The Horton–Rogers–Lapwood Problem Revisited: The Effect of Pressure Work

The problem of the onset of convective roll instabilities in a horizontal porous layer with isothermal boundaries at unequal temperatures, well known as the Horton–Rogers–Lapwood problem, is revisited including the effect of pressure work and viscous dissipation in the local energy balance. A linear stability analysis of rolls disturbances is performed. The analysis shows that, while the contribution of viscous dissipation is ineffective, the contribution of the pressure work may be important. The condition of marginal stability is investigated by adopting two solution procedures: method of weighted residuals and explicit Runge–Kutta method. The pressure work term in the energy balance yields an increase of the value of the Darcy–Rayleigh number at marginal stability. In other words, the effect of pressure work is a stabilizing one. Furthermore, while the critical value of the Darcy– Rayleigh number may be considerably affected by the pressure work contribution, the critical value of the wave number is affected only in rather extreme cases, i.e. for very high values of the Gebhart number. A nonlinear stability analysis is also performed pointing out that the joint effects of viscous dissipation and pressure work result in a reduction of the excess Nusselt number due to convection, when the Darcy–Rayleigh number is replaced by the superadiabatic Darcy–Rayleigh number.     

Combined effect of thermal modulation and rotation on the onset of stationary convection in a porous layer

The stability of a fluid-saturated horizontal rotating porous layer subjected to time-periodic temperature modulation is investigated when the condition for the principle of exchange of stabilities is valid. The linear stability analysis is used to study the effect of infinitesimal disturbances. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Darcy–Rayleigh number and wavenumber. The shift in critical Darcy–Rayleigh number is calculated as a function of frequency of modulation, Taylor number, and Darcy–Prandtl number. It is established that the convection can be advanced by the low frequency in-phase and lower-wall temperature modulation, where as delayed by the out-of-phase modulation. The effect of Taylor number and Darcy–Prandtl number on the stability of the system is also discussed. We found that by proper tuning of modulation frequency, Taylor number, and Darcy–Prandtl number it is possible to advance or delay the onset of convection.     

Instability of Hadley–Prats Flow with Viscous Heating in a Horizontal Porous Layer

The onset of convective rolls instability in a horizontal porous layer subject to a basic temperature gradient inclined with respect to gravity is investigated. The basic velocity has a linear profile with a non-vanishing mass flow rate, i.e., it is the superposition of a Hadley-type flow and a uniform flow. The influence of the viscous heating contribution on the critical conditions for the onset of the instability is assessed. There are four governing parameters: a horizontal Rayleigh number and a vertical Rayleigh number defining the intensity of the inclined temperature gradient, a Péclet number associated with the basic horizontal flow rate, and a Gebhart number associated with the viscous dissipation effect. The critical wave number and the critical vertical Rayleigh number are evaluated for assigned values of the horizontal Rayleigh number, of the Péclet number, and of the Gebhart number. The linear stability analysis is performed with reference either to transverse or to longitudinal roll disturbances. It is shown that generally the longitudinal rolls represent the preferred mode of instability.     

Stability of Flow with Viscous Dissipation in a Horizontal Porous Layer with an Open Boundary Having a Prescribed Temperature Gradient

A buoyancy-induced stationary flow with viscous dissipation in a horizontal porous layer is investigated. The lower boundary surface is impermeable and subject to a uniform heat flux. The upper open boundary has a prescribed, linearly varying, temperature distribution. The buoyancy-induced basic velocity profile is parallel and non-uniform. The linear stability of this basic solution is analysed numerically by solving the disturbance equations for oblique rolls arbitrarily oriented with respect to the basic velocity field. The onset conditions of thermal instability are governed by the Rayleigh number associated with the prescribed wall heat flux at the lower boundary, by the horizontal Rayleigh number associated with the imposed temperature gradient on the upper open boundary, and by the Gebhart number associated with the effect of viscous dissipation. The critical value of the Rayleigh number for the onset of the thermal instability is evaluated as a function of the horizontal Rayleigh number and of the Gebhart number. It is shown that the longitudinal rolls, having axis parallel to the basic velocity, are the most unstable in all the cases examined. Moreover, the imposed horizontal temperature gradient tends to stabilise the basic flow, while the viscous dissipation turns out to have a destabilising effect.     

Mixed convection with viscous dissipation in an inclined porous channel with isoflux impermeable walls

Combined forced and free convection flow in a fluid saturated inclined plane channel is investigated by taking into account the effect of viscous dissipation. Steady parallel flow is considered assuming that the temperature gradient in the parallel flow direction is constant, and the channel walls are subject to uniform symmetric heat fluxes. Two possible formulations of the Darcy–Boussinesq scheme are considered, based on two different choices of the reference temperature for modelling buoyancy. The first choice is a constant temperature, while the second is a streamwise changing temperature. It is shown that both approaches substantially agree in the formulation of the balance equations for the range of values of the Darcy–Rayleigh number such that viscous dissipation is important. The boundary value problem is solved analytically for any tilt angle, revealing that it admits dual solutions for assigned values of the governing parameters. The rather important effect of viscous dissipation in the special case of adiabatic channel walls is outlined. E. Magyari is on leave from Institute of Building Technology, ETH—Zürich     

Effect of anisotropy of porous media on the onset of buoyancy-driven convection

A theoretical analysis of buoyancy-driven instability under transient basic fields is conducted in an initially quiescent, fluid-saturated, horizontal porous layer. Darcy’s law is used to explain characteristics of fluid motion and the anisotropy of permeability is considered. Under the Boussinesq approximation and the principle of exchange of stabilities, the stability equations are derived by using the linear stability theory and the energy method. The linear stability equations are analyzed numerically by using the frozen-time model and the linear amplification theory and the global stability limits are obtained numerically from the energy method. For the various anisotropic ratios, the critical times are predicted as a function of the Darcy–Rayleigh number and the critical Darcy–Rayleigh number is also obtained. The present predictions are compared each another and with existing theoretical ones.     

Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid Past a Vertical Plate Immersed in a Porous Medium

The effects of viscous dissipation on unsteady free convection from an isothermal vertical flat plate in a fluid saturated porous medium are examined numerically. The Darcy–Brinkman–Forchheimer model is employed to describe the flow field. A new model of viscous dissipation is used for the Darcy–Brinkman–Forchheimer model of porous media. The simultaneous development of the momentum and thermal boundary layers are obtained by using a finite difference method. Boundary layer and Boussinesq approximation have been incorporated. Numerical calculations are carried out for various parameters entering into the problem. Velocity and temperature profiles as well as local friction factor and local Nusselt number are shown graphically. It is found that as time approaches infinity, the values of friction factor and heat transfer coefficient approach steady state.     

The Effect of Strong Heterogeneity and Strong Throughflow on the Onset of Convection in a Porous Medium: Non-Periodic Global Variation

The effect of strong throughflow and strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is investigated. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. A computer package has been extended to deal with the case of vertical throughflow.     

Effect of Thermal/Gravity Modulation on the Onset of Convection in a Maxwell Fluid Saturated Porous Layer

The effect of thermal/gravity modulation on the onset of convection in a Maxwell fluid saturated porous layer is investigated by a linear stability analysis. Modified Darcy–Maxwell model is used to describe the fluid motion. The regular perturbation method based on the small amplitude of modulation is employed to compute the critical Rayleigh number and the corresponding wavenumber. The stability of the system characterized by a correction Rayleigh number is calculated as a function of the viscoelastic parameter, Darcy–Prandtl number, normalized porosity, and the frequency of modulation. It is found that the low frequency symmetric thermal modulation is destabilizing while moderate and high frequency symmetric modulation is always stabilizing. The asymmetric modulation and lower wall temperature modulations are, in general, stabilizing while the system becomes unstable for large values of Darcy–Prandtl number and for small frequencies. It is shown that in general the gravity modulation produces a stabilizing effect on the onset of convection for moderate and high frequency. The small frequency gravity modulation is found to have destabilizing effect on the stability of the system.     

Effect of Vertical Modulation on the Onset of Filtration Convection

The present paper examines the effect of vertical harmonic vibration on the onset of convection in an infinite horizontal layer of fluid saturating a porous medium. A constant temperature distribution is assigned on the rigid boundaries, so that there exists a vertical temperature gradient. The mathematical model is described by equations of filtration convection in the Darcy–Oberbeck–Boussinesq approximation. The linear stability analysis for the quasi-equilibrium solution is performed using Floquet theory. Employment of the method of continued fractions allows derivation of the dispersion equation for the Floquet exponent σ in an explicit form. The neutral curves of the Rayleigh number Ra versus horizontal wave number α for the synchronous and subharmonic resonant modes are constructed for different values of frequency Ω and amplitude A of vibration. Asymptotic formulas for these curves are derived for large values of Ω using the method of averaging, and, for small values of Ω, using the WKB method. It is shown that, at some finite frequencies of vibration, there exist regions of parametric instability. Investigations carried out in the paper demonstrate that, depending on the governing parameters of the problem, vertical vibration can significantly affect the stability of the system by increasing or decreasing its susceptibility to convection.      

The Effect of Strong Heterogeneity and Strong Throughflow on the Onset of Convection in a Porous Medium: Periodic and Localized Variation

The effect of strong throughflow and strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is investigated with the aid of a computer package. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. Previous work on the case of non-periodic global variation is now extended to the case of either periodic variation or localized variation.     

The Onset of Convection in a Heterogeneous Porous Medium with Vertical Throughflow

The effect of vertical heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a non-uniform basic temperature gradient resulting from vertical throughflow, is studied analytically using linear stability theory. It is found that, to first order, a linear variation of the reciprocal of permeability with depth has no effect on the critical value of the Rayleigh number Ra _c based on the harmonic mean of the permeability, but a quadratic variation increasing in the upwards direction leads to a reduction in Ra _c.     

Linear Stability of the Double-Diffusive Convection in a Horizontal Porous Layer with Open Top: Soret and Viscous Dissipation Effects

The linear stability of the double-diffusive convection in a horizontal porous layer is studied considering the upper boundary to be open. A horizontal temperature gradient is applied along the upper boundary. It is assumed that the viscous dissipation and Soret effect are significant in the medium. The governing parameters are horizontal Rayleigh number (\(Ra_\mathrm{H}\)), solutal Rayleigh number (\(Ra_\mathrm{S}\)), Lewis number (Le), Gebhart number (Ge) and Soret parameter (Sr). The Rayleigh number (Ra) corresponding to the applied heat flux at the bottom boundary is considered as the eigenvalue. The influence of the solutal gradient caused due to the thermal diffusion on the double-diffusive instability is investigated by varying the Soret parameter. A horizontal basic flow is induced by the applied horizontal temperature gradient. The stability of this basic flow is analyzed by calculating the critical Rayleigh number (\(Ra_\mathrm{cr}\)) using the Runge–Kutta scheme accompanied by the Shooting method. The longitudinal rolls are more unstable except for some special cases. The Soret parameter has a significant effect on the stability of the flow when the upper boundary is at constant pressure. The critical Rayleigh number is decreasing in the presence of viscous dissipation except for some positive values of the Soret parameter. How a change in Soret parameter is attributing to the convective rolls is presented.     

Onset of Convection in a Porous Medium with Strong Vertical Throughflow

The problem for determining the critical Rayleigh number for the onset of convection in a horizontal porous layer with vertical throughflow is re-examined with the aim of obtaining analytical formulas applicable in the cases of weak and strong throughflow. For the case of strong throughflow an asymptotic analysis is performed.     

Effect of Thermal Non-Equilibrium on Convective Instability in a Ferromagnetic Fluid-Saturated Porous Medium

The effect of local thermal non-equilibrium (LTNE) on the onset of thermomagnetic convection in a ferromagnetic fluid-saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated. A modified Forchheimer-extended Darcy equation is employed to describe the flow in the porous medium, and a two-field model is used for temperature representing the solid and fluid phases separately. It is found that both the critical Darcy–Rayleigh number and the corresponding wave number are modified by the LTNE effects. Asymptotic solutions for both small and large values of scaled interphase heat transfer coefficient H _t are presented and compared with those computed numerically. An excellent agreement is obtained between the asymptotic and the numerical results. Besides, the influence of magnetic parameters on the instability of the system is also discussed. The available results in the literature are recovered as particular cases from the present study.     

Estimating the Hydraulic Conductivity of Two-Dimensional Fracture Networks Using Network Geometric Properties

Based on a modified Darcy–Brinkman–Maxwell model, a linear stability analysis of a Maxwell fluid in a horizontal porous layer heated from below by a constant flux is carried out. The non-oscillatory instability and oscillatory instability with different hydrodynamic boundaries such as rigid and free surfaces at the bottom are studied. Compared with the rigid surface cases, onset of fluid motion due to non-oscillatory instability and oscillatory instability is found to occur both more easily for the system with a free bottom surface. The critical Rayleigh number for onset of fluid motion due to non-oscillatory instability is lower with a constant flux heating bottom than with an isothermal heating bottom, but the result due to oscillatory instability is in contrast. The effects of the Darcy number, the relaxation time, and the Prandtl number on the critical Rayleigh number are also discussed.     

The Effects of Combined Horizontal and Vertical Heterogeneity on the Onset of Convection in a Porous Medium with Vertical Throughflow

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, with weak vertical throughflow, are studied analytically for the case of weak heterogeneity. It is found that when the boundary conditions at the upper and lower boundaries are symmetric, the throughflow magnitude and the permeability and conductivity gradients enter the expression for the critical Rayleigh number at second order. The throughflow on its own is stabilizing but the combination of throughflow and heterogeneity may be either stabilizing or destabilizing.     

Linear and nonlinear stability analysis of binary Maxwell fluid convection in a porous medium

The stability analysis of the quiescent state in a Maxwell fluid-saturated densely packed porous medium subject to vertical concentration and temperature gradients is presented. A single phase model with local thermal equilibrium between the porous matrix and the Maxwell fluid is assumed. The critical Darcy–Rayleigh numbers and the corresponding wave numbers for the onset of stationary and oscillatory convection are determined. A Lorenz like system is obtained for weakly nonlinear stability analysis.     

Rotating Brinkman–Lapwood Convection with Modulation

The purpose of this article is to analyze, theoretically, the effect of modulation on rotating Brinkman–Lapwood convection, i.e., buoyancy-driven convection in a sparse porous medium subjected to rotation. Darcy–Brinkman momentum equation with Coriolis term has been used to describe the flow. The system is considered rotating about an axis with non-uniform rotation speed. In particular, we assume that the rotation speed is varying sinusoidally with time. A linear stability analysis has been performed to find the critical Rayleigh number in modulated case. The effect of modulated rotation speed is found to have a stabilizing effect on the onset of convection for different values of modulation frequency and the other physical parameters involved.     

Stability of thermal convection between non-uniformly heated plates

The effect of horizontal as well as vertical temperature gradients on the stability of natural convection in a thin horizontal layer of viscous, incompressible fluid is studied on the basis of linear theory. The boundaries are taken to be rigid, perfectly thermally conducting, having prescribed temperatures and the horizontal temperature gradient is assumed to be small. It is found that for Prandtl number greater than 0.13, the critical Rayleigh number is always larger than that for the corresponding Benard problem. The preferred mode of disturbance is stationary and will be a transverse roll (having axes normal to the basic flow) or a longitudinal roll (having axes aligned in the direction of the basic flow) depending on whether the Prandtl number is less or larger than 1.7. Finally, it is shown that the instability is of thermal origin.