Modeling a swimming fish with an initial boundary value problem: Unsteady maneuvers of an elastic plate with internal force generation

In order to model unsteady maneuvers in swimming fish, we develop an initial-boundary value problem for a fourth-order hyperbolic partial differential equation in which the fishʹs body is treated as an inhomogeneous elastic plate. The model is derived from the three-dimensional equations of elastic dynamics, and is essentially a simple variant of the classical Kirchhoff model for a dynamic plate. The model incorporates body forces generating moment to simulate muscle force generation in fish. The initial-boundary value problem is reduced to a beam model in one spatial dimension and formulated computationally using finite differences. Interaction with the surrounding water is represented by nonlinear viscous damping. Two example applications using simple but physically reasonable physiological parameters are presented and interpreted. One models the acceleration from rest to steady swimming, the other a rapid turn from rest.