Optimal energy-utilization ratio for long-distance cruising of a model fish

The efficiency of total energy utilization and its optimization for long-distance migration of fish have attracted much attention in the past. This paper presents theoretical and computational research, clarifying the above well-known classic questions. Here, we specify the energy-utilization ratio (f(η)) as a scale of cruising efficiency, which consists of the swimming speed over the sum of the standard metabolic rate and the energy consumption rate of muscle activities per unit mass. Theoretical formulation of the function f(η) is made and it is shown that based on a basic dimensional analysis, the main dimensionless parameters for our simplified model are the Reynolds number (Re) and the dimensionless quantity of the standard metabolic rate per unit mass (R(pm)). The swimming speed and the hydrodynamic power output in various conditions can be computed by solving the coupled Navier-Stokes equations and the fish locomotion dynamic equations. Again, the energy consumption rate of muscle activities can be estimated by the quotient of dividing the hydrodynamic power by the muscle efficiency studied by previous researchers. The present results show the following: (1) When the value of f(η) attains a maximum, the dimensionless parameter R(pm) keeps almost constant for the same fish species in different sizes. (2) In the above cases, the tail beat period is an exponential function of the fish body length when cruising is optimal, e.g., the optimal tail beat period of Sockeye salmon is approximately proportional to the body length to the power of 0.78. Again, the larger fish's ability of long-distance cruising is more excellent than that of smaller fish. (3) The optimal swimming speed we obtained is consistent with previous researchers' estimations.