Squirmer

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 ${\displaystyle u_{r}(R,\theta )=0}$.

Shaker, ${\displaystyle \beta =-\infty }$	Pusher, ${\displaystyle \beta =-5}$	Neutral, ${\displaystyle \beta =0}$	Puller, ${\displaystyle \beta =5}$	Shaker, ${\displaystyle \beta =\infty }$	Passive particle
Shaker, ${\displaystyle \beta =-\infty }$	Pusher, ${\displaystyle \beta =-5}$	Neutral, ${\displaystyle \beta =0}$	Puller, ${\displaystyle \beta =5}$	Shaker, ${\displaystyle \beta =\infty }$	Passive particle
Velocity field of squirmer and passive particle (top row: lab frame, bottom row: swimmer frame)

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]

Swimmer Type	pusher	neutral swimmer	puller	shaker	passive particle
Squirmer Parameter	${\displaystyle \beta <0}$	${\displaystyle \beta =0}$	${\displaystyle \beta >0}$	${\displaystyle \beta =\pm \infty }$
Decay of Velocity Far Field	${\displaystyle \mathbf {u} \propto 1/r^{2}}$	${\displaystyle \mathbf {u} \propto 1/r^{3}}$	${\displaystyle \mathbf {u} \propto 1/r^{2}}$	${\displaystyle \mathbf {u} \propto 1/r^{2}}$	${\displaystyle \mathbf {u} \propto 1/r}$
Biological Example	E.Coli	Paramecium	Chlamydomonas reinhardtii

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 contraty, 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.