squirmer

Here we give the flow field of a squirmer in the case of a non-deformable axisymmetric spherical squirmer (radius $R$). 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_1P_1(\cos\theta)+\sum_{n=2}^{\infty}\left(\frac{R^{n+2}}{r^{n+2}}-\frac{R^n}{r^n}\right)B_nP_n(\cos\theta)\;,

u_{\theta}(r,\theta)=\frac 2 3 \left(\frac{R^3}{2r^3}+1\right)B_1V_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_nV_n(\cos\theta)\;.

Here $B\_n$ are constant coefficients, $P\_n(\backslash cos\backslash theta)$ are Legendre polynomials, and $V\_n(\backslash cos\backslash theta)=\backslash frac\{-2\}\{n(n+1)\}\backslash partial\_\{\backslash theta\}P\_n(\backslash cos\backslash theta)$.

One finds $P\_1(\backslash cos\backslash theta)=\backslash cos\backslash theta,\; P\_2(\backslash cos\backslash theta)=\backslash tfrac\; 1\; 2\; (3\backslash cos^2\backslash theta-1),\; \backslash dots,\; V\_1(\backslash cos\backslash theta)=\backslash sin\backslash theta,\; V\_2(\backslash cos\backslash theta)=\; \backslash tfrac\{1\}\{2\}\; \backslash sin\; 2\backslash theta,\; \backslash dots$.

The expressions above are in the frame of the moving particle. At the interface one finds $u\_\{\backslash theta\}(R,\backslash theta)=\backslash sum\_\{n=1\}^\{\backslash infty\}\; B\_nV\_n$ and $u\_r(R,\backslash theta)=0$.

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

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_nP_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_nV_n(\cos\theta)\;.

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

The series above are often truncated at $n=2$ in the study of far field flow, $r\backslash gg\; R$. Within that approximation, $u\_\{\backslash theta\}(R,\backslash theta)=B\_1\backslash sin\backslash theta+\backslash tfrac\; 1\; 2\; B\_2\; \backslash sin\; 2\; \backslash theta$, with squirmer parameter $\backslash beta=B\_2/|B\_1|$. The first mode $n=1$ characterizes a hydrodynamic source dipole with decay $\backslash propto\; 1/r^3$ (and with that the swimming speed $U$). The second mode $n=2$ corresponds to a hydrodynamic stresslet or force dipole with decay $\backslash propto\; 1/r^2$.{{cite book|title=Low Reynolds number hydrodynamics|last1=Happel|first1=John|last2=Brenner|first2=Howard|series=Mechanics of fluids and transport processes |year=1981|volume=1 |issn=0921-3805|doi=10.1007/978-94-009-8352-6|isbn=978-90-247-2877-0 }} Thus, $\backslash beta$ gives the ratio of both contributions and the direction of the force dipole. $\backslash beta$ is used to categorize microswimmers into pushers, pullers and neutral swimmers.{{cite journal|last1=Downton|first1=Matthew T|last2=Stark|first2=Holger|title=Simulation of a model microswimmer|journal=Journal of Physics: Condensed Matter|volume=21|issue=20|year=2009|pages=204101|issn=0953-8984|doi=10.1088/0953-8984/21/20/204101|pmid=21825510 |bibcode=2009JPCM...21t4101D|s2cid=35850530 }}

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

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Category:Fluid dynamics