Biofluiddynamics of balistiform and gymnotiform locomotion. Part 2. The pressure distribution arising in two-dimensional irrotational flow from a general symmetrical motion of a flexible flat plate normal to itself

Abstract

When, for an otherwise unbounded fluid, the unique irrotational flow compatible with the instantaneous motion of an immersed body has been calculated, it is straightforward to deduce the pressure field from the unsteady form of Bernoulli's equation if the body is rigid. On the other hand, if the body is flexible, a somewhat subtle analysis is required to determine the time derivative of velocity potential, ∂ϕ/∂t, which occurs in that equation. This is because no simple relationship exists between the instantaneous form of ϕ and its form at a nearby instant.

In the case of two-dimensional flow, however, the two forms of ϕ for a flexible body may be related, not in general by a simple translational and/or rotational mapping as for a rigid-body motion, but by a conformal mapping. The example of a flexible flat plate is used here to illustrate this approach to calculating the pressure field.

In the analysis of balistiform motion by elongated-body theory (Lighthill & Blake 1990), one part of the propulsive force on the fish has magnitude equal to P, the area integral of the pressure field just described. This area integral is shown in §3 below to take a simple form $\overline{U}M$− E in terms of the flow's momentum M and kinetic energy E per unit length and a certain weighted average $\overline{U}$ of the plate's velocity normal to itself. Although, in the case of motile fins attached to a rigid body of much greater depth, M was found (Lighthill & Blake 1990) to take an enhanced value, no such enhancement is found either for the product $\overline{U}M$ or for E, so that P itself is also not enhanced. For the relevance of these findings to the efficiency of balistiform motion, see Lighthill & Blake (1990).