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An analytical solution for the flow field of a shear flow over a rectangularcavity containing a second immiscible fluid is derived. While flow of asingle-phase fluid over a cavity is a standard case investigated in fluiddynamics, flow over a cavity which is filled with a second immiscible fluid,has received little attention. The flow filed inside the cavity is consideredto define a boundary condition for the outer flow which takes the form of aNavier slip condition with locally varying slip length. The slip-lengthfunction is determined from the related problem of lid-driven cavity flow.Based on the Stokes equations and complex analysis it is then possible toderive a closed analytical expression for the flow field over the cavity forboth the transversal and the longitudinal case. The result is a comparativelysimple function, which displays the dependence of the flow field on the cavitygeometry and the medium filling the cavity. The analytically computed flowfield agrees well with results obtained from a numerical solution of theNavier-Stokes equations. The studies presented in this article are ofconsiderable practical relevance, for example for the flow oversuperhydrophobic surfaces.

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