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Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries. (English) Zbl 1195.35239
Following Bridgman’s treatise, the authors consider incompressible fluid whose viscosity depends on the pressure, the temperature, and the shear rate, which is contained in a bounded connected set \(\omega\in{\mathbb R}^3\). They regard the velocity, pressure, temperature, specific entropy, heat flux, and the extra stress as unknown functions, and formulate the governing equations they should satisfy. The equations represent the balance of mass, momentum, and energy. They formulate a corresponding initial-boundary condition, and give a notion of a generalized solution. It means that the solution should belong to an appropriate function space. Then, they give an existence result for such a generalized solution. The existence is proved by means of an approximate Faedo-Galerkin method, using two parameters.

MSC:
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76A05 Non-Newtonian fluids
76M10 Finite element methods applied to problems in fluid mechanics
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