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

*N.A. Patankar, P. Singh, D.D. Joseph, R. Glowinski, T.-W. Pan*

ABSTRACT: 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+W x r; u
being the velocity of the fluid at a point in the particle domain; U and W 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 particulate flows developed by Singh, Joseph,
Hesla, Glowinski and Pan (1999). The code is verified by comparing results with
other simulations and experiments.

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*Last updated October 16, 2000*