A new formulation of the distributed Lagrange multiplier/fictitious domain method for particulate flows

A Lagrange-multiplier-based fictitious-domain method (DLM) for the direct numerical simulation of rigid particulate flows in a Newtonian fluid was presented by Glowinski, Pan, Hesla and Joseph (1999). An important feature of this finite element based method is that the flow in the particle domain is constrained to be a rigid body motion by using a well-chosen field of Lagrange multipliers. The constraint of rigid body motion is represented by u = U+ω×r; u being the velocity of the fluid at a point in the particle domain; U and ω are the translational and angular velocities of the particle, respectively; and r is the position vector of the point with respect to the center of mass of the particle. The fluid-particle motion is treated implicitly using a combined weak formulation in which the mutual forces cancel. This formulation together with the above equation of constraint gives an algorithm that requires extra conditions on the space of the distributed Lagrange multipliers when the density of the fluid and the particles match. In view of the above issue a new formulation of the Lagrange-multiplier-based fictitious-domain method for particulate flow is presented in this paper. In this approach the deformation rate tensor within the particle domain is constrained to be zero at points in the fluid occupied by rigid solids. This formulation shows that the state of stress inside a rigid body depends on the velocity field similar to pressure in an incompressible fluid. The new formulation is implemented by modifying the DLM code for two dimensional