Transient and persistent Marangoni–Bénard convection in the presence of surfactants

A liquid film subject to a transverse temperature gradient, with cool fluid above and hot fluid below, can erupt into a periodic flow caused by the Marangoni–Bénard instability, a thermocapillary flow driven by the decrease in surface tension with temperature. A unit cell in this periodic flow has an upwelling of hot fluid at its center, and downward flow at the cool edges. There is renewed interest in this flow field, which has been adopted as a means to create periodic structures from particles suspended within evaporating films. Surfactants can alter the onset conditions for the flow through either surfactant-related Marangoni stresses or through increased thermocapillary coupling. We study the effects of an insoluble surfactant on the transients and steady behavior of this flow in a two-dimensional simulation of the non-linearly coupled heat, surfactant transport, and momentum equations in a unit cell of the flow field for the circumstance in which convective effects are strong compared to surface diffusion. This is the usual case in experiment, which cannot be accessed by the perturbative schemes typically adopted in stability analyses. Typically, surface tension reduces with surfactant concentration. For this case, at low surface concentration, surfactant is swept to the outer edges of the unit cell, and the interface can be divided into a surfactant-covered region with zero velocity, and a surfactant-free region of width λfree. Flow persists in the surfactant free region if λfree corresponds to a linearly unstable wavelength. That is, remarkably, the steady response of the non-linear system can be related to classical marginal stability results from a linear stability analysis, provided the wavelength is replaced by λfree. Insoluble surfactants in coexisting liquid expanded and liquid condensed surface states can promote thermocapillary flows, as surface tension is highly coupled to temperature, and decoupled from concentration in two phase coexistence. For such surfactants, linearly unstable conditions are readily established. The flow will rapidly occur and will subsequently self-quench. Surfactants will re-spread, setting conditions for auto-oscillatory behavior.