Abstract
In this paper, the instability resulting from small perturbations of the Darcy–Bénard 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.
Highlights
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The Darcy–Bénard problem designates the study of the onset of convection in a horizontal porous layer saturated by a fluid and bounded by parallel plane walls at different temperatures
The aim of this paper is a revisitation of the Darcy–Bénard problem from the perspective of spatial stability analysis
Summary
The usual approach to the solution of the Darcy–Bénard problem is the linear stability analysis This analysis is usually carried out by testing how spatially periodic and time–. A different method for the linear stability analysis of stationary shear flows is based on a different class of perturbation modes, namely time–periodic and spatially–developing Fourier modes [2,10,11,12]. The aim of this paper is a revisitation of the Darcy–Bénard problem from the perspective of spatial stability analysis. The latter is a shorthand for the method based on the spatially–developing Fourier modes of perturbation having a periodic dependence on time. The predictions of the spatial stability analysis for the Darcy–Bénard problem are presented and discussed showing that Fourier modes exponentially–growing in space along their direction of propagation exist for every positive value of the Rayleigh number
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