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]

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.

${\displaystyle u_r(r,\theta )=\frac{2}{3}(\frac{R^3}{r^3}-1)B_1{P}_1(\cos \theta )+{\displaystyle \sum _{n=2}^{\mathrm{\infty }}}(\frac{R^{n+2}}{r^{n+2}}-\frac{R^n}{r^n})B_n{P}_n(\cos \theta ),}$
${\displaystyle u_{\theta }(r,\theta )=\frac{2}{3}(\frac{R^3}{2r^3}+1)B_1{V}_1(\cos \theta )+{\displaystyle \sum _{n=2}^{\mathrm{\infty }}}\frac{1}{2}(n\frac{R^{n+2}}{r^{n+2}}+(2-n)\frac{R^n}{r^n})B_n{V}_n(\cos \theta ).}$

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 ${\displaystyle P_1(\cos \theta )=\cos \theta ,P_2(\cos \theta )=\frac{1}{2}(3{\cos }^2\theta -1),\dots ,V_1(\cos \theta )=\sin \theta ,V_2(\cos \theta )=\frac{1}{2}\sin 2\theta ,\dots }$.
The expressions above are in the frame of the moving particle. At the interface one finds ${\displaystyle u_{\theta }(R,\theta )={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{B}_n{V}_n}$ and ${\displaystyle u_r(R,\theta )=0}$.

Velocity field of squirmer and passive particle (top row: lab frame, bottom row: swimmer frame, ${\displaystyle \beta =B_2/|B_1|}$ ).

By using the Lorentz Reciprocal Theorem, one finds the velocity vector of the particle ${\displaystyle \mathbf{𝐔}=-\frac{1}{2}\int \mathbf{𝐮}(R,\theta )\sin \theta \mathrm{d}\theta =\frac{2}{3}{B}_1{\mathbf{𝐞}}_z}$. The flow in a fixed lab frame is given by ${\displaystyle {\mathbf{𝐮}}^L=\mathbf{𝐮}+\mathbf{𝐔}}$:

${\displaystyle u_r^L(r,\theta )=\frac{R^3}{r^3}U{P}_1(\cos \theta )+{\displaystyle \sum _{n=2}^{\mathrm{\infty }}}(\frac{R^{n+2}}{r^{n+2}}-\frac{R^n}{r^n})B_n{P}_n(\cos \theta ),}$
${\displaystyle u_{\theta }^L(r,\theta )=\frac{R^3}{2r^3}U{V}_1(\cos \theta )+{\displaystyle \sum _{n=2}^{\mathrm{\infty }}}\frac{1}{2}(n\frac{R^{n+2}}{r^{n+2}}+(2-n)\frac{R^n}{r^n})B_n{V}_n(\cos \theta ).}$

with swimming speed ${\displaystyle U=|\mathbf{𝐔}|}$. Note, that ${\displaystyle \underset{r\to \mathrm{\infty }}{\lim }{\mathbf{𝐮}}^L=0}$ and ${\displaystyle u_r^L(R,\theta )\ne 0}$.

The series above are often truncated at ${\displaystyle n=2}$ in the study of far field flow, ${\displaystyle r\gg R}$. Within that approximation, ${\displaystyle u_{\theta }(R,\theta )=B_1\sin \theta +\frac{1}{2}{B}_2\sin 2\theta }$, 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.

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