  Fish typically swim by periodic bending of their bodies. Bending seems to
follow a universal rule; it occurs at about one-third from the posterior end of
the fish body with a maximum bending angle of about $30^o$. However, the
hydrodynamic mechanisms that shaped this convergent design and its potential
benefit to fish in terms of swimming speed and efficiency are not well
understood. It is also unclear to what extent this bending is active or follows
passively from the interaction of a flexible posterior with the fluid
environment. Here, we use a self-propelled two-link model, with fluid-structure
interactions described in the context of the vortex sheet method, to analyze
the effects of both active and passive body bending on the swimming
performance. We find that passive bending is more efficient but could reduce
swimming speed compared to rigid flapping, but the addition of active bending
could enhance both speed and efficiency. Importantly, we find that the phase
difference between the posterior and anterior sections of the body is an
important kinematic factor that influences performance, and that active
antiphase flexion, consistent with the passive flexion phase, can
simultaneously enhance speed and efficiency in a region of the design space
that overlaps with biological observations. Our results are consistent with the
hypothesis that fish that actively bend their bodies in a fashion that exploits
passive hydrodynamics can at once improve speed and efficiency.
