Finite Perturbations by Yield Stress of the Constitutive Relations of Nonlinear Viscous Media

Flows of incompressible continuous media with tensor-linear constitutive relations and an arbitrary scalar nonlinearity are investigated in the cases of absence of the yield stress (nonlinear-viscous liquids) and its presence (viscoplastic media with nonlinear viscosity). The occurrence of a finite yield stress of a medium is interpreted as a finite perturbation of the scalar dependence linking the stress intensity and the strain rate. A one-parameter family of such perturbed dependencies is proposed. As a test problem, a one-dimensional problem of stationary shear flow of a flat layer on an inclined plane in the field of gravity is given. The maximum velocities and consumptions are compared for different perturbation parameters. It is shown that, in the bounds thus arising, the sign of the convexity of the material function that specifies some nonlinear viscosity plays a great role.