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]

## Contents

* 1 Velocity field in particle frame
* 2 Swimming speed and lab frame
* 3 Structure of the flow and squirmer parameter
* 4 See also
* 5 References

## 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.

ur(r,θ)=23(R3r3−1)B1P1(cos⁡θ)+∑n=2∞(Rn+2rn+2−Rnrn)BnPn(cos⁡θ),
uθ(r,θ)=23(R32r3+1)B1V1(cos⁡θ)+∑n=2∞12(nRn+2rn+2+(2−n)Rnrn)BnVn(cos⁡θ).

Here Bn are constant coefficients, Pn(cos⁡θ) are Legendre polynomials, and Vn(cos⁡θ)=−2n(n+1)∂θPn(cos⁡θ).
One finds P1(cos⁡θ)=cos⁡θ,P2(cos⁡θ)=12(3cos2θ−1),…,V1(cos⁡θ)=sin⁡θ,V2(cos⁡θ)=12sin⁡2θ,….
The expressions above are in the frame of the moving particle. At the interface one finds uθ(R,θ)=∑n=1∞BnVn and ur(R,θ)=0.

Shaker, β=−∞ | Pusher, β=−5 | Neutral, β=0 | Puller, β=5 | Shaker, β=∞ | Passive particle

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Shaker, β=−∞ | Pusher, β=−5 | Neutral, β=0 | Puller, β=5 | Shaker, β=∞ | Passive particle
Velocity field of squirmer and passive particle (top row: lab frame, bottom row: swimmer frame, β=B2/|B1| ).

## Swimming speed and lab frame

By using the Lorentz Reciprocal Theorem, one finds the velocity vector of the particle 𝐔=−12∫𝐮(R,θ)sin⁡θdθ=23B1𝐞z. The flow in a fixed lab frame is given by 𝐮L=𝐮+𝐔:

urL(r,θ)=R3r3UP1(cos⁡θ)+∑n=2∞(Rn+2rn+2−Rnrn)BnPn(cos⁡θ),
uθL(r,θ)=R32r3UV1(cos⁡θ)+∑n=2∞12(nRn+2rn+2+(2−n)Rnrn)BnVn(cos⁡θ).

with swimming speed U=|𝐔|. Note, that limr->∞𝐮L=0 and urL(R,θ)≠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≫R. Within that approximation, uθ(R,θ)=B1sin⁡θ+12B2sin⁡2θ, with squirmer parameter β=B2/|B1|. The first mode n=1 characterizes a hydrodynamic source dipole with decay ∝1/r3 (and with that the swimming speed U). The second mode n=2 corresponds to a hydrodynamic stresslet or force dipole with decay ∝1/r2.[4] Thus, β gives the ratio of both contributions and the direction of the force dipole. β is used to categorize microswimmers into pushers, pullers and neutral swimmers.[5]

Swimmer Type | pusher | neutral swimmer | puller | shaker | passive particle

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Squirmer Parameter | β<0 | β=0 | β>0 | β=±∞ |
Decay of Velocity Far Field | 𝐮∝1/r2 | 𝐮∝1/r3 | 𝐮∝1/r2 | 𝐮∝1/r2 | 𝐮∝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 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.

## See also

* Protist locomotion

## References

1. ↑ 1.0 1.1 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. ↑ 2.0 2.1 Blake, J. R. (1971). "A spherical envelope approach to ciliary propulsion". Journal of Fluid Mechanics 46 (1): 199–208. doi:10.1017/S002211207100048X. ISSN 0022-1120. Bibcode: 1971JFM....46..199B.
3. ↑ 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. PMID 23944457. Bibcode: 2013PhRvE..88a2301B.
4. ↑ Happel, John; Brenner, Howard (1981). Low Reynolds number hydrodynamics. Mechanics of fluids and transport processes. 1. doi:10.1007/978-94-009-8352-6. ISBN 978-90-247-2877-0.
5. ↑ Downton, Matthew T; Stark, Holger (2009). "Simulation of a model microswimmer". Journal of Physics: Condensed Matter 21 (20). doi:10.1088/0953-8984/21/20/204101. ISSN 0953-8984. PMID 21825510. Bibcode: 2009JPCM...21t4101D.

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