caa a22 4500 463163478 CHVBK 20180406164754.0 cr unu---uuuuu 170326e20070401xx s 000 0 eng 10.1007/s11242-006-9035-x doi (NATIONALLICENCE)springer-10.1007/s11242-006-9035-x The onset of convection in a shallow box occupied by a heterogeneous porous medium with constant flux boundaries [Elektronische Daten] [D. Nield, A. Kuznetsov] The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, are studied analytically using linear stability theory for the case of weak heterogeneity. Attention is focused on the case of constant flux upper and lower boundaries, a case for which the critical horizontal wavenumber is zero, and attention is also concentrated on the case of a shallow layer. It is found that the effect of such heterogeneity on the critical value of the Rayleigh number Ra based on mean properties is of second order if the properties vary in a piecewise constant or linear fashion. The effects of horizontal heterogeneity and vertical heterogeneity are then comparable once the aspect ratio is taken into account, and to a first approximation are independent. The combination of permeability heterogeneity and conductivity heterogeneity can be either stabilizing or destabilizing for the present case. Springer Science+Business Media B.V., 2006 Natural convection nationallicence Heterogeneity nationallicence Instability nationallicence Horton-Rogers-Lapwood problem nationallicence A : Aspect ratio (height to width) nationallicence c : Specific heat nationallicence H : Height of the enclosure nationallicence k : k */k 0 nationallicence k * : Overall (effective) thermal conductivity nationallicence k 0 : Mean value of k *(x *,y *) nationallicence K : K */K 0 nationallicence K * : Permeability nationallicence K 0 : Mean value of K *(x *,y *) nationallicence L : Width of the enclosure nationallicence P : Dimensionless pressure, $$\frac{(\rho c)_{\rm f} K_0}{\mu k_0}P^* $$ nationallicence P * : Pressure nationallicence Ra : Rayleigh number, $$\frac{(\rho c)_{\rm f} \rho_0 g\beta K_0 L(T_1 - T_0)}{\mu k_0}$$ nationallicence t * : Time nationallicence t : Dimensionless time, $$\frac{k_0}{(\rho c)_{\rm m} L^2}t^* $$ nationallicence T * : Temperature nationallicence T 0 : Temperature at the upper boundary nationallicence T 1 : Temperature at the lower boundary nationallicence u : Dimensionless horizontal velocity, $$\frac{(\rho c)_{\rm m} L}{k_0}u^* $$ nationallicence u * : Vector of Darcy velocity, (u *,v *) nationallicence v : Dimensionless vertical velocity, $$\frac{(\rho c)_{\rm m} L}{k_0}v^* $$ nationallicence x : Dimensionless horizontal coordinate, x */L nationallicence x * : Horizontal coordinate nationallicence y : Dimensionless upward vertical coordinate, y */H nationallicence y * : Upward vertical coordinate nationallicence Greek symbols nationallicence β : Fluid volumetric expansion coefficient nationallicence θ : Dimensionless temperature, $$\frac{T^* -T_0}{T_1 -T_0}$$ nationallicence μ : Fluid viscosity nationallicence ρ : Density nationallicence ρ0 : Fluid density at temperature T 0 nationallicence σ : Heat capacity ratio, $$\frac{(\rho c)_{\rm m}}{(\rho c)_{\rm f}}$$ nationallicence ψ : Streamfunction defined by Eqs. (10a,b) nationallicence Subscripts nationallicence f : Fluid nationallicence m : Overall porous medium nationallicence Superscripts nationallicence * : Dimensional variable nationallicence Nield D. Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland, New Zealand aut Kuznetsov A. Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, 27695-7910, Raleigh, NC, USA aut Transport in Porous Media Kluwer Academic Publishers 67/3(2007-04-01), 441-451 0169-3913 67:3<441 2007 67 11242 https://doi.org/10.1007/s11242-006-9035-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11242-006-9035-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Nield D. Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland, New Zealand aut NATIONALLICENCE 700 1- Kuznetsov A. Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, 27695-7910, Raleigh, NC, USA aut NATIONALLICENCE 773 0- Transport in Porous Media Kluwer Academic Publishers 67/3(2007-04-01), 441-451 0169-3913 67:3<441 2007 67 11242 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer