  Recognising that surfactants can impede the drag reduction resulting from
superhydrophobic surfaces (SHSs), we investigate the impact of spatio-temporal
fluctuations in surfactant concentration on the drag-reduction properties of
SHSs. We model the unsteady transport of soluble surfactant in a channel flow
bounded by two SHSs. The flow is laminar, pressure-driven, and the SHSs are
periodic in the streamwise and spanwise directions. We assume that the channel
length is much longer than the streamwise period, the streamwise period is much
longer than the channel height and spanwise period, and 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-flux transport over the length of
the channel, which is coupled to a quasi-steady advection-diffusion equation
for surfactant transport over individual plastrons. As diffusion over the
length of the channel is typically small, the surfactant flux is governed by a
nonlinear wave equation. In the fundamental case of the transport of a bolus of
surfactant, we predict its propagation speed and describe its nonlinear
evolution via interaction with the SHS. The propagation speed can fall below
the average streamwise velocity as the surfactant adsorbs and rigidifies the
plastrons. Smaller concentrations of surfactant are advected faster than larger
ones, so that wave-steepening effects can lead to shock formation in the
surfactant-flux distribution. Our asymptotic results reveal how unsteady
surfactant transport can affect the spatio-temporal evolution of the slip
velocity, drag reduction and effective slip length in SHS channels.
