  We study shear-driven liquid flow over a planar surface with an embedded
gas-filled groove, with the gas-liquid interface protruding slightly above or
below the planar surface. The flow direction is along the groove, taken to be
much longer than wide, and the gas-liquid interface is assumed to be covered by
an incompressible surface fluid, representing a surfactant phase. Using the
incompressiblity condition for the surface fluid, the equations of motion and
corresponding boundary conditions for the liquid phase are obtained by
minimizing the dissipation rate. Assuming a moderate deformation of the
interface, a domain perturbation technique with the maximal deformation as the
small parameter is employed. The Stokes equation in the liquid phase under
corresponding boundary conditions is solved to second order in the deformation
using the Keldysh-Sedov formalism. The obtained analytical results are compared
with numerical calculations of the same problem, allowing an assessment of the
limits of validity of the expansion. While on a planar gas-liquid interface no
flow is induced, a recirculating flow is observed on an interface protruding
slightly above or below the planar surface. The study sheds light onto the
mobility of curved gas-liquid interfaces in the presence of surfactants acting
as an incompressible surface fluid.
