Comments on Reply to Comments on “Non-Darcian Forced Convection Flow of Viscous Dissipating Fluid Over a Flat Plate Embedded in a Porous Medium”

This note is a comment on the reply by Aydin and Kaya (Transp Porous Media (to appear), 2008b) to comments by Rees and Magyari (Transp Porous Media (to appear), 2008) on an article by Aydin and Kaya (Transp Porous Media (to appear), 2008a) concerning the combined effects of viscous dissipation and surface mass flux on the forced convection boundary-layer flow in a saturated porous medium modeled by the Brinkman equation. It is argued that the statements by Rees and Magyari are in fact appropriate. The thermal boundary condition imposed at the edge of the boundary-layer by Aydin and Kaya is incompatible with the energy equation, and thus the results of their paper are inconsistent with the physics of the situation. The attempts of Aydin and Kaya to justify their paper are flawed by an inappropriate assumption and calculations with an insufficiently large parameter.     

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.     

The Effect of Vertical Throughflow on the Onset of Convection in a Porous Medium in a Rectangular Box

The effect of vertical throughflow on the onset of convection in a rectangular box occupied by a saturated porous medium uniformly heated from below, is studied using linear stability theory. It is found that, for small values of the throughflow, the stabilizing effect of the throughflow and the stabilizing effect of the confining lateral walls of the box are approximately independent of each other.     

A Further Note on the Onset of Convection in a Layer of a Porous Medium Saturated by a Non-Newtonian Fluid of Power-Law Type

The singular behavior of the Horton–Rogers–Lapwood problem when a Newtonian fluid is replaced by a standard power-law fluid is investigated further. Using weakly nonlinear stability theory, an estimate is made of the amplitude of convection at which the convection is initiated (in the case of a fluid with index n > 1) or levels off (in the case of a fluid with n < 1).     

A Note on the Onset of Convection in a Layer of a Porous Medium Saturated by a Non-Newtonian Nanofluid of Power-Law Type

The published literature on convection patterns arising in natural convective flow in an inclined porous layer is surveyed, and the situation with respect to the occurrence of polyhedral cells is clarified. These are observed in experiments but are not predicted by the simple classical theory nor are they observed in simulations based on that simple theory.     

The Onset of Convection in a Heterogeneous Porous Medium with Transient Temperature Profile

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 transient heating or otherwise, is studied analytically using linear stability theory. Two particular situations, corresponding to instantaneous bottom heating and constant-rate bottom heating, are studied. Estimates of the timescale for the development of convection instability are obtained.     

Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: Brinkman Model

The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Brinkman model is employed. Three cases of free–free, rigid–rigid, and rigid–free boundaries are considered. The analysis reveals that for a typical nanofluid (with large Lewis number), the prime effect of the nanofluids is via a buoyancy effect coupled with the conservation of nanoparticles, whereas the contribution of nanoparticles to the thermal energy equation is a second-order effect. It is found that the critical thermal Rayleigh number can be reduced or increased by a substantial amount, depending on whether the basic nanoparticle distribution is top-heavy or bottom-heavy, by the presence of the nanoparticles. Oscillatory instability is possible in the case of a bottom-heavy nanoparticle distribution.     

Effect of Local Thermal Non-equilibrium on the Onset of Convection in a Porous Medium Layer Saturated by a Nanofluid

The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is analytically studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Darcy model is employed. The effect of local thermal non-equilibrium among the particle, fluid, and solid-matrix phases is investigated using a three-temperature model. The analysis reveals that in some circumstances the effect of LTNE can be significant, but for a typical dilute nanofluid (with large Lewis number and with small particle-to-fluid heat capacity ratio) the effect is small.