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| The study of the dynamic surface tension of surfactant solutions at the air-water interface has been revealed an interesting issue since it plays an important role in several biological, biochemical and industrial processes. When a new surface is formed in a surfactant solution, surfactant molecules migrate from the bulk of the solution to the air-water interface and, consequently, they vary its surface properties.
This process is modeled by the partial diffusion equation in one spatial dimension, together with suitable initial and boundary conditions, being the unknowns both the surface and bulk concentrations. Moreover, in order to close the problem, we consider an adsorption model, that is coupled to the system of equations as a boundary condition at the subsurface. There are two families of models for describing the adsorption dynamics: the diffusion-controlled models and the mixed kinetic-diffusion ones.
In this work, we focus on a diffusion-controlled model considering the well-known Langmuir isotherm, for which we study the existence and uniqueness of weak solution. The existence is obtained by using the Rothe method, an intermediate problem, a priori error estimates and passing to the limit, and the uniqueness is proved integrating in time the weak equations and using adequate test functions. |
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