  Surfactants can have a detrimental effect on the drag reduction in shear flow
over superhydrophobic surfaces in Cassie state. While surfactant-free
gas-liquid interfaces are often well approximated as shear-free, surfactants
can impede the flow by stacking up in front of obstacles. We study shear-flow
along an array of narrow gas-filled grooves of finite length embedded in an
otherwise planar surface, with the gas-liquid interface protruding slightly
above or below the plane. Assuming immiscible surfactants forming an
incompressible, inviscid surfactant phase at the gas-liquid interfaces we
employ a recently proposed model [Baier and Hardt, J.Fluid Mech., 949 (2022)]
for addressing this situation. Using a domain perturbation technique together
with the Lorentz reciprocal theorem we obtain the slip length characterizing
the flow over such surfaces to second order in the maximal interface deflection
as a small parameter. We find that within the range of moderate interface
deflections studied, the slip length for flow over such surfaces is negative
(positive) for surfaces protruding above (below) the surface and is much
smaller than for flow over a corresponding surfactant-free interface. Thus,
contrary to expectations of reduced drag in flow over superhydrophobic surfaces
in Cassie state, surfactant covered interfaces can even be detrimental for drag
reduction in the limit where surfactants act as an effectively incompressible
surface-fluid. This has important implications for the appropriate design of
superhydrophobic surfaces for reducing flow resistance.
