Convection due to the selective absorption of radiation in a porous medium

A model for convection due to the selective absorption of radiation in a fluid saturated porous medium is investigated. The model is based on a similar one introduced for a viscous fluid by Krishnamurti [x]. Employing this adapted model we show the growth rate for the linearised system is real. A linear instability analysis is performed. Global stability thresholds are also found using nonlinear energy theory. An excellent agreement is found between the linear instability and nonlinear stability Rayleigh numbers, so that the region of potential subcritical instabilities is very small, demonstrating that the linear theory accurately emulates the physics of the onset of convection. Received February 10, 2003 / Accepted February 10, 2003/ Published online May 9, 2003 / B. Straughan     

Convection induced by the selective absorption of radiation for the Brinkman model

Convection induced by the selective absorption of radiation in a porous medium is studied analytically and numerically using the Brinkman model. Both linear instability analysis and nonlinear stability analysis are employed. The thresholds show excellent agreement so that the region of potential subcritical instabilities is very small, demonstrating that linear theory is accurate enough to predict the onset of convective motion. A surprising result shows that the critical Rayleigh number increases linearly as (Darcy number x Brinkman coefficient / dynamic viscosity of the fluid) increases.Received: 6 May 2003, Accepted: 26 May 2003     

Convection due to the selective absorption of radiation in a porous medium

Continuum Mech. Thermodyn. (2003) 15: 451-462 Digital Object Identifier (DOI) 10.1007/s00161-003-0125-5 Published online September 12, 2003-© Springer-Verlag 2003 Due to a technical error, the present contribution has been published twice in this journal. This article has already appeared in Volume 15 Number 3 (June 2003) and should be cited accordingly. Springer-Verlag wishes to apologize to its customers and readers for this mistake.Published online: 27 Oktober 2003     

Conditional and unconditional nonlinear stability for convection induced by absorption of radiation in a porous medium

Convection induced by the selective absorption of radiation is investigated, for the case of an internal heat source that is modelled quadratically with respect to concentration. The growth rate for the linearised system is shown to be real, and a linear instability analysis is performed. To establish conditional and unconditional nonlinear stability results, both the Darcy and Forchheimer models are employed to describe fluid flow. Due to the presence of significant regions of potential subcritical instabilities, the results indicate that linear theory may only be accurate enough to predict the onset of convective motion when the model for the internal heat source is predominantly linear.Received: 6 May 2003, Accepted: 9 August 2003, Published online: 12 December 2003     

Transition to instability of a phase interface in a porous medium in the isothermal approximation

The stability of the phase interface in geothermal systems is considered in the isothermal approximation with allowance for capillary effects. The dispersion relation is obtained and the domains of stability and instability of steady-state vertical flows are found. Possible types of transition to instability, namely, transitions with the most unstable mode corresponding to zero and infinite wavenumbers or to all wavenumbers simultaneously, are described. In the first case the nonlinear Kolmogorov-Petrovskii-Piskunov equation describing the evolution of a narrow strip of weakly unstable modes on the stability threshold is derived. The effect of the parameters of the system on its stability is investigated.     

Three dimensional simulation of unstable immiscible displacement in the porous medium

I.IntroductionThedisplacementofimmisciblefluidsiswidelyutilizedinenhancedoilrecovery.ThestabilityoftheinterfaceisveryimportantandpeopIehavepaidmoreandmoreattentiontoit,becauseitrelatestotheefficiencyofoilrecovery.Manypapersreportedthecharacteristicsoftheu…     

On the Chebyshev collocation spectral approach to stability of fluid flow in a porous medium

In this paper, the temporal development of small disturbances in a pressure‐driven fluid flow through a channel filled with a saturated porous medium is investigated. The Brinkman flow model is employed in order to obtain the basic flow velocity distribution. Under normal mode assumption, the linearized governing equations for disturbances yield a fourth‐order eigenvalue problem, which reduces to the well‐known Orr–Sommerfeld equation in some limiting cases solved numerically by a spectral collocation technique with expansions in Chebyshev polynomials. The critical Reynolds number Re_c, the critical wave number α_c, and the critical wave speed c_c are obtained for a wide range of the porous medium shape factor parameter S. It is found that a decrease in porous medium permeability has a stabilizing effect on the fluid flow. Copyright © 2008 John Wiley & Sons, Ltd.     

Effect of vibration on the stability of a plane displacement front in a porous medium

The effect of linearly polarized vibration on the stability of a plane displacement front in a porous medium is studied. The problem of the stability of the motion of a plane displacement front traveling at a constant velocity U under the action of vibration normal to the front is considered. It is shown that under the action of vibration the dynamics of the plane displacement front can be described by the Mathieu equation with a dissipative term. Using the standard averaging method, in the case of high-frequency vibration it is revealed that vibration can only increase the stability of the system. It is found that the vibration stabilizes the plane displacement front with respect to part of the perturbation spectrum.     

Spin-up in a saturated porous medium

It is shown that, on the Brinkman model, spin-up is confined to boundary layers whose thickness is of order k ^1/2, and the spin-up is established in a time of order k/, where k, , and denote permeability, density, porosity and dynamic viscosity, respectively.     

Nonlinear convection in a non-Darcy porous medium

In this paper, the natural convection in a non-Darcy porous medium is studied using a temperature-concentration-dependent density relation. The effect of the two parameters responsible for the nonlinear convection is analyzed for different values of the inertial parameter, dispersion parameters, Rayleigh number, Lewis number, Soret number, and Dufour number. In the aiding buoyancy, the tangential velocity increases steeply with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter when the inertial effect is zero. However, when the inertial effect is non-zero, the effect of the nonlinear temperature parameter and the nonlinear concentration parameter on the tangential velocity is marginal. The concentration distribution varies appreciably and spreads in different ranges for different values of the double dispersion parameters, the inertial effect parameter, and also for the parameters which control the nonlinear temperature and the nonlinear concentration. Heat and mass transfer varies extensively with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter depending on Dacry and non-Darcy porous media. The variation in heat and mass transfer when all the effects, i.e., the inertial effect, double dispersion ef- fects, and Soret and Dufour effects, are simultaneously zero and non-zero. The combined effects of the nonlinear temperature parameter, the nonlinear concentration parameter and buoyancy are analyzed. The effect of the nonlinear temperature parameter and the nonlinear concentration parameter and also the cross diffusion effects on heat and mass transfer are observed to be more in Darcy porous media compared with those in non- Darcy porous media. In the opposing buoyancy, the effect of the temperature parameter is to increase the heat and mass transfer rate, whereas that of the concentration parameter is to decrease.     

On the interaction of water waves and seabed by the porous medium model

The interaction of water waves and seabed is studied by using Yamamoto's model, which takes into account the deformation of soil skeletal frame, compressibility of pore fluid flow as well as the Coulumb friction. When analyzing the propagation of three kinds of stress waves in seabed, a simplified dispersion relation and a specific damping formula are derived. The problem of seabed stability is further treated analytically based on the Mohr-Coulomb theory. The theory is finally applied to the coastal problems in the Lian-Yun Harbour and compared with observations and measurements in soil-wave tank with satisfactory results. The project supported by the National Science Foundation of China     

The horizontal intrusion layer of melt in a saturated porous medium

This article presents a theory of how the melt region advances as an intrusion layer along the top boundary of a solid phase-change material that is heated from the side. The phase-change material fills the pores of a solid matrix. We show that the thickness of the horizontal melt layer increases as x^3/5, where x is the horizontal distance measured by from the leading edge of the layer. The total length of the intrusion layer increases as t^3/4, and as T_max^5/4. Finite-difference simulations of convection melting in the Darcy-Rayleigh number range of 200–800 agree with the theoretical results. We also show that in a rectangular porous medium heated from the side, the size of the entire melt region is dominated by the melting contributed by the horizontal intrusion layer, if the time is great enough so that the group (Ste Fo)^3/4 is greater than 1.     

Thermohaline instability in a rotating anisotropic porous medium

The onset of thermohaline convection in an anisotropic rotating porous layer of infinite horizontal extent is investigated. Numerical computations are made assuming horizontal isotropy in permeability. It is observed that (i) for certain parameters a bottom-heavy arrangement destabilises a rotating anisotropic porous layer, (ii) the lower the anisotropy parameter, the higher the range of bottom-heavy solute gradient for which there is destabilisation, (iii) increase in the anisotropy parameter stabilises the system, and (iv) for some values of the parameters rotation destabilises the system, though in general, it has a stabilising effect.     

Plane steady-state incompressible fluid flow through a porous medium with a nonlinear resistance laws in the presence of a sink

Plane nonlinear fluid flows through a porous medium which simulate a sink located at the same distance from the roof and floor of the stratum for two nonlinear flow laws are constructed. The following flow laws are taken: a power law and a law of special form reducing to analytic functions in the hodograph plane.     

Investigation of convection in a horizontal cylindrical layer of a saturated porous medium

The structures of the convective motions and the nature of the heat transfer in a horizontal cylindrical layer are studied numerically for the Forchheimer model of a porous medium in the Boussinesq approximation. New asymmetric solutions of the equations of convection flow through a porous medium are found. Their development, domains of existence, and stability are investigated. One consists of a multivortex structure with asymmetric vortices in the near-polar region. Another asymmetric solution is realized at large Grashof numbers in the form of a convective plume deflected from the vertical. The threshold Grashof number of formation of the asymmetric motions depends on the Prandtl number and the cylindrical layer thickness.     

Viscous coupling in a model porous medium geometry: Effect of fluid contact area

We study the effect of fluid contact area on viscous coupling in the parallel flow of immiscible fluids in a porous media geometry. We consider flow on opposite sides of a planar interface, consisting of alternating solid and open (slit) segments. We use the analytical solution of Tio and Sadhal [15] to derive explicit expressions for viscous coupling in terms of the fractional area of contact between the fluids and the viscosity ratio,M. ForM=1, the coefficient matrix obtained is symmetric showing that Onsager's relations are satisfied. In this case, the resulting viscous coupling is typically very small, in agreement with recent experimental results. Lattice gas simulations forM=1 using theBGK model support the theoretical results and show that viscous coupling further diminishes as the wall thickness increases. Assuming the same configuration, analytical results are next derived forM1. The results confirm an existing reciprocity relation between the off-diagonal terms. Viscous coupling remains small.     

Numerical and analytical investigations of thermosolutal instability in rotating Rivlin-Ericksen fluid in porous medium with Hall current

Numerical and analytical investigations of the thermosolutal instability in a viscoelastic Rivlin-Ericksen fluid are carried out in the presence of a uniform vertical magnetic field to include the Hall current with a uniform angular velocity in a porous medium. For stationary convection, the stable solute gradient parameter and the rotation have stabilizing effects on the system, whereas the magnetic field and the medium permeability have stabilizing or destabilizing effects on the system under certain conditions. The Hall current in the presence of rotation has stabilizing effects for sufficiently large Taylor numbers, whereas in the absence of rotation, the Hall current always has destabilizing effects. These effects have also been shown graphically. The viscoelastic effects disappear for stationary convection. The stable solute parameter, the rotation, the medium permeability, the magnetic field parameter, the Hall current, and the vis-coelasticity introduce oscillatory modes into the system, which are non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.     

The inertial effect on the natural convection flow within a fluid-saturated porous medium

The general momentum equation for fluid flow within a porous medium is supposedly valid for any fluid-porous medium configuration. One of the main concerns of using the general equations refers to the inclusion of both inertia terms, namely, the convective inertia term and the Forchheimer term. In this study, we go beyond the important discussion about the correctness of including both terms in the general momentum equations by focusing upon the effect of the convective inertia term on the heat transfer results. The fluid-porous medium system considered here is a cavity bounded by solid surfaces with vertical walls maintained at constant but different temperatures. The natural convection problem is solved numerically, and the results are compared with a general theory developed by using the method of scale analysis. It is demonstrated that the convective inertia term effect upon the heat transfer results is minor for 0.01 ≤ Pr ≤ 1, 10 ≤ Ra_D ≤ 10⁴, 10⁻⁸ ≤ Da ≤ 10⁻², and porosities 0.4 and 0.8. It is also shown that, contrary to the general belief, the convective inertial effect upon the heat transfer within the cavity is minimized when the Prandtl number is reduced.     

Thermal instability of a porous medium with relaxation and inertia in the presence of Hall effects

Summary  The thermal instability of a Rivlin–Ericksen fluid in a porous medium is considered in the presence of a uniform vertical magnetic field to include the effect of Hall currents. For the case of stationary convection, the magnetic field has a stabilizing effect on the system, whereas the Hall current has a destabilizing effect on the system. The medium permeability has both stabilizing and destabilizing effects, depending on the Hall parameter M. The kinematic viscoelasticity has no effect on stationary convection. Graphs have been plotted by giving numerical values to the parameters, to depict the stability characteristics. The magnetic field (and corresponding Hall currents) introduces oscillatory modes in the system, which would be nonexistent in their absence. The sufficient conditions for the nonexistence of overstability are also obtained. Received 20 May 1999; accepted for publication 8 March 2000