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]
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 .
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 .
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 Type | pusher | neutral swimmer | puller | shaker | passive particle |
Squirmer Parameter | |||||
Decay of Velocity Far Field | |||||
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.
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]
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 .
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 .
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 Type | pusher | neutral swimmer | puller | shaker | passive particle |
Squirmer Parameter | |||||
Decay of Velocity Far Field | |||||
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.