The Long-Time Dynamics of Two Hydrodynamically-Coupled Swimming Cells |
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Authors: | Sébastien Michelin Eric Lauga |
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Institution: | (1) Department of Physics, University of Arizona, 1118 E. 4th St., Tucson, AZ 85721, USA;(2) Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118, USA;(3) Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK |
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Abstract: | Swimming microorganisms such as bacteria or spermatozoa are typically found in dense suspensions, and exhibit collective modes
of locomotion qualitatively different from that displayed by isolated cells. In the dilute limit where fluid-mediated interactions
can be treated rigorously, the long-time hydrodynamics of a collection of cells result from interactions with many other cells,
and as such typically eludes an analytical approach. Here, we consider the only case where such problem can be treated rigorously
analytically, namely when the cells have spatially confined trajectories, such as the spermatozoa of some marine invertebrates.
We consider two spherical cells swimming, when isolated, with arbitrary circular trajectories, and derive the long-time kinematics
of their relative locomotion. We show that in the dilute limit where the cells are much further away than their size, and
the size of their circular motion, a separation of time scale occurs between a fast (intrinsic) swimming time, and a slow
time where hydrodynamic interactions lead to change in the relative position and orientation of the swimmers. We perform a
multiple-scale analysis and derive the effective dynamical system—of dimension two—describing the long-time behavior of the
pair of cells. We show that the system displays one type of equilibrium, and two types of rotational equilibrium, all of which
are found to be unstable. A detailed mathematical analysis of the dynamical systems further allows us to show that only two
cell-cell behaviors are possible in the limit of t→∞, either the cells are attracted to each other (possibly monotonically), or they are repelled (possibly monotonically as
well), which we confirm with numerical computations. Our analysis shows therefore that, even in the dilute limit, hydrodynamic
interactions lead to new modes of cell-cell locomotion. |
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