  When surface-active molecules are released at a liquid interface, their
spreading dynamics is controlled by Marangoni flows. Though such Marangoni
spreading was investigated in different limits, exact solutions remain very
few. Here we consider the spreading of an insoluble surfactant along the
interface of a deep fluid layer. For two-dimensional Stokes flows, it was
recently shown that the non-linear transport problem can be exactly mapped to a
complex Burgers equation [Crowdy, SIAM J. Appl. Math. 81, 2526 (2021)]. We
first present a very simple derivation of this equation. We then provide fully
explicit solutions and find that varying the initial surfactant distribution -
pulse, hole, or periodic - results in distinct spreading behaviors. By
obtaining the fundamental solution, we also discuss the influence of surface
diffusion. We identify situations where spreading can be described as an
effective diffusion process but observe that this approximation is not
generally valid. Finally, the case of a three-dimensional flow with axial
symmetry is briefly considered. Our findings should provide reference solutions
for Marangoni spreading, that may be tested experimentally with fluorescent or
photoswitchable surfactants.
