Hauptseminar Soft Matter SS 2019/Multipole expansion of flow fields The squirmer model

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Multipole expansion of flow fields: The squirmer model
Michael Maihöfer
Mihail Popescu


All microswimmers, regardless of the mechanism of propulsion, create disturbance flows in the suspending fluid.

In analogy with the case of electrostatics or diffusion, the steady state Stokes equations (most relevant for micro-swimmers), which are linear, admit fundamental solutions (Green's function), often denoted also as "singularities", describing the flows induced by point sources [1]. A few of them (the point force and force-doublet, the source and source doublet) will be derived and discussed [2,3].

While any Stokes hydrodynamic flow produced by a microswimmer can be written as a series representation in the fundamental solutions, accounting for all the details of a specific swimmer (e.g., a microorganism whose surface is covered by beating of cilia) can be a daunting task. The "squirmer" denotes the basic model introduced by Lighthill [4] and Blake [5] to describe the motion at low Reynolds numbers of (nearly) spherical micro-organisms which achieve motility through Newtonian liquids via a "surface actuation", possessing axial symmetry, of the fluid. The presentation will succinctly cover, using Brenner's derivation of the general axisymmetric solution of the incompressible Stokes equations in spherical coordinate [1], the calculation of the velocity of the swimmer, as well as of the induced hydrodynamic flow, for the general squirmer model [6]; the notions of "pusher", "puller", "neutral", "shaker" squirmer will also be discussed.


  1. J. Happel and H. Brenner, Low Reynolds number hydrodynamics (Noordhoff Int. Pub., Leyden, The Netherlands, 1973), Ch. 1, 3-5, 4.1-4.7, 6.1-6.5, 7
  2. J.R. Blake and A.T. Chwang, Fundamental singularities of viscous flow. Part I, J. Eng. Math. 8, 23 (1974)
  3. S.E. Spagnolie and E. Lauga, Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far field approximations, J. Fluid. Mech. 700, 105 (2012)
  4. M. J. Lighthill, Commun. Pure App. Math. 5, 109 (1952)
  5. J. R. Blake, J. Fluid Mech. 46, 199 (1971)
  6. M.N. Popescu et al, Eur. J. Phys. E 41, 145 (2018)