Effect of Internal Heat Generation on the Onset of Marangoni Convection in a Fluid Layer Overlying a Layer of an Anisotropic Porous Medium

Linear stability analysis has been performed to investigate the effect of internal heat generation on the criterion for the onset of Marangoni convection in a two-layer system comprising an incompressible fluid-saturated anisotropic porous layer over which lies a layer of the same fluid. The upper non-deformable free surface and the lower rigid surface are assumed to be insulated to temperature perturbations. The fluid flow in the porous layer is governed by the modified Darcy equation and the Beavers–Joseph empirical slip condition is employed at the interface between the two layers. The resulting eigenvalue problem is solved exactly. Besides, analytical expression for the critical Marangoni number is also obtained by using regular perturbation technique with wave number as a perturbation parameter. The effect of internal heating in the porous layer alone exhibits more stabilizing effect on the system compared to its presence in both fluid and porous layers and the system is least stable if the internal heating is in fluid layer alone. It is found that an increase in the value of mechanical anisotropy parameter is to hasten the onset of Marangoni convection while an opposite trend is noticed with increasing thermal anisotropy parameter. Besides, the possibilities of controlling (suppress or augment) Marangoni convection is discussed in detail.     

Throughflow Effects on Penetrative Convection in Superposed Fluid and Porous Layers

The effect of vertical throughflow on the onset of penetrative convection simulated via internal heating in a two-layer system in which a layer of fluid overlies and saturates a layer of porous medium is studied. Flow in the porous medium is governed by Forchheimer-extended Darcy equation, and Beavers?CJoseph slip condition is applied at the interface between the fluid and the porous layers. The boundaries are considered to be rigid, however permeable, and insulated to temperature perturbations. The eigenvalue problem is solved using a regular perturbation technique with wave number as a perturbation parameter. The ratio of fluid layer thickness to porous layer thickness, ??, the direction of throughflow, and the presence of volumetric internal heat source in fluid and/or porous layer play a decisive role on the stability characteristics of the system. In addition, the influence of Prandtl number arising due to throughflow is also emphasized on the stability of the system. It is observed that both stabilizing and destabilizing factors can be enhanced because of the simultaneous presence of a volumetric heat source and vertical throughflow so that a more precise control (suppress or augment) of thermal convective instability in a layer of fluid or porous medium is possible.     

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.     

On the stability of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid

The stability of the conduction regime of natural convection in a porous vertical slab saturated with an Oldroyd-B fluid has been studied. A modified Darcy’s law is utilized to describe the flow in a porous medium. The eigenvalue problem is solved using Chebyshev collocation method and the critical Darcy–Rayleigh number with respect to the wave number is extracted for different values of physical parameters. Despite the basic state being the same for Newtonian and Oldroyd-B fluids, it is observed that the basic flow is unstable for viscoelastic fluids—a result of contrast compared to Newtonian as well as for power-law fluids. It is found that the viscoelasticity parameters exhibit both stabilizing and destabilizing influence on the system. Increase in the value of strain retardation parameter \(\Lambda _2 \) portrays stabilizing influence on the system while increasing stress relaxation parameter \(\Lambda _1\) displays an opposite trend. Also, the effect of increasing ratio of heat capacities is to delay the onset of instability. The results for Maxwell fluid obtained as a particular case from the present study indicate that the system is more unstable compared to Oldroyd-B fluid.     

Effect of Thermal Modulation on the Onset of Convection in Walters B Viscoelastic Fluid-Saturated Porous Medium

The linear stability of Walters B viscoelastic fluid-saturated horizontal porous layer is examined theoretically when the walls of the porous layer are subjected to time-periodic temperature modulation. Three types of boundary temperature modulations are considered namely, symmetric, asymmetric, and only the lower wall temperature is modulated while the upper wall is held at constant temperature. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Rayleigh number and the corresponding wave number. The shift in critical Rayleigh number is calculated as a function of modulation frequency, viscoelastic parameter, and Prandtl number. The effect of all three types of modulations is found to be destabilizing as compared to the unmodulated system. This result is in contrast to the system with other types of fluids. Besides, the influence of physical parameters on the control of convective instability of the system is discussed.     

多孔介质内黏弹性流体的热对流稳定性研究

基于修正的Darcy模型, 介绍了多孔介质内黏弹性流体热对流稳定性研究的现状和主要进展. 通过线性稳定性理论, 分析计算多孔介质几何形状(水平多孔介质层、多孔圆柱以及多孔方腔)、热边界条件(底部等温加热、底部等热流加热、底部对流换热以及顶部自由开口边界)、黏弹性流体的流动模型(Darcy-Jeffrey, Darcy-Brinkman-Oldroyd以及Darcy-Brinkman -Maxwell模型)、局部热非平衡效应以及旋转效应对黏弹性流体热对流失稳的临界Rayleigh数的影响. 利用弱非线性分析方法, 揭示失稳临界点附近热对流流动的分叉情况, 以及失稳临界点附近黏弹性流体换热Nusselt数的解析表达式. 采用数值模拟方法, 研究高Rayleigh数下黏弹性流体换热Nusselt数和流场的演化规律,分析各参数对黏弹性流体热对流失稳和对流换热速率的影响.主要结果:(1)流体的黏弹性能够促进振荡对流的发生;(2)旋转效应、流体与多孔介质间的传热能够抑制黏弹性流体的热对流失稳;(3)在临界Rayleigh数附近,静态对流分叉解是超临界稳定的, 而振荡对流分叉可能是超临界或者亚临界的,主要取决于流体的黏弹性参数、Prandtl数以及Darcy数;(4)随着Rayleigh数的增加,热对流的流场从单个涡胞逐渐演化为多个不规则单元涡胞, 最后发展为混沌状态.      

Stability of steady plane-parallel convective motion of a chemically active medium

A study is made of a vertical plane layer of reacting fluid whose boundaries are maintained at constant equal temperatures. As a result of heating due to a chemical reaction of zeroth order taking place in the fluid a steady plane-parallel convective flow develops in the layer, and if the internal heat release is sufficiently intense this can become unstable. The linear stability of this motion has hitherto been considered only in the hydro-dynamic formulation [1], in which one can ignore the thermal perturbations and their influence on the development of the hydrodynamic perturbations (the region of small Prandtl numbers). In the present paper, the stability boundary is determined for arbitrary values of the Prandtl number and the Frank-Kamenetskii parameter FK characterizing the steady plane-parallel regime. An important difference between this flow and the types of convective motion hitherto studied [2] is that the basic planeparallel flow of the reacting medium is possible only in a definite range of the parameter FK: At values of the parameter exceeding a critical value, there is a thermal explosion — abrupt strong heating of the fluid. This is due to the essentially nonlinear dependence of the heat release of a chemical reaction on the temperature.     

Nonlinear convection induced by inclined thermal and solutal gradients with mass flow

The energy method is developed for the convection problem induced by inclined thermal and solutal gradients, with horizontal mass flow, in a horizontal layer of a saturated porous medium. A non-linear stability analysis is performed and compound matrix method is employed for numerical calculations. For representative parameter values the critical vertical thermal Rayleigh number and wave number are calculated. It is noted that the effect of horizontal thermal or solutal gradient is to switch from stabilizing to destabilizing as their magnitude increases, for zero or small values of mass flow rate. For higher values of mass flow rate the effect is always destabilizing. It is also noted that the horizontal concentration gradient has a stronger destabilizing effect as compared to the horizontal temperature gradient. Received March 18, 2000     

Stability of free convection in a rotating porous layer distant from the axis of rotation

The stability and onset of convection in a rotating fluid saturated porous layer subject to a centrifugal body force and placed at an offset distance from the center of rotation is investigated analytically. The marginal stability criterion is established in terms of a critical centrifugal Rayleigh number and a critical wave number for different values of the parameter representing the dimensionless offset distance from the center of rotation. At the limit of an infinite distance from the center of rotation the results are identical to the convection resulting from heating a porous layer from below subject to the gravitational body force. At the other limit, when the parameter controlling the offset distance approaches zero, the results converge to previously found solutions for the convection in a porous layer adjacent to the axis of rotation. The results provide the stability map for all positive values of the parameter controlling the offset distance from the center of rotation, hence bridging the gap between the two extreme limit cases.     

Transient free convection with mass transfer MHD micropolar fluid in a porous plate by the network method

The transient problem of coupled heat and mass transfer of a micropolar fluid in magneto‐hydrodynamic free convection from a vertical infinite porous plate with an exponentially decaying heat generating considering the viscous dissipation and ohmic heating effects is studied. Joule heating must be considered when the viscous dissipation and the Prandtl number are large. The non‐dimensional equations for the conservation of mass, momentum, energy and concentration are solved by means a numerical technique based on electric analogy (network simulation method). This method provides the numerical response of the system by running the network in circuit resolution software with the solution to both transient and steady‐state problems at the same time, and its programming does not require manipulation of the sophisticated mathematical software that is inherent in other numerical methods. The effects of the material parameters, viscous dissipation, internal generation and Joule heating on velocity, angular momentum and temperature fields across the boundary layer are investigated. In addition, the skin‐friction coefficient, couple stress coefficient, Nusselt number and Sherwood number are shown in tabular form. The numerical results for velocity and temperature distributions of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. Copyright © 2007 John Wiley & Sons, Ltd.     

Linear and Nonlinear Stability Analysis of a Horton–Rogers–Lapwood Problem with an Internal Heat Source and Brinkman Effects

The effect of internal heat source on convection in a layer of fluid in a porous medium was analyzed using linear and nonlinear analysis, and boundaries are assumed to be stress-free and isothermal. Normal mode technique is used for linear analysis, and energy method is used for nonlinear stability analysis. The presence of heat generation leads to the possibility of the existence of a subcritical instability. Effects of increase of Darcy–Brinkman number and internal heat parameter on critical Rayleigh numbers were analyzed numerically using Chebyshev pseudospectral method.     

Thermal Instability in a Horizontal Porous Channel with Horizontal Through Flow and Symmetric Wall Heat Fluxes

The linear thermoconvective instability of the basic parallel flow in a plane and horizontal porous channel is investigated. The boundary walls are assumed to be impermeable and subject to symmetric and uniform heat fluxes. The wall heat fluxes produce either a net heating or a net cooling of the fluid saturated porous medium. A horizontal mass flow rate is externally impressed leading to a stationary basic state with a temperature gradient inclined to the vertical. A region of possibly unstable thermal stratification exists either in the lower half-channel (boundary heating), or in the upper half-channel (boundary cooling). The convective instability of the basic flow is governed by the Rayleigh number and by the Péclet number. In the case of boundary heating, the thermal instability arises when the Rayleigh number exceeds its critical value, that depends on the Péclet number. The change of the critical Rayleigh number as a function of the Péclet number is determined numerically for arbitrary normal modes oblique to the basic flow direction. The most dangerous modes are the longitudinal rolls, with a wave vector perpendicular to the basic velocity. There exists a minimum value of the Péclet number, 19.1971, below which no linear instability is detected.     

Convective heat transfer in porous media with columnar structure

An analysis is made of convective heat transport, produced by uniform heating from below, in a horizontal layer of a porous medium consisting of vertical slabs or columns of different permeabilities. Estimates of the heat flux are made on the assumption that flow in one column does not interact with flow in adjacent columns. The results are compared with those for a homogeneous layer, for which previous work is reviewed. It is found that an inhomogeneous layer transports less heat than a homogeneous layer for which the mean Rayleigh number is the same, if the Rayleigh number is supercritical throughout the layer. If the Rayleigh number is subcritical in part of the layer, the inhomogeneous layer may transport more heat than the equivalent homogeneous layer.     

Convective Roll Instabilities of Vertical Throughflow with Viscous Dissipation in a Horizontal Porous Layer

The vertical throughflow with viscous dissipation in a horizontal porous layer is studied. The horizontal plane boundaries are assumed to be isothermal with unequal temperatures and bottom heating. A basic stationary solution of the governing equations with a uniform vertical velocity field (throughflow) is determined. The temperature field in the basic solution depends only on the vertical coordinate. Departures from the linear heat conduction profile are displayed by the temperature distribution due to the forced convection effect and to the viscous dissipation effect. A linear stability analysis of the basic solution is carried out in order to determine the conditions for the onset of convective rolls. The critical values of the wave number and of the Darcy–Rayleigh number are determined numerically by the fourth-order Runge–Kutta method. It is shown that, although generally weak, the effect of viscous dissipation yields an increase of the critical value of the Darcy–Rayleigh number for downward throughflow and a decrease in the case of upward throughflow. Finally, the limiting case of a vanishing boundary temperature difference is discussed.     

The stability of a flow in a side-heated vertical layer with internal heat sources

The linear stability of a combined convective flow in a side-heated plane vertical layer of incompressible fluid is considered. The layer contains uniformly distributed heat sources. The steadystate flow is a superposition of the Gershuni convective flow and a flow induced by the action of the internal heat sources. The stability maps of the combined flow are constructed for different values of the Prandtl number. It is found that the internal sources damp the action of the mechanisms of the Gershuni flow crisis. The lateral heating may result in both the stabilization and the destabilization of the flow caused by the action of internal heat sources.     

Conjugate Natural Convection in a Porous Enclosure with Non-Uniform Heat Generation

Effects of a conductive wall on natural convection in a square porous enclosure having internal heating at a rate proportional to a power of temperature difference is studied numerically in this article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (0 ???Ra ???1000), the internal heating and the local exponent parameters (0 ????? ???5), (1 ????? ???3), the wall to porous thermal conductivity ratio (0.44 ???Kr ???9.9) and the ratio of wall thickness to its width (0.02 ???D ???0.5). The results are presented to show the effect of these parameters on the fluid flow and heat transfer characteristics. It is found a strong internal heating can generate significant maximum fluid temperature more than the conductive solid wall. Increasing value thermal conductivity ratio and/or decreasing the thickness of solid wall can increase the maximum fluid temperature. It is also found that at very low Rayleigh number, the heat transfer across the porous enclosure remain stable for any values of the thermal conductivity ratio.     

On non-linear magneto convection in a mushy layer with joule heating

This paper studies the flow pattern of non-linear magneto convection that can be realized in a horizontal mushy layer and in the presence of joule heating, which is the amount of heat produced by the induced magnetic field. We consider the appropriate system of equations and the associated boundary conditions for the flow in the mushy layer subjected to a vertical magnetic field of uniform strength. Under certain assumptions and conditions, we determine the stable finite-amplitude solutions of the resulting system using a perturbation approach and stability analysis. We find, in particular, that for a wide range of values of the joule heating parameter and sufficiently small amplitude of the flow, the only stable convective flow is in the form of subcritical down-hexagons with down-flow at the cells' centers and up-flow at the cells' boundaries. This result is in sharp contrast to the case in the absence of joule heating where instead the subcritical up-hexagons with up-flow at the cells' centers and down-flow at the cells' boundaries can be stable. In the presence of joule heating the stable subcritical down-hexagons were found to be enhanced with increasing the strength of the externally imposed magnetic field.     

Forced convection of a power-law fluid in a porous channel – numerical solutions

A detailed numerical study of laminar forced convection in a porous channel which contains a fibrous medium saturated with a power-law fluid was performed. Hydrodynamic and heat transfer results are presented for a configuration that has uniform heat flux or uniform temperature heating at the walls. The flow in the porous medium was modeled using the modified Brinkman-Forchheimer-extended Darcy model for power law fluids in which the non-Darcy effects of inertia and boundary were considered. Parametric studies were conducted to examine the effects of Darcy number, power law index, inertia parameter and Prandtl number. The results indicate that when the power law index is decreased, the velocity gradient near the walls increases but these effects are reduced gradually as the Darcy number decreases until the Darcy regime (Da≤10⁻⁶) is reached in which case the effects of power law index become negligible. As the power law index is decreased, the thermal boundary layer thickness decreases significantly only in the non-Darcy regime. Consequently, as the power law index decreases, the fully developed Nusselt number increases considerably in the non-Darcy regime whereas in the Darcy regime the change in Nusselt number is very small. As the Prandtl number increases, the local Nusselt number increases and this effect is more significant for shear thinning fluids (n<1.0). Received on 2 March 1998     

Double-diffusive convection induced by selective absorption of radiation in a fluid overlying a porous layer

A linear instability analysis for the inception of double-diffusive convection with a concentration based internal heat source is presented. The system encompasses a layer of fluid which lies above a porous layer saturated with the same fluid. Detailed stability characteristics results are presented for key physical parameters including the solute Rayleigh number, depth ratio of the fluid to porous layer and strength of radiative heating.     

Flow and heat transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet subject to a transverse magnetic field

An analysis is performed for flow and heat transfer of a steady laminar boundary layer flow of an electrically conducting fluid of second grade in a porous medium subject to a transverse uniform magnetic field past a semi-infinite stretching sheet with power-law surface temperature or power-law surface heat flux. The effects of viscous dissipation, internal heat generation of absorption and work done due to deformation are considered in the energy equation. The variations of surface temperature gradient for the prescribed surface temperature case (PST) and surface temperature for the prescribed heat flux case (PHF) with various parameters are tabulated. The asymptotic expansions of the solutions for large Prandtl number are also given for the two heating conditions. It is shown that, when the Eckert number is large enough, the heat flow may transfer from the fluid to the wall rather than from the wall to the fluid when Eckert number is small. A physical explanation is given for this phenomenon.