Talk abstract:

Models of Undulatory Fish Swimming

T.J. Pedley
University of Cambridge

G.I. Taylor
University of Cambridge

and

S.J. Hill
Department of Applied Mathematics
University of Leeds, U.K.

Many fish swim by stimulating their muscles so as to cause waves of displacement to propagate down their bodies. The reactive force exerted on the water generates the thrust required for propulsion. Mathematical modelling of how a fish swims requires that the external "biofluiddynamics'' be coupled to the internal mechanics of its muscles and other tissues.

The best-known theory for the hydrodynamics of undulatory fish swimming is Lighthill's highly successful elongated-body theory [1, 2], in which the curvature of the fish is assumed small and the effect on the fish of the vortex wake is neglected. Cheng et al [3] did not make these simplifications in developing their vortex lattice panel method, but the fish was assumed to be infinitely thin and the undulations of small amplitude. Input to such models is the observed undulation of the fish (here the saithe, Pollachius virens) during steady state swimming, though a small rigid-body motion has to be added to ensure that the hydrodynamic lateral force and torque balance the fish's body inertia at all times (the "recoil correction" [1]).

In our new model Cheng's approach is extended to large amplitude. The "fish" is infinitely thin, but account is taken of the wake's influence on the fish body; the main difficulty is tracking the non-planar vortex wake as it rolls up. At large amplitude, the recoil correction also requires an estimate of the viscous drag, which is assessed using classical boundary layer theory. For the internal mechanics, the fish is modelled as an active bending beam [4], and from the predicted hydrodynamic load, the distribution of bending moment generated by the muscles can be deduced. Results are presented in the form of (a) visualisations of the wake (in two and three dimensions), (b) plots of the force distribution along the fish, and (c) the distribution of muscle bending moment. It is found that the wave of muscle activation must propagate much more quickly than the bending wave; this is observed in vivo.

References

1. Lighthill, M.J., 1960. Note on the swimming of slender fish. J. Fluid Mech. 9: 305-317.
2. Lighthill, J., 1975. Mathematical Biofluiddynamics, SIAM.
3. Cheng, J.-Y., Zhuang, L.-X. & Tong, B.-G., 1991. Analysis of swimming three-dimensional waving plates. J. Fluid Mech. 232: 341-355.
4. Hess, F. & Videler, J.J., 1984. Fast continuous swimming of saithe ( Pollachius virens): a dynamic analysis of bending moments and muscle power. J. Exp. Biol. 109: 229-251.

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1998-1999 Mathematics in Biology