Thermally unstable throughflow of a power–law fluid in a vertical porous cylinder with arbitrary cross–section

Abstract

The present paper investigates how the cross–sectional shape of a vertical porous cylinder affects the onset of thermoconvective instability of the Rayleigh–Bénard 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, Ra, the Péclet number, Pe, and the power–law index, n. Results are presented in the form of marginal stability curves with Ra plotted as a function of the cylinder aspect ratio, by assuming different values of Pe and n. The critical values of Ra are also computed. Results show that the critical Rayleigh number Rac for instability depends only on Pe 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.