Longitudinal thermocapillary slip about a dilute periodic mattress of protruding bubbles

A common realization of superhydrophobic surfaces comprises of a periodic array of cylindrical bubbles which are trapped in a periodically grooved solid substrate. We consider the thermocapillary animation of liquid motion by a macroscopic temperature gradient which is longitudinally applied over such a bubble mattress. Assuming a linear variation of the interfacial tension with the temperature, at slope \sigma T, we seek the effective velocity slip attained by the liquid at large distances away from the mattress. We focus upon the dilute limit, where the groove width 2c is small compared with the array period 2l. The requisite velocity slip in the applied-gradient direction, determined by a local analysis about a single bubble, is provided by the approximation \begin{align} \pi \frac{G\sigmaT c^2}{\mu l} I(\alpha), \end{align∗}wherein G is the applied-gradient magnitude, \mu is the liquid viscosity and I(\alpha) , a non-monotonic function of the protrusion angle \alpha , is provided by the quadrature, \begin{align} I(\alpha) = \frac{2}{\sin\alpha} \int 0^\infty\frac{\sinh s\alpha}{ \cosh s(\pi-\alpha) \sinh s \pi} \, \textrm{d} s.