### Nonisothermal flows of viscoelastic fluids: Thermodynamics, analysis and numerical simulation

#### Abstract

Dependent on the time scale of the flow, viscoelastic materials mainly show viscous or elastic behaviour. In recent years much progress has been made with the numerical simulation of flows of viscoelastic materials. For example, it has become possible to calculate stationary flows with high Deborah numbers. Mostly this was restricted to isothermal flows. However, in practice many flows are nonisothermal. On the one hand due to the external cooling or heating, on the other hand due the internal production of heat, for example by the dissipation of mechanical energy.

Besides the usual balances of mass, linear momentum, angular momentum and energy, some equations are needed to describe the specific behaviour of the polymeric material. These so-called constitutive equations are necessary for the stress, the heat flux and the internal energy. General equations for these quantities will be derived from the thermodynamics, where internal (tensor) variables will be used to describe the relaxation phenomena. These internal tensor variables are a measure for the elastic deformation of a polymeric fluid. The resulting equations include the differential models for the stress that are commonly used in the literature. The resulting equation for the heat flux is an extension of the well-known Fourier law. The heat conduction tensor may depend on the (elastic) deformation of the material. With this dependence the experimentally observed anisotropy of the heat conduction tensor may be described, i.e.\ an increasing thermal conductivity in the direction of orientation and a decreasing thermal conductivity in the direction perpendicular to the orientation. Finally the balance of internal energy may be written as a temperature equation with the help of the thermodynamics. For this, it is important to distinguish the irreversible (dissipative) and reversible processes. This is possible with the help of the balance of entropy. Particularly, the temperature dependence of the density and the shear modulus appear to be important. Although these coefficients are relatively weak functions of the temperature, they do have a large influence on the reversible processes in the temperature equation. The temperature dependence of the density results in a cooling during expansion and a heating during compression. The temperature dependence of the shear modulus determines whether the reversible energy is stored as internal energy (energy elastic) or as entropy (entropy elastic). If the storage of energy is entropy elastic, a part of the temperature changes are reversible. If the free energy, or equivalently the stress model, of a viscoelastic fluid is known, it is possible to relate all thermodynamic coefficients in the temperature equation to experimental data in equilibrium, for example the thermal expansion coefficient and the heat capacity at constant pressure. The consequences for viscoelastic materials will be elucidated by means of the `neo-Hookean' model, which is often used in the literature. On the basis of experimental data from the literature the order of magnitude of the temperature dependences of the thermodynamic coefficients will be reviewed. Furthermore, an overview of nonisothermal rheological experiments will be included which indicate that the temperature history is important. Through dimensional analysis it will be checked which temperature effects may be important in shear flows.

It is too difficult to solve the complete, coupled system of equations analytically. When the coefficients are independent of the temperature, however, analysis of the equations is possible for steady flows. In a shear flow and a uniaxial elongation flow the behaviour of some well-known differential models will be worked out for high deformation rates. With these results the consequences will be examined for a simple model, with constant coefficients, for the anisotropy of the heat conduction tensor. The simple model does not give a qualitative agreement with the experimental data for all stress models. Only if the stress model has a second normal stress difference, the decrease of the thermal conductivity perpendicular to the flow can be described. If the coefficients are assumed to depend on the invariants the qualitative agreement is still possible for the other stress models.

The system of nonisothermal equations has been implemented in a computer program for isothermal flows of viscoelastic fluids. The equations of motion and the temperature equation have been solved by using a finite element method. The stresses have been calculated by a streamline integration method. For the numerical solution process a number of problems arise. For the temperature equation a standard upwind method (SUPG) has been used to avoid too much grid refinement for convection-dominated flows. Another problem arises when the mechanical dissipation has to be calculated. For this, the inverse of the internal deformation tensor is needed. Although theoretically the internal deformation tensors are positive definite, they may become indefinite due to numerical errors. To avoid large numerical errors in the mechanical dissipation, a positive lower bound of the determinant of the internal deformation tensor is needed then. A method has been developed to determine the theoretical lower bound of the determinant. For many models a positive lower bound can be obtained. The outflow boundary condition also requires some extra attention. In convection-dominated flows it is often necessary to prescribe an approximation of the normal stress due to the normal stress differences of viscoelastic fluids. A Dirichlet boundary condition or a constant pressure at the outflow does not work then.

For the numerical calculations two different polymer melts will be taken: a polyethylene (LDPE) and a polystyrene (PS 678E) melt. These fluids show a different behaviour for the same type of flow. The viscosity of the polystyrene melt is a much stronger function of the temperature than for the polyethylene melt. On the other hand the anisotropy of the heat conduction tensor of polystyrene is much smaller than for polyethylene. For the numerical calculations the influence of the mechanical dissipation, the anisotropy of the heat conduction tensor, the cooling due to the thermal expansion and the temperature dependence of the shear modulus will be examined in more detail. Firstly the simple case of a fully developed axisymmetrical pipe flow will be discussed. Except the mechanical dissipation, the cooling due to the thermal expansion and the anisotropy of the heat conduction tensor, may be important, particularly for high flow rates (Brinkman numbers). For LDPE the decrease of the thermal conductivity perpendicular to the flow may cause a considerable increase of the temperature. Next a Graetz--Nusselt problem, with a temperature jump on the wall, will be examined. Due to the dominance of the convection, the mechanical dissipation and the cooling due to the thermal expansion are less important. The solution is mainly determined by convection and diffusion. For LDPE the anisotropy of the heat conduction tensor may become very large. This is mainly caused by the large values of the thermal conductivity parallel to the flow. Finally the flow through a 4:1 contraction will be considered. For this problem a vortex may arise in the entry corner. For LDPE the internal production of heat is not very important (due to the dominance of the convection). For PS, however, the internal heat production may be important, particularly for the vortex intensity. If the wall is cooled from the contraction, the magnitude of the vortex also changes, particularly for PS. Although the difference between the mechanical dissipation and the stress work are large near the contraction, the influence on the temperature distribution is relatively small due to the dominance of the convection. For LDPE the anisotropy of the heat conduction tensor also appears to be important. Besides a decrease in the direction to the flow, the thermal conductivity perpendicular to the flow may become very large.

#### Keywords

temperature equation; differential stress models; thermodynamics; (positive) dissipation; energy and entropy elasticity; compressibility; non-equilibrium heat capacity and thermal expansion; anisotropic heat conduction; dissipation; viscoelasticity; numerical simulation; combined finite element / streamline integration; outflow boundary conditions