Growth rates for the linearized motion of fluid interfaces away from equilibrium

Abstract We consider the motion of a two‐dimensional interface separating an inviscid, incompressible, irrotational fluid, influenced by gravity, from a region of zero density. We show that under certain conditions the equations of motion, linearized about a presumed time‐dependent solution, are wellposed; that is, linear disturbances have a bounded rate of growth. If surface tension is neglected, the linear equations are well‐posed provided the underlying exact motion satisfies a condition on the acceleration of the interface relative to gravity, similar to the criterion formulated by G. I. Taylor. If surface tension is included, the linear equations are well‐posed without qualifications, whether the fluid is above or below the interface. An interesting qualitative structure is found for the linear equations. A Lagrangian approach is used, like that of numerical work such as 3, except that the interface is assumed horizontal at infinity. Certain integral equations which occur, involving double layer potentials, are shown to be solvable in the present case. © 1993 John Wiley & Sons, Inc.