Mathematics in Industrial Problems pp 166-185 | Cite as

# Coping with complex boundaries

## Abstract

There are many applications in materials science involving boundary value problems in which the boundaries have complicated shapes. Examples include the computation of the capacity and translational friction coefficients of objects having general shape, the discharge of Poiseuille flow through a pipe of arbitrary cross section, and the calculation of “virial coefficients” which describe the leading order concentration dependence of the effective material properties of suspensions or composites containing a small concentration of complex shaped particles. Moreover, the scattering of small objects by electromagnetic (radar, visible light) and pressure (acoustic) waves involves mathematical problems identical to the virial coefficient calculations which have many practical applications corresponding to cases where the scattering particles have elaborate structure (e.g., snow flakes). Direct analytical approaches based on traditional differential equations methods are often not effective in dealing with this class of problems. On April 15, 1994 Jack F. Douglas from National Institute for Standards and Technology (NIST) presented a different approach based on recasting this kind of boundary value problem in terms of an averaging over random walk paths. He showed through a sequence of examples how this point of view leads to viable numerical and analytical treatment of complex boundaries. He described recent work and presented open problems.

## Keywords

Random Walk Effective Property Brownian Particle Poiseuille Flow Complex Boundary## Preview

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