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Modelling of Non-Isothermal Viscoelastic Flows 1 ELSEVIER J. Non-Newtonian Fluid Mech., 68 (1997) 205-224 Modelling of non-isothermal viscoelastic flows 1 Gerrit W.M. Peters *, Frank P.T. Baaijens Centre for Polymers and Composites, Faculty of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands Received 25 November 1995; revised 29 April 1996 Abstract The modelling of non-isothermal flow of viscoelastic materials using differential constitutive equations is investi- gated. The approach is based on the concept of a slip tensor describing the nonaffine motion of the stress-carrying structure of the fluid. By specifying a slip tensor and the elastic behaviour of the structure, the constitutive model is determined. This slip tensor also appears in the energy equation and thus, for a given constitutive model, the form of the energy equation is known. Besides the partitioning between dissipated and elastically stored energy, also the difference between entropy and energy elasticity is discussed. Numerical simulations are based on a stabilized Discontinuous Galerkin method to solve the mass, momentum and constitutive equations and a Streamline Upwind Petrov-Galerkin formulation to solve the energy equation. Coupling is achieved by a fixed point iteration. The flow around a confined cylinder is investigated, showing differences between viscous and viscoelastic modelling, and between limiting cases of viscoelastic modelling. © 1997 Elsevier Science B,V. Keywords: Discontinuous Galerkin; FENE; Non-isothermal; Slip tensor; SUPG; Viscoelastic 1. Introduction Although most of the research on the flow of viscoelastic fluids concerns isothermal cases, many flows of practical interest in polymer melt processing are non-isothermal. The combina- tion of high viscosities of polymeric melts and high deformation rates results in the transforma- tion of large amounts of mechanical energy into heat, and therefore in a temperature rise of the material. This phenomenon is, for instance, used in extruders where viscous dissipation is used to enhance melting of the material. Furthermore, the effect of high cooling rates, as obtained during injection moulding as the melt hits the wall of the mould, also needs to be accounted for. In this process the material also deforms close to Tg, where the rheological properties are most sensitive to thermal changes. For viscoelastic materials the stress not only depends on deforma- * Corresponding author. t Paper presented at the Polymer Melt Rheology Conference, University of Wales, Aberystwyth, 3-6 September 1995. 0377-0257/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S0377-0257(96)01 51 I-X 206 G.W.M. Peters, F.P.T. Baaijens /J. Non-Newtonian Fluid Mech. 68 (1997) 205-224 tion and deformation history, but also on temperature and temperature history. Therefore, temperature should be an independent variable in the constitutive equations for the stress tensor. The temperature dependence of linear viscoelastic properties, e.g. the relaxation times, has been described by the principle of time-temperature superposition. This principle states that all characteristic times of the material depend in the same way on temperature and, therefore, can be described with one function of the temperature (aT). Of all proposed expressions (empirical and semi-empirical) the Williams-Landel-Ferry (WLF) function is the most widely used. However, the time-temperature superposition principle holds for rheological properties at different constant temperatures; it does not describe the consequences of temperature changes in time and space. One way to address this problem is the concept of a pseudo-time which accounts for the effect of temperature variations on the rheological behaviour [1,2], as used in the simulation of non-isothermal flows by Luo and Tanner [3]. The validity of this postulate is still the subject of discussion and open for experimental confirmation [4]. Another way to deal with this problem is the use of molecular theories. Marrucci formulated constitutive equations for non-isothermal flows starting from the kinetic theory for dilute solutions of Hookean dumbbells [5] and Wiest extended this to bead-spring chains [6]. Wiest and Phan-Thien extended the Curtiss-Bird theory to non-isothermal flows [7], and Wiest examined the non-isothermal variant of the Giesekus and other related equations. These molecular based theories give constitutive equations that differ from those obtained from the pseudo-time postulate. Another difficulty with solving non-isothermal viscoelastic flows comes from the energy equation. Usually it is assumed that the internal energy of fluids is a function of the temperature only, but this is not a proper assumption for viscoelastic fluids. Braun and Friedrich [8] and Ko and Lodge [9] studied this problem for viscometric flows. In order to specify which part of the supplied mechanical power is dissipated and which part is accumulated as elastic energy, the energy equation needs to be reconsidered. The two limiting cases are ideal-elastic and viscous material behaviours. Deformation of elastic materials is a reversible process: mechanical energy is stored and can be released into mechanical energy again. Deformation of purely viscous materials is irreversible: mechanical energy is entirely dissipated. For viscoelastic materials, mechanical energy is partly stored as elastic energy and partly dissipated. In addition to the correct partitioning of elastically stored and dissipated energy, two different ways of storing elastic energy in a viscoelastic material have to be distinguished: entropy and (internal) energy elasticity [10]. The first is related to the entropy part of the Helmholtz free energy, the second to the internal energy part. Anisotropic heat conduction, for which the thermal conductivity tensor is related to kinematic quantities [11] will not be considered here. We will focus on the development of constitutive equations of the differential type for non-isothermal flows and related problems with the energy equation. Although it is felt that there is an urgent need for experimental data, it is thought that first of all a theoretical framework is needed, even though relatively simple, that gives the possibility to explore these experiments. They are much too complicated to be tackled starting from an experimental point of departure. Numerical simulations, showing the consequences of different experimental conditions and theoretical assumptions are indispensable for the design and interpretation of these experiments. G.W.M. Peters, F.P.T. Baaijens / J. Non-Newtonian Fluid Mech. 68 (1997)205-224 207 To demonstrate this, the flow of a polymer melt around a confined cylinder is analyzed. This is a benchmark flow extensively studied by our group [12-14]. It is one of the complex flows that is supplementary to the simple shear flows, commonly used for testing constitutive equations. Simple shear flows do not contain enough information on the fluid rheology to ensure reliable predictions in more complex flows. Until now it has been assumed that these flows are isothermal. However, one should be cautious about sometimes unexpected [15,16], non-isothermal effects. It is believed that mapping of stress and velocity (and the deformation rate derived from that), and at the same time the mapping of the temperature field, in space and time, for a well-defined flow, will assist in understanding non-isothermal viscoelastic flows and in the confirmation of theoretical assumptions. Results of complex flows will be used for adaptation of viscoelastic constitutive equations in a systematic way, while taking into account, if necessary, non-isother- mal effects. One of the main reasons why such an approach might work at this time is the recent availability of codes for numerical simulation of viscoelastic flows that become faster and more reliable, so that different algorithms converge to the same solution for (some) benchmark problems [17]. The numerical methods used here have been tested by comparing results with other published work and with experimental results from flows of polymer solutions and polymer melts [12,13]. In this paper, first of all constitutive modelling of the Cauchy stress is discussed. The modelling summarizes many of the existing constitutive models [18] and offers the possibility to extend these models in a systematic way. Next, the Clausius-Duhem inequality is explored and viscoelastic behaviour is incorporated into the energy equation. Thereafter, the non-isothermal flow of a polymer melt around a confined cylinder is analyzed numerically. 2. Viscoelastic constitutive equations The structure of a fluid is represented with connector vectors/~i. The subscript i is introduced to distinguish between different parts of the structure. Differences have their origin not only in the behaviour of the elements themselves, but also in different interactions of the elements with their surroundings. From the principle of virtual work it follows that the elastic stress contribution to the Cauchy stress tensor of elements of type i is given by Kramer's expression for the average stress tensor [19,20]: tre i = ~v(fiiRi), (1) where v is the number density of particles per unit volume, f" the connector force, and (.) the average over the non-equilibrium configuration distribution function ~(/~i) of the vectors /~i- The factor ~ is a force-reducing factor, introduced to capture models like Larson's partial extending model. The connector force j~ is considered to be a nonlinear
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