Reiner-Rivlin Media

An important class of non-Newtonian fluids is formed by isotropic rheological stable media whose stress tensor [ry] is a continuous function of the shear rate tensor [e,j] and is independent of the other kinematic and dynamic variables. One can rigorously prove that the most general rheological model satisfying these conditions is the following nonlinear model of a viscous non-Newtonian Stokes medium [19]  

In the case of an incompressible fluid, the first invariant is zero, I - div v = 0. For simple one- and two-dimensional flows (such as flows in thin films, longitudinal flow in a tube, and tangential flow between concentric cylinders), the third invariant h is identically zero. 

The scalar functions p and e determine various rheological models of non-Newtonian media. For example, the case p = const and s = 0 corresponds to the 

If we choose the coefficients p and s in (6.1.7) to be nonzero constants, then we arrive at the Reiner-Rivlin model, which additively combines the Newton model with a tensor-quadratic component. In this case the constants p and e are called, respectively, the shear and the dilatational (transverse) viscosity. Equation (6.1.7) permits one to give a qualitative description of specific features of the mechanical behavior of viscoelastic fluids, in particular, the Weissenberg effect (a fluid rises along a rotating shaft instead of flowing away under the action of the centrifugal force). 

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