Convective Instability in a Horizontal Porous Channel with Permeable and Conducting Side Boundaries

The stability analysis of the motionless state of a horizontal porous channel with rectangular cross-section and saturated by a fluid is developed. The heating from below is modelled by a uniform flux, while the top wall is assumed to be isothermal. The side boundaries are considered as permeable and perfectly conducting. The linear stability of the basic state is studied for the normal mode perturbations. The principle of exchange of stabilities is proved, so that only stationary normal modes need to be considered in the stability analysis. The eigenvalue problem for the neutral stability condition is solved analytically, and a closed-form dispersion relation is obtained for the neutral stability. The Darcy-Rayleigh number is expressed as an implicit function of the longitudinal wave number and of the aspect ratio. The critical wave number and the critical Darcy-Rayleigh number are evaluated for different aspect ratios. The preferred modes under critical conditions are detected. It is found that the selected patterns of instability at the critical Rayleigh number are two-dimensional, for slender or square cross-sections of the channel. On the other hand, instability is three dimensional when the critical width-to-height ratio, 1.350517, is exceeded. Eventually, the effects of a finite longitudinal length of the channel are discussed