Finite element study of a heated thin fluid layer including surfactant effect
This investigation deals with modeling and numerical approximation of thermocapillary and surfactant effects in long wavelength evolution of thin liquid layers. We consider several aspects including: model development, scaling and perturbation analysis, variational formulations, finite element approximation and solution strategies for these problems. Issues related to treating the fluid volume constraint within this variational finite element setting are also considered. A linear stability analysis of the nonlinear thermocapillary problem is developed and its implications are explored numerically over the related parameter space. In the inclined plane case for the thermocapillary problem, it is observed in the time dependent finite element solutions that a slight inclination of the system may give rise to premature onset of instability. We also extend the treatment of the physical problem to include an insoluble surfactant monolayer and develop a supporting perturbation analysis. The resulting more complex model involves coupling to an additional transport equation for surfactant concentration on the surface. We develop the variational formulation, finite element implementation and stability analysis of this coupled system. We characterize the stability behavior into four parametric regions based on linear stability analysis and verify the behavior using finite element approximation of time dependent solutions on one and two-dimensional spatial domains.