Spreading of a diffusive soluble surfactant in a deep layer with inertial effects

In this study, we conduct a numerical analysis of the spreading dynamics of a soluble and diffusive surfactant in a deep layer of Newtonian fluid. We investigate three different initial conditions for the surfactant distributions: a Gaussian pulse, a Cauchy hole, and a periodic distribution. We solved the momentum equations in the Stokes limit (Re→0) and also considering inertial effects (Re≫1), where Re denotes the Reynolds number. Furthermore, we include the convective-diffusion equation for both nondiffusive (Pe_{s} and Pe→∞) and diffusive (finite Pe_{s} and Pe) surfactants, with Pe_{s} and Pe representing surface and bulk Péclet numbers, respectively. The governing equations are coupled through a tangential stress balance at the interface, where a nonlinear Langmuir equation of state relates interfacial surface tension to surfactant concentration. The temporal evolution of the initial surfactant distribution is primarily influenced by several dimensionless parameters: the bulk and surface Péclet numbers, the Biot number (Bi), the solubility parameter (β), and the dimensionless surfactant depletion depth (α). Our findings indicate that the initial surfactant concentration is diminished due to diffusivity, and its decay toward a homogeneous state is significantly affected by solubility. The adsorption and desorption processes operate as an equilibrium mechanism that tends to homogenize the surface surfactant concentration.