given in Fig.8 for ε = 0.5 and for dif- coelastic flow between eccentric cylinders in Dris and ferent Br values. We observe a decrease in stress values Shaqfeh (1998), where this effect has been explained by

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Table 4. Average bulk Nusselt modulus at stationary curved wall for different Brinkman numbers and for changing elongational (ε) effect (Re = 0.5 , ωr = 0.3 , Pe = 900 , We = 1.0 ). ε Br = 1 Br = 10 Br = 50 Br = 100 0.3 3.2174 7.0747 8.7645 13.7454 0.4 3.1439 5.8409 9.1122 13.8547 0.5 3.0462 6.4898 9.4409 13.9614 0.6 2.9422 5.6591 9.6401 13.9796

Table 5. Average bulk Nusselt modulus at the stationary curved wall for different Brinkman numbers and for changing elastic (We ) effect ( Re = 0.5 , ωr = 0.3 , Pe = 900 , ε = 0.5 ). We Br = 10 Br = 50 Br = 100 1.0 6.4898 9.4409 13.9614 1.5 7.6847 14.4218 14.6132 2.0 8.1507 14.3364 14.4148 2.5 9.1741 13.7945 17.6621 3.0 9.1767 13.2900 19.8090

() Fig. 10. We effect on normal stress Tθθ distributions along ening and viscous dissipation are analyzed in terms of heat transverse direction for non-isothermal viscoelastic PTT flow, transfer properties in the cavity. is used to (Br=50, Re = 0.5 , ωr = 0.3 , ε = 0.5 ) (a) along moving lid (b) characterize the heat transfer between walls and fluid, and along stationary curved wall. defined as Nu= rihk⁄ . The heat transfer coefficient at the wall, h, is defined with respect to the wall heat flux Table 3. Maximum temperature values and locations for different qw = hT()b –Ti , where Tb , the bulk temperature is given as, ε ω values at Re = 0.5 , We = 1.0 , Pe = 900 , Br = 10 , r = 0.3 . r o ρ ∫r uθTrd ε Tmax ()xT ,yT i max max Tb = ------(12) ro ∫ ρuθdr 0.3 5.6834 (1.2151,0.03831) ri 0.4 5.9280 (1.2247,0.03985) The average Nusselt number along the moving lid and 0.5 5.9552 (1.2347,0.04018) stationary curved wall for different Brinkman numbers 0.6 6.0923 (1.2350,0.04020) (Br ) are tabulated for different values of elongational parameter (ε ) and Weissenberg number (We ) in Tables 4 and 5 respectively. It is observed in Table 4 that along the downstream of polymeric normal stresses and stationary curved wall, where the higher constant tem- lack of time for the polymer to react to local velocity gra- perature is imposed, heat transfer is enhanced with dients as We is increased. decreasing the elongational parameter ε for Br = 1 and In Table 3, maximum temperature values and their loca- Br = 10 , however at high viscous dissipation (Br = 50 and tions in the polar cavity are given for different values of Br = 100 ) the enhancement is denoted for increasing the the elongational parameter ε at Re = 0.5 , We = 1.0 , elongational parameter. ωr = 0.3 and Br = 10 . It is observed that viscous heating In Table 5 it is observed that along the stationary curved increases with increasing shear thinning/strain softening wall, heat transfer is enhanced with increasing We num- effects. The location of the maximum temperature is shift- ber for all Brinkman numbers. These behaviors can be ing away from the moving lid in the direction of lid compared to the analytical results given by Pinho and motion. Temperature gradients are observed to decrease Coelho (2006), where heat transfer is considered in vis- with strain softening/shear thinning, i.e. the decrease in coelastic concentric annular pipe flow for simplified PTT elongational parameter ε. model. It should also be noted that, in their work, the authors use a one-dimensional spatial approximation and 4.3. The effect of thermoelasticity on heat transfer assume temperature independent properties (for relaxation characteristics time and viscosity ratio). They observe a reduction in heat 2 The combined effects of elasticity, strain hardening-soft- transfer with the increase in combined parameter εWe

284 Korea-Australia Rheology J., Vol. 24, No. 4 (2012) Isothermal and non-isothermal viscoelastic flow of PTT fluid in lid-driven polar cavity under low viscous dissipation, however this trend is found Pe : Péclet number to be reversed for high viscous dissipation. Our numerical r : Radial direction results for the full model with temperature dependent ri : Inner radius properties, seem to indicate a similar behavior for the ro : Outer radius effect of elongational (ε ) parameter, in lid-driven polar Re : Reynolds number cavity flow. However, we note that the heat transfer is t : Non-dimensional time enhanced by increasing elasticity (We number) in both T : Non-dimensional temperature ˜ low and high viscous dissipation cases. These results sug- T : Dimensional temperature gest that the main mechanism for the change in heat trans- Tb : Bulk temperature fer behavior at low and high viscous dissipation is related Ti : Inner wall temperature to elongational effects and shear thinning. To : Outer wall temperature T : Viscoelastic extra-stress tensor 5. Conclusion Trr,,Tθθ Trθ : Viscoelastic extra-stress tensor components V : Velocity vector Isothermal and non-isothermal flows of viscoelastic PTT V0 : Characteristic velocity fluids have been studied numerically for the lid-driven (ur,uθ ) : Velocity components in r and θ directions polar cavity geometry with cavity angle of θp = π ⁄2 and We : Weissenberg number gap ratio of ri = 1 , ro = 1.75 . The results have been com- (xy, ) : Cartesian coordinates pared to Newtonian case and differences in flow structure and temperature/stress fields have been revealed under Greek Symbols elastic/elongational effects at low inertia. The effects of low and high viscous dissipation have been demonstrated. α : Stress temperature dependency parameter It has been found that elasticity and strain softening/shear β : Viscosity ratio thinning lead to central vortex shifts in opposite direc- (∆θ,∆r ) : Space increment tions. For isothermal and non-isothermal cases, growth in ε : Elongational parameter size/strength of upstream stationary corner vortex is θ : Transverse direction observed under strain hardening. This vortex is reduced in θp : Polar cavity angle size/strength with increased viscous dissipation. Shear and λ : Relaxation time normal stress components and their gradients decrease in µ0 : Total viscosity calculated at reference temperature the region close to moving lid under increased viscous dis- µ : Total viscosity sipation, which introduces an increase in temperature and µs : Solvent viscosity its gradient as well. Viscous heating is observed to µp : Zero shear rate polymer viscosity increase with strain softening/shear thinning. ρ : Density Heat transfer at stationary curved wall is observed to ψ : Streamfunction decrease with increased elongational effects at low vis- ω : Vorticity cous dissipation. However this behavior is reversed at ωr : Retardation parameter high viscous dissipation where the heat transfer is enhanced with increasing elongational effects. Also, the References heat transfer at this same location increases with elasticity for both low and high viscous dissipation. Al- Mubaiyedh, U.A., R. Sureshkumar, and B. Khomami, 2002, The effect of viscous heating on the stability of Taylor-Couette List of Symbols flow, J. Fluid Mech. 462, 111-132. Coelho, P.M., F.T. Pinho, and P.J. Oliveira, 2003, Thermal entry Br : Brinkman number flow for a viscoelastic fluid: Graetz problem for the PTT model, Int. J. Heat Mass Tran. 46, 3865-3880. Cp : Heat capacity Darbandi, M. and S. Vakilipour, 2008, Developing implicit pres- c1,c2 : constants of WLF model D : Rate of deformation tensor sure weighted upwinding scheme to calculate steady and unsteady flows on unstructured grids, Int. J. Numer. Meth. E : Relative difference norm Fluid 56, 115-141. k : Thermal conductivity Dris, I. and E.S.G. Shaqfeh, 1998, Flow of a viscoleastic fluid L : Velocity gradient between eccentric cylinders: impact on flow stability, J. Non- Nu : Nusselt number Newtonian Fluid Mech. 80, 59-87. Nr : Number of grid points in r direction Fattal, R. and R. Kupferman, 2005, Time-dependent simulation Nθ : Number of grid points in θ direction of viscoelastic flows at high Weissenberg number using the p : Pressure

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