Flow stability of a Grad-model liquid layer on an inclined plane

Abstract

A study is presented of the flow of stability of a Grad-model liquid layer [1, 2] flowing over an inclined plane under the influence of the gravity force. It is assumed that at every point of the considered material continuum, along with the conventional velocity vector v, there is defined an angular velocity vector ω, the internal moment stresses are negligibly small, and in the general case the force stress tensor τkj is asymmetric. The model is characterized by the usual Newtonian viscosity η, the Newtonian “rolling” viscosity ηr, and the relaxation time τ=ρ J/4 ηr, where J is a scalar constant of the medium with dimensions of moment of inertia per unit mass, ρ is the density. It is assumed that the medium is incompressible, the coefficients η, ηr, J are constant [2]. The exact solution of the equations of motion, corresponding to flow of a layer with a plane surface, coincides with the solution of the Navier-Stokes equations in the case of flow of a layer of Newtonian fluid. The equations for three-dimensional periodic disturbances differ considerably from the corresponding equations for the problem of the flow stability of a layer of a Newtonian medium. It is shown that the Squire theorem is valid for parallel flows of a Grad liquid. The flow stability of the layer with respect to long-wave disturbances is studied using the method of sequential approximations suggested in [3, 4].