  Laminar flow over a bubble mattress is expected to experience a significant
reduction in friction since the individual surfaces of the bubbles are
shear-free. However, if the bubbles are sufficiently curved, their protrusion
into the fluid and along the flow direction can lead to an increase in friction
as was recently demonstrated experimentally and computationally. We provide in
this paper a simple model for this result. We consider a shear flow at low
Reynolds number past a two-dimensional array of bubbles, and calculate
analytically the effective slip length of the surface as function of the bubble
geometry in the dilute limit. Our model is able to reproduce quantitatively the
relationship between effective friction and bubble geometry obtained in
numerical computations, and in particular: (a) The asymmetry in friction
between convex and concave bubbles, and (b) the existence of a geometric
transition from reduced to enhanced friction at a critical bubble protrusion
angle.
