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The model of the fish
This site presents a mechanical model for a fish that propels itself primarily by undulating its body. The model assumes that the undulations are relatively small in comparison with the length of the fish, and that the fishcan be viewed as an elastic continuum. That the fish is a continuum implies that we ignore the anatomical structure of the fish, regarding it as a indecomposable body. That the fish is elastic implies that it generates forces that are proportional to the amount it deforms itself. These assumptions greatly simplify the real world situation, and allow us to construct a mathematical model that is reasonable to attempt to solve. While the assumptions may seem extreme, the results seem to agree with experiment for fish classified as carangiformby biologists. This includes many different types of fish; in particular, tuna and sunfish are considered
carangiform.
The simplfying assumptions allow us to treat the fish in the same way we treat a metal beam. Mathematical models of beams a very common and have a long history, principally because of their use in engineering and architecture. The most important difference between a beam and a fish (for our purposes) is that while a beam is stiff along its entire length, a fish is stiff in front (the anterior) but loses its stiffness at its tail (the posterior). Thus our model is essentially that of a dynamic (moving) beam that loses stiffness.
The model of the water
Mathematical models of water are very common, but even the simplest models of the motion of water are very mathematically sophisticated. More complicated models of the motion of water can only be solved approximately by careful use of computers. In order to simplify our task, we make an important assumption that the water is inviscid. This means that water doesn't "stick" to things the way oil, honey, and other viscous liquids do. While this assumption seems justified by comparison, in fact water does have some viscosity. Try pouring all the water out of a full glass and you will see that while most of the water leaves the glass quickly, there is a thin film that remains on the walls of the glass and takes some time to drip out. This is viscosity at work, and both theorists and researchers have ample evidence that waters viscosity is very important in the way fish swim. Nonetheless, we have made this simplification to render the problem mathematically tenable. So far as we know, it is the first mathematical model the integrates the kinematics (laws of motion) the fish and the water.
Feedback to
RobRoot@Lafayette.edu
