Abstract
The effect of local thermal non-equilibrium on the onset of double-diffusive convection in a porous medium consisting of two horizontal layers is studied analytically. Linear stability theory is applied. Variations of permeability, fluid conductivity, solutal diffusivity, solid conductivity, interphase heat transfer coefficient, and porosity are considered. It is found that with the introduction of double diffusion, the heterogeneity of porosity now has a major effect, comparable to the effects of heterogeneity of permeability and fluid conductivity. The general results are obtained by using a one-term Galerkin approximation. We validate this approximation by comparing these results with those obtained by using a highly accurate numerical solver. We thus established the accuracy of a one-term Galerkin approximation for stability analysis of a complicated convection problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- \(a \) :
-
Dimensionless horizontal wavenumber
- D :
-
d/dz
- \(D_{S}\) :
-
Solutal diffusivity
- \(C\) :
-
Dimensionless concentration, \(\frac{C^{*}-C_0 }{\Delta C}\)
- \(C^{*}\) :
-
Concentration
- \(C_{0}\) :
-
Concentration at the upper boundary
- \(h\) :
-
Interphase heat transfer coefficient (incorporating the specific surface area) between the fluid and solid particles
- \(\hat{{h}}\) :
-
Parameter defined in Eq. (17)
- \(h_{r}\) :
-
Interphase heat transfer coefficient ratio, \(h_{2}/h_{1}\)
- \(g\) :
-
Gravitational acceleration
- g :
-
Gravitational acceleration vector
- \(H\) :
-
Dimensional layer depth
- \(k\) :
-
Thermal conductivity of the porous medium
- \(\hat{{k}}_f \) :
-
Parameter defined in Eq. (17)
- \(k_{fr}\) :
-
Conductivity ratio for the fluid, \(k_{f2}/k_{f1}\)
- \(\hat{{k}}_s \) :
-
Parameter defined in Eq. (17)
- \(k_{sr}\) :
-
Conductivity ratio for the solid, \(k_{s2}/k_{s1}\)
- \(K \) :
-
Permeability of the porous medium
- \(K_{r}\) :
-
Permeability ratio, \(K_{2}/K_{1}\)
- \(\hat{{K}}\) :
-
Parameter defined in Eq. (17)
- Le:
-
Lewis number, \(\frac{k_{f1} }{(\rho c)_f D_S }\)
- \(N\) :
-
Interphase heat transfer parameter, \(\frac{h_1 H^{2}}{\phi _1 k_{f1} }\)
- \(N_{G}\) :
-
Number of terms in Galerkin approximation
- \(P \) :
-
Dimensionless pressure, \(\frac{(\rho c)_f K_1 }{\mu k_{f1} }P^{*}\)
- \(P^{*}\) :
-
Pressure, excess over hydrostatic
- Ra:
-
Thermal Rayleigh number, \(\frac{\rho _0 g\beta K_1 H\Delta T}{\mu k_{f1} /(\rho c)_f }\)
- \(\hbox {Ra}_\mathrm{D}\) :
-
Solutal Rayleigh number, \(\frac{\rho _0 g\beta _S K_1 H\Delta C}{\mu D_S }\)
- \(t\) :
-
Dimensionless time,\(\frac{k_{f1} }{(\rho c)_f H^{2}}t^{*}\)
- \(t^{*} \) :
-
Time
- \(T\) :
-
Dimensionless temperature, \(\frac{T^{*}-T_0 }{\Delta T}\)
- \(T^{*}\) :
-
Temperature
- \(T_{0 }\) :
-
Temperature at the upper boundary
- \((u,v,w)\) :
-
Dimensionless velocity components, \(\frac{(\rho c)_f H}{k_{f1} }(u^{*},v^{*},w^{*})\)
- \(\mathbf{u}^{*}\) :
-
Darcy velocity, \((u^{*},v^{*},w^{*})\)
- \((x,y,z)\) :
-
Dimensionless Cartesian coordinates, \((x^{*},y^{*},z^{*})/H\); \(z \) is the vertically upward coordinate
- \((x^{*},y^{*},z^{*})\) :
-
Cartesian coordinates
- \(\alpha \) :
-
Modified thermal diffusivity ratio, \(\frac{(\rho c)_{s1}}{(\rho c)_{f1} }\frac{k_{f1} }{k_{s1} }\)
- \(\beta \) :
-
Volumetric thermal expansion coefficient of the fluid
- \(\beta _{S}\) :
-
Volumetric solutal expansion coefficient of the fluid
- \(\gamma \) :
-
Modified thermal conductivity ratio, \(\frac{\phi _1 k_{f1} }{(1-\phi _1 )k_{s1} }\)
- \(\delta \) :
-
Dimensionless layer depth ratio (interface position)
- \(\hat{{\delta }}\) :
-
Parameter defined in Eq. (17)
- \(\delta _{r}\) :
-
Inverse solid fraction ratio,\(\frac{1-\phi _1}{1-\phi _2 }\)
- \(\Delta C\) :
-
Concentration difference
- \(\Delta T\) :
-
Temperature difference
- \(\varepsilon \) :
-
Dimensionless small quantity
- \(\hat{{\varepsilon }}\) :
-
Parameter defined in Eq. (17)
- \(\varepsilon _r\) :
-
Solid heat capacity ratio, \(\frac{(\rho c)_{s2}}{(\rho c)_{s1} }\)
- \(\mu \) :
-
Viscosity of the fluid
- \(\rho _{0 }\) :
-
Fluid density at temperature \(T_{0}\)
- \(\rho _{f}\) :
-
Fluid density
- \(\left( {\rho c} \right) _f \) :
-
Heat capacity of the fluid
- \(\left( {\rho c} \right) _m \) :
-
Effective heat capacity of the porous medium
- \((\rho c)_s \) :
-
Heat capacity of the solid
- \(\phi \) :
-
Porosity
- \(\phi _{r}\) :
-
Porosity ratio, \(\phi _{2}\)/\(\phi _{1}\)
- \(\hat{{\phi }}\) :
-
Parameter defined in Eq. (17)
- \(B \) :
-
Basic state
- \(c\) :
-
Critical value
- \(f \) :
-
Fluid phase
- \(m\) :
-
Effective property for the porous medium
- \(r \) :
-
Relative quantity
- \(s \) :
-
Solid phase
- \(S\) :
-
Solute
- 1:
-
The region \(0\le z^{*}<\delta H\)
- 2:
-
The region \(\delta H\le z^{*}\le H\)
- \(^{\prime }\) :
-
Perturbation variable
- \(*\) :
-
Dimensional variable
References
Barletta, A., Storesletten, L.: Thermoconvective instabilities in an inclined porous channel heated from below. Int. J. Heat Mass Transf. 54, 2724–2733 (2011)
Nield, D.A.: Effects of local thermal nonequilibrium in steady convective processes in a saturated porous medium: forced convection in a channel. J. Porous Media 1, 181–186 (1998)
Nield, D.A.: A note on local thermal non-equilibrium in porous media near boundaries and interfaces. Transp. Porous Media 95, 581–584 (2012)
Nield, D.A., Bejan, A.: Convection in Porous Media, 4th edn. Springer, New York (2013)
Nield, D.A., Kuznetsov, A.V.: Local thermal non-equilibrium and heterogeneity effects on the onset of convection in a layered porous medium. Transp. Porous Media 102, 1–13 (2014)
Platten, J.K., Legros, J.C.: Convection in Liquids. Springer, Berlin (1984)
Postelnicu, A., Rees, D.A.S.: The onset of Darcy-Brinkman convection in a porous layer using a thermal nonequilibrium model—Part I: Stress free boundaries. Int. J. Energy Res. 27, 961–973 (2003)
Rees, D.A.S., Bassom, A.P.: The onset of Darcy–Bénard convection in an inclined layer heated from below. Acta Mech. 144, 103–118 (2000)
Straughan, B.: Stability and Wave Motion in Porous Media. Springer-Verlag, New York (2008)
Vadasz, P.: Heat conduction in nanofluid suspensions. ASME J. Heat Transf. 128, 465–477 (2006)
Acknowledgments
AVK gratefully acknowledges the support of the Alexander von Humboldt Foundation though the Humboldt Research Award.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nield, D.A., Kuznetsov, A.V., Barletta, A. et al. The Effects of Double Diffusion and Local Thermal Non-equilibrium on the Onset of Convection in a Layered Porous Medium: Non-oscillatory Instability. Transp Porous Med 107, 261–279 (2015). https://doi.org/10.1007/s11242-014-0436-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-014-0436-y