Computation of impinging droplet with soluble surfactants and dynamic contact angle

An arbitrary Lagrangian--Eulerian (ALE) finite element scheme for computations of soluble surfactant droplet impingement on a horizontal surface is presented. The numerical scheme solves the time-dependent Navier--Stokes equations for the fluid flow, scalar convection-diffusion equation for the surfactant transport in the bulk phase, and simultaneously, surface evolution equations for the surfactants on the free surface and on the liquid-solid interface. The effects of surfactants on the flow dynamics are included into the model through the surfactant-dependent surface tension and dynamic contact angle. In particular, the dynamic contact angle $(\theta_d)$ of the droplet is defined in terms of the surfactant concentration at the contact line and the equilibrium contact angle $(\theta_e^0)$ of the clean surface using the nonlinear equation of state for surface tension. Further, the surface forces are included in the model as the surface divergence of the surface stress tensor that allows to incorporate the Marangoni effects without calculating the surface gradient of the surfactant concentration on the free surface. In addition to a mesh convergence study and validation of the numerical results with experiments, the effects of adsorption and desorption surfactant coefficients on the flow dynamics in wetting, partially wetting and non-wetting droplets are studied in detail. It is observed that the effect of surfactants are more in wetting droplets than in the non-wetting droplets. Further, the presence of surfactants at the contact line reduces the equilibrium contact angle further when $\theta_e^0$ is less than $90^\circ$, and increases it further when $\theta_e^0$ is greater than $90^\circ$. The presence of surfactants has no effect on the contact angle, when $\theta_e^0 =90^\circ$. The numerical study clearly demonstrates that the surfactant-dependent contact angle ...