Slip length for longitudinal shear flow over an arbitrary-protrusion-angle bubble mattress: the small-solid-fraction singularity

We study the effective slip length for unidirectional flow over a superhydrophobic mattress of bubbles in the small-solid-fraction limit is an element of << 1. Using scaling arguments and utilising an ideal-flow analogy we elucidate the singularity of the slip length as is an element of -> 0: relative to the periodicity it scales as log(1/is an element of) for protrusion angles 0 <= alpha < pi/2 and as is an element of(-1/2) for 0 < pi/2 - alpha = O(is an element of(1/2))We continue with a detailed asymptotic analysis using the method of matched asymptotic expansions, where 'inner' solutions valid close to the solid segments are matched with 'outer' solutions valid on the scale of the periodicity, where the bubbles protruding from the solid grooves appear to touch. The analysis yields asymptotic expansions for the effective slip length in each of the protrusion-angle regimes. These expansions overlap for intermediate protrusion angles, which allows us to form a uniformly valid approximation for arbitrary protrusion angles 0 <= alpha <= pi/2. We thereby explicitly describe the transition with increasing protrusion angle from a logarithmic to an algebraic small-solid-fraction slip-length singularity.