Abstract
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.
Highlights
In the last decades, many authors investigated the onset of convection in a fluid saturated porous medium heated from below, starting from the pioneering papers by Horton and Rogers [1], and by Lapwood [2]
A closed-form expression is obtained for the implicit neutral stability relation between the Darcy-Rayleigh number and the longitudinal wave number
If we do not specify the modal number n and the transverse aspect ratio s, the parameter bn is nothing but a redefined wave number that can vary continuously over the set of positive real numbers
Summary
Many authors investigated the onset of convection in a fluid saturated porous medium heated from below, starting from the pioneering papers by Horton and Rogers [1], and by Lapwood [2]. Nowadays, this topic is termed, in a wide sense, either Horton-RogersLapwood problem, or Darcy-Bénard problem. Various types of thermal boundary conditions, involving all possible combinations of isothermal and isoflux conditions, were analysed. The condition of a finite lateral width of the channel was analysed as well. The case of a horizontal porous channel with rectangular cross-section was studied by Sutton [8], Beck [9], and further explored by Nilsen and Storesletten [10]
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