Some uniqueness and stability results in the theory of micropolar solid–fluid mixture

Abstract

This paper is devoted to the study of a binary homogeneous mixture of an isotropic micropolar linear elastic solid with an incompressible micropolar viscous fluid. Assuming that the internal energy of the solid and the dissipation energy are positive definite forms, some uniqueness and continuous data dependence results are presented. Also, some estimates which describe the time behaviour of solution are established, provided the above two energies are positive definite forms. These estimates are used to prove that for the linearized equations, in the absence of body loads and for null boundary conditions, the solution is asymptotically stable. Then a uniqueness result under mild assumptions on the constitutive constants is given using the Lagrange–Brun identities method. These mathematical results prove that the approach of a micropolar solid–fluid mixture is in agreement with physical expectations.