Squirmer Explained

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 ). These expressions are given in a spherical coordinate system.u _{r(r,\theta) = (2)/(3)} \left( (R ^{3})/(r ^{3}) -1 \right)B _{1P} _{1(\cos\theta)+\sum} \left( ^{inf ty} _{n =2} (R ^{n+2})/(- r ^{n+2}) (R ^{n})/(r ^{n}) \right)B _{nP} _{n(\cos\theta) ,}
u _{\theta} (r,\theta) = (2)/(\left( 3) (R ^{3})/(2 r ^{3}) +1 \right)B _{1V} _{1(\cos\theta)+\sum} ^{inf ty} ^{n =2} (1)/(+ ( 2- n) 2 \left(n (R ^{n+2})/(r ^{n+2})) (R ^{n})/(r ^{n}) \right)B _{nV} _{n(\cos\theta) .}

Here

B _{n} are constant coefficients, P _{n(\cos\theta)} are Legendre polynomials, and V _{n(\cos\theta) = (-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),} ... , V _{1(\cos\theta)=\sin\theta,} V _{2(\cos\theta)=} \tfrac{ 1 }{ 2 } \sin 2 \theta, ... .
The expressions above are in the frame of the moving particle. At the interface one finds u _{\theta} (R,\theta) = \sum ^{inf ty} _{n =1} B _{nV} _{n} and u _{r(R,\theta)=0} .

Swimming speed and lab frame

By using the Lorentz Reciprocal Theorem, one finds the velocity vector of the particle

U =- \tfrac{ 1 }{ 2 } \int u (R,\theta)\sin\theta d \theta = \tfrac 2 3 B _{1} e _{z} . The flow in a fixed lab frame is given by u ^{L=u+U} :u ^{L(r,\theta) = (R ^{3})/(r ^{3})} _{r} UP _{1(\cos\theta)+\sum} \left( ^{inf ty} _{n =2} (R ^{n+2})/(- r ^{n+2}) (R ^{n})/(r ^{n}) \right)B _{nP} _{n(\cos\theta) ,}
u ^{L(r,\theta) = (R ^{3})/(2 r ^{3})} _{\theta} UV _{1(\cos\theta)+\sum} ^{inf ty} ^{n =2} (1)/(+ ( 2- n) 2 \left(n (R ^{n+2})/(r ^{n+2})) (R ^{n})/(r ^{n}) \right)B _{nV} _{n(\cos\theta) .}

with swimming speed

U = | U | . Note, that \lim _{r → infty} u ^{L=0} and u ^{L} _{r(R,\theta) ≠} 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 inf ty
Decay of Velocity Far Field	u \propto 1 /r ^{2}	u \propto 1 /r ^{3}	u \propto 1 /r ^{2}	u \propto 1 /r ^{2}	u \propto 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

Notes and References

1. Lighthill. M. J.. On the squirming motion of nearly spherical deformable bodies through liquids at very small reynolds numbers. Communications on Pure and Applied Mathematics. 5. 2. 1952. 109–118. 0010-3640. 10.1002/cpa.3160050201.
2. Blake. J. R.. A spherical envelope approach to ciliary propulsion. Journal of Fluid Mechanics. 46. 1. 1971. 199–208. 0022-1120. 10.1017/S002211207100048X. 1971JFM....46..199B. 122519123 .
3. Bickel. Thomas. Majee. Arghya. Würger. Alois. Flow pattern in the vicinity of self-propelling hot Janus particles. Physical Review E. 88. 1. 2013. 1539-3755. 10.1103/PhysRevE.88.012301. 1401.7311. 2013PhRvE..88a2301B. 23944457. 012301. 36558271 .
4. Book: Low Reynolds number hydrodynamics. Happel. John. Brenner. Howard. Mechanics of fluids and transport processes . 1981. 1 . 0921-3805. 10.1007/978-94-009-8352-6. 978-90-247-2877-0 .
5. Downton. Matthew T. Stark. Holger. Simulation of a model microswimmer. Journal of Physics: Condensed Matter. 21. 20. 2009. 204101. 0953-8984. 10.1088/0953-8984/21/20/204101. 21825510 . 2009JPCM...21t4101D. 35850530 .