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Generalizations of the Navier-Stokes fluid from a new perspective. (English) Zbl 1231.76073

Summary: In this paper we study incompressible fluids described by constitutive equations from a different perspective, than that usually adopted, namely that of expressing kinematical quantities in terms of the stress. Such a representation is the appropriate way to express fluids like the classical Bingham fluid or fluids whose material moduli depend on the pressure. We consider models wherein the symmetric part of the velocity gradient is given by a “power-law” of the stress. This stress power-law model automatically satisfies the constraint of incompressibility without our having to introduce a Lagrange multiplier to enforce the constraint. The model also includes the classical incompressible Navier-Stokes model as a special subclass. We compare the stress power-law model with the classical power-law models and we show that the stress power-law model can, for certain parameter values, exhibit qualitatively different response characteristics than the classical power-law models and — on the other hand — it can be, for certain parameter values, used as a substitute for the classical power-law models. Using a stress power-law model we study several steady flow problems and obtain exact analytical solutions, and we argue that the possibility to obtain an exact analytical solution suggests, among others, that using these models provides an interesting alternative to the classical power-law models for which reasonable exact analytical solutions cannot be obtained. Finally, we discuss the issue of the choice of boundary conditions, and we show that the choice of boundary conditions has, at least for one of the problems that we study, a profound impact on the solvability of the boundary value problem.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
76A99 Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena
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