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

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The Squirmer model and the Boundary Element Method for Phoretic Swimmers
Miru Lee
Mihail Popescu


The "squirmer" denotes the basic model introduced by Lighthill [1] and Blake [2] to describe the motion at low Reynolds numbers of (nearly) spherical micro-organisms through Newtonian liquids.

In brief, the model can be formulated as follows. Consider a spherical particle of radius R suspended in an unbounded, incompressible Newtonian fluid of mass density \rho and viscosity \mu, which far-away from the particle is quiescent. The particle is assumed to be neutrally buoyant and there are no external forces acting upon the particle. Furthermore, owing to some internal mechanisms, the particle has an intrinsic axis of polar symmetry \mathbf{p}, and induces flow of the surrounding fluid via a prescribed axisymmetric distribution of velocity at its surface. As a result, the particle translates through the liquid with velocity \mathbf{U} along the direction \mathbf{p}. At the same time, the particle induces a hydrodynamic flow of the surrounding liquid. The velocity of the particle is assumed to be such that the corresponding Reynolds number \mathrm{Re} := \rho R |U| /\mu \ll 1.

When one deals with swimmers of complex shapes, or phoretic particles for which the distribution of slip velocity on the surface can be influenced by the presence of nearby interfaces, it is difficult to find analytic solutions for the equations (Stokes and, for phoretic swimmers, Laplace) governing the dynamics. Thus numerical methods have to be employed in such situations.

The boundary-element method (BEM) is a numerical method for solving partial differential equations based on integral representations of the solution [3]. 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, and thus represents a natural choice in numerical studies of the steady-state motion of simple models of self-phoretic particles. 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.'

This topic will therefore cover the following two main points:

  1. Analysis of the squirmer model for:
    1. Brenner's derivation of the general axisymmetric solution of the incompressible Stokes equations in spherical coordinate [4].
    2. Calculation of the velocity U and of the flow for the general squirmer model of Lighthill and Blake. Discussion of the reduced (far-field) models (pusher, puller, neutral, shaker)
  2. The formulation of the BEM for the Laplace and Stokes equations, as applied to the case of a self-diffusiophoretic Janus particle with boundary conditions of a constant-flux activity and a phoretic slip [5].


  1. M. J. Lighthill, Commun. Pure App. Math. 5, 109 (1952).
  2. J. R. Blake, J. Fluid Mech. 46, 199 (1971).
  3. C. Pozrikidis, A Practical Guide to BOUNDARYELEMENT METHODS with the Software Library BEMLIB (CHAPMAN & HALL/CRC, Boca Raton, 2002), Ch. 4, 5, 7.
  4. J. Happel and H. Brenner, Low Reynolds number hydrodynamics (Noordhoff Int. Pub., Leyden, The Netherlands, 1973), Ch. 4-23.
  5. 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).