Quiet swimming at low Reynolds number

The stresslet provides a simple model of the flow created by a small, freely swimming and neutrally buoyant aquatic organism and shows that the far field fluid disturbance created by such an organism in general decays as one over distance squared. Here we discuss a quieter swimming mode that eliminates the stresslet component of the flow and leads to a faster spatial decay of the fluid disturbance described by a force quadrupole that decays as one over distance cubed. Motivated by recent experimental results on fluid disturbances due to small aquatic organisms, we demonstrate that a three-Stokeslet model of a swimming organism which uses breast stroke type kinematics is an example of such a quiet swimmer. We show that the fluid disturbance in both the near field and the far field is significantly reduced by appropriately arranging the propulsion apparatus, and we find that the far field power laws are valid surprisingly close to the organism. Finally, we discuss point force models as a general framework for hypothesis generation and experimental exploration of fluid mediated predator-prey interactions in the planktonic world.

Figure 1

Planktonic breast stroke swimmers across taxonomic groups and sizes: (a) Chlamydomonas reinhardtii, a flagellate (image courtesy of Knut Drescher); (b) Mesodinium rubrum, a ciliate; (c) a nauplius (juvenile) of Acartia tonsa, a copepod; and (d) Podon intermedius, a cladoceran. The Reynolds numbers of the swimmers are approximately ${10}^{-3}, 0.1$, 10, and 10, respectively.

Figure 2

Three-Stokeslet model of a breast stroke swimming aquatic organism. The red vectors represent the point forces.

Figure 3

Velocity fields in the three-Stokeslet model: (left) $\alpha =1$, (middle) $\alpha =0.1$, and (right) $\alpha =0$. The thick red arrows represent the point forces and the green dots the stagnation points on the $z$ axis. The flow fields are shown as thin black vectors and blue streamline segments.

Figure 4

Magnitude of the velocity in the three-Stokeslet model (blue) on the positive $x$ axis (top) and the positive $z$ axis (bottom) with $\alpha =1$ (left), $\alpha =0.1$ (middle), and $\alpha =0$ (right). Also shown are the far field approximations (7) and (8) for $\alpha =1$ (red, $r^{-2}$ decay) and $\alpha =0$ (green, $r^{-3}$ decay).

Figure 5

Micro-organisms with multiple flagella. In all cases the swimming direction is upwards, and the red vectors represent approximately the forces. The flagella function with either a flagellar beat, f, or a ciliary beat, c. Protists with (a) two pairs of flagella that beat out of phase, e.g., Carteria, (b) a leading and a trailing flagellum, e.g., Nephroselmis, and (c) all four flagella pushing steadily, e.g., Cymbomonas [13]. (d) A haptophyte with two flagella and a haptonema, h, in the front, e.g., Chrysochromulina [12].