INTERNATIONAL JOURNAL OF c  2007 Institute for Scientific NUMERICAL ANALYSIS AND MODELING Computing and Information Volume 4, Number 3-4, Pages 425–440 FINITE ELEMENT APPROXIMATION OF THE NON-ISOTHERMAL STOKES-OLDROYD EQUATIONS CHRISTOPHER COX, HYESUK LEE, AND DAVID SZURLEY Dedicated to Max Gunzburger on the occasion of his 60th birthday. Abstract. We consider the Stokes-Oldroyd equations, defined here as the Stokes equations with the Newtonian constitutive equation explicitly included. Thus a polymer-like stress tensor is included so that the dependent variable structure of a viscoelastic model is in place. The energy equation is cou- pled with the mass, momentum, and constitutive equations through the use of temperature-dependent viscosity terms in both the constitutive model and the momentum equation. Earlier works assumed temperature-dependent constitu- tive (polymer) and Newtonian (solvent) viscosities when describing the model equations, but made the simplifying assumption of a constant solvent viscosity when carrying out analysis and computations; we assume no such simplifica- tion. Our analysis coupled with numerical solution of the problem with both temperature-dependent viscosities distinguishes this work from earlier efforts. Key Words. viscous fluid, non-isothermal, finite elements, Stokes-Oldroyd. 1. Introduction Viscoelastic flows occur in a variety of applications, including polymer process- ing. The complexity of the governing equations and the physical domains makes analysis of the mathematical models and the associated numerical methods espe- cially difficult. Current efforts to model viscoelastic flows often revolve around the solution of a (modified) Stokes problem, [5]. The isothermal linear elasticity equations, modified in form to have the same dependent variable structure as the equations governing viscoelastic flows, is analyzed along with a numerical solu- tion in [2]. The Stokes problem is a special case (the incompressible limit) of the equations considered in that work. The purpose of this paper is to analyze the finite element solution of the non- isothermal Stokes problem, modified similarly as in [2]. Thermodynamics play a prominent role in many viscoelastic flow scenarios, especially in polymer process- ing. Realistic models must ultimately include temperature dependence, since flow characteristics such as viscosity vary widely as temperature varies within normal operating constraints, [1]. The rest of this paper is outlined as follows. The governing equations are pre- sented in the next section, with particular attention given to the manner in which temperature dependence is expressed. Details regarding the weak formulation and Received by the editors February 21, 2006 and, in revised form, February 23, 2006. 2000 Mathematics Subject Classification. 65N30. This research was supported in part by the ERC program of the National Science Foundation under Award Number EEC-9731680 through the Center for Advanced Engineering Fibers and Films at Clemson University. 425