Effective longitudinal slip over grooves encapsulated by a nearly inviscid lubricant

We calculate the effective slip length for a rectangularly grooved periodic surface encapsulated (i.e., fully wetted) by a lubricant fluid and subjected to exterior shear flow parallel to the grooves. Our focus is the limit of a nearly-inviscid lubricant, where the ratio $\mu$ of the lubricant viscosity to that of the exterior fluid is small. This limit is singular for an encapsulated surface, indicating a dominant lubricant-flow effect - a stark contrast to superhydrophobic surfaces where the role of the lubricant is typically negligible.