An integral equation method for closely interacting surfactant‐covered droplets in wall‐confined Stokes flow

Abstract

Summary A highly accurate method for simulating surfactant‐covered droplets in two‐dimensional Stokes flow with solid boundaries is presented. The method handles both periodic channel flows of arbitrary shape and stationary solid constrictions. A boundary integral method together with a special quadrature scheme is applied to solve the Stokes equations to high accuracy, also for closely interacting droplets. The problem is considered in a periodic setting and Ewald decompositions for the Stokeslet and stresslet are derived. Computations are accelerated using the spectral Ewald method. The time evolution is handled with a fourth‐order, adaptive, implicit‐explicit time‐stepping scheme. The numerical method is tested through several convergence studies and other challenging examples and is shown to handle drops in close proximity both to other drops and solid objects to high accuracy.

An accurate integral equation method for Stokes flow with piecewise smooth boundaries

“…This causes the numerical errors from standard Gauss-Legendre quadrature to grow quite large when evaluating the integrals at any point close to a boundary. To handle this, the integrals are treated in the manner described in [32]. In short, the main idea is similar to that explained in Sect.…”

“…This quantity has no physical meaning beyond imposing a pressure gradient. For a channel with flat parallel walls, a simple alternative would be to add a predetermined background flow, as done in [32].…”

“…The real space sum can be truncated in real space, while the Fourier sum can be truncated in Fourier space. The splits for the two-dimensional Stokeslet and stresslet, as well as truncation estimates are derived in [32]. Here we list the results.…”

“…They can be thought of as the zero Reynolds number limit of the Navier-Stokes equations. Of the many applications of the Stokes equations, they are often used to model particle suspensions including solid particles [4,19], drops [25,31,32], or vesicles [33]. Even for non-zero Reynolds numbers, the Stokes equations often describe very well the flow near a solid boundary, and can therefore be used to derive effective slip boundary conditions for problems at higher Reynolds numbers [1,5].…”

“…For particulate flows, such models are useful because they allow for the computation of various time averaged quantities over a relatively small reference cell, without the need to simulate an unfeasibly large domain. BIEs have been successfully applied to such problems [24,32,41].…”