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

薄膜二元混合物的简化模型是以前开发和润滑近似的基础上。表面张力被认为是温度和浓度的函数。增加了浓度的2D简化方程（质量守恒方程）。我们已经通过启发式论证假定温度是垂直坐标上的线性函数，并且可以忽略对流和扩散项。利用这些假设，该模型很好地符合基于纯流体的薄膜方程的理论。空间维度减少了一个，因此计算时间大大减少，允许在三维空间中计算图案的形成和相当长的演化时间。为了系统地研究和证明到目前为止所做的近似，我们决定使用完整的三维能量和浓度方程系统作为下一步。在所有进行的模拟中，发现温度是垂直坐标的几乎完全线性函数。这使我们能够消除温度方程，并将系统简化为两个控制方程。使用这个系统，线性稳定性分析显示了一个轻微的修改值，其中一个获得导电状态和单调或振荡不稳定性。然而，问题的物理性质并没有本质上的改变。使用非线性模拟，类似于简化模型，可以通过粗化具有类似孤子形状的单个静止或行进液滴来获得。

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.