Stability of an Unsteady Shear of Bingham Medium in the Plane Layer

Abstract

This work studies the plane-parallel unsteady shear of a homogeneous two-constant viscoplastic Bingham medium in the infinite layer. It was assumed that the axial velocity of the flow as a function of one spatial coordinate and time is known from the solution to the classical one-dimensional unsteady problem. The time variation in the thicknesses of the possible rigid zones is considered; their boundaries are parallel to the boundaries of the layer. The two-dimensional plane perturbations are superposed upon the main flow. The problem in terms of perturbations reduces to one linearized equation for the amplitude of the function of the flow with the corresponding set of four boundary conditions, and several variants of such quadruples are studied here. With the method of integral relationships, the problem reduces to the minimization problem of ratios of quadratic functionals depending on time in the space H₂(a;b), where a and b are functions of time determined by the motion of the rigid zones in the main flow. For different variants of imposition of the boundary conditions, the generalized Friedrichs inequalities are proved, and the sufficient integral estimates of stability are derived in which the Reynolds and Saint-Venant numbers and the maximum shear velocity in thickness in the main flow play a role. The dependence of the obtained estimates on the viscous and plastic properties of the medium is discussed.