The flow field around a freely swimming copepod in steady motion. Part I: Theoretical analysis

Abstract

The three-dimensional flow field around a free-swimming copepod in steady motion was studied theoretically. This study was based on coupling the Navier-Stokes equations with the dynamic equations for an idealized body of a copepod. To allow analytical solutions to the flow field, three simplifications were made: (a) to simulate the effect of the beating movement of the cephalic appendages, a force-field was added to the Navier-Stokes equations, (b) to linearize the problem, Stokes flow was used, and (c) to simplify the morphologies of the copepods, a spherical body shape was assumed. Analytical solutions were derived for five steady motions: (1) hovering, (2) sinking, (3) upwards swimming, (4) backwards swimming and (5) forwards swimming. The results show that the geometry of the flow field around a freely swimming copepod varies significantly with the different swimming behaviours. When a copepod hovers in the water, or swims very slowly, it generates a wide, cone-shaped flow field. In contrast, when a copepod sinks, or swims fast, the flow geometry is not cone-shaped, but cylindrical, narrow and long. These results are consistent with published observations on live copepods. It is shown that the differences in the flow geometry with the different swimming behaviours are due to the relative importance between the two factors in generating the flow field: the copepod's swimming motion and the requirement to counterbalance the copepod's excess weight. The results also highlight the importance of considering freely swimming copepods as self-propelled rather than as towed bodies. Self-propelled' means a freely swimming copepod must gain thrust from the surrounding water in order to counterbalance the dragforce by water and its excess weight. Regardless of swimming behaviours and velocities, the far-field velocity field decays to that of the velocity field generated by a point force of magnitude equal to the copepod's excess weight in an infinite domain. On the other hand, using the towed body model yields a flow field with much different far- and nearfield flow characteristics. Hence, the towed body model is inherently unable to reproduce fundamental characteristics of the flow field around a freely swimming copepod.