This paper is an analytical study on natural two-dimensional convection in horizontal rectangular channels filled by isotropic and anisotropic porous media. The channel walls, assumed to be impermeable and perfectly heat conducting, are nonuniformly heated to establish a linear temperature distribution in the vertical direction. We derive the critical Rayleigh numbers for the onset of convection and examine the steady flow patterns at moderately supercritical Rayleigh numbers. The stability properties of these flow patterns are examined against two-dimensional perturbations using a weakly nonlinear theory.

We investigate the onset of Rayleigh–Bénard convection in a horizontal porous layer with anisotropic permeability. The permeability is transversely isotropic, whereas the orientation of the longitudinal principal axes is arbitrary. This is sufficient to achieve qualitatively new flow patterns with a tilted plane of motion or tilted lateral cell walls. The critical Rayleigh number and wavenumber at marginal stability are calculated. There are two different types of convection cells (rolls): (i) the plane of motion is tilted, whereas the lateral cell walls are vertical; (ii) the plane of motion is vertical, whereas the lateral cell walls are tilted as well as curved. It turns out that type (i) occurs when the transverse permeability is larger than the longitudinal permeability, and for the converse case type (ii) is preferred.

The vertical throughflow with viscous dissipation in a horizontal porous layer is studied. The horizontal plane boundaries are assumed to be isothermal with unequal temperatures and bottom heating. A basic stationary solution of the governing equations with a uniform vertical velocity field (throughflow) is determined. The temperature field in the basic solution depends only on the vertical coordinate. Departures from the linear heat conduction profile are displayed by the temperature distribution due to the forced convection effect and to the viscous dissipation effect. A linear stability analysis of the basic solution is carried out in order to determine the conditions for the onset of convective rolls. The critical values of the wave number and of the Darcy-Rayleigh number are determined numerically by the fourth-order Runge-Kutta method. It is shown that, although generally weak, the effect of viscous dissipation yields an increase of the critical value of the Darcy-Rayleigh number for downward throughflow and a decrease in the case of upward throughflow. Finally, the limiting case of a vanishing boundary temperature difference is discussed.

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A necessary condition for linear stability of steady inviscid helical gas flows is found by the generalized progressing-wave expansion method. The criterion obtained is compared with the known Richardson number criteria giving sufficient conditions for stability.