Mathematics: Numerical Analysis and Scientific Computing
Finite Volume Methods for Advection Diffusion on Moving Interfaces and Application on Surfactant Driven Thin Film Flow
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The book is divided into two parts. First, we present two finite volumes schemes for the discretization of convection-diffusion-reaction problems on moving surfaces. The first scheme extends the two points flux approximation finite volumes on moving surfaces. The second scheme presents a finite volume scheme of type O-method. Here, we construct around the mesh vertices a linear approximation of the solution to the given problem using the unknowns located at cells' centers. A suitable flux continuity on cells interfaces in incorporated. The method allows also the construction of a second order ...
The book is divided into two parts. First, we present two finite volumes schemes for the discretization of convection-diffusion-reaction problems on moving surfaces. The first scheme extends the two points flux approximation finite volumes on moving surfaces. The second scheme presents a finite volume scheme of type O-method. Here, we construct around the mesh vertices a linear approximation of the solution to the given problem using the unknowns located at cells' centers. A suitable flux continuity on cells interfaces in incorporated. The method allows also the construction of a second order upwind for convection operators; which makes the overall scheme second order in space. Next, we model the flow of a surfactant driven thin-film. Here, the use of tensor theory combined with lubrication approximation helps to reduce the Navier-Stokes equations describing the flow of the thin-film in three dimensions to a fourth order equation stated on the moving curved surface whose unknown is the film height. The surfactant, assumed to be insoluble, is modeled by a convection-diffusion equation on the fluid-air interface. We simulate the coupled system using an interface tracking method.
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