A proof that convection in a porous vertical slab may be unstable

Abstract

The stability of the natural convection parallel flow in a vertical porous slab is reconsidered, by reformulating the problem originally solved in Gill’s classical paper of 1969 (J. Fluid Mech., vol. 35, pp. 545–547). A comparison is made between the set of boundary conditions where the slab boundaries are assumed to be isothermal and impermeable (Model A), and the set of boundary conditions where the boundaries are modelled as isothermal and permeable (Model B). It is shown that Gill’s proof of linear stability for Model A cannot be extended to Model B. The question about the stability/instability of the basic flow is examined by carrying out a numerical solution of the stability eigenvalue problem. It is shown that, with Model B, the natural convection parallel flow in the basic state becomes unstable when the Darcy–Rayleigh number is larger than 197.081. The normal modes selected at onset of instability are transverse rolls. Direct numerical simulations of the nonlinear regime of instability are carried out.