A homogenization approach to the effect of surfactant concentration and interfacial slip on the flow past viscous drops

Abstract

In this paper, the collective motion of viscous drops is investigated in terms of a porous medium setting. We assume that the interface of each of the drops is fully covered with a stagnant layer of surfactant in an arbitrary unsteady Stokes flow with low surface Péclet number limit. The effect of the interfacial slip coefficient on the behavior of the flow fields is also considered. The modeling of the physical processes leads to a system of parabolic partial differential equations (PDEs) with interfacial boundary conditions. We have proposed a model in a micro–macro scale setting. We have shown the existence of a unique positive weak solution of such system. We have also obtained a homogenized (upscaled) model of the system via two-scale convergence and boundary unfolding method.

Keywords:

• Stoke's equations
• surfactant concentration at the interface
• flow past a viscous drops
• system of parabolic equations
• well-posedness
• Rothe's method
• homogenization
• two-scale convergence
• periodic unfolding

AMS subject classifications:

• 35K57
• 35K55
• 76S05
• 47J35
• 76D05
• 76D45
• 76R50

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For $n\in \mathbb{N}$, ${\mathbb{R}}^n:=\left\{\right(x_1,x_2,\dots ,x_n):x_i\in \mathbb{R}\text{ for }1\le i\le n\}$. Note that the symbols $\mathbb{R},\mathbb{N},\mathbb{Z}$ etc. denote the usual set of all reals, positive integers, integers.