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 ).[1][2] These expressions are given in a spherical coordinate system.


Here are constant coefficients, are Legendre polynomials, and .
One finds .
The expressions above are in the frame of the moving particle. At the interface one finds and .

Shaker,
Pusher,
Neutral,
Puller,
Shaker,
Passive particle
Shaker,
Pusher,
Neutral,
Puller,
Shaker,
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 . The flow in a fixed lab frame is given by :


with swimming speed . Note, that and .

Structure of the flow and squirmer parameter

The series above are often truncated at in the study of far field flow, . Within that approximation, , with squirmer parameter . The first mode characterizes a hydrodynamic source dipole with decay (and with that the swimming speed ). The second mode corresponds to a hydrodynamic stresslet or force dipole with decay .[4] Thus, gives the ratio of both contributions and the direction of the force dipole. is used to categorize microswimmers into pushers, pullers and neutral swimmers.[5]

Swimmer Typepusherneutral swimmerpullershakerpassive particle
Squirmer Parameter
Decay of Velocity Far Field
Biological ExampleE.ColiParameciumChlamydomonas 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 contraty, 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.

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.
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