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Squirmer

Model in fluid dynamics From Wikipedia, the free encyclopedia

Squirmer
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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]

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Spherical microswimmer in Stokes flow
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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 .

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Shaker,
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Pusher,
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Neutral,
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Puller,
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Shaker,
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Passive particle
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Shaker,
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Pusher,
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Neutral,
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Puller,
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Shaker,
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Passive particle
Velocity field of squirmer and passive particle (top row: lab frame, bottom row: swimmer frame, ).
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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 .

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Structure of the flow and squirmer parameter

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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 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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See also

References

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