Physical Interpretation and Mathematical Properties of the Stress-DLM Formulation for Rigid Particulate Flows

Patankar et al. presented a new formulation of the Lagrange-multiplier-based fictitious-domain method (DLM) for the direct numerical simulation of rigid particulate flows. The idea is to assume that the entire fluid-particle domain is some fluid and then to constrain the particle domain to move with a rigid motion by setting the deformation-rate tensor equal to zero. The constraint gives a vector Lagrange multiplier field in the particle domain. This approach is usually referred to as the stress-DLM formulation. We first discuss a physical interpretation of this formulation where we see that, with an appropriate choice of the constraint equations, the vector Lagrange multiplier field is in fact the displacement field of a linear elastic body. We theoretically investigate the existence and uniqueness properties of the stress-DLM formulation by considering a model problem. We find that the velocity and the Lagrange multiplier can be represented by an equal-order interpolation scheme in a finite element formulation. This is unlike the incompressibility constraint where equal order interpolation of pressure and velocity can lead to spurious modes.