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
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
.
File:Shaker pusher.png
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File:Shaker puller.png
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File:Shaker pusher.png
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File:Shaker puller.png
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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 stokeslet 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 Type | pusher | neutral swimmer | puller | shaker | passive particle |
Squirmer Parameter | ![]() |
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Decay of Velocity Far Field | ![]() |
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Biological Example | E.Coli | Paramecium | Chlamydomonas reinhardtii |
As can be seen in the figures above, the (lab frame) velocity field of the passive particle corresponds to a monopole. Furthermore, the mode corresponds to a dipole (see case
) and the
mode corresponds to a quadrupole (see cases
).
References
- ↑ 1.0 1.1 Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 2.0 2.1 Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.