The orientation of spheroidal microorganisms swimming in a flow field

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Abstract
This paper shows how to calculate local equilibrium orientations of inhomogeneous spheroidal particles placed in a flow field. The results can be applied either to dilute suspensions of inert particles or to swimming microorganisms; illustrative examples are chosen with the latter appli­cation in mind. The centre of mass of a particle is displaced from the geometric centre C along the axis of symmetry, and the orientation of this axis (represented by the unit vector p ) is determined from the balance between the gravitational couple, non-zero when p is not vertical, and the viscous couple exerted by the surrounding fluid. Fluid and particle inertia are neglected. ‘Local equilibrium’ means that p is stationary in a suitable frame of reference, which may be the laboratory frame or one rotating rigidly relative to it, at the values of fluid velocity, vorticity and rate of strain evaluated at C in the absence of the particle. It is also shown how to determine the stability of local equilibria. Stable equilibrium values of p are calculated explicitly for a number of experimentally realizable flow fields, including vertical Poiseuille flow in a pipe, conical sink flow, two-dimensional straining and shearing flows in a vertical plane, and the wake of a falling sphere. The analysis is particularly simple for spherical particles, when the local rate of strain does not contribute to the viscous couple. The results have implications for laboratory manipulation of the trajectories of swimming algae, and for the develop­ment of collective behaviour and the existence of critical phenomena in suspensions of them.

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