In this thesis, we present the equations governing the conservation of momentum and mass of an incompressible mixture of two fluids with different viscosities and densities contained in a vertical slab, then introduce a numerical method for approximating the velocity and pressure of said mixture, in which the velocity and pressure is updated in time by first computing a tentative update to the velocity based on a finite difference using the previous velocity and pressure, then computing a correctional pressure term used to both correct the tentative velocity so that it is divergence-free as well as update the pressure field.

Using key concepts from the theory of functional analysis, we show first the existence and uniqueness of the variational problems involved with the previously introduced methods, then the theoretic convergence of an appropriate finite volume scheme.

Lastly, we use a software implementation of the described methods and schemes to perform numerical investigations on the evolution of such a mixture. In particular, we investigate the possibility of said mixture reaching an equilibrium in which the two fluids are completely separated, with one fluid lying entirely on top of the other.