The onset of Darcy–Bénard convection in an unlimited horizontal porous layer is studied theoretically. The thermomechanical boundary conditions of Dirichlet or Neumann type at the lower and upper plane are switched from one type to another, at certain values of the horizontal x-coordinate. A semi-infinite portion of the lower boundary is defined as thermally conducting and impermeable, while the remaining boundary is open and with given heat flux. At the upper boundary, the same thermomechanical conditions are applied, but with a relative spatial displacement L and in the opposite spatial order. A domain of local destabilization around the origin is generated between the lines of discontinuity x=±L/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x = \pm \,L/2$$\end{document}. The marginal state of convection is triggered centrally, while it is penetrative in the domains exterior to the central domain. The onset problem is solved numerically, with a general 3D mode of disturbance, but 2D disturbances are preferred in most cases. The critical Rayleigh number is given as a function of the dimensionless gap width L and the wavenumber k in the y direction along the lines of discontinuity in the boundary conditions. An asymptotic formula for 2D penetrative eigenfunctions is shown to be in agreement with the numerical results.

中文翻译：

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水平多孔层中对流的横向渗透开始

理论上研究了无限水平多孔层中达西-贝纳德对流的发生。在水平 x 坐标的特定值处，下平面和上平面的 Dirichlet 或 Neumann 型热机械边界条件从一种类型切换到另一种类型。下边界的半无限部分被定义为导热且不可渗透的，而其余边界是开放的并且具有给定的热通量。在上边界，应用相同的热机械条件，但具有相对空间位移 L 和相反的空间顺序。在不连续 x=±L/2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\ usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x = \pm \,L/2$$\end{document}。边缘对流状态是集中触发的，而它在中心域外部的域中具有穿透性。起始问题通过数值解决，使用一般的 3D 扰动模式，但在大多数情况下首选 2D 扰动。临界瑞利数作为无量纲间隙宽度 L 和沿边界条件中不连续线在 y 方向上的波数 k 的函数给出。