Scheid Benoit, Kalliadasis Serafim, Ruyer-Quil Christian, Colinet Pierre
TIPS-Génie chimique, Université Libre de Bruxelles C. P. 165/67, Avenue F.D. Roosevelt 50, 1050 Bruxelles, Belgium.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Dec;78(6 Pt 2):066311. doi: 10.1103/PhysRevE.78.066311. Epub 2008 Dec 22.
We study three-dimensional wave patterns on the surface of a film flowing down a uniformly heated wall. Our starting point is a model of four evolution equations for the film thickness h , the interfacial temperature theta , and the streamwise and spanwise flow rates, q and p , respectively, obtained by combining a gradient expansion with a weighted residual projection. This model is shown to be robust and accurate in describing the competition between hydrodynamic waves and thermocapillary Marangoni effects for a wide range of parameters. For small Reynolds numbers, i.e., in the "drag-gravity regime," we observe regularly spaced rivulets aligned with the flow and preventing the development of hydrodynamic waves. The wavelength of the developed rivulet structures is found to closely match the one of the most amplified mode predicted by linear theory. For larger Reynolds numbers, i.e., in the "drag-inertia regime," the situation is similar to the isothermal case and no rivulets are observed. Between these two regimes we observe a complex behavior for the hydrodynamic and thermocapillary modes with the presence of rivulets channeling quasi-two-dimensional waves of larger amplitude and phase speed than those observed in isothermal conditions, leading possibly to solitarylike waves. Two subregions are identified depending on the topology of the rivulet structures that can be either "ridgelike" or "groovelike." A regime map is further proposed that highlights the influence of the Reynolds and the Marangoni numbers on the rivulet structures. Interestingly, this map is found to be related to the variations of amplitude and speed of the two-dimensional solitary-wave solutions of the model. Finally, the heat transfer enhancement due to the increase of interfacial area in the presence of rivulet structures is shown to be significant.
我们研究了沿均匀加热壁面流动的液膜表面的三维波动模式。我们的出发点是一个由四个演化方程组成的模型，分别用于描述液膜厚度(h)、界面温度(\theta)以及流向和展向流速(q)和(p)，该模型是通过将梯度展开与加权残差投影相结合得到的。结果表明，该模型在描述广泛参数范围内的流体动力学波与热毛细马兰戈尼效应之间的竞争时具有鲁棒性和准确性。对于小雷诺数，即在“曳力-重力 regime”中，我们观察到与流动方向对齐的规则间隔的细流，这些细流阻止了流体动力学波的发展。发现所形成的细流结构的波长与线性理论预测的最放大模式的波长密切匹配。对于较大的雷诺数，即在“曳力-惯性 regime”中，情况与等温情况相似，未观察到细流。在这两个 regime 之间，我们观察到流体动力学和热毛细模式的复杂行为，存在细流引导比等温条件下观察到的更大振幅和相速度的准二维波，这可能导致类孤立波。根据细流结构的拓扑结构可分为“脊状”或“槽状”，确定了两个子区域。进一步提出了一个 regime 图，突出了雷诺数和马兰戈尼数对细流结构的影响。有趣的是，发现该图与模型二维孤立波解的振幅和速度变化有关。最后，结果表明，在存在细流结构的情况下，由于界面面积增加而导致的传热增强是显著的。
