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
). 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,  |
|
Shaker,  |
Pusher,  |
Neutral,  |
Puller,  |
Shaker,  |
|
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
. 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.
Swimmer Type |
pusher |
neutral swimmer |
puller |
shaker |
passive particle
|
Squirmer Parameter |
 |
 |
 |
 |
|
Decay of Velocity Far Field |
 |
 |
 |
 |
|
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
).