D. A. Nield

Surface tension and buoyancy effects in cellular convection

The cells observed by Benard (1901) when a horizontal layer of fluid is heated from below were explained by Rayleigh (1916) in terms of buoyancy, and by Pearson (1958) in terms of surface tension. These rival theories are now combined. Linear perturbation techniques are used to derive a sixth-order differential equation subject to six boundary conditions. A Fourier series method has been used to obtain the eigenvalue equation for the case where the lower boundary surface is a rigid conductor and the upper free surface is subject to a general thermal condition. Numerical results are presented. It was found that the two agencies causing instability reinforce one another and are tightly coupled. Cells formed by surface tension are approximately the same size as those formed by buoyancy. Benards experiments are briefly discussed.

The thermohaline Rayleigh-Jeffreys problem

The onset of convection induced by thermal and solute concentration gradients, in a horizontal layer of a viscous fluid, is studied by means of linear stability analysis. A Fourier series method is used to obtain the eigenvalue equation, which involves a thermal Rayleigh number R and an analogous solute Rayleigh number S , for a general set of boundary conditions. Numerical solutions are obtained for selected cases. Both oscillatory and monotonic instability are considered, but only the latter is treated in detail. The former can occur when a strongly stabilizing solvent gradient is opposed by a destablizing thermal gradient. When the same boundary equations are required to be satisfied by the temperature and concentration perturbations, the monotonic stability boundary curve in the ( R, S )-plane is a straight line. Otherwise this curve is concave towards the origin. For certain combinations of boundary conditions the critical value of R does not depend on S (for some range of S ) or vice versa. This situation pertains when the critical horizontal wave-number is zero. A general discussion of the possibility and significance of convection at ‘zero’ wave-number (single convection cell) is presented in an appendix.

The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface

Abstract This paper is a critique of the ability of the Brinkman–Forchheimer equation to adequately model flow in a porous medium and at a porous-medium/clear-fluid interface. It is demonstrated that certain terms in the equation as commonly used require modification, and that there is a difficulty when using this equation to deal with a stress boundary condition.

Thermally developing forced convection in a porous medium: parallel plate channel with walls at uniform temperature, with axial conduction and viscous dissipation effects

Abstract A modified Graetz methodology is applied to investigate the thermal development of forced convection in a parallel plate channel filled by a saturated porous medium, with walls held at uniform temperature, and with the effects of axial conduction and viscous dissipation included. The Brinkman model is employed. The analysis leads to expressions for the local Nusselt number, as a function of the dimensionless longitudinal coordinate and other parameters (Darcy number, Peclet number, Brinkman number).

The boundary correction for the Rayleigh-Darcy problem: limitations of the Brinkman equation

The no-slip condition on rigid boundaries necessitates a correction to the critical value of the Rayleigh–Darcy number for the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below. It is shown that the use of the Brinkman equation to obtain this correction is not justified, because of the limitations of that equation. These limitations are discussed in detail. An alternative procedure, based on a model in which the porous medium is sandwiched between two fluid layers, and the Beavers–Joseph boundary condition is applied at the interfaces, is described, and an expression for the correction is obtained. It is found that the correction can be of either sign, depending on the relative magnitudes of the parameters involved.

Onset of convection in a fluid layer overlying a layer of a porous medium

A linear stability analysis is applied to a system consisting of a horizontal fluid layer overlying a layer of a porous medium saturated with the same fluid, with uniform heating from below. Surface-tension effects at a deformable upper surface are allowed for. The solution is obtained for constant-flux thermal boundary conditions.

Resolution of a Paradox Involving Viscous Dissipation and Nonlinear Drag in a Porous Medium

The modelling of viscous dissipation in a porous medium saturated by an incompressible fluid is discussed, for the case of Darcy, Forchheimer and Brinkman models. An apparent paradox relating to the effect of inertial effects on viscous dissipation is resolved, and some wider aspects of resistance to flow (concerning quadratic drag and cubic drag) in a porous medium are discussed. Criteria are given for the importance or otherwise of viscous dissipation in various situations.

Forced convection in a fluid-saturated porous-medium channel with isothermal or isoflux boundaries

We present a fresh theoretical analysis of fully developed forced convection in a fluid-saturated porous-medium channel bounded by parallel plates, with imposed uniform heat flux or isothermal condition at the plates. As a preliminary step, we obtain an ‘exact’ solution of the Brinkman-Forchheimer extension of Darcys momentum equation for flow in the channel. This uniformly valid solution permits a unified treatment of forced convection heat transfer, provides the means for a deeper explanation of the physical phenomena, and also leads to results which are valid for highly porous materials of current practical importance.

Effect of local thermal non-equilibrium on thermally developing forced convection in a porous medium

Abstract The classical Graetz methodology is applied to investigate the effect of local thermal non-equilibrium on the thermal development of forced convection in a parallel-plate channel filled by a saturated porous medium, with walls held at constant temperature. The Brinkman model is employed. The analysis leads to an expression for the local Nusselt number, as a function of the dimensionless longitudinal coordinate, the Peclet number, the Darcy number, the solid–fluid heat exchange parameter, the solid/fluid thermal conductivity ratio, and the porosity.

Thermally Developing Forced Convection in a Porous Medium: Circular Duct with Walls at Constant Temperature, with Longitudinal Conduction and Viscous Dissipation Effects

A modified Graetz methodology is applied to investigate the thermal development of forced convection in a circular duct filled by a saturated porous medium, with walls held at constant temperature, and with the effects of longitudinal conduction and viscous dissipation included. The Brinkman model is employed. The analysis leads to expressions for the local Nusselt number, as a function of the dimensionless longitudinal coordinate and other parameters (Darcy number, Péclet number, Brinkman number).