Slip length for longitudinal shear flow over a dilute periodic mattress of protruding bubbles

An analytical formula for the frictional slip length associated with transverse shear flow over a bubble mattress comprising a dilute periodic array of parallel circular-arc grooves protruding into the fluid has recently been presented by Davis and Lauga [Phys. Fluids 21, 011701 (2009)]. This letter derives an analytical formula for the slip length associated with longitudinal shear flow over the same surface. The formula is in excellent agreement with a phenomenological result based on finite element simulations given by Teo and Khoo [Microfluid. Nanofluid. 9, 499 (2010)].