Instantaneous Stokes flow in a conical apex aligned with gravity and bounded by a stress-free surface

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 degrees 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 degrees 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 apex to obtain insight into how the morphology of the cone tip may evolve. It is found that when 2 theta(0) < 180 degrees the included angle of the cone is maintained, but when 2 theta(0) > 180 degrees the corner becomes either rounded or cusped, depending on the flow in the far field. Our results are compared with those obtained by Betelu et al. [Phys. Fluids, 8 (1996), pp. 2269-2274] for the analogous problem of two-dimensional Stokes flow in a wedge with stress-free surfaces under the action of gravity parallel to the bisecting plane.