Partially dissipative systems with applications to fluid-solid interaction problems

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The interactions of viscous incompressible fluids with (rigid or elastic) solids are ubiquitous in engineering, geophysical and biological applications. The long-term goal of our research program is to provide new analytical techniques for a systematic investigation of stability and long-time behaviour of fluid-solids interacting systems. In this proposal, we will focus on several multiphysics fluid-rigid body interaction problems that can be mathematically cast as "partially dissipative" systems of differential equations. A nonlinear evolution equation is said to be partially dissipative if the following two properties are satisfied: (P1) there exists an energy functional that is non-increasing along the trajectories; and (P2) there exists a vector field (depending on the solution) that is conserved at all times. The short-term objective is to characterize the long-time behaviour of a large class of solutions to three related fluid-rigid body interaction problems. The governing equations are partially dissipative and feature the coupling of the Navier-Stokes equations with the balance of momentums for the solids in presence of magnetic fields and/or heat transfer. The following topics will establish a stimulating research programme for the training of 3 PhD, 2 Master's and 2 NSERC USRA students: (1) Asymptotic stability of rigid body motions for solids with a fluid-filled gap. Consider a system of two solids, one contained inside the other, and with the gap between them completely filled by a viscous incompressible fluid. We will study the asymptotic stability of permanent rotations (i.e., steady states given by the system rotating as a whole rigid body with constant angular velocity). (2) Long-time behaviour of fluid-filled rigid bodies in a magnetic field. We consider a rigid body with a cavity completely filled by an electrically conducting fluid. We will investigate whether motions of the coupled fluid-solid system are able to generate and sustain a magnetic field within the system. (3) Long-time dynamics of rigid bodies with a fluid-filled gap subject to both temperature gradients and a gravitational field. The objective is to determine whether solid rotations inhibit the formation of fluid convecting motions at large times. Every problem proposed above is characterized by having different sources of dissipation while keeping a conservative component. From a mathematical point of view, these problems are examples of abstract nonlinear evolution equations satisfying (P1) and (P2). Existing theories concerning stability and long-time behaviour of solutions are applicable mainly to solutions in a strong functional setting. The far-reaching impact is to provide a novel theory for the long-time behaviour of weak solutions to partially dissipative systems. Besides their mathematical significance, problems (1)-(3) provide phenomenological models for applications in geophysics and engineering.