Spatially Developing Modes: The Darcy–Bénard Problem Revisited

In this paper, the instability resulting from small perturbations of the Darcy-Benard system is explored. An analysis based on time-periodic and spatially developing Fourier modes is adopted. The system under examination is a horizontal porous layer saturated by a fluid. The two impermeable and isothermal plane boundaries are considered to have different temperatures, so that the porous layer is heated from below. The spatial instability for the system is defined by taking into account both the spatial growth rate of the perturbation modes and their propagation direction. A comparison with the neutral stability condition determined by using the classical spatially periodic and time-evolving Fourier modes is performed. Finally, the physical meaning of the concept of spatial instability is discussed. In contrast to the classical analysis, based on spatially periodic modes, the spatial instability analysis, involving time-periodic Fourier modes, is found to lead to the conclusion that instability occurs whenever the Rayleigh number is positive.