Self-similar regimes in the viscous Marangoni spreading of surfactant on a deep subphase

The spreading of an insoluble surfactant on a fluid interface is a fundamental fluid mechanics problem that has been investigated extensively due to its implications for flows in biology, the environment, and in technological applications. While many different regimes have been studied theoretically, analytical progress has proven challenging in the limit of a deep subphase at low Reynolds and high Péclet numbers, due to the non-local nature of the coupling between the interfacial velocity field and the concentration of surfactant [Thess, Phys. Rev. Lett. (1995)]. Recently, the problem was shown to be equivalent to the complex Burgers equation [Crowdy, SIAM J. Appl. Math. (2021)], paving the way to exact solutions obtained through the method of characteristics [Bickel, Phys. Rev. E (2022)]. Here, we study the self-similarity of the problem, providing further insights into its structure. First, a phase-plane formalism is used to identify different self-similar regimes. In the case of a pulse of outward-spreading surfactant, we find a globally valid solution exhibiting self-similarity of the first kind. On the other hand, surfactant-free holes collapsing inward exhibit self-similarity of the second kind, which only holds locally. We find that the exponents in the power-law behavior of these hole solutions can be obtained exactly using stability arguments, and distinguish two different power-law exponents that hold depending on the nature of the initial condition. We also discuss the importance of higher-order effects like surface diffusion and endogenous surfactant on these idealized solutions.