Free Convection of a Bingham Fluid in a Differentially-Heated Porous Cavity:The Effect of a Square Grid Microstructure

We examine how a square-grid microstructure affects the manner in which a Bingham fluid is convected in a sidewall-heated rectangular porous cavity. When the porous microstructure is isotropic, flow arises only when the Darcy–Rayleigh number is higher than a critical value, and this corresponds to when buoyancy forces are sufficient to overcome the yield threshold of the Bingham fluid. In such cases, the flow domain consists of a flowing region and stagnant regions within which there is no flow. Here, we consider a special case where the constituent pores form a square grid pattern. First, we use a network model to write down the appropriate macroscopic momentum equations as a Darcy–Bingham law for this microstructure. Then detailed computations are used to determine strongly nonlinear states. It is found that the flow splits naturally into four different regions: (i) full flow, (ii) no-flow, (iii) flow solely in the horizontal direction and (iv) flow solely in the vertical direction. The variations in the rate of heat transfer and the strength of the flow with the three governing parameters, the Darcy–Rayleigh number, Ra, the Rees–Bingham number, Rb, and the aspect ratio, A, are obtained

Unsteady thermal boundary layer flows of a Bingham fluid in a porous medium following a sudden change in surface heat flux

We consider the effect of suddenly applying a uniform heat flux to a vertical wall bounding a porous medium which is saturated by a Bingham fluid. We consider both an infinite porous domain and a vertical channel of finite width. Initially, the evolving temperature field provides too little buoyancy force to overcome the yield threshold of the fluid. For the infinite domain convection will always eventually arise, but this does not necessarily happen in the vertical channel. We show (i) how the presence of yield surfaces alters the classical results for Newtonian flows and (ii) the manner in which the locations of the yield surfaces change as time progresses

Thermal receptivity of free convective flow from a heated vertical surface: linear waves

Numerical techniques are used to study the receptivity to small-amplitude thermal disturbances of the boundary layer flow of air which is induced by a heated vertical flat plate. The fully elliptic nonlinear, time-dependent Navier–Stokes and energy equations are first solved to determine the steady state boundary-layer flow, while a linearised version of the same code is used to determine the stability characteristics. In particular we investigate (i) the ultimate fate of a localised thermal disturbance placed in the region near the leading edge and (ii) the effect of small-scale surface temperature oscillations as means of understanding the stability characteristics of the boundary layer. We show that there is a favoured frequency of excitation for the time-periodic disturbance which maximises the local response in terms of the local rate of heat transfer. However the magnitude of the favoured frequency depends on precisely how far from the leading edge the local response is measured. We also find that the instability is advective in nature and that the response of the boundary layer consists of a starting transient which eventually leaves the computational domain, leaving behind the large-time time-periodic asymptotic state. Our detailed numerical results are compared with those obtained using parallel flow theory

The effect of conducting boundaries on weakly nonlinear Darcy–Bénard convection

We consider convection in a uniform fluid-saturated porous layer which is bounded by conducting plates and heated from below. The primary aim is to determine the identity of the postcritical convection planform as a function of the thicknesses and conductivities of the bounding plates relative to that of the porous layer. This work complements and extends an early paper by Riahi (J Fluid Mech 129:153–171, 1983) who considered a situation where the porous layer is bounded by infinitely thick conducting media.We present regions in parameter space wherein convection in the form of rolls is unstable and within which cells with square planform form the preferred pattern

Local thermal non-equilibrium effects in the Darcy-Benard instability with isoflux boundary conditions

none2siThe Darcy-Bénard problem with constant heat flux boundary conditions is studied in a regime where the fluid and solid phases are in local thermal non-equilibrium. The onset conditions for convective instability in the plane porous layer are investigated using a linear stability analysis. Constant heat flux boundary conditions are formulated according to the Amiri-Vafai-Kuzay Model A, where the boundary walls are assumed as impermeable and with a high thermal conductance. The normal mode analysis of the perturbations imposed on the basic state leads to a one-dimensional eigenvalue problem, solved numerically to determine the neutral stability condition. Analytical solutions are found for the limit of small wave numbers, and in the regime where the conductivity of the solid phase is much larger than the conductivity of the fluid phase. A comparison with the corresponding results under conditions of local thermal equilibrium is carried out. The critical conditions for the onset of convection correspond to a zero wave number only when the inter-phase heat transfer coefficient is sufficiently large. Otherwise, the critical conditions correspond to a nonzero wave number.mixedA. Barletta; D. A. S. ReesA. Barletta; D. A. S. Ree

The effect of conducting sidewalls on the onset of convection in a porous cavity

Abstract In this short paper we consider the effect of the presence of conducting sidewalls of finite thickness on the onset of convection in a two-dimensional porous cavity. Two cases are considered where the outer boundaries of the sidewalls are either perfectly insulating or perfectly conducting. A unified theory is presented which combines both these cases, and the stability properties of the overall system is found to undergo a full transition from that of the classical Darcy-Bénard problem to that of the degenerate system studied in detail by Rees and Tyvand (2004)

The onset of convection in horizontally partitioned porous layers

In this paper, the onset of convection in a horizontally partitioned porous layer is investigated. Two identical sublayers are separated by a thin impermeable barrier. There exists a background horizontal flow in one of the layers or, equivalently, flows of half that strength in each sublayer but in opposite directions. A linearised stability analysis is performed where the horizontal component of the disturbance is factored into separate Fourier modes, leaving an ordinary differential eigenvalue problem for the critical Darcy-Rayleigh number as a function of the wavenumber. The dispersion relation is derived and the neutral stability curves are obtained for a wide range of horizontal flow rates. The presence of the horizontal flow alters the morphology of the neutral curves from that which occurs when there is no flow and travelling modes may arise. We also determine the condition under which the most dangerous disturbance changes from a stationary mode to travelling mode. Some three-dimensional aspects are also considered. (C) 2011 American Institute of Physics. [doi:10.1063/1.3589864

Weakly nonlinear convection in a porous layer with multiple horizontal partitions

We consider convection in a horizontally uniform fluid-saturated porous layer which is heated from below and which is split into a number of identical sublayers by impermeable and infinitesimally thin horizontal partitions. Rees and Genc (Int J Heat Mass Transfer 54:3081-3089, 2010) determined the onset criterion by means of a detailed analytical and numerical study of the corresponding dispersion relation and showed that this layered system behaves like the single-sublayer constant-heat-flux Darcy-Benard problem when the number of sublayers becomes large. The aim of the present work is to use a weakly nonlinear analysis to determine whether the layered system also shares the property of the single-sublayer constant-heat-flux Darcy-Benard problem by having square cells, as opposed to rolls, as the preferred planform for convection.We consider convection in a horizontally uniform fluid-saturated porous layerwhich is heated from below and which is split into a number of identical sublayers byimpermeable and infinitesimally thin horizontal partitions. Rees and Gen&ccedil; (Int J Heat MassTransfer 54:3081&ndash;3089,&nbsp;&nbsp;2010) determined the onset criterion by means of a detailed analyticaland numerical study of the corresponding dispersion relation and showed that this layeredsystem behaves like the single-sublayer constant-heat-flux Darcy&ndash;B&eacute;nard problem when thenumber of sublayers becomes large. The aim of the present work is to use a weakly nonlinearanalysis to determine whether the layered system also shares the property of the singlesublayerconstant-heat-flux Darcy&ndash;B&eacute;nard problem by having square cells, as opposed torolls, as the preferred planform for convection.</p

The Effect of Internal and External Heating on the Free Convective Flow of a Bingham Fluid in a Vertical Porous Channel

We study the steady free convective flow of a Bingham fluid in a porous channel where heat is supplied by both differential heating of the sidewalls and by means of a uniform internal heat generation. The detailed temperature profile is governing by an external and an internal Darcy-Rayleigh number. The presence of the Bingham fluid is characterised by means of a body force threshold as given by the Rees-Bingham number. The resulting flow field may then exhibit between two and four yield surfaces depending on the balance of magnitudes of the three nondimensional parameters. Some indication is given of how the locations of the yield surfaces evolve with the relative strength of the Darcy-Rayleigh numbers and the Rees-Bingham number. Finally, parameter space is delimited into those regions within which the different types of flow and stagnation patterns arise

The effect of local thermal non-equilibrium on forced convection boundary layer flow from a heated surface in porous media

A steady two-dimensional forced convective thermal boundary layer flow in a porous medium is studied. It is assumed that the solid matrix and fluid phase which comprise the porous medium are subject to local thermal non-equilibrium conditions, and therefore two heat transport equations are adopted, one for each phase. When the basic flow velocity is sufficiently high, the thermal fields may be described accurately using the boundary layer approximation, and the resulting parabolic system is analysed both analytically and numerically. Local thermal non-equilibrium effects are found to be at their strongest near the leading edge, but these decrease with distance from the leading edge and local thermal equilibrium is attained at large distances