  In this paper, the instability of layered two-phase flows caused by the
presence of a soluble surfactant (or a surface active solute) is studied. The
fluids have different viscosities, but are density matched to focus on
Marangoni effects. The fluids flow between two flat plates, which are
maintained at different solute concentrations. This establishes a constant flux
of solute from one fluid to the other in the base state. A linear stability
analysis is performed, using a combination of asymptotic and numerical methods.
In the creeping flow regime, Marangoni stresses destabilize the flow, provided
a concentration gradient is maintained across the fluids. One long wave and two
short wave Marangoni instability modes arise, in different regions of parameter
space. A well-defined condition for the long wave instability is determined in
terms of the viscosity and thickness ratios of the fluids, and the direction of
mass transfer. Energy budget calculations show that the Marangoni stresses that
drive long and short wave instabilities have distinct origins. The former is
caused by interface deformation while the latter is associated with convection
by the disturbance flow. Consequently, even when the interface is non-deforming
(in the large interfacial tension limit), the flow can become unstable to short
wave disturbances. On increasing $Re$, the viscosity-induced interfacial
instability comes into play. This mode is shown to either suppress or enhance
the Marangoni instability, depending on the viscosity and thickness ratios.
This analysis is relevant to applications such as solvent extraction in
microchannels, in which a surface-active solute is transferred between fluids
in parallel stratified flow. It is also applicable to the thermocapillary
problem of layered flow between heated plates.
