  In this paper we consider a slab of viscous incompressible fluid bounded
above by a free boundary, bounded below by a flat rigid interface, and acted on
by gravity. The unique equilibrium is a flat slab of quiescent fluid. It is
well-known that equilibria are asymptotically stable but that the rate of decay
to equilibrium depends heavily on whether or not surface tension forces are
accounted for at the free interface. The aim of the paper is to better
understand the decay rate by studying a generalization of the linearized
dynamics in which the surface tension operator is replaced by a more general
fractional-order differential operator, which allows us to continuously
interpolate between the case without surface tension and the case with surface
tension. We study the decay of the linearized problem in terms of the choice of
the generalized operator and in terms of the horizontal cross-section. In the
case of a periodic cross-section we identify a critical order of the
differential operator at which the decay rate transitions from almost
exponential to exponential.

Analysis of PDEs