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Analytical expressions for the flow field as well as for the effective sliplength of a shear flow over a surface with periodic rectangular grooves arederived. The primary fluid is in the Cassie state with the grooves being filledwith a secondary immiscible fluid. The coupling of both fluids is reflected ina locally varying slip distribution along the fluid-fluid interface, whichmodels the effect of the secondary fluid on the outer flow. The obtainedclosed-form analytical expressions for the flow field and effective slip lengthof the primary fluid explicitly contain the influence of the viscosities of thetwo fluids as well as the magnitude of the local slip, which is a function ofthe surface geometry. They agree well with results from numerical computationsof the full geometry. The analytical expressions allow investigating theinfluence of the viscous stresses inside the secondary fluid for arbitrarygeometries of the rectangular grooves. For classic superhydrophobic surfaces,the deviations in the effective slip length compared to the case of inviscidgas flow are are pointed out. Another important finding with respect to anaccurate modeling of flow over microstructured surfaces is that the local sliplength of a grooved surface is anisotropic.

Influence of the enclosed fluid on the flow over a microstructured surface in the Cassie state