Hydrodynamic Interaction between Two Swimmers at Low Reynolds Number

We investigate the hydrodynamic interactions between micro-organisms swimming at low Reynolds number. By considering simple model swimmers, and combining analytic and numerical approaches, we investigate the time-averaged flow field around a swimmer. At short distances the swimmer behaves like a pump. At large distances the velocity field depends on whether the swimming stroke is invariant under a combined time-reversal and parity transformation. We then consider two swimmers and find that the interaction between them consists of two parts: a passive term, independent of the motion of the second swimmer, and an active term resulting from the simultaneous swimming action of both swimmers. The swimmer-swimmer interaction is a complicated function of their relative displacement, orientation, and phase, leading to motion that can be attractive, repulsive, or oscillatory.

Figure 1

The cyclic motion of the linear three sphere swimmer [18]. The swimmer goes through the four steps (i)–(iv) and returns to its original shape, but is displaced by a distance $\Delta$ to the right.

Figure 2

The flow field around a swimmer, averaged over one swimming cycle. The direction of swimming is from left to right and the flow is axisymmetric about the swimmer. The magnitude of velocities are plotted on a logarithmic scale to enable both the near sphere and far field behavior to be observed simultaneously. The parameters used were $a=0.1D$, ${\xi }^B=0.2D$, and ${\xi }^F=0.4D$.

Figure 3

The function $\Phi$ describing the effect of the relative phase of the swimmers on the interactions. The parameters used were the following: black line ${\xi }^B={\xi }^F=0.3D$, curve to the left of this (red), ${\xi }^B=0.4D$, ${\xi }^F=0.2D$, and curve to the right of this (blue), ${\xi }^B=0.2D$, ${\xi }^F=0.4D$.

Figure 4

Long time behavior of two swimmers, $A$ and $B$, (a) in phase, and (b) $\pi$ out of phase. (i)–(vi) illustrate a selection of trajectories representing attractive (A), repulsive (R), oscillatory (O), parallel (P), expanding oscillatory (OE), and contracting oscillatory (OC) behavior, respectively.