Longitudinal and transverse flow over a cavity containing a second immiscible fluid

An analytical solution for the low-Reynolds-number flow field of a shear flow over a rectangular cavity containing a second immiscible fluid is derived. While flow of a single-phase fluid over a cavity is a standard case investigated in fluid dynamics, flow over a cavity that is filled with a second immiscible fluid has received little attention. The flow field inside the cavity is considered to define a boundary condition for the outer flow, which takes the form of a Navier slip condition with locally varying slip length. The slip-length function is determined heuristically from the related problem of lid-driven cavity flow. Based on the Stokes equations and complex analysis, it is then possible to derive a closed analytical expression for the flow field over the cavity for both the transverse and the longitudinal case. The result is a comparatively simple function, which displays the dependence of the flow field on the cavity geometry and the medium filling the cavity. The analytically computed expression agrees well with results obtained from a numerical solution of the Navier-Stokes equations.