Waves and instabilities of viscoelastic fluid film flowing down an inclined wavy bottom.
Waves and instabilities of viscoelastic fluid film flowing down an inclined wavy bottom.
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DOI：
10.1103/physreve.102.023117
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发表时间：
2020-08
期刊：
影响因子：
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通讯作者：
S. Mukhopadhyay;A. Mukhopadhyay
中科院分区：
文献类型：
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作者：
S. Mukhopadhyay;A. Mukhopadhyay
本文用解析和数值方法分析了粘弹性流体薄膜沿中等陡度的倾斜波浪底流下时波浪的演化和流体动力学不稳定性。经典的长波展开法已被用来制定一个非线性的发展方程的自由表面。本文采用简正波方法从空间和时间两个角度对线性稳定性进行了分析。利用多重尺度法导出了一个Ginzburg-Landau型非线性方程，研究了该方程的弱非线性稳定解。两个重要的波族，发现并沿着详细讨论了演化系统的行波解，得到了γ_{1}和γ_{2}。用Scikit-FDif进行了时间相关的数值研究。整个调查主要是进行一般周期性的底部，然后讨论了正弦地形的一个特定的案例研究的详细结果。实例研究表明，底部陡度在线性区域中起着双重作用。在上坡区域，增加阻力具有稳定效果，而在下坡区域则相反。而粘弹性参数Γ在线性和非线性区域中都具有整个区域的失稳效应。通过弱非线性分析，超临界和亚临界解决方案都是可能的。值得注意的是，对于固定的Γ和ε，在下坡区域而不是上坡区域，无条件区减小，而爆炸区增加。如果我们增加Γ并保持τ不变，在特定区域也会出现同样的现象。行波解揭示了一个事实，即为了得到γ_{1}族波，我们需要将雷诺数增加得比γ_{2}族波所处的雷诺数稍大一些.非线性表面方程的时空演化表明存在不同类型的有限振幅永久波.
Evolution of waves and hydrodynamic instabilities of a thin viscoelastic fluid film flowing down an inclined wavy bottom of moderate steepness have been analyzed analytically and numerically. The classical long-wave expansion method has been used to formulate a nonlinear evolution equation for the development of the free surface. A normal-mode approach has been adopted to discuss the linear stability analysis from the viewpoint of the spatial and temporal study. The method of multiple scales is used to derive a Ginzburg-Landau-type nonlinear equation for studying the weakly nonlinear stability solutions. Two significant wave families, viz., γ_{1} and γ_{2}, are found and discussed in detail along with the traveling wave solution of the evolution system. A time-dependent numerical study is performed with Scikit-FDif. The entire investigation is conducted primarily for a general periodic bottom, and the detailed results of a particular case study of sinusoidal topography are then discussed. The case study reveals that the bottom steepness ζ plays a dual role in the linear regime. Increasing ζ has a stabilizing effect in the uphill region, and the opposite occurs in the downhill region. While the viscoelastic parameter Γ has a destabilizing effect throughout the domain in both the linear and the nonlinear regime. Both supercritical and subcritical solutions are possible through a weakly nonlinear analysis. It is interesting to note that the unconditional zone decreases and the explosive zone increases in the downhill region rather than the uphill region for a fixed Γ and ζ. The same phenomena occur in a particular region if we increase Γ and keep ζ fixed. The traveling wave solution reveals the fact that to get the γ_{1} family of waves we need to increase the Reynolds number a bit more than the value at which the γ_{2} family of waves is found. The spatiotemporal evolution of the nonlinear surface equation indicates that different kinds of finite-amplitude permanent waves exist.