Fluid-particle flow: a symmetric formulation

In this Note, we present a numerical method to simulate the motion of solid particles in a moving viscous fluid. The fluid is supposed to be Newtonian and incompressible. The Arbitrary Lagrangian Eulerian formulation of the Navier-Stokes equations is discretized at the first order in time, as are the equations for the solid bodies. The advection term is taken into account by a method of characteristics. The variational formulation of the coupled problem is then established, and the boundary integrals expressing the hydrodynamical forces are eliminated. By introduction of an appropriate Finite Element approximation, a symmetric linear system is obtained. This system is solved by an inexact Uzawa algorithm, preconditionned by a Laplace operator with Neumann boundary conditions on the pressure. Numerical results are presented, for 2 and 100 particles: The Reynolds number in both cases is of the order of 100.