Boundary Integral Methods for Multicomponent Fluids and Multiphase Materials

Abstract

We present a brief review of the application of boundary integral methods in two dimensions to multicomponent fluid flows and multiphase problems in materials science. We focus on the recent development and outcomes of methods which accurately and efficiently include surface tension. In fluid flows, we examine the effects of surface tension on the Kelvin-Helmholtz and Rayleigh-Taylor instabilities in inviscid fluids, the generation of capillary waves on the free surface, and problems in Hele-Shaw flows involving pattern formation through the Saffman-Taylor instability, pattern selection, and singularity formation. In materials science, we discuss microstructure evolution in diffusional phase transformations, and the effects of the competition between surface and elastic energies on microstructure morphology. A common link between these different physical phenomena is the utility of an analysis of the appropriate equations of motion at small spatial scales to develop accurate and efficient time-stepping methods.

Stress Calculations on Multiply Connected Domains

“…The unknown density is the limiting value of´Þ µ on itself. The representations (5,20) have previously, and successfully, been used in other elastostatic contexts [13,14,16,40]. In Sections 5 and 6 we shall extend their use further.…”

“…This may complicate post-processing [25]. In this section we derive more flexible and efficient integral equations for the hole problem, based on the representations (5,20) for¨´Þµ and ©´Þµ. The unknown density is the limiting value of´Þ µ on itself.…”

“…Starting with the representations (5,20) for¨´Þµ and ©´Þµ, we demand that the traction on be zero so that the conditions (16,17) hold. This leads to the integral equatioń…”

“…The inclusion ¼ has moduli ¾ and ¾ . Sherman [38] derived an integral equation for this problem based on continuity of the integral of traction and of the displacement at , and on a representation for ´Þµ and ´Þµ related to our representations (5,20) for¨´Þµ and ©´Þµ by partial integration. For our density ª´Þµ,…”

On surface tension modelling using the level set method

“…Others are the front-tracking method [5,6], the boundary integral method [7], and the volume-of-fluid (VOF) method [8][9][10]. Unfortunately, it does not appear possible at the present time to assert which of the methods is generally superior, as each appears to have its own strengths and weaknesses.…”