On the free surface of a viscous fluid motion

Abstract

We consider a container of fluid with a rod inserted in the centre. As the rod rotates the surface of the fluid forms a curved surface whose shape correctly balances the forces of gravity, internal stress, atmospheric pressure, and surface tension. The surface of the fluid is depressed in the neighbourhood of the rod in the case of a Newtonian fluid, but the fluid may climb along the rod in the case of non-Newtonian fluids. In recent work by D. D. Joseph and R. S. Fosdick this free surface problem has been treated quantitatively by virtue of a formal perturbation series in which the solution is developed in powers of the angular velocity of the rod. The purpose of the present paper is to give a rigorous proof of convergence in the case of a Newtonian fluid. The method here may possibly be applied to other free surface problems - for example, surface waves of a viscous fluid. The convergence proof provides, to our knowledge, the first rigorous existence theorem for a free surface problem in the theory of viscous fluids. The proof also raises a novel problem in the theory of elliptic systems of partial differential equations. By means of the implicit function theorem, the question of convergence is reduced to that of obtaining a priori estimates for an elliptic boundary value problem. That problem is formulated on a domain with a ridge where the fluid surface meets the rod. In addition, the type of boundary conditions prescribed differ on the free surface and on the rod. In general, the solution to such a mixed boundary value problem is not smooth at the ridge, even if the boundary data is smooth. The solution will be smooth, however, if the boundary data satisfy certain consistency conditions which make it compatible with the given set of partial differential equations. The consistency conditions in question here are a set of linear relations between various derivatives of the boundary and inhomogeneous data. It is shown that if one makes the assumption that the wetting angle of the surface is zero, the consistency conditions are invariant under the full nonlinear equations of the free surface problem. This makes it possible to consider the original problem on a smaller function space - namely the subclass of functions satisfying the appropriate consistency conditions - and in this subclass one can apply the implicit function theorem and obtain the required a priori estimates. The solution thus obtained is regular up to the ridge.