Surface structures on thin fluid layers of a binary mixture in confined geometries

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A simplified model for a thin film binary mixture was previously developed and based on lubrication approximation. The surface tension was considered as a function of both temperature and concentration. A 2D simplified equation for the concentration (mass conservation equation) was added. We have supposed by heuristic arguments that the temperature was a linear function on the vertical coordinate and that convective and diffusion terms can be neglected. Using these assumptions the model fits well into the theory based on the thin film equation for pure fluids. The spatial dimension is reduced by one and therefore the computing time considerably decreases, allowing for the computation of pattern formation in three dimensions and for rather long evolution times.In order to study and justify the approximations done so far systematically, we decided to use the complete system of the 3D energy and concentration equations as a next step. In all the simulations performed the temperature was found as an almost perfectly linear function of the vertical coordinate. This permits us to eliminate the temperature equation and to reduce the system to two governing equations. Using this system, linear stability analysis shows a slight modification of the values for which one obtains conductive state and monotonic or oscillatory instabilities. However the physics of the problem is not essentially modified. Using nonlinear simulations, like for the simplified model one can obtain through coarsening one single stationary or traveling drop having a soliton-like shape.