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Finite element approximation of the non-isothermal Stokes-Oldroyd equations. (English) Zbl 1242.76013
Summary: 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.
MSC:
76A10Viscoelastic fluids
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
76D07Stokes and related (Oseen, etc.) flows
76M10Finite element methods (fluid mechanics)