The linear stability of buoyant parallel flow in a vertical porous layer with an annular cross section is investigated. The vertical cylindrical boundaries are kept at different uniform temperatures, and they are assumed to be impermeable. The emergence of linear instability by convection cells is excluded on the basis of a numerical solution of the linearised governing equations. This result extends to the annular geometry the well-known Gill’s theorem regarding the impossibility of convective instability in a vertical porous plane slab whose boundaries are impermeable and isothermal with different temperatures. The extension of Gill’s theorem to the annular domain is approached numerically by evaluating the growth rate of normal mode perturbations and showing that its sign is negative, which means asymptotic stability of the basic flow. A concurring argument supporting the absence of linear instability arises from the investigation of cases where the impermeability condition at the vertical boundaries is relaxed and a partial permeability is modelled through Robin boundary conditions for the pressure. With partially permeable boundaries, an instability emerges which takes the form of axisymmetric normal modes. Then, as the boundary permeability is reduced towards zero, the critical Rayleigh number becomes infinite.

中文翻译：

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环形截面垂直多孔层平行流稳定性研究

研究了具有环形截面的垂直多孔层中浮力平行流的线性稳定性。垂直圆柱边界保持在不同的均匀温度下，并且假定它们是不可渗透的。基于线性化控制方程的数值解，排除了对流单元引起的线性不稳定性的出现。该结果扩展到环形几何形状，即众所周知的 Gill 定理，即关于垂直多孔平板中不可能发生对流不稳定性，其边界是不可渗透的，并且在不同温度下等温。通过评估常模扰动的增长率并显示其符号为负，这意味着基本流动的渐近稳定性，可以数值地接近吉尔定理到环形域的扩展。支持不存在线性不稳定性的一致论点来自对放松垂直边界的不渗透条件和通过压力的罗宾边界条件模拟部分渗透率的情况的调查。对于部分可渗透边界，会出现以轴对称法向模式形式出现的不稳定性。然后，随着边界渗透率向零减小，临界瑞利数变为无穷大。