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 Reynolds number 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.