On the thermal instability in a horizontal rectangular porous channel heated from below by a constant flux

The onset of thermoconvective instability in a rectangular horizontal channel filled with a fluid-saturated porous medium is studied. The channel is heated from below with a constant flux. The top wall is maintained at a uniform constant temperature, while the lateral boundaries are permeable and perfectly conducting. The stability of the basic motionless state is analysed with respect to small-amplitude disturbances. The eigenvalue problem for the neutral stability condition is solved analytically for the normal modes. A closed-form expression is obtained for the implicit neutral stability relation between the Darcy-Rayleigh number and the longitudinal wave number. The critical condition, viz. the absolute minimum of the Darcy-Rayleigh number for the instability, is determined for different aspect ratios of the rectangular cross-section. The preferred modes under critical conditions are detected. It is found that the selected patterns of instability at the critical Rayleigh number may be 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.