We combine a general formulation of microswimmer equations of motion with a numerical bead-shell model to calculate the hydrodynamic interactions with the fluid, from which the swimming speed, power, and efficiency are extracted. From this framework, a generalized Scallop theorem emerges. The applicability to arbitrary shapes allows for the optimization of the efficiency with respect to the swimmer geometry. We apply this scheme to “three-body swimmers” of various shapes and find that the efficiency is characterized by the single-body friction coefficient in the long-arm regime, while in the short-arm regime the minimal approachable distance becomes the determining factor. Next, we apply this scheme to a biologically inspired set of swimmers that propel using a rotating helical flagellum. Interestingly, we find two distinct optimal shapes, one of which is fundamentally different from the shapes observed in nature (e.g., bacteria).
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We find for the three-body swimmers and for the helical flagellum swimmers, such that and hence , from which Eq. (3) follows. In general, the linear relation stays valid, but the expression for is more involved.
There are also hydrodynamic simulation studies that capture the dynamics of the E. coli bacterium accurately.57–59
