# Hauptseminar Active Matter SS 2017/Squirmer Model

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Date
2017-05-09
Topic
The Squirmer Model
Speaker
tba
Tutor
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 ${\displaystyle R}$ suspended in an unbounded, incompressible Newtonian fluid of mass density ${\displaystyle \rho }$ and viscosity ${\displaystyle \mu }$, 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 ${\displaystyle \mathbf {p} }$, 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 ${\displaystyle \mathbf {U} }$ along the direction ${\displaystyle \mathbf {p} }$. 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 ${\displaystyle \mathrm {Re} :=\rho R|U|/\mu \ll 1}$.

The topic will cover the followings:

1. Brenner's derivation of the general axisymmetric solution of the incompressible Stokes equations in spherical coordinate [3].
2. 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)
3. Mapping of basic ("constant flux") models of chemically active Janus colloids to effective squirmers.

## Literature

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