  We have developed a theoretical analysis to systematically study the
late-time evolution of the Rayleigh-Taylor instability in a finite-sized
spatial domain. The nonlinear dynamics of fluids with similar and contrasting
densities are considered for two-dimensional flows driven by sustained
acceleration. The flows are periodic in the plane normal to the direction of
acceleration and have no external mass sources. Group theory analysis is
applied to accurately account for the mode coupling. Asymptotic nonlinear
solutions are found to describe the inter-facial dynamics far from and near the
boundaries. The influence of the size of the domain on the diagnostic
parameters of the flow is identified. In particular, it is shown that in a
finite-sized domain the flow is slower compared to the spatially extended case.
The direct link between the multiplicity of solutions and the inter-facial
shear function is explored. It is suggested that the inter-facial shear
function acts as a natural parameter to the family of analytic solutions.
