We now take up the mechanical principles underlying Eulerian swimming of a thin, flexible creature, with the aim of understanding morphology in terms of the mechanisms of propulsion. For summaries of the related biology see Lighthill (1975, chap. 2) and Wu (1971a).

Small-perturbation theory of slender fish

Many fish change shape rather gradually along the anterior–posterior axis. (There are, however, many exceptions: for example, angelfish.) As a first approximation, we consider the swimming of slender, neutrally bouyant fish, By slender we mean, among other things, that the cross-sectional area of the body changes slowly along its length. Notation and terminology are summarized in Figure 10.1.

Necessary conditions for the validity of slender-body theory are such as to ensure that velocity perturbations caused by the fish are a small fraction of its swimming speed. It is certainly sufficient that the body surface S be smooth and that tangent planes always make a small angle with the x axis. However, it turns out that if these conditions are exactly met and the cross-sectional area is zero at the extremities, the theory predicts zero mean thrust. Fortunately, the model also allows us to treat “slender” fish with sharp downstream edges, at the caudal fin, for example, even though s(x) may be discontinuous there. Also, it is not necessary that surface slope always be small, at the nose of the fish, for example, provided that this occurs over at most a fraction of its length.