It is a well-known fact that the onset of Rayleigh–Benard convection occurs via a long-wavelength instability when the horizontal boundaries are thermally insulated. The aim of this paper is to quantify the exact dimensions of a cylinder of rectangular cross-section wherein stable three-dimensional Rayleigh–Benard convection sets in via a long-wavelength instability from the motionless state at the same value of the critical Rayleigh number as the corresponding horizontally unbounded problem when the bounding horizontal walls have infinite thermal conductance. Hence, we consider three-dimensional Rayleigh–Benard convection in a cell of infinite extent in the x-direction, confined between two vertical walls located at $$y= \pm H$$ and horizontal boundaries located at $$z=0$$ and $$z=d$$ . Our analysis predicts the existence of the sought stable state for experimental velocity boundary conditions at the vertical walls provided the aspect ratio $$\delta = H/d$$ takes a certain value. In the limit $$H \rightarrow \infty $$ , we retrieve the stability characteristics of the horizontally unbounded problem. As expected, the analysis predicts two counter-rotating rolls aligned along the y-direction of period $$2 \pi /\delta $$ equal to the period of the roll in the y-direction of the corresponding unbounded problem. A long-scale asymptotic analysis leads to the derivation of an evolution partial differential equation (PDE) that is fourth order in space and contains a single bifurcation parameter. The PDE, valid for a specific value of $$\delta $$ , is analyzed analytically and numerically as function of the bifurcation parameter and for a variety of velocity boundary conditions at the vertical walls to seek the stable steady-state solutions. The same analysis is also extended to the case of convection in a fluid-saturated porous medium.

中文翻译：

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存在狄利克雷热边界条件下的三维稳定长波长对流

众所周知，当水平边界隔热时，瑞利-贝纳德对流的开始是通过长波长不稳定性发生的。本文的目的是量化矩形横截面圆柱体的精确尺寸，其中稳定的三维瑞利-贝纳德对流通过从静止状态的长波长不稳定性以与临界瑞利数相同的值作为当边界水平壁具有无限热导时，相应的水平无界问题。因此，我们考虑在 x 方向无限延伸的单元中的三维瑞利-贝纳对流，限制在位于 $$y= \pm H$$ 的两个垂直壁和位于 $$z=0$ 的水平边界之间$ 和 $$z=d$$ 。我们的分析预测，如果纵横比 $$\delta = H/d$$ 取某个值，则在垂直壁上存在实验速度边界条件所寻求的稳定状态。在极限 $$H \rightarrow \infty $$ 中，我们检索了水平无界问题的稳定性特征。正如预期的那样，该分析预测两个反向旋转的滚动沿 y 方向排列的周期 $$2 \pi /\delta $$ 等于相应无界问题的 y 方向滚动周期。长尺度渐近分析导出了空间四阶演化偏微分方程 (PDE)，并包含单个分岔参数。PDE，对于 $$\delta $$ 的特定值有效，分析和数值分析作为分岔参数和垂直壁上的各种速度边界条件的函数，以寻求稳定的稳态解。同样的分析也扩展到流体饱和多孔介质中的对流情况。