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.[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]

Spherical microswimmer 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 ${\displaystyle R}$).[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)

Swimming speed and lab 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 ${\displaystyle u_{r}^{L}(R,\theta )\neq 0}$.

Structure of the flow and squirmer parameter

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

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

References

1. 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.
2. Blake, J. R. (1971). "A spherical envelope approach to ciliary propulsion". Journal of Fluid Mechanics. 46 (01): 199. Bibcode:1971JFM....46..199B. doi:10.1017/S002211207100048X. ISSN 0022-1120.
3. Bickel, Thomas; Majee, Arghya; Würger, Alois (2013). "Flow pattern in the vicinity of self-propelling hot Janus particles". Physical Review E. 88 (1): 012301. arXiv:1401.7311. Bibcode:2013PhRvE..88a2301B. doi:10.1103/PhysRevE.88.012301. ISSN 1539-3755. PMID 23944457.
4. Happel, John; Brenner, Howard (1981). "Low Reynolds number hydrodynamics". doi:10.1007/978-94-009-8352-6. ISSN 0921-3805. Cite journal requires |journal= (help)
5. Downton, Matthew T; Stark, Holger (2009). "Simulation of a model microswimmer". Journal of Physics: Condensed Matter. 21 (20): 204101. Bibcode:2009JPCM...21t4101D. doi:10.1088/0953-8984/21/20/204101. ISSN 0953-8984.