Horizontal thermal convection in a porous medium (original) (raw)
Related papers
Journal of Mathematical Fluid Mechanics, 2013
The problem is considered of thermal convection in a saturated porous medium contained in an infinite vertical channel with differentially heated sidewalls. The theory employed allows for different solid and fluid temperatures in the matrix. Nonlinear energy stability theory is used to derive a Rayleigh number threshold below which convection will not occur no matter how large the initial data. A generalized nonlinear analysis is also given which shows convection cannot occur for any Rayleigh number provided the initial data is suitably restricted.
Global stability for thermal convection in a fluid overlying a highly porous material
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009
This paper investigates the instability thresholds and global nonlinear stability bounds for thermal convection in a fluid overlying a highly porous material. A two-layer approach is adopted, where the Darcy–Brinkman equation is employed to describe the fluid flow in the porous medium. An excellent agreement is found between the linear instability and unconditional nonlinear stability thresholds, demonstrating that the linear theory accurately emulates the physics of the onset of convection.
Unconditional nonlinear stability for convection in a porous medium with vertical throughflow
Acta Mechanica, 2007
Linear and nonlinear stability analyses of vertical throughflow in a fluid saturated porous layer, which is modelled using a cubic Forchheimer model, are studied. To ensure unconditional nonlinear results are obtainable, and to avoid the loss of key terms, a weighted functional is used in the energy analysis. The linear instability and nonlinear stability thresholds show considerable agreement when the vertical throughflow is small, although there is substantial deterioration of this agreement as the vertical throughflow increases.
STABILITY ANALYSIS OF DOUBLE-DIFFUSIVE CONVECTION IN POROUS ENCLOSURES
In this paper a linear stability analysis is presented for the problem of thermosolutal convection in a rectangular cell filled with a Brinkman (sparsely packed) porous medium saturated by a binary fluid. Transverse gradients of heat and solute are applied on the vertical walls while the horizontal ones are adiabatic and impermeable. When the buoyancy forces induced by the thermal and solutal effects are opposing each other and of equal intensity an equilibrium solution, corresponding to the rest state, is possible. The critical parameters for the onset of motion are determined using numerical procedures based on the Galerkin and finite element methods. The boundaries defining the regions of stationary and oscillatory instabilities are delineated. The Brinkman-extended momentum equation is used to model the flow in the porous medium. Results for a pure viscous fluid and a Darcy (densely packed) porous medium emerge from the present analysis as limiting cases.
Iraqi Journal of Science, 2022
The linear instability and nonlinear stability analyses are performed for the model of bidispersive local thermal non-equilibrium flow. The effect of local thermal non-equilibrium on the onset of convection in a bidispersive porous medium of Darcy type is investigated. The temperatures in the macropores and micropores are allowed to be different. The effects of various interaction parameters on the stability of the system are discussed. In particular, the effects of the porosity modified conductivity ratio parameters, and , with the inter-phase momentum transfer parameters and, on the onset of thermal Convection are also considered. Furthermore, the nonlinear stability boundary is found to be below the linear instability threshold. The numerical results are presented for free-free boundary conditions.
Stability analysis of thermosolutal convection in a vertical packed porous enclosure
Physics of Fluids, 2002
A linear stability theory is performed to investigate the stability of the quiescent state and fully developed thermosolutal convection within a vertical porous enclosure subject to horizontal opposing gradients of temperature and solute. The fluid motion is modeled using the unsteady form of Darcy's law coupled with energy and species conservation equations. The effect of different thermal and solutal boundary conditions is considered. The linearized governing equations are solved numerically using a finite element method. The thresholds for oscillatory and stationary convection are determined as functions of the governing parameters. It is concluded that the porosity and the acceleration parameter of the porous medium have a strong effect on the onset of overstability for a confined enclosure and on the wave number for an infinite enclosure. The stability analysis of fully developed flows within a slender enclosure reveals that an increase in the porosity and the acceleration parameter of the porous media delays the appearance of oscillatory finite amplitude flows. A nonlinear numerical solution is also computed by solving the full governing equations using a finite element method. Within the overstable regime, nonlinear traveling waves exist within slender enclosures, subject to Dirichlet thermal and solutal boundary conditions.
Journal of Non-Equilibrium Thermodynamics, 2022
The internal heat source and reaction effects on the onset of thermosolutal convection in a local thermal non-equilibrium porous medium are examined, where the temperature of the fluid and the solid skeleton may differ. The linear instability and nonlinear stability theories of Darcy-Brinkman type with fixed boundary condition are carried out where the layer is heated and salted from below. The D 2 Chebyshev tau technique is used to calculate the associated system of equations subject to the boundary conditions for both theories. Three different types of internal heat source function are considered, the first type increases across the layer, while the second decreases, and the third type heats and cools in a nonuniform way. The effect of different parameters on the Rayleigh number is depicted graphically. Moreover, the results detect that utilizing the internal heat source, reaction, and non-equilibrium have pronounced effects in determining the convection stability and instability thresholds.
On the evolution of thermal disturbances during natural convection in a porous medium
Journal of Fluid Mechanics, 1980
In natural convection in a porous medium heated from below, the convective flow in two dimensions becomes unsteady above a certain critical Rayleigh number and exhibits a fluctuating or oscillatory behaviour (depending on the confinement in the horizontal dimension). This fluctuating behaviour is due to a combination of the instability of the thermal boundary layers a t horizontal boundaries together with a 'triggering' effect of earlier disturbances. The point of the origin of the instability of the thermal boundary layer appears to play a dominant role in determining the regularity of the fluctuating flow. This numerical study investigates the importance of this point of evolution and concludes that there may exist more than one oscillatory mode of convection, depending on its position. The investigation focuses on the symmetry of the flow and demonstrates that with stable and accurate numerical schemes, an artificial symmetry may be imposed in trhe absence of realistic physical noise. If an initially symmetric perturbation is imposed the flow retains an essentially symmetric flow pattern with a high degree of regularity in the oscillatory behaviour. The imposition of an asymmetric perturbation results in a degradation of regularity. The appearance of the symmetric, regularly oscillatory flow is characterized by a symmetric (and stationary) arrangement of the points of origin of the instability of the upper and lower thermal boundary layers; in the case of the irregular oscillations the points of origin are not symmetric and their locations are not fixed.
Stability analysis of double diffusive convection in a vertical brinkman porous enclosure
International Communications in Heat and Mass Transfer, 1998
In this paper a linear stability analysis is presented for the problem of thermosolutal convection in a rectangular cell filled with a Brinkman (sparsely packed) porous medium saturated by a binary fluid. Transverse gradients of heat and solute are applied on the vertical walls while the horizontal ones are adiabatic and impermeable. When the buoyancy forces induced by the thermal and solutal effects are opposing each other and of equal intensity an equilibrium solution, corresponding to the rest state, is possible. The critical parameters for the onset of motion are determined using numerical procedures based on the Galerkin and finite element methods. The boundaries defining the regions of stationary and oscillatory instabilities are delineated. The Brinkman-extended momentum equation is used to model the flow in the porous medium. Results for a pure viscous fluid and a Darcy (densely packed) porous medium emerge from the present analysis as limiting cases.
Journal of Mathematical Analysis and Applications, 2007
A new approach to nonlinear L 2 -stability for double diffusive convection in porous media is given. An auxiliary system Σ of PDEs and two functionals V , W are introduced. Denoting by L and N the linear and nonlinear operators involved in Σ, it is shown that Σ-solutions are linearly linked to the dynamic perturbations, and that V and W depend directly on L-eigenvalues, while (along Σ) dV dt and dW dt not only depend directly on L-eigenvalues but also are independent of N . The nonlinear L 2 -stability (instability) of the rest state is reduced to the stability (instability) of the zero solution of a linear system of ODEs. Necessary and sufficient conditions for general, global L 2 -stability (i.e. absence of regions of subcritical instabilities for any Rayleigh number) are obtained, and these are extended to cover the presence of a uniform rotation about the vertical axis.