The steady oblique path of buoyancy-driven disks and spheres

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 Re-SO which does not depend upon the body-to-fluid density ratio and is distinct from the critical Reynolds number Re-SS 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 Re-SO slightly lower than Re-SS, in agreement with available numerical studies.