Abstract

One of the most-studied unsteady locomotor behaviors exhibited by fishes is the c-start escape response. Although the kinematics of these responses have been studied extensively and two well-defined kinematic stages have been documented, only a few studies have focused on hydrodynamic patterns generated by fishes executing escape behaviors. Previous work has shown that escape responses by bluegill sunfish generate three distinct vortex rings, each with central orthogonal jet flows, and here we extend this conclusion to two other species: stickleback and mosquitofish. Jet #1 is formed by the tail during Stage 1, and moves in the same direction as Stage-2 movement of the fish, thereby reducing final escape-velocity but also rotating the fish. Jet #2, in contrast, moves approximately opposite to the final direction of the fish's motion and contains the bulk of the total fluid-momentum powering the escape response. Jet #3 forms during Stage 2 in the mid-body region and moves in a direction approximately perpendicular to jets 1 and 2, across the direction of movement of the body. In this study, we used a mechanical controller to impulsively move passively flexible plastic panels of three different stiffnesses in heave, pitch, and heave + pitch motions to study the effects of stiffness on unsteady hydrodynamics of escape. We were able to produce kinematics very similar to those of fish c-starts and also to reproduce the 3-jet hydrodynamic pattern of the c-start using a panel of medium flexural stiffness and the combined heave + pitch motion. This medium-stiffness panel matched the measured stiffness of the near-tail region of fish bodies. This motion also produced positive power when the panel straightened during stage 2 of the escape response. More flexible and stiffer panels resulted in non-biological kinematics and patterns of flow for all motions. The use of simple flexible models with a mechanical controller and program of fish-like motion is a promising approach for studying unsteady behaviors of fish which can be difficult to manipulate experimentally in live animals.

Free electrons have a uniform kinetic energy density (KED), which evolves into a spatially varying quantity as the electrons respond to the gradual imposition of an external potential. In this paper and a companion paper, we examine two sets of functionals for describing the local, non-negative KED that emerges after such a perturbation. In this paper, we emphasize potential functionals, deriving the first-and secondorder deviations from the free-electron KED as functionals of the perturbing potential, also reconsidering the analogous functionals for the local density of states and the electron density. (In the second paper, we use these results to re-express the KED response in terms of functionals of the induced electron density.) We develop reciprocal-space formulations of the response kernels to complement previously known real-space forms. The first-order function is straightforward to obtain, but the second-order function requires considerable effort. To manage the derivations, we relate the KED response to that of the one-electron Green function, and then examine the latter in detail. Finally, we provide extensive validation of the derived response functions based on asymptotic analysis of an integral representation, numerical integration of the same generating integral, and application to the linear potential model.