Figure 2. Efficient propulsion by an actuated, flexible fish model (top right image, the length of the scale bar is 3 cm) emerges when the yaw and heave of the head are coupled with the correct phase angle.

(a) Force production of the model (lower right plot) was evaluated as a function of oscillation frequency and phase difference between yaw and heave at 0.8 L s⁻¹. The heat map denotes the magnitude of force averaged over one tail-beat cycle. Negative values (blue) indicate a region where the fluid resistance was greater than the propulsive force generated by the model (drag). Positive values (red) indicate where the propulsive force of the model was greater than the fluid resistance (thrust). In a steadily swimming fish, there is no net force acting on the body (that is, thrust equals drag). In our experiments, this condition corresponded to the C-shaped region delineated by the dashed lines. The new heat-map plot on the left shows that within this region, the cost of transport differs as a function of oscillation frequency and phase difference. Low (blue) and high (red) cost of transport denote high- and low-propulsive efficiency, respectively. The locations of minimum and maximum values are shown (white circles). (b) The model displays very different kinematics depending on which phase difference and oscillation frequency values it adopts. At 110° (blue line, high-propulsive efficiency) it is similar to the amplitude and phase envelope of a live fish (black line). As in fish, body amplitude of the physical model increases posteriorly and the mechanical body wave is initiated at the head and travels down the body with a constant velocity. This is indicated by a linear increase in phase values down the body. At 270° (red line, low propulsive efficiency), the kinematics departs substantially from a live fish. Amplitude values are normalized to the maximum tail beat amplitude. Phase values are normalized to the phase difference between head and the tail.