赫尔肖流
本征函数
不稳定性
物理
波长
波数
特征向量
非线性系统
线性稳定性
平面(几何)
缩放比例
水动力稳定性
机械
数学分析
光学
雷诺数
数学
几何学
湍流
量子力学
作者
F. Graf,Eckart Meiburg,Carlos Härtel
标识
DOI:10.1017/s0022112001006516
摘要
We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number ( Ra ) ‘Hele-Shaw’ instability mode, along with a high- Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989). In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra . For large Ra , the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.
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