Here we give the flow field of a squirmer in the case of a non-deformable axisymmetric spherical squirmer (radius ${\displaystyle R}$ ).^[1][2] These expressions are given in a spherical coordinate system.

${\displaystyle u_{r}(r,\theta )={\frac {2}{3}}\left({\frac {R^{3}}{r^{3}}}-1\right)B_{1}P_{1}(\cos \theta )+\sum _{n=2}^{\infty }\left({\frac {R^{n+2}}{r^{n+2}}}-{\frac {R^{n}}{r^{n}}}\right)B_{n}P_{n}(\cos \theta )\;,}$ 
${\displaystyle u_{\theta }(r,\theta )={\frac {2}{3}}\left({\frac {R^{3}}{2r^{3}}}+1\right)B_{1}V_{1}(\cos \theta )+\sum _{n=2}^{\infty }{\frac {1}{2}}\left(n{\frac {R^{n+2}}{r^{n+2}}}+(2-n){\frac {R^{n}}{r^{n}}}\right)B_{n}V_{n}(\cos \theta )\;.}$ 

Here ${\displaystyle B_{n}}$  are constant coefficients, ${\displaystyle P_{n}(\cos \theta )}$  are Legendre polynomials, and ${\displaystyle V_{n}(\cos \theta )={\frac {-2}{n(n+1)}}\partial _{\theta }P_{n}(\cos \theta )}$ .
One finds ${\displaystyle P_{1}(\cos \theta )=\cos \theta ,P_{2}(\cos \theta )={\tfrac {1}{2}}(3\cos ^{2}\theta -1),\dots ,V_{1}(\cos \theta )=\sin \theta ,V_{2}(\cos \theta )={\tfrac {1}{2}}\sin 2\theta ,\dots }$ .
The expressions above are in the frame of the moving particle. At the interface one finds ${\displaystyle u_{\theta }(R,\theta )=\sum _{n=1}^{\infty }B_{n}V_{n}}$  and ${\displaystyle u_{r}(R,\theta )=0}$ .

By using the Lorentz Reciprocal Theorem, one finds the velocity vector of the particle ${\displaystyle \mathbf {U} =-{\tfrac {1}{2}}\int \mathbf {u} (R,\theta )\sin \theta \mathrm {d} \theta ={\tfrac {2}{3}}B_{1}\mathbf {e} _{z}}$ . The flow in a fixed lab frame is given by ${\displaystyle \mathbf {u} ^{L}=\mathbf {u} +\mathbf {U} }$ :

${\displaystyle u_{r}^{L}(r,\theta )={\frac {R^{3}}{r^{3}}}UP_{1}(\cos \theta )+\sum _{n=2}^{\infty }\left({\frac {R^{n+2}}{r^{n+2}}}-{\frac {R^{n}}{r^{n}}}\right)B_{n}P_{n}(\cos \theta )\;,}$ 
${\displaystyle u_{\theta }^{L}(r,\theta )={\frac {R^{3}}{2r^{3}}}UV_{1}(\cos \theta )+\sum _{n=2}^{\infty }{\frac {1}{2}}\left(n{\frac {R^{n+2}}{r^{n+2}}}+(2-n){\frac {R^{n}}{r^{n}}}\right)B_{n}V_{n}(\cos \theta )\;.}$ 

with swimming speed ${\displaystyle U=|\mathbf {U} |}$ . Note, that ${\displaystyle \lim _{r\rightarrow \infty }\mathbf {u} ^{L}=0}$  and ${\displaystyle u_{r}^{L}(R,\theta )\neq 0}$ .

The series above are often truncated at ${\displaystyle n=2}$  in the study of far field flow, ${\displaystyle r\gg R}$ . Within that approximation, ${\displaystyle u_{\theta }(R,\theta )=B_{1}\sin \theta +{\tfrac {1}{2}}B_{2}\sin 2\theta }$ , with squirmer parameter ${\displaystyle \beta =B_{2}/|B_{1}|}$ . The first mode ${\displaystyle n=1}$  characterizes a hydrodynamic source dipole with decay ${\displaystyle \propto 1/r^{3}}$  (and with that the swimming speed ${\displaystyle U}$ ). The second mode ${\displaystyle n=2}$  corresponds to a hydrodynamic stresslet or force dipole with decay ${\displaystyle \propto 1/r^{2}}$ .^[4] Thus, ${\displaystyle \beta }$  gives the ratio of both contributions and the direction of the force dipole. ${\displaystyle \beta }$  is used to categorize microswimmers into pushers, pullers and neutral swimmers.^[5]

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.

• Protist locomotion