Instantaneous Stokes Flow in a Conical Apex Aligned with Gravity and Bounded by a Stress-Free Surface

Abstract

The instantaneous viscous motion near the apex of a fluid cone of included angle $2\theta_0$ with stress-free boundary conditions is studied in the limit of zero Reynolds number. Gravity acts parallel to the conical axis, and surface tension is neglected. When $2\theta_0 > 134.6^{\circ}$ the dominant term of the solution close to the apex is independent of gravity and depends only on the far-field boundary conditions; the leading behavior is thus a self-similar solution of the second kind. In this case the instantaneous flow is akin to the steady flow in a rigid cone reported by Liu and Joseph [SIAM J. Appl. Math., 34 (1978), pp. 286--296] who showed that toroidal eddies appear below a critical apex angle. For $2\theta_0 < 134.6^{\circ}$ the flow close to the apex is dominated by gravity and represents a self-similar solution of the first kind. The complete eigenvalue problem is solved and example streamline patterns are presented. We estimate the short-time behavior of flow in the neighborhood of the ape...