Abstract The present paper investigates how the cross–sectional shape of a vertical porous cylinder affects the onset of thermoconvective instability of the Rayleigh–Benard type. The fluid saturating the porous medium is assumed to be a non–Newtonian power–law fluid. A linear stability analysis of the vertical thorughflow is carried out. Three special shapes of the cylinder cross–section are analysed: square, circular and elliptical. The effect of changing the power–law index is investigated. The stability of a steady base state with vertical throughflow is analysed. The resulting stability problem is a differential eigenvalue problem that is solved numerically through the shooting method. The dimensionless numbers here considered are the non–Newtonian version of the Darcy–Rayleigh number, R a , the Peclet number, P e , and the power–law index, n. Results are presented in the form of marginal stability curves with R a plotted as a function of the cylinder aspect ratio, by assuming different values of P e and n. The critical values of R a are also computed. Results show that the critical Rayleigh number R a c for instability depends only on P e and n, and is independent of the shape of the cylinder cross–section. The geometry of the sidewall just contributes the selection of the allowed wavenumbers.

中文翻译：

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具有任意横截面的垂直多孔圆柱体中幂律流体的热不稳定通流

摘要 本文研究了垂直多孔圆柱体的横截面形状如何影响瑞利-贝纳德型热对流不稳定性的开始。饱和多孔介质的流体被假定为非牛顿幂律流体。对垂直通流进行了线性稳定性分析。分析了圆柱横截面的三种特殊形状：方形、圆形和椭圆形。研究了改变幂律指数的效果。分析了具有垂直通流的稳态基态的稳定性。由此产生的稳定性问题是一个微分特征值问题，通过射击方法进行数值求解。这里考虑的无量纲数是 Darcy-Rayleigh 数 R a 、佩克莱特数 Pe 和幂律指数 n 的非牛顿版本。结果以边际稳定性曲线的形式呈现，其中 R a 绘制为圆柱纵横比的函数，假设 P e 和 n 的值不同。还计算了 R a 的临界值。结果表明，不稳定的临界瑞利数 R ac 仅取决于 P e 和 n，与圆柱横截面的形状无关。侧壁的几何形状仅有助于选择允许的波数。