LACK OF COLLISION IN A SIMPLIFIED 1D MODEL FOR FLUID–SOLID INTERACTION

Abstract

In this paper we consider a simplified model for fluid–solid interaction in one space dimension. The fluid is assumed to be governed by the viscous Burgers equation. It is coupled with a finite number of solid masses in the form of point particles, which share the velocity of the fluid and are accelerated by the jump in velocity gradient of the fluid on both sides, which replaces here the standard pressure jump of Navier–Stokes models. We prove global existence and uniqueness of solutions. This requires proving that the solid particles never collide in finite time, a key fact that follows from suitable a priori estimates together with uniqueness results for ordinary differential equations. We also describe the asymptotic behavior of solutions as t → ∞, extending previous results established for a single solid mass. The evolution of the relative position of the particles is examined in terms of the strength of the convection term. The possible 2D analogues of these results in the context of Navier–Stokes equations are open problems.