Fig 1.
(A) and (B) are the meshes without body deformation for the eel and mackerel, respectively. Flow fields in the middle coronal planes (z = 0) of the eel (C) and mackerel (D) models in the laboratory frame. The arrows indicate the velocity direction and the colors represent the magnitude of the velocity. Only flow speeds greater than 0.02 in nondimensionalized units are shown in color.
Fig 2.
Spatiotemporal distribution of the fore-aft force (Fx) on the eel (A) and the mackerel (B) for two periods.
Negative values indicate thrust, because the swimming direction is in the -x direction.
Fig 3.
(A & C,) Spatiotemporal distribution of the lateral force on the body for two periods from the simulation. The dashed black line indicates a zero-crossing (phase) of the force. (B & D) Comparison of the phase of the lateral force along the body from CFD (solid black line, the same as the dashed lines in A and C) with the phase of the negation of the velocity and the phase of the negation of the acceleration. A 2π term is added or subtracted to ensure continuity.
Fig 4.
Spatiotemporal distribution of the torque on the body in two periods for the eel (A) and the mackerel (B).
The solid and dashed lines indicate the maximum and minimum curvatures, respectively. The same information is illustrated in S3 and S4 Videos.
Fig 5.
Internal power (PT(s, t)) distribution for the eel (A) and the mackerel (C) and the work done over a cycle by muscles along the bodies of the eel (B) and the mackerel (D).
The dashed line in (C) indicates the zero-crossing of the torque in the mackerel (Fig 4B). The two black dots in (C) indicate an example time instant when two points on the body have the opposite sign of power but the same sign of torque. The solid blue lines in (B & D) represent the work by integrating only the positive values in (A & C), and the dashed-dotted red lines represent the work (W) by integrating both positive and negative values. The dotted green lines represent the positive work W+ when body elasticity is considered (Fig 7B & 7D). The cyan dashed lines represent the work done to the fluid by the integration of PF(s, t).
Fig 6.
Spatiotemporal distributions of the torque (T) and power (P) along the body when pure resistive forces (A for T and B for P) or pure reactive forces (C for T and D for P) are considered.
All values are normalized to the respective maximum values in each subfigure. The solid and dashed lines indicate the maximum and minimum curvatures, respectively. The insets are the distribution of the work () done by the internal torque. The gray lines in the insets indicate 0 to guide the eye. In the calculation, the body is uniform, the undulation amplitude is uniform and infinitesimal, and the wavelength is the same as the body length. See S3 Appendix for details of the derivation.
Fig 7.
Torque (A) and power (B) distributions when the elasticity of the body is considered for the eel and torque (C) and power (D) distributions for the mackerel.
The dashed lines indicate zero-crossings of the torque.
Fig 8.
Torque (A) and power (B) distributions when the viscosity of the body is considered for the eel and torque (C) and power (D) distributions for the mackerel.
The solid and dashed lines indicate the maximum and minimum curvatures, respectively.
Fig 9.
The energy transfer by tendons.
(A) Diagram of the tendon model. The muscles and tendon on the left side are drawn as semitransparent for clarity. (B) The torques and time derivative of curvatures as a function of time at the two joints and the torque generated by the tendon muscle. The vertical lengths of the shaded regions indicate the magnitudes of the respective torques. The bars at the bottom indicate the predicted muscle activation periods when the tendons are used. The blue and black bars combined predict the muscle activation at joint 1, and the red bar predicts the muscle activation at joint 2. (C) The total power generated by the muscles with and without the tendons and the power from the tendon muscle. The gray lines indicate the negative power on the tendon muscle but is counted as 0 in computing Ptotal. The stage indices are labeled at the tops of (B) and (C).