Impact of insoluble surfactants on the flow of a liquid over a surface containing gas-filled grooves perpendicular to the flow direction

A gas-liquid interface can become immobilized when surfactants gather upstream of an obstacle, with Marangoni stress balancing viscous shear, effectively rendering the surface rigid. This effect, well-known for influencing the rate of ascend of small air bubbles in a liquid, has also been proposed to crucially influence the flows near partially wetted superhydrophobic surfaces in Cassie state. To investigate this, we analytically and numerically study laminar shear flow of an incompressible liquid over a surface with periodic gas inclusions and insoluble surfactants attached to the gas-liquid interface. Assuming advection-dominated transport of the surfactants along the interface, a semi-analytical expression for the fraction of the gas-liquid interface immobilized by surfactants is found. The resulting effective slip on the surface is compared with the ideal situation of flow across a surface not contaminated by surfactants.