The steady oblique path of buoyancy-driven disks and spheres

Abstract

We consider the steady motion of disks of various thicknesses in a weakly viscous flow, in the case where the angle of incidence $\alpha$ (defined as that between the disk axis and its velocity) is small. We derive the structure of the steady flow past the body and the associated hydrodynamic force and torque through a weakly nonlinear expansion of the flow with respect to $\alpha$. When buoyancy drives the body motion, we obtain a solution corresponding to an oblique path with a non-zero incidence by requiring the torque to vanish and the hydrodynamic and net buoyancy forces to balance each other. This oblique solution is shown to arise through a bifurcation at a critical Reynolds number ReSO which does not depend upon the body-to-fluid density ratio and is distinct from the critical Reynolds number ReSS corresponding to the steady bifurcation of the flow past the body held fixed with $\alpha=0$. We then apply the same approach to the related problem of a sphere that weakly rotates about an axis perpendicular to its path and show that an oblique path sets in at a critical Reynolds number ReSO slightly lower than ReSS , in agreement with available numerical studies.