The onset of Prandtl-Darcy-Prats convection in a horizontal porous layer

We consider the effect of finite Prandtl–Darcy numbers of the onset of convection
in a porous layer heated isothermally from below and which is subject to a horizontal pressure gradient. A dispersion relation is found which relates the critical Darcy–Rayleigh number and the induced phase speed of the cells to the wavenumber and the imposed Péclet and Prandtl–Darcy numbers. Exact numerical solutions are given and these are supplemented by asymptotic solutions for both large and small values of the governing nondimensional parameters. The classical value of the critical Darcy–Rayleigh number is 4π^2, and we show that this value increases whenever the Péclet number is nonzero and the Prandtl–Darcy number is finite simultaneously. The corresponding wavenumber is always less than π and the phase speed of the convection cells is always smaller than the background flux velocity.