Interfacial Dynamics for Stokes Flow

Abstract

Theoretical and computational aspects of the method of interfacial dynamics for flow at vanishing Reynolds number are considered. The mathematical formulation relies on the boundary-integral representation that expresses the flow in terms of distributions of Stokes-flow singularities over the interfaces. The densities of the distributions are identified with the jump in the hydrodynamic traction due to interfacial in-plane and transverse tensions, the interfacial velocity, or the strength of a hydrodynamic potential. The numerical procedure involves describing the interfaces in terms of interfacial marker points that reproduce the evolving shapes of the interfaces by global or local interpolation; solving integral equations of the second kind for the interfacial velocity or for the density of a hydrodynamic potential; and computing the motion of the marker points while simultaneously updating interfacial fields relevant to the dynamics, including the concentration of a surfactant and the position of interfacial point particles at an equilibrium configuration. Interfaces exhibiting isotropic tension, elastic tensions, viscous, and incompressible behavior are considered. The mathematical modeling of the tensions and bending moments developing over interfaces with a membrane-like constitution is discussed in the context of the theory of thin shells. To facilitate the numerical implementation, the coupling of the interfacial mechanics to the hydrodynamics by means of interface force and torque balances is formulated in global Cartesian coordinates. Recent progress in the implementation of boundary-element methods is reviewed, and areas for further research are identified.