Variational analysis of the Langmuir–Hinshelwood dynamic mixed-kinetic adsorption model

The paper deals with the mathematical and numerical analysis of surfactants behavior at the air–water interface, by using a mixed-kinetic adsorption model which is known as the Langmuir–Hinshelwood model and which is consistent with the classical Langmuir isotherm at equilibrium. The model consists of a coupled nonlinear system of a parabolic equation for the surfactant concentration and an ordinary differential equation for the surface concentration. The existence and uniqueness of the weak solution is proved by applying a result for the linear parabolic equation and fixed-point techniques. Then, fully discrete approximations are introduced by using a finite element method and a hybrid combination of both backward and forward Euler schemes. An a priori error estimate result is proved from which, under adequate additional regularity conditions, the linear convergence of the algorithm is derived. Finally, numerical simulations are presented to demonstrate the accuracy of the algorithm and the behavior of the solution for a commercially available surfactant.