Oscillatory Marangoni flow in a deep layer of a Carreau fluid

We study the spreading dynamics of an insoluble and non-diffusive surfactant on the free surface of a deep layer of a Carreau fluid. When a non-uniform distribution of surfactant is imposed on the free surface of an initially motionless fluid, variations in surface tension are induced, causing the fluid motion from regions of lower surface tension to those of higher tension. Such movement redistributes the surfactant concentration until a uniform condition is reached. The two-dimensional momentum and convection-diffusion equations are employed to determine the hydrodynamics in the deep fluid layer and the evolution of the surfactant on the fluid surface. The modified vorticity-stream function formulation is used to solve numerically the hydrodynamic field, where the non-Newtonian dependent variables of the problem are decomposed into a Newtonian part and a non-Newtonian contribution. One of the most critical variables in practical applications regarding this phenomenon concerns the time required for the surfactant to reach a uniform distribution over the interface. The results show that the dimensionless parameters that control the decay of the variations in surfactant concentration in time are the Reynolds number Re, the fluid behavior index n, and ϵ, reflecting the influence of the inertia of the fluid. The results show that the temporal decay of the surfactant concentration on the fluid surface increases significantly for high values of the Reynolds number, it is more attenuated in pseudoplastic fluids than in Newtonian fluids, and the surfactant concentration decay exhibits asymmetric oscillations when inertial effects increase.