Hauptseminar Active Matter SS 2017/Boundary Element Method for Phoretic Swimmers

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Boundary Element Method for Phoretic Swimmers
Will Uspal


The boundary-element method (BEM) is a numerical method for solving partial differential equations based on integral representations of the solution. The method is well suited for solving linear, elliptic, homogeneous partial differential equations governing boundary-value problems, such as Laplace’s equation, Helmholtz’s equation, or Stokes' equations of creeping flow. In BEM the solution is expressed in terms of so-called boundary distributions of fundamental solutions — i.e., the Green’s functions expressing the field due to a localized source — of the particular differential equation considered. These initially unknown distributions are then computed such as to satisfy the specified boundary conditions of the problem.

The BEM is therefore a natural choice in studies of the steady-state motion of simple models of chemically active self-phoretic particles, for which the motion is described by suitable solutions of, e.g., Laplace's equation governing the distribution of "chemical" and of Stokes' equations governing the hydrodynamics. The topic will cover the formulation of the BEM for the Laplace and Stokes equations, as well as that of a regularized BEM formulation, as applied to the case of a self-diffusiophoretic Janus particle. The case of constant-flux activity on the surface of the particle, and the limit of a thin boundary layer that can be modeled with a phoretic slip boundary condition, will be considered. Furthermore, the BEM has the distinct advantage that it can easily be adapted to study the effect of confining geometries, provided an appropriate Green's function can be found. These aspects are illustrated using two benchmark examples: a particle moving in unbounded fluid, and a particle moving in the vicinity of a planar wall.


  1. C. Pozrikidis, A Practical Guide to BOUNDARYELEMENT METHODS with the Software Library BEMLIB (CHAPMAN & HALL/CRC, Boca Raton, 2002), Ch. 4, 5, 7.
  2. W.E. Uspal, M.N. Popescu, S. Dietrich, and M. Tasinkevych, Self-propulsion of a catalytically active particle near a planar wall: from reflection to sliding and hovering, Soft Matter 11, 434 (2015).
  3. Q. Sun, E. Klaseboer, B.C. Khoo, and D.Y.C. Chan, A robust and non-singular formulation of the boundary integral method for the potential problem, Eng. Anal. Bound. Elem. 43, 117 (2014); ibid, Boundary regularized integral equation formulation of Stokes flow, Phys. Fluids 27, 023102 (2015).