While the world’s fastest human reaches a speed of about 10 km/h in water and the fastest submarine about 80 km/h, swordfish easily exceed 100 km/h when hunting. How do fish swim so fast? This innocent question has kept the scientific community in check for 50 years. In 2018, the team at Tingyu Mingfrom the Beijing Computational Science Research Center (China), solved the problemmodeling the movement of the fish as faithfully as possible.
Until then, the community was divided between two theories. The first, proposed in 1952 by the physicist Geoffrey Taylor and the second, by the mathematician James Lighthill in 1960. The key difference between the two is the type of force generated by the animal which, for its authors, explains the swimming mechanism. For Taylor, it is a resistive force, which acts in the opposite direction to the body’s motion, but is directly related to velocity. For Lighthill, it is a reactive force, which acts in the opposite direction to an action force and is linked to acceleration. It may seem like a subtle difference, but it is key to understanding fish propulsion and artificially reproducing it.
Taylor’s theory – or resistive force theory – considered that the impulse comes from the interaction of the surface of the fish with the water. Water is a viscous fluid and therefore generates resistance to movement. His idea was that the skin of the fish, which he divided into small segments, generates resistance, but, as each segment undulates, the resistance is greater in the direction perpendicular to the body than in the direction parallel to it. The result – which can be seen in the image – is a push in a parallel direction, or forward.
Lighthill’s theory is somewhat more complicated. Imagine a ball that moves submerged in water. The viscosity of the water, in this case, manifests itself in the form of small blocks of fluid that are dragged along, forming eddies that travel in the opposite direction to which the ball moves, as can be seen in Figure 2. Consequently, a pattern known as street of vortices or whirlpools of Von Karman. In real conditions many vortices appear, of different sizes, but, on average, the situation is very similar, as can be seen in Figure 3.
In this pattern the eddies spin in the wake of the ball, so they generate a force, or moment, in the sense that it moves. There is a conservation law which indicates that in a closed system (where there are no external forces) the sum of the linear moments is always constant. Therefore, in order for the total amount of linear momentum to be conserved, this force is extracted from the displacement of the ball itself, which causes it to slow down.
For this reason, submarines need powerful nuclear engines to be able to move at high speeds over long distances, overcoming the viscosity of the water. However, fish are able to use this same principle to their advantage. With their tail, they exchange the position of the vortices and, thus, a resulting force is generated in the opposite direction to which the fish travels. The result is a force that pushes the fish, as if it were a motor formed by the combined action of small vortices, as can be seen in Figure 4. This type of force is known as reactive force.
To find out which of the two theories was the most appropriate, Tingyu Ming and his collaborators created a three-dimensional computational hydrodynamic model of two types of fish: eel swimmers, such as eels, and carangiform swimmers, such as mackerel or tuna. The former undulate their entire body, while the latter only bend the rear half. The team used actual fish motion data to calibrate their models to calculate the force generated by these fish.
This very precise modeling showed that both theories are correct: depending on the type of fish and even the part of the animal’s body that we are considering, the description is given by one or the other. For example, for both mackerel and eel, resistive forces are more important in the middle of the body, but reactive forces are more important near the tails.
The work showed that fish propulsion is much more complex than previously thought, and presumably just as difficult to reproduce artificially. Although research in this regard is very incipient, these advances represent a theoretical basis for future development of faster submarines, more efficient underwater vehicles that allow us to explore the ocean in a less invasive way, or even methods of travel that generate less noise pollution – which it is one of the causes of biodiversity loss in the world’s seas.
William Garcia Sanchez He is a predoctoral researcher at the ICMAT.
Agate Timon G. Longoria is coordinator of the ICMAT Mathematical Culture Unit.
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems”.
Edition and coordination: Agate A. Timón G Longoria (ICMAT).
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