The interface between two miscible solutions in porous media and Hele-Shaw cells (two glass plates separated by a thin gap) in a gravity field can destabilise due to buoyancy-driven and double-diffusive effects. In this paper the conditions for instability to arise are presented within an analytical framework by considering the eigenvalue problem based on the tools used extensively by Chandrasekhar. The model considered here is Darcy’s law coupled to evolution equations for the concentrations of different solutes. We have shown that, when there is an interval in the spatial domain where the first derivative of the base-state density profile is negative, the flows are unstable to stationary or oscillatory modes. Whereas for base-state density profiles that are strictly monotonically increasing downwards such that the first derivative of the base-state density profile is positive throughout the domain (for instance, when a lighter solution containing a species A overlies a denser solution containing another species B), a necessary and sufficient condition for instability is the presence of a point on either side of the initial interface where the second derivative of the base-state density profile is zero such that it changes sign. In such regimes the instability arises as non-oscillatory modes (real eigenvalues). The neutral stability curve, which delimits the stable from the unstable regime, that follows from the discussion presented here along with the other results are in agreement with earlier observations made using numerical computations. The analytical approach adopted in this work could be extended to other instabilities arising in porous media.

中文翻译：

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多孔介质中浮力驱动流动的不稳定性

由于浮力驱动和双扩散效应，多孔介质中的两种混溶溶液与重力场中的 Hele-Shaw 池（两块玻璃板由一个薄间隙隔开）之间的界面可能会不稳定。在本文中，通过考虑基于 Chandrasekhar 广泛使用的工具的特征值问题，在分析框架内提出了不稳定性出现的条件。这里考虑的模型是与不同溶质浓度的演化方程耦合的达西定律。我们已经证明，当空间域中存在基态密度剖面的一阶导数为负的区间时，流动对平稳或振荡模式不稳定。而对于严格单调向下增加的基态密度分布，使得基态密度分布的一阶导数在整个域中为正（例如，当包含物质 A 的较轻溶液覆盖包含另一种物质 B 的密度较大的溶液时)，不稳定的必要和充分条件是在初始界面的任一侧存在一个点，其中基态密度分布的二阶导数为零，因此它改变了符号。在这种情况下，不稳定性以非振荡模式（实特征值）的形式出现。中性稳定性曲线将稳定状态与不稳定状态区分开来，遵循此处介绍的讨论以及其他结果，与使用数值计算进行的早期观察一致。