Squirmer


The squirmer is a model for a spherical microswimmer swimming in Stokes flow. The squirmer model was introduced by James Lighthill in 1952 and refined and used to model Paramecium by John Blake in 1971.
Blake used the squirmer model to describe the flow generated by a carpet of beating short filaments called cilia on the surface of Paramecium. Today, the squirmer is a standard model for the study of self-propelled particles, such as Janus particles, in Stokes flow.

Velocity field in particle frame

Here we give the flow field of a squirmer in the case of a non-deformable axisymmetric spherical squirmer. These expressions are given in a spherical coordinate system.


Here are constant coefficients, are Legendre polynomials, and.

One finds.

The expressions above are in the frame of the moving particle. At the interface one finds and.

Swimming speed and lab frame

By using the Lorentz Reciprocal Theorem, one finds the velocity vector of the particle. The flow in a fixed lab frame is given by :


with swimming speed. Note, that and.

Structure of the flow and squirmer parameter

The series above are often truncated at in the study of far field flow,. Within that approximation,, with squirmer parameter. The first mode characterizes a hydrodynamic source dipole with decay . The second mode corresponds to a hydrodynamic stresslet or force dipole with decay. Thus, gives the ratio of both contributions and the direction of the force dipole. is used to categorize microswimmers into pushers, pullers and neutral swimmers.
Swimmer Typepusherneutral swimmerpullershakerpassive particle
Squirmer Parameter
Decay of Velocity Far Field
Biological ExampleE.ColiParameciumChlamydomonas reinhardtii

As can be seen in the figures above, the velocity field of the passive particle corresponds to a monopole. Furthermore, the mode corresponds to a dipole and the mode corresponds to a quadrupole.