## Suspensions

Jerry Brackbill (Los Alamos National Laboratory) and I developed a numerical method to solve equations for a liquid containing solid suspended particles [1]. Suspensions are common, and occur in blood flow, transport of slurries, movements of sediments in river beds, and advanced composites manufacturing. The numerical approach is a modification of Peskin's [2] immersed boundary method, combined with features from a particle-in-cell code [3-4]. Equations of motion for the fluid are solved using a finite difference method on a computational grid, with suspended particles replaced by a force density in the fluid equations. The force density is computed by discretizing stress-strain constitutive equations for the solid, using data provided by a collection of material points which mark regions in the fluid occupied by solids. The method retains the advantages of the immersed-boundary method in that one set of equations holds in the entire computational domain and the complication of internal boundaries is eliminated. Also, computational work is linear in the number of material points, so the method has the potential to handle a large number of particles. Our new addition is the manner in which the force density is constructed, with material properties of the suspended solids clearly specified by stress-strain constitutive equations. Since the material points are not physically or logically connected, particles of any size and shape are easily treated. Unlike most other methods for simulating suspensions, this method is not restricted to Stokes' flow and, in principle, the suspending fluid can be non-Newtonian.

A two-dimensional implementation of the method has been used to establish validity of this formulation by comparing numerical results for elastic vibrations, and particle settling in viscous fluids, to theory and analysis [1]. Part of this work was supported by a Sandia-University Research Project grant and an NSF grant.

## References

• [1] D. Sulsky and J.U. Brackbill, A Numerical Method for Suspension Flow, J. Comput. Phys. 96: 339-368 (1991).
• [2] C. S. Peskin, Numerical Analysis of Blood Flow in the Heart, J. Comput. Phys. 25: 220-252 (1977).
• [3] J. U. Brackbill and H. M. Ruppel, FLIP: A Method for Adaptively Zoned, Particle-in-Cell Calculations in Two Dimensions, J. Comput. Phys., 65: 314-343 (1986).
• [4] J. U. Brackbill, D. B. Kothe and H. M. Ruppel, FLIP: A Low-Dissipation, Particle-in-Cell Method for Fluid Flow, Comput. Phys. Comm. 48: 25-38 (1988).

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Last updated: September, 1998
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