The onset of double diffusive convection in a sparsely packed porous layer using a thermal non-equilibrium model

We examine the effect of local thermal non-equilibrium on double diffusive convection in a fluid-saturated sparsely packed porous layer heated from below and cooled from above, using both linear and nonlinear stability analyses. The Brinkman model is employed as the momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for the energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that a small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusion that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, ratio of diffusivities, Vadasz number and Darcy number on the stability of the system is investigated. The nonlinear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out.     

On the stability of double diffusive convection in a porous layer with throughflow

Summary The effect of throughflow on the stability of double diffusive convection in a porous layer is investigated for different types of hydrodynamic boundary conditions. The lower and upper boundaries are assumed to be insulating to temperature and concentration perturbations. The resulting eigenvalue problem is solved by the Galerkin technique. The curvature of the basic temperature as well as solute concentration gradients significantly affects the stability of the system. It is observed that, for a suitable choice of parametric values, Hopf bifurcation occurs always prior to direct bifurcation, and the throughflow alters the nature of bifurcation. In contrast to the single component system, it is found that throughflow is (a) destabilizing even if the lower and upper boundaries are of the same type, and (b) stabilizing as well as destabilizing, irrespective of its direction, when the boundaries are of different types.     

Triple diffusive convection in porous media

The presence of more than one chemical dissolved in fluid mixtures is very often requested for describing natural phenomena (contaminant transport, underground water flow, acid rain effects, worming of the stratosphere). In the present paper, a triple convective-diffusive fluid mixture saturating a porous horizontal layer, heated from below and salted from above and below, is studied. In closed forms, conditions sufficient for (i) inhibiting the onset of convection and (ii) guaranteeing the global nonlinear stability of the thermal conduction solution and the absence of subcritical instabilities are obtained.     

Boundary domain integral method for the study of double diffusive natural convection in porous media

The main purpose of this paper is to present a boundary domain integral method (BDIM) for the solution of natural convection in porous media driven by combining thermal and solutal buoyancy forces. The Brinkman extension of the classical Darcy equation is used for the momentum conservation equation. The numerical scheme was tested on a natural convection problem within a square porous cavity, where different temperature and concentration values are applied on the vertical walls, while the horizontal walls are adiabatic and impermeable. The results for different governing parameters (Rayleigh number, Darcy number, buoyancy ratio and Lewis number) are presented and compared with published work. There is a good agreement between those results obtained using the presented numerical scheme and reported studies using other numerical methods.     

Maximum density effects on natural convection in a vertical annulus filled with a non-Darcy porous medium

Summary This article numerically studies the problem of natural convection in a porous medium saturated with cold water, under density inversion, within a vertical annulus. In modeling the flow in the porous medium the non-Darcy effects, which include the Forchheimer inertia and Brinkman viscous effects are taken into account. The governing equations are solved numerically by the finite difference method using the modified strongly implicit procedure. The effects of the inversion parameter _m , radius ratioR ^*, aspect ratioAR, Forchheimer inertia parameter Fc/Pr, and Darcy number parameter Da on the heat transfer and fluid flow characteristics are discussed in detail. Results show that both the inversion parameter and radius ratio have a significant influence on the flow structure and heat transfer rate in the annulus. It is also found that the mean Nusselt number decreases as the Forchheimer inertia parameter or the Darcy number increases. Moreover, the results obtained here are also compared and favorably agree with numerical results and with experimental data.Nomenclature AR aspect ratio,H/L - d particle diameter - Da Darcy number,K/(L ²) - Fc Forchheimer number,K/L - g gravitational acceleration - H annulus height - K permeability - K transport property defined in Eq. (5) - L gap width,r _o –r _i - Nu _i , Nu _o local Nusselt number of the inner and outer cylinders, respectively - mean Nusselt number of the inner and outer cylinders, respectively - p pressure - Pr Prandtl number, / - q constant in Eq. (9) - r radial coordinate - R dimensionless radial coordinate,(r–r _i )/L - r _i radius of inner cylinder - r _o radius of outer cylinder - R ^* radius ratio,(r _o –r _i )/r _i - Ra Rayleigh number,K _m gL(T _h –T _c ) ^q / - T dimensional temperature - T _c dimensional temperature of inner cylinder - T _h dimensional temperature of outer cylinder - T _m temperature corresponding to the density maximum, 4.029325°C - u, v Darcian velocity components inr andz directions, respectively     

Non-Darcian effects on forced convection heat transfer over a flat plate in a highly porous medium

Summary The effects of inertia forces and the distance from the leading edge of the plate on the velocity and temperature fields as well as on the skin friction and heat transfer coefficients in the boundary layer flow over a semi-infinite flat plate embedded in a saturated porous medium of high porosity are studied. It is shown that the inertia forces have a significant influence on the flow characteristics in this problem.     

Double diffusive convection in a viscoelastic fluid-saturated porous layer using a thermal non-equilibrium model

The stability of a binary viscoelastic fluid-saturated porous layer which is heated from below is studied, where the fluid and solid phases are not in local thermal equilibrium. The modified Darcy-Oldroyd model is employed as a momentum equation, with the fluid and solid phase temperature fields modelled separately. It is found that the inter-phase heat transfer coefficient has a significant effect on the stability of the system. Competition between the processes of viscous relaxation and thermal diffusion cause the convection to set in through oscillatory rather than stationary instability, with the viscoelastic parameters inhibiting the onset of convection. In the case of weakly non-linear theory, both steady and unsteady cases are considered. In the unsteady case the transient behaviour of the Nusselt and Sherwood numbers is investigated. The effect of thermal non-equilibrium on heat and mass transfer is also discussed.     

Free convection effects on the oscillatory flow of a couple stress fluid through a porous medium

Summary Effects of free convection currents on the oscillatory flow of a polar fluid through a porous medium, which is bounded by a vertical plane surface of constant temperature, have been studied. The surface absorbs the fluid with a constant suction and the free stream velocity oscillates about a constant mean value. Analytical expressions for the velocity and the angular velocity fields have been obtained, using the regular perturbation technique. The effects of Grashof numberG; material parameters and ; Prandtl numberP; permeability parameterK and frequency parametern on the velocity and the angular velocity are discussed. The effects of cooling and heating of a polar fluid compared to a Newtonian fluid have also been discussed. The velocity of a polar fluid is found to decrease as compared to the Newtonian fluid.List of symbols C _p specific heat at constant pressure - g acceleration due to gravity - G Grashof number - K ⁺ permeability of the porous medium - K dimensionless permeability - P Prandtl number - t ⁺ time - t dimensionless time - T _w ⁺ mean temperature of the surface - T ⁺ temperature of the fluid - T ⁺ temperature of the fluid away from the surface - density of the fluid - viscosity - _r rotational viscosity - C _a ,C _d coefficients of couple stress viscosities - I a scalar constant of dimension equal to that of the moment of inertia of unit mass - x ⁺,y ⁺ coordinate system - u ⁺,v ⁺ velocity components in thex ⁺ andy ⁺ directions - u dimensionless velocity in thex ⁺-direction - ⁺ angular velocity component - dimensionless angular velocity - n ⁺ frequency of oscillations - n dimensionless frequency - perturbation parameter - U a constant velocity - u ₀ mean velocity - u ₁ fluctuating part of the velocity - ⁰ mean angular velocity - ¹ fluctuating part of the angular velocity - T ₀ mean temperature - T ₁ fluctuating part of the temperature - ⁰ coefficient of the volume expansion - kinematic viscosity - ^r rotational kinematic viscosity - , material parameters characterizing the polarity of the fluid - v ₀ suction velocity - density of the fluid far from the surface - y dimensionless coordinate normal to the surface     

Maximum density effects on natural convection from a discrete heater in a cavity filled with a porous medium

Summary. A numerical study of natural convection in a two-dimensional porous cavity saturated with water which possesses a density maximum in the vicinity of 3.98°C is carried out in the present paper. It is assumed that one of the cavity vertical walls is heated differentially by an isothermal discrete heater. The other vertical wall is cooled to a constant temperature, while the horizontal walls are adiabatic. Non-Boussinesq and Darcy models are used in the mathematical formulations. The effects of the location of the center of the discrete isothermal heater, the length of the heater and the aspect ratio of the porous cavity are studied for a wide range of modified Rayleigh numbers (50 Ra1000). For long heater and low Ra, it is required to place the heater in the middle of the vertical wall in order to get maximum heat transfer. For short heater and high Ra, fixing the heater in the upper half of the vertical wall leads to an enhancement of the heat transfer. When the aspect ratio A=0.5, the heater is far from the cold wall and hence more thermal resistance in this case reduces the average Nusselt number. On the other hand, higher values of average Nusselt numbers are found for specified Ra and L when the aspect ratio A is increased.     

Mixed convection boundary-layer on a vertical cylinder embedded in a saturated porous medium

Summary The mixed convection boundary layer on a vertical circular cylinder embedded in a saturated porous medium is considered. It is found that the flow depends on the parameter =R _a /P _e whereR _a andP _e are the Rayleigh number and Peclet number respectively. gives the ratio of the velocity scale for free convection to that for the forced convection. When is small the solution is, to a first approximation, obtained by a known heat conduction problem. The flow near the leading edge is considered and it is shown that a solution is possible only for ₀, ₀–1.354, and that a stable finite-difference solution away from the leading edge can be obtained only if –1; with <–1 there is a region of reversed flow near the cylinder. The finite-difference scheme is unable to give a satisfactory solution at very large distances from the leading edge, and to overcome this difficulty a simple approximate solution is developed. This solution shows that at large distances along the cyclinder, forced convection eventually becomes the dominant mechanism for heat transfer. This is also confirmed by an asymptotic solution of the full boundarylayer problem.Nomenclature a radius of cylinder - g acceleration of gravity - K permeability of the porous medium - N _u non-dimensional Nusselt number - r radial coordinate - non-dimensionalr=r/a - R _a Rayleigh number=(g T)Ka/ - P _e Peclet number=U ₀ a/ - T temperature - T _w temperature of the cylinder (constant) - T ₀ temperature of the ambient fluid (constant) - T temperature difference=T _w –T ₀ - u Darcy's law velocity in thex direction - U ₀ velocity of the outer flow - v Darcy's law velocity in ther-direction - x coordinate measuring distance along the cylinder - X non-dimensionalx,=x(aP _e )^–1 - equivalent thermal diffusivity - coefficient of thermal expansion - ratio of free to forced convection=R _a /P _e - viscosity of the convective fluid - density of the ambient fluid - non-dimensional temperature - stream function With 2 Figures     

On dual solutions occurring in mixed convection in a porous medium

Summary The dual solutions to an equation, which arose previously in mixed convection in a porous medium, occuring for the parameter in the range 0 < < ₀ are considered. It is shown that the lower branch of solutions terminates at =0 with an essential singularity. It is also shown that both branches of solutions bifurcate out of the single solution at =₀ with an amplitude proportional to (₀-)^1/2. Then, by considering a simple time-dependent problem, it is shown that the upper branch of solutions is stable and the lower branch unstable, with the change in temporal stability at =₀ being equivalent to the bifurcation at that point.     

Thermal convection and nonlinear effects of a superfluid3He-4He mixture in a porous medium

The convective instability of superfluid³He-⁴He mixtures in porous media is investigated. The general hydrodynamic equations are derived and reduced to a single nonlinear equation for a scalar field. The superfluid mixtures in a porous medium have a constant⁴He chemical potential and behave essentially like a classical fluid in a porous medium. Two-fluid effects are calculated both at the onset of steady convection and the subsequent boundary of instability. The shift of critical Rayleigh number is about 1% or less at the onset of convection, but can be as large as 20% or more at the instability boundary for some regions aroundT 1 K. This two-fluid shift is quite large compared to the corresponding 0.001% shift at the onset of convection for bulk superfluid³He-⁴He mixtures.     

Hydromagnetic convection flow through a porous medium in axially varying pipe

The hydromagnetic mixed convection flow through a porous medium in a pipe of varying radius in a uniform axial magnetic field is analyzed. The pipe wall is maintained at a prescribed nonuniform temperature. The governing equations are solved analytically to obtain the velocity, temperature, and induced magnetic field. Their behaviors are evaluated for different variations in the governing parameters. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 79, No. 4, pp. 96–104, July–August, 2006.     

Mixed convection boundary layer flow on a vertical surface in a saturated porous medium

Summary The flow of a uniform stream past an impermeable vertical surface embedded in a saturated porous medium and which is supplying heat to the porous medium at a constant rate is considered. The cases when the flow and the buoyancy forces are in the same direction and when they are in opposite direction are discussed. In the former case, the flow develops from mainly forced convection near the leading edge to mainly free convection far downstream. Series solutions are derived in both cases and a numerical solution of the equations is used to describe the flow in the intermediate region. In the latter case, the numerical solution indicates that the flow separates downstream of the leading edge and the nature of the solution near this separation point is discussed.     

Influence of radiation on mixed convection over a wedge in non-Darcy porous medium

The influences of radiation on mixed convection flow of an optically dense viscous fluid along an isothermal wedge embedded in non-Darcy porous medium, in the presence of heat source/sink are numerically investigated. The entire mixed convection regime is covered by a single parameter χ from the pure free convection limit (χ=0) to the pure forced convection limit (χ=1). Forchheimer’s extension is employed to describe the fluid flow in the porous medium and the Rosseland diffusion approximation is considered to describe the radiative heat flux in the energy equation. The governing equations, including internal heat source/sink, are first transformed into a dimensionless form by the nonsimilar transformation and then solved by the Keller box method. The effect of the radiation parameter, mixed convection parameter, Forchheimer number and heat source/sink parameter on the velocity and temperature profiles as well as on the local Nusselt number is presented and analyzed. The results are compared with those known from the literature and excellent agreement between the results is obtained.     

Effect of cross diffusion on double diffusive convection in the presence of horizontal gradients

The effect of cross diffusion namely Soret coefficient and Dufour coefficient on the double diffusive convection in an unbounded vertically stratified two component system with compensating horizontal thermal and solute gradients is investigated in this paper. The conditions for the onset of stationary instability and oscillatory instability are established by the use of normal mode analysis. The effect of various physical parameters on the stability of the system is shown graphically. It is shown that the maximum growth rate of instability, the slope of the wave front and the wave number depend on both Soret and Dufour parameters. We also found the flux ratio corresponding to the maximum growth rate. Some of the known results of the former problems are deduced as special cases.