Abstract
A method for determining Stokes flow around particles near a wall or in a thin film bounded by a wall on one side and a nondeformable gas-liquid interface on the other side is developed. The no-slip boundary conditions at the wall are satisfied by constructing an image system based on Lamb’s multipoles. Earlier results for the image systems for the flow due to a point force or a force dipole are extended to image systems for force or source multipoles of arbitrary orders. For the case of a film, the image system consists of an infinite series of multipoles on both sides of the film. Accurate evaluation of the flow due to these images is discussed, including the use of Shanks transforms. The method is applied to several problems including chains of particles, radially expanding particles, drops, and porous particles.
Highlights
The problem of determining velocity field around particles in close proximity of a wall or a gas-liquid interface arises in the analysis of many physical and biological phenomena including cell adhesion, slurry transport, and particulate flows in microfluidic devices
There is considerable interest in simulating biofilms1–3 formed by bacteria such as Escherichia coli
The use of the ek transform reduces the computational time substantiallynote that, as mentioned earlier, the number of multipoles required by the image system increases with the number of reflections, and the computational cost of cn roughly increases in proportion to n and the total computing time of Green’s function increases as N2, N being the total number of reflections, we were initially surprised to find that the error did not decrease exponentially with p, as was the case in the number of series reviewed by Shanks
Summary
The problem of determining velocity field around particles in close proximity of a wall or a gas-liquid interface arises in the analysis of many physical and biological phenomena including cell adhesion, slurry transport, and particulate flows in microfluidic devices. Understanding these phenomena require numerical simulations of particle-wall and particle-particle interactions involving hundreds or even thousands of particles. The method is used to solve a number of simple flow problems involving either a single particle, a chain of particles, or particles in a thin liquid film bounded on one side by a wall and by a gas-liquid interface on the other side.
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