Flow stability and Prigogine theorem for the stationary flow of a viscoplastic body

Abstract

A slow flow of a nonlinear viscoplastic body on whose boundary the velocities of travel are given has been investigated for stability. We have also proved the Prigogine theory on minimum entropy production in the case of the isothermal special stationary flow of such a body.Keywords: slow flow of a viscoplastic body, integral estimate, special stationary flows, minimum entropy production.Introduction. In [1,2], it was proved that the slow flow of a linear viscous body on whose boundary the velocities of travel are given is insensitive to small perturbations of these velocities. In [2], it was also shown that in the case where the flow of such a body is isothermal and stationary, the entropy production is minimum. In [3,4], for partial processes of uniform deformation of a strip and a circular cylinder the stability of a slow flow of nonlinear viscoplastic bodies with respect to perturbations of state variables generated by small perturbations of the body boundaries was investigated. However, the question on the applicability of the Prigogine theorem on minimum entropy production for a viscoplastic body to these processes was not considered there.In the present work, we investigate a slow flow of a nonlinear viscoplastic body for arbitrary processes of deformation in which the velocities of travel on the body boundary are known. It is shown that these processes are insensitive, in the integral sense, to small perturbations of the velocities of travel. The special stationary flows of a viscoplastic body in which for isothermal conditions the entropy production is minimum have been determined. Systems of Equations for the Main Flow of a Viscoplastic Body and Perturbations of This Flow.We consider a slow flow of a viscoplastic body for which in the equations of motion