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Fluids of differential type: Critical review and thermodynamic analysis. (English) Zbl 0899.76062

Summary: Thermodynamics, in the form of a dissipation inequality and the commonly accepted idea that the stored energy should have an extremum in equilibrium, is used to find restrictions on the response functions for the stress and the stored energy in incompressible fluids of differential type. For those special fluids of differential type known as grade \(n\) fluids these thermodynamic restrictions have been a source of some controversy and much confusion not withstanding the fact that they are in complete harmony with results achieved by either linear or nonlinear stability analysis. In order to clarify th issues that seem to underlie this controversy, we provide an extended analysis of the genesis and development of fluids of differential type. As part of our analysis, we will show that certain ideas of flow retardation and model approximation have been consistently misinterpreted. Additionally, we establish several new results concerning the thermodynamics of these materials. A special application of our results reveals that work of D. D. Joseph and M. Renardy on the instability of the rest state for certain, very special grade \(n\) fluids is in fact inapplicable to all those grade \(n\) fluids that are consistent with thermodynamics.

MSC:

76A99 Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena
82D15 Statistical mechanics of liquids
80-XX Classical thermodynamics, heat transfer
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