The effect of surface tension on the motion of an isolated fluid body

Abstract

Consider a body made f rom a uniform, incompressible, linear viscous fluid. The body may be deformed in an arbitrary way, by the application of tractions at its boundary and by the application of body forces, up until some instant of time, but thereafter we suppose it to be isolated in the sense that the applied tractions and body forces are removed and cease to act at all subsequent times. Our aim is to investigate the motion of the body after isolation, or, more specifically, to investigate the effect of the presence or absence of surface tension on the motion after isolation. To this end we prove two results. We prove first of all that if there is no surface tension and if the angular momentum, which must be constant after isolation, does not vanish then either the diameter of the body grows unboundedly or the mean pressure ultimately becomes negative. Since the action of surface tension is to oppose the growth of the surface area of the body we might suspect that this result cannot be generally true when the body has non-vanishing surface tension, at least for values of the angular momentum which are not too large. We shall establish that there is substance to this suspicion by showing, and this is our second result, that if the angular momentum is not too large, in a sense which will be made precise, then there are isolated motions of the body in which the diameter remains constant and the pressure is always positive. The mechanical properties of the body are determined by the constant values M, V,/~, a of its mass, its volume, its viscosity and its surface tension, respectively. I t will be assumed that, in the motions with which we are concerned, the region B(t) , occupied by the body at the time t, is a compact three-dimensional manifoldwith-boundary, whose boundary aB(t) is a class C 2 two-dimensional manifold with unit outward normal n(x, t) at xeaB(t). Notwithstanding our use of the symbol V for the constant total volume, we will use it also to denote the volume measure on B(t). The area measure on aB(t) is denoted by A and A(t) is the area of aB(t). We need certain vector identities (2), (3) and (4) involving the normal n; before we can formulate them we must extend n (., t) to a class C 1 unit vector field defined on a neighbourhood of the boundary. To do this let e > 0 be a positive number and let U~(t) be the open neighbourhood