James J. Feng, Chun Liu, Jie Shen & Pengtao Yue
IMA Volumes in Mathematics and Its Applications, Vol. 141,
pp. 1-26
(Modeling of Soft
Matter,
eds. M.-C. T. Calderer and E. Terentjev)
Springer, New York
(2005)
Abstract - The use of a phase field to
describe interfacial phenomena has a long and fruitful tradition. There
are two key ingredients to the method: the transformation of Lagrangian
description of geometric motions to Eulerian description framework, and
the employment of the energetic variational procedure to derive the
coupled systems. Several groups have used this theoretical framework to
approximate Navier-Stokes systems for two-phase flows. Recently, we
have adapted the method to simulate interfacial dynamics in blends of
microstructured complex fluids. This review has two objectives. The
first is to give a more or less self-contained exposition of the
method. We will briefly review the literature, present the governing
equations and discuss a numerical scheme based on different numerical
schemes, such as spectral methods. The second objective is to elucidate
the subtleties of the model that need to be handled properly for
certain applications. These points, rarely discussed in the literature,
are essential for a realistic representation of the physics and a
successful numerical implementation. The advantages and limitations of
the method will be illustrated by numerical examples. We hope that this
review will encourage readers whose applications may potentially
benefit from a similar approach to explore it further.