Model in fluid dynamics
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 (radius formula_1). These expressions are given in a spherical coordinate system.
formula_2
formula_3
Here formula_4 are constant coefficients, formula_5 are Legendre polynomials, and formula_6. 
One finds formula_7.
The expressions above are in the frame of the moving particle. At the interface one finds formula_8 and formula_9.
Swimming speed and lab frame.
By using the Lorentz Reciprocal Theorem, one finds the velocity vector of the particle formula_10. The flow in a fixed lab frame is given by formula_11:
formula_12
formula_13
with swimming speed formula_14. Note, that formula_15 and formula_16.
Structure of the flow and squirmer parameter.
The series above are often truncated at formula_17 in the study of far field flow, formula_18. Within that approximation, formula_19, with squirmer parameter formula_20. The first mode formula_21 characterizes a hydrodynamic source dipole with decay formula_22 (and with that the swimming speed formula_23). The second mode formula_17 corresponds to a hydrodynamic stresslet or force dipole with decay formula_25. Thus, formula_26 gives the ratio of both contributions and the direction of the force dipole. formula_26 is used to categorize microswimmers into pushers, pullers and neutral swimmers.
The above figures show the velocity field in the lab frame and in the particle-fixed frame. The hydrodynamic dipole and quadrupole fields of the squirmer model result from surface stresses, due to beating cilia on bacteria, or chemical reactions or thermal non-equilibrium on Janus particles. The squirmer is force-free. On the contrary, the velocity field of the passive particle results from an external force, its far-field corresponds to a "stokeslet" or hydrodynamic monopole. A force-free passive particle doesn't move and doesn't create any flow field. 
References.
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