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

Abstract

The equations of incompressible micropolar fluid flow are a coupled system of vector differential equations involving the two basic vectors, viz. the velocity q̄ and the microrotation v̄ of the fluid elements. Let D = D ( t) be a bounded region in space, and let a flow velocity and a microrotation be prescribed at each point of the boundary of D( t). Assume that D( t) as well as the assigned velocity and microrotation vectors depend periodically on the time t and that the condition (2 μ+ k) j−4 a ⩽ 0 is satisfied (equation (25) in the text). Further assumptions are that (i) to every continuous initial distribution of the flow fields over D, there corresponds a solution of the field equations for all time t ⩾ 0 satisfying the prescribed boundary conditions; (ii) there is one solution for which the Reynolds numbers Re, Rm satisfy the condition Re 2 + Rm 2 < 80 and this solution is equicontinuous in x ̄ = (x,y,z) for all t. Then there exists a unique, stable, periodic solution of the micropolar flow equations in D( t) taking the prescribed values on the boundary. The proof of the theorem rests on a formula describing the rate of decay of the kinetic energy of the difference of two micropolar flows in the domain subject to the same boundary conditions.