Click here to flash read.
Having a finite interfacial thickness, the phase-field models supply a way to
model the fluid interfaces, which allows the calculations of the interface
movements and deformations on the fixed grids. Such modeling is applied to the
computation of two-phase incompressible Stokes flows in this paper, leading to
a system of Stokes-Cahn-Hilliard equations. The Stokes equation is modified by
adding the continuum force $ - c \nabla w $, where $ c $ is the order parameter
and $ w $ is the chemical potential of $ c $. Similarly, the advection effects
are modeled by addition of the term $ \vec{u} \cdot \nabla c $ in the
Cahn-Hilliard equation. We hereby discuss how the solutions to the above
equations approach the original sharp interface Stokes equation as the
interfacial thickness $ \varepsilon$ tends to zero. We start with a microscopic
model and then the homogenized or upscaled version to the same from author's
previous work, cf. \cite{lakhmara2022}, where the analysis and homogenization
of the system have been performed in detail. Further, we perform the numerical
computations to compare the outcome of the effective model with the original
heterogeneous microscale model.
No creative common's license