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Hydrodynamic interaction of two swimming model micro-organisms

  • TAKUJI ISHIKAWA (a1), M. P. SIMMONDS (a2) and T. J. PEDLEY (a2)

Abstract

In order to understand the rheological and transport properties of a suspension of swimming micro-organisms, it is necessary to analyse the fluid-dynamical interaction of pairs of such swimming cells. In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, referred to as a squirmer. The centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). The effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The interaction of two squirmers is calculated analytically for the limits of small and large separations and is also calculated numerically using a boundary-element method. The analytical and the numerical results for the translational–rotational velocities and for the stresslet of two squirmers correspond very well. We sought to generate a database for an interacting pair of squirmers from which one can easily predict the motion of a collection of squirmers. The behaviour of two interacting squirmers is discussed phenomenologically, too. The results for the trajectories of two squirmers show that first the squirmers attract each other, then they change their orientation dramatically when they are in near contact and finally they separate from each other. The effect of bottom-heaviness is considerable. Restricting the trajectories to two dimensions is shown to give misleading results. Some movies of interacting squirmers are available with the online version of the paper.

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JFM classification

Type Description Title
VIDEO
Movies

Ishikawa et al. supplementary movie
Movie 1. The movie shows the hydrodynamic interactions, calculated numerically using a boundary element method, between two swimming micro-organisms modelled as squirming spheres (‘squirmers’) with prescribed tangential surface velocity. Effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. In this movie, the ratio of second mode squirming to first mode squirming, β = 1; the initial conditions are that  the angle between the orientation vector of the two squirmers, θ0 = π, and the displacement (defined in figure 17 in the main text), δy = 1.

 Video (2.1 MB)
2.1 MB
VIDEO
Movies

Ishikawa et al. supplementary movie
Movie 2. As movie 1 but for β = 5, under initial conditions θ0 = π and δy = 1.

 Video (2.1 MB)
2.1 MB
VIDEO
Movies

Ishikawa et al. supplementary movie
Movie 3. As movie 1 but for β = 5, under initial conditions θ0 = π/2 and δy = 1.

 Video (3.6 MB)
3.6 MB
VIDEO
Movies

Ishikawa et al. supplementary movie
Movie 4. As movie 1 but for β = 5 under initial conditions θ0 = π/2 and δy = -1.

 Video (3.0 MB)
3.0 MB
VIDEO
Movies

Ishikawa et al. supplementary movie
Movie 5. As movie 1 but for β = 5 under initial conditions θ0 = π/4 and δy = -2.

 Video (2.5 MB)
2.5 MB
VIDEO
Movies

Ishikawa et al. supplementary movie
Movie 6. As movie 1 but for  β = 1 under initial conditions θ0 = 0 and δy = 1.

 Video (2.2 MB)
2.2 MB
VIDEO
Movies

Ishikawa et al. supplementary movie
Movie 7. As movie 1 but for β = 5 under initial conditions θ0 = 0 and δy = 1.

 Video (3.7 MB)
3.7 MB

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