Hu J, Yin X Y, Ben Hadid H, Henry D
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Feb;77(2 Pt 2):026302. doi: 10.1103/PhysRevE.77.026302. Epub 2008 Feb 7.
A detailed temporal and spatiotemporal stability analysis of two-layer falling films with density and viscosity stratification is performed by using the Chebyshev collocation method to solve the full system of linear stability equations. From the neutral curves Re(k) for the surface mode and the interface mode of instability, obtained for different density ratios gamma of the upper layer to the lower layer, it is found that smaller density ratios make the surface mode and the short-wave interface mode much more stable, and can even make the short-wave interfacial instability disappear. Moreover, through the study of the local growth rates of the spatiotemporal instability as a function of the ray velocity V , it is found that for not too small incline angles like theta=0.2, the two-layer flow is always convectively unstable, and there is a transition between long- and short-wave instabilities which is determined by the Briggs-Bers collision criterion. Due to the existence of the absolute Rayleigh-Taylor instability for gamma>0 and theta=0, a transition from convective to absolute instability can be detected at small incline angles, and the corresponding boundary curves are plotted for different Reynolds numbers, viscosity ratios, and incline angles. It is found that there exists a limit Reynolds number above which the two-layer film flow can only be convectively unstable for a fixed small incline angle. The spatial amplification properties of the convective waves are finally presented for both surface and interface modes.
通过使用切比雪夫配置法求解线性稳定性方程组的完整系统，对具有密度和粘度分层的双层降膜进行了详细的时间和时空稳定性分析。从针对上层与下层不同密度比γ获得的表面模式和界面不稳定模式的中性曲线Re(k)发现，较小的密度比会使表面模式和短波界面模式更加稳定，甚至能使短波界面不稳定消失。此外，通过研究时空不稳定性的局部增长率作为射线速度V的函数发现，对于像θ = 0.2这样不太小的倾斜角，双层流总是对流不稳定的，并且长波和短波不稳定性之间存在由布里格斯 - 贝斯碰撞准则决定的转变。由于对于γ > 0和θ = 0存在绝对瑞利 - 泰勒不稳定性，在小倾斜角处可以检测到从对流不稳定到绝对不稳定的转变，并针对不同的雷诺数、粘度比和倾斜角绘制了相应的边界曲线。发现存在一个极限雷诺数，高于该数时，对于固定的小倾斜角，双层膜流只能是对流不稳定的。最后给出了表面和界面模式对流波的空间放大特性。
