Biofluiddynamics of balistiform and gymnotiform locomotion. Part 4. Short-wavelength limitations on momentum enhancement

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Abstract
Elongated-body theory, used by Lighthill & Blake (1990) to investigate fish locomotion by undulatory movements of median fins, and to demonstrate momentum enhancement in the case when motile fins are attached to a rigid fish body of far greater depth, approximates local fluid motions by solutions of the two-dimensional Laplace equation. Here, a better local approximation (equation (2) below) to the three-dimensional Laplace equation for fluid motions of undulatory type is used to investigate the possibility of short-wavelength limitations on momentum enhancement. In an extreme case (fish bodies of very small width and very large depth) when on elongated-body theory the momentum enhancement factor β is predicted to become very large, short-wavelength considerations are shown to place a specific upper limit on β (see figure 2). In more general cases, this upper limit should perhaps be regarded as coexisting with other upper limits associated with either nonzero width or finite depth of fish body.Short-wavelength limitations on momentum enhancement are of some biological interest as implying the existence not only of advantages (including a reduction in body drag) but also of some competing disadvantages (limitations in propulsive force) arising from progressive reductions in the wavelength of fin undulations.