Existence and stability of equilibria of free-falling plates

Steady gliding and diving motions in animals such as birds may be loosely modeled by considering flat plates falling freely through fluid. We investigate equilibrium solutions of a quasi-steady two-dimensional nonlinear model of thin rectangular plates subject to gravitational and fluidic forces at intermediate Reynolds numbers. We begin by proving the existence and uniqueness of such equilibrium states for a given set of fixed dimensionless geometric parameters. We then examine a broad range of these equilibria through linear stability analysis, and present phase diagrams showing a highly complex structure of stable and unstable regions including multiple Hopf bifurcation boundaries. We then verify these findings via the full nonlinear model. We identify a new flight mode in addition to those already studied in previous literature, and demonstrate its existence experimentally. Finally, we propose a necessary but insufficient condition for divergent stability based on the sign of the derivative of the aerodynamic center of pressure, as well as several other factors that substantially contribute to the stability. These results are highly generalized to flat plates with arbitrary geometry and density, as well as to fluids of arbitrary densities. We also highlight some connections to real examples of steady animal flight.