Squirmer

Spherical microswimmer in Stokes flow

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.^[1] ^[2] 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.^[3]

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 $R$ ).^[1]^[2] These expressions are given in a spherical coordinate system.

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

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

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

Swimming speed and lab frame

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

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

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

Structure of the flow and squirmer parameter

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

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

As can be seen in the figures above, the (lab frame) velocity field of the passive particle corresponds to a monopole. Furthermore, the $B_{1}$ mode corresponds to a dipole (see case $\beta =0$ ) and the $B_{2}$ mode corresponds to a quadrupole (see cases $\beta \neq 0$ ).

References

1 2 Lighthill, M. J. (1952). "On the squirming motion of nearly spherical deformable bodies through liquids at very small reynolds numbers". Communications on Pure and Applied Mathematics. 5 (2): 109–118. doi:10.1002/cpa.3160050201. ISSN 0010-3640.
1 2 Blake, J. R. (1971). "A spherical envelope approach to ciliary propulsion". Journal of Fluid Mechanics. 46 (01): 199. doi:10.1017/S002211207100048X. ISSN 0022-1120.
↑ Bickel, Thomas; Majee, Arghya; Würger, Alois (2013). "Flow pattern in the vicinity of self-propelling hot Janus particles". Physical Review E. 88 (1). doi:10.1103/PhysRevE.88.012301. ISSN 1539-3755.
↑ Happel, John; Brenner, Howard (1981). "Low Reynolds number hydrodynamics". doi:10.1007/978-94-009-8352-6. ISSN 0921-3805.
↑ Downton, Matthew T; Stark, Holger (2009). "Simulation of a model microswimmer". Journal of Physics: Condensed Matter. 21 (20): 204101. doi:10.1088/0953-8984/21/20/204101. ISSN 0953-8984.

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