Abstract

This paper reports a breakdown in linear stability theory under conditions of neutral stability that is deduced by an examination of exponential modes of the form $h\approx{{e}^{i(kx-\omega t)}}$, where $h$ is a response to a disturbance, $k$ is a real wavenumber and $\omega (k)$ is a wavelength-dependent complex frequency. In a previous paper, King et al. (2016, Stability of algebraically unstable dispersive flows. Phys. Rev. Fluids, 1, 073604) demonstrates that when Im$[\omega (k)]=0$ for all $k$, it is possible for a system response to grow or damp algebraically as $h\approx{{t}^{s}}$ where $s$ is a fractional power. The growth is deduced through an asymptotic analysis of the Fourier integral that inherently invokes the superposition of an infinite number of modes. In this paper, the more typical case associated with the transition from stability to instability is examined in which Im$[\omega (k)]=0$ for a single mode (i.e. for one value of $k$) at neutral stability. Two partial differential equation systems are examined, one that has been constructed to elucidate key features of the stability threshold, and a second that models the well-studied problem of rectilinear Newtonian flow down an inclined plane. In both cases, algebraic growth/decay is deduced at the neutral stability boundary, and the propagation features of the responses are examined.

中文翻译：

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勘误到：多孔介质和谐波在多孔弹性体中传播的理论

摘要

本文报告了在中性稳定性条件下线性稳定性理论的分解，该分解是通过检查以下形式的指数模式得出的：$ h \ approx {{e} ^ {i（kx- \ omega t）}} $，其中h $是对扰动的响应，$ k $是实波数，$ \ω（k）$是与波长相关的复数频率。在以前的论文中，金等人。（2016，代数不稳定色散流的稳定性。，1，073604）证明，当对于所有$ k $，Im $ [\ omega（k）] = 0 $时，系统响应可能随着$ h \ approx {{t} ^ {s }} $其中$ s $是小数幂。增长是通过对傅立叶积分的渐近分析得出的，该分析固有地调用了无限多个模的叠加。在本文中，研究了与从稳定到不稳定的过渡相关的更典型情况，其中对于中性稳定状态下的单一模式（即$ k $的一个值），Im $ [ω（k）] = 0 $。研究了两个偏微分方程系统，一个方程组已被构造为阐明稳定性阈值的关键特征，第二个方程组已为充分研究的直线状牛顿流沿斜面的流动问题建模。在这两种情况下