  Guided by computation, we theoretically calculate the steady flow driven by
Marangoni stress due to surfactant introduced on a fluid interface at a
constant rate. Two separate extreme cases, where the surfactant dynamics is
dominated by the adsorbed phase or the dissolved phase are considered. We focus
on the case where the size of the surfactant source is much smaller than the
size of the fluid domain, and the resulting Marangoni stress overwhelms viscous
forces so that the flow is strongest in a boundary layer close to the
interface. We derive the resulting flow in a region much larger than the
surfactant source but smaller than the domain size is described by
approximating it with a self-similar profile. The radially outward component of
fluid velocity decays with the radial distance $r$ as $r^{-3/5}$ when the
surfactant spreads in an adsorbed phase, and as $r^{-1}$ when it spreads in a
dissolved phase. Universal flow profiles that are independent of the system
parameters emerge in both the cases. Three hydrodynamic signatures are
identified to distinguish between the two cases and verify the applicability of
our analysis with successively stringent tests.
