Abstract
The onset of Darcy–Bénard convection in an unlimited horizontal porous layer is studied theoretically. The thermomechanical boundary conditions of Dirichlet or Neumann type at the lower and upper plane are switched from one type to another, at certain values of the horizontal x-coordinate. A semi-infinite portion of the lower boundary is defined as thermally conducting and impermeable, while the remaining boundary is open and with given heat flux. At the upper boundary, the same thermomechanical conditions are applied, but with a relative spatial displacement L and in the opposite spatial order. A domain of local destabilization around the origin is generated between the lines of discontinuity x = pm ,L/2. The marginal state of convection is triggered centrally, while it is penetrative in the domains exterior to the central domain. The onset problem is solved numerically, with a general 3D mode of disturbance, but 2D disturbances are preferred in most cases. The critical Rayleigh number is given as a function of the dimensionless gap width L and the wavenumber k in the y direction along the lines of discontinuity in the boundary conditions. An asymptotic formula for 2D penetrative eigenfunctions is shown to be in agreement with the numerical results.
Highlights
Natural convection arising by buoyancy-driven instability in a porous medium heated from below is usually called Darcy–Bénard convection
A simple physical model is introduced for the convection onset in a homogeneous and isotropic porous layer of infinite horizontal extent, heated uniformly from below
We have investigated the effects of switching the thermomechanical boundary conditions from Dirichlet to Neumann type at one location at the upper boundary, and oppositely at one location at the lower boundary, generating a transition zone with displacement length L between the two locations of switching boundary conditions
Summary
Natural convection arising by buoyancy-driven instability in a porous medium heated from below is usually called Darcy–Bénard convection. Tyvand et al (2019) solved the onset problem for a 2D porous rectangle with boundary conditions that defy analytical solutions. A spatially concentrated instability is surrounded by penetrative convection laterally, beyond the two central borderlines where the boundary conditions switch. McKibbin (1986) started the investigation of convection instability which is penetrative horizontally He added physical realism for geothermal models by considering vertical porous layers with different properties. He maintained the confinement between two horizontal planes of the HRL problem and found locally triggered instability, penetrating laterally into locally stable domains. Ahmad and Rees (2016) considered the Laplacian thermal fields in solid blocks surrounding a porous box with convection onset. We will investigate how the onset criterion depends on the width of the unstable domain, and we will investigate the spatial decay of the thermomechanical field penetrating into the surrounding stable domains
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