When open-cut mines are eventually abandoned, they leave a large hole with sloping sides. The hole fills with rain water, and there is also contaminated run-off from surrounding land, that moves through the rock and eventually through the sloping sides of the abandoned mine. This paper considers a two-dimensional unsteady model motivated by this leaching flow through the rock and into the rain-water reservoir. The stability of the interface between the two fluids is analysed in the inviscid limit. A viscous Boussinesq model is also presented, and a closed-form solution is presented to this problem, after it has been linearized in a manner consistent with Boussinesq theory. That solution suggests that the interfacial zone is effectively neutrally stable as it evolves in time. However, an asymptotic theory in the interfacial region shows the interface to be unstable. In addition, the nonlinear Boussinesq model is solved using a spectral method. Interfacial travelling waves and roll-up are observed and discussed, and compared against the predictions of asymptotic Boussinesq theory.

中文翻译：

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倾斜边界上致密渗流层的不稳定性

当露天矿最终被废弃时，它们会留下一个侧面倾斜的大洞。这个洞里充满了雨水，还有来自周围土地的污染径流，这些径流穿过岩石，最终穿过废弃矿井的斜坡。本文考虑了一个二维非定常模型，该模型由通过岩石进入雨水库的浸出流驱动。在无粘性极限下分析两种流体界面的稳定性。还提出了粘性 Boussinesq 模型，并在以符合 Boussinesq 理论的方式对其进行线性化后，为该问题提供了一个封闭形式的解决方案。该解决方案表明，随着时间的推移，界面区域实际上是中性稳定的。然而，界面区域的渐近理论表明界面不稳定。此外，非线性 Boussinesq 模型使用谱方法求解。观察和讨论了界面行波和卷起，并与渐近 Boussinesq 理论的预测进行了比较。