Unsteady evolution of slip and drag in surfactant-contaminated superhydrophobic channels

We examine the unsteady transport of soluble surfactant in a laminar channel flow bounded between two superhydrophobic surfaces (SHSs) that are periodic in the streamwise and spanwise directions. We assume that the channel length is much larger than the streamwise period, the streamwise period is much larger than the channel height and spanwise period, and that bulk diffusion is sufficiently strong for cross-channel concentration gradients to be small. By combining long-wave and homogenisation theories, we derive an unsteady advection-diffusion equation for surfactant transport over long lengthscales and slow timescales, which is coupled to a quasi-steady advection-diffusion equation for surfactant transport over each plastron. We predict the propagation speed of disturbances to the surfactant concentration field and describe the nonlinear evolution of disturbances via interaction with the SHS. The propagation speed of disturbances falls below the average streamwise velocity as the surfactant becomes less soluble, via coupling between bulk and interfacial transport. We show that wave-steepening effects can lead to shock formation in the surfactant flux distribution. These findings reveal the spatio-temporal evolution of the slip velocity and predict the drag reduction for microchannel applications.