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The steady oblique path of buoyancy-driven disks and spheres. (English) Zbl 1275.76064

Summary: 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.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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[1] DOI: 10.1016/j.jfluidstructs.2011.03.013 · doi:10.1016/j.jfluidstructs.2011.03.013
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