Finite element approximation of the non-isothermal Stokes-Oldroyd equations

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 coupled 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 constitutive (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 simplification. Our analysis coupled with numerical solution of the problem with both temperature-dependent viscosities distinguishes this work from earlier efforts.