Shankar V, Sahu Akhilesh K
Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208 016, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Jan;73(1 Pt 2):016301. doi: 10.1103/PhysRevE.73.016301. Epub 2006 Jan 10.
The linear stability of a liquid layer flowing down an inclined plane lined with a deformable, viscoelastic solid layer is analyzed in order to determine the effect of the elastohydrodynamic coupling between the liquid flow and solid deformation on the free-surface instability in the liquid layer. The stability of this two-layer system is characterized by two qualitatively different interfacial instability modes: In the absence of the deformable solid layer, the free surface of the liquid film undergoes a long-wave instability due to fluid inertia. With the presence of the deformable solid layer, the interface between the fluid and the solid undergoes a finite-wavelength instability when the deformable solid becomes sufficiently soft. The effect of the solid layer deformability on the free-surface instability of the liquid film flow is analyzed using a long-wave asymptotic analysis. The asymptotic results show that for a fixed Reynolds number and inclination angle, the free-surface instability is completely suppressed in the long-wave limit when the nondimensional (inverse) solid elasticity parameter Gamma=Va eta/(GR)increases beyond a critical value. Here, Va is the average velocity of the liquid film flow, eta is the viscosity of the liquid, G is the shear modulus of the solid layer, and is R the thickness of the liquid layer. The predictions of the asymptotic analysis are verified and extended to finite wavelengths using a numerical solution, and this indicates that the suppression of the free-surface instability indeed continues to finite wavelength disturbances. Further increase of Gamma is found to have two consequences: first, the interface between the liquid and the deformable solid layer could become unstable at finite wavelengths; second, the free-surface interfacial mode could also become unstable at finite wavelengths due to an increase in solid layer deformability. However, our numerical results demonstrate that, for a given average velocity, there exists a sufficient window in the value of shear modulus G where both the unstable modes are absent at all wavelengths. Our study therefore suggests that soft solid layer coatings could potentially provide a passive method of suppressing free-surface instabilities in liquid film flows.
分析了沿衬有可变形粘弹性固体层的倾斜平面流动的液层的线性稳定性，以确定液流与固体变形之间的弹性流体动力耦合对液层自由表面不稳定性的影响。这个两层系统的稳定性由两种性质不同的界面不稳定性模式来表征：在没有可变形固体层的情况下，液膜的自由表面由于流体惯性而经历长波不稳定性。当存在可变形固体层时，当可变形固体变得足够软时，流体与固体之间的界面会经历有限波长的不稳定性。使用长波渐近分析来分析固体层可变形性对液膜流动自由表面不稳定性的影响。渐近结果表明，对于固定的雷诺数和倾斜角，当无量纲（倒数）固体弹性参数Gamma=Va eta/(GR)超过临界值时，自由表面不稳定性在长波极限中被完全抑制。这里，Va是液膜流动的平均速度，eta是液体的粘度，G是固体层的剪切模量，R是液层的厚度。使用数值解验证了渐近分析的预测并将其扩展到有限波长，这表明自由表面不稳定性的抑制确实持续到有限波长扰动。发现Gamma的进一步增加有两个后果：第一，液体与可变形固体层之间的界面在有限波长处可能变得不稳定；第二，由于固体层可变形性的增加，自由表面界面模式在有限波长处也可能变得不稳定。然而，我们的数值结果表明，对于给定的平均速度，在剪切模量G的值中存在一个足够的窗口，在所有波长下两种不稳定模式都不存在。因此，我们的研究表明，软固体层涂层可能潜在地提供一种抑制液膜流动中自由表面不稳定性的被动方法。
