# Difference between revisions of "Hauptseminar Active Matter SS 2017/Squirmer Model"

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In brief, the model can be formulated as follows. Consider a spherical particle of radius | In brief, the model can be formulated as follows. Consider a spherical particle of radius | ||

− | + | <math>R</math> suspended in an unbounded, incompressible Newtonian fluid of mass density <math>\rho</math> and | |

− | viscosity | + | viscosity <math>\mu</math>, which far-away from the particle is quiescent. The particle is assumed |

to be neutrally buoyant and there are no external forces acting upon the particle. | to be neutrally buoyant and there are no external forces acting upon the particle. | ||

Furthermore, owing to some internal mechanisms, the particle has an intrinsic axis of polar | Furthermore, owing to some internal mechanisms, the particle has an intrinsic axis of polar | ||

− | symmetry | + | symmetry <math>\mathbf{p}</math>, and induces flow of the surrounding fluid via a prescribed axisymmetric distribution |

− | of velocity at its surface. As a result, the particle translates through the liquid with velocity | + | of velocity at its surface. As a result, the particle translates through the liquid with velocity <math>\mathbf{U}</math> |

− | along the direction | + | along the direction <math>\mathbf{p}</math>. At the same time, the particle induces a hydrodynamic flow of the |

surrounding liquid. The velocity of the particle is assumed to be such that the | surrounding liquid. The velocity of the particle is assumed to be such that the | ||

− | corresponding Reynolds number | + | corresponding Reynolds number <math>\mathrm{Re} := \rho R |U| /\mu \ll 1</math>. |

The topic will cover the followings: | The topic will cover the followings: | ||

− | + | <ol style="list-style-type:lower-alpha"> | |

− | equations in spherical coordinate [3]. | + | <li> Brenner's derivation of the general axisymmetric solution of the incompressible Stokes equations in spherical coordinate [3].</li> |

− | + | <li> Calculation of the velocity U and of the flow for the general squirmer model of Lighthill and Blake. Discussion of the reduced (far-field) models (pusher, puller, neutral, shaker)</li> | |

− | Lighthill and Blake. Discussion of the reduced (far-field) models (pusher, puller, neutral, | + | <li> Mapping of basic ("constant flux") models of chemically active Janus colloids to effective squirmers.</li> |

− | shaker) | + | </ol> |

− | |||

== Literature == | == Literature == |

## Revision as of 14:20, 11 January 2017

- Datum
- 2017-05-09
- Thema
- Squirmer model
- Vortragender
- tba
- Betreuer
- Mihail Popescu

## Contents

The "squirmer" denotes the basic model introduced by Lighthill [1] and Blake [2] to describe the motion at low Reynolds numbers of (nearly) spherical micro-organisms through Newtonian liquids.

In brief, the model can be formulated as follows. Consider a spherical particle of radius suspended in an unbounded, incompressible Newtonian fluid of mass density and viscosity , which far-away from the particle is quiescent. The particle is assumed to be neutrally buoyant and there are no external forces acting upon the particle. Furthermore, owing to some internal mechanisms, the particle has an intrinsic axis of polar symmetry , and induces flow of the surrounding fluid via a prescribed axisymmetric distribution of velocity at its surface. As a result, the particle translates through the liquid with velocity along the direction . At the same time, the particle induces a hydrodynamic flow of the surrounding liquid. The velocity of the particle is assumed to be such that the corresponding Reynolds number .

The topic will cover the followings:

- Brenner's derivation of the general axisymmetric solution of the incompressible Stokes equations in spherical coordinate [3].
- Calculation of the velocity U and of the flow for the general squirmer model of Lighthill and Blake. Discussion of the reduced (far-field) models (pusher, puller, neutral, shaker)
- Mapping of basic ("constant flux") models of chemically active Janus colloids to effective squirmers.

## Literature

- M. J. Lighthill, Commun. Pure App. Math. 5, 109 (1952).
- J. R. Blake, J. Fluid Mech. 46, 199 (1971).
- J. Happel and H. Brenner, Low Reynolds number hydrodynamics (Noordhoff Int. Pub., Leyden, The Netherlands, 1973), Ch. 4-23.
- S. Michelin and E. Lauga, J. Fluid Mech. 747, 572 (2014).