Existence of periodic solutions of the equations of incompressible microstretch fluid flow

Abstract

The flow of incompressible microstretch fluid is governed by a system of differential equations involving the velocity vector q , the microprotation vector v and the scalar v representing the microstretch of the fluid element. Let R = R(t) be a bounded domain in space and let the field ( q , v , v) be prescribed at each point of the boundary ∂R(t). If the domain R(t) and the boundary data depend periodically on the time t, it is shown that under some assumptions on the initial distribution of the flow fields and the material constants of the fluid, there exists a unique, stable, periodic solution of the microstretch flow equations in R(t), taking the prescribed values on the boundary ∂R(t) (Theorem 2 of the paper). The proof rests on some relations describing the rate of decay of the energy functionals corresponding to the difference of two microstretch flows in the domain that have the same density and gyration parameters and are subject to the same boundary conditions.