  This paper concerns the dynamics of a layer of incompressible viscous fluid
lying above a rigid plane and with an upper boundary given by a free surface.
The fluid is subject to a constant external force with a horizontal component,
which arises in modeling the motion of such a fluid down an inclined plane,
after a coordinate change. We consider the problem both with and without
surface tension for horizontally periodic flows. This problem gives rise to
shear-flow equilibrium solutions, and the main thrust of this paper is to study
the asymptotic stability of the equilibria in certain parameter regimes. We
prove that there exists a parameter regime in which sufficiently small
perturbations of the equilibrium at time $t=0$ give rise to global-in-time
solutions that return to equilibrium exponentially in the case with surface
tension and almost exponentially in the case without surface tension. We also
establish a vanishing surface tension limit, which connects the solutions with
and without surface tension.
