Abstract
The effect of local thermal non-equilibrium on the onset of double-diffusive convection in a porous medium consisting of two horizontal layers, each internally heated, is studied analytically. Linear stability theory is applied. Variations of permeability, fluid thermal conductivity, solid thermal conductivity, heat source strength in the solid and fluid phases, concentration source strength, interphase heat transfer coefficient and porosity are considered. In addition to the major effects from heterogeneity of permeability, fluid thermal conductivity and heat source strength in the fluid phase as with single-diffusive convection, it is now found that major effects arise from heterogeneity of solutal source strength and porosity. We used two different methods to obtain our results. Analytical results that readily show the effects of parameter variations were obtained by using a low-term Galerkin approximation, which was validated by using a highly accurate numerical solver. Since for a problem with large number of parameters simple analytical results are highly desirable, the quantification of the accuracy of a low-term Galerkin approximation presented in our paper is quite important.
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Abbreviations
- a :
-
Dimensionless horizontal wavenumber
- C :
-
Dimensionless concentration, \(\frac{(\rho c)_\mathrm{f} \rho _0 g\beta _\mathrm{C} K_1 H}{\mu k_\mathrm{f1} }(C^{*} -C_0)\)
- \(C^{*}\) :
-
Solute concentration
- \(C_{0}\) :
-
Solute concentration at each of the upper and lower boundaries
- D :
-
d/dz
- \(D_\mathrm{C}\) :
-
Solutal diffusivity
- h :
-
Interface heat transfer coefficient (incorporating the specific surface area) between the fluid and solid particles
- \(\hat{{h}}\) :
-
Parameter defined in Eq. (18)
- \(h_\mathrm{r}\) :
-
Interface heat transfer coefficient ratio, \(h_{2}/h_{1}\)
- g :
-
Gravitational acceleration
- g :
-
Gravitational acceleration vector
- H :
-
Dimensional layer depth
- k :
-
Thermal conductivity
- \(k_\mathrm{f}\) :
-
Thermal conductivity of the fluid phase
- \(\hat{{k}}_\mathrm{f}\) :
-
Parameter defined in Eq. (18)
- \(k_\mathrm{fr}\) :
-
Fluid thermal conductivity ratio, \(k_\mathrm{f2}/k_\mathrm{f1}\)
- \(k_\mathrm{s}\) :
-
Thermal conductivity of the solid phase
- \(\hat{{k}}_\mathrm{s}\) :
-
Parameter defined in Eq. (18)
- \(k_\mathrm{sr}\) :
-
Solid thermal conductivity ratio, \(k_\mathrm{s2}/k_\mathrm{s1}\)
- K :
-
Permeability of the porous medium
- \(K_\mathrm{r}\) :
-
Permeability ratio, \(K_{2}/K_{1}\)
- \(\hat{{K}}\) :
-
Parameter defined in Eq. (18)
- Le :
-
Lewis number, \(\frac{k_\mathrm{f1} }{(\rho c)_\mathrm{f} D_\mathrm{C}}\)
- N :
-
Interface heat transfer parameter, \(\frac{h_1 H^{2}}{\phi _1 k_\mathrm{f1}}\)
- P :
-
Dimensionless pressure, \(\frac{(\rho c)_\mathrm{f} K_1 }{\mu k_\mathrm{f1} }P^{*}\)
- \(P^{*}\) :
-
Pressure, excess over hydrostatic
- Q :
-
Volumetric heat source strength
- \(Q_\mathrm{C}\) :
-
Volumetric solute source strength
- \(\hat{{Q}}_\mathrm{C}\) :
-
Parameter defined in Eq. (18)
- \(Q_\mathrm{Cr}\) :
-
Solute source ratio, \(Q_\mathrm{C2}/Q_\mathrm{C1}\)
- \(\hat{{Q}}_\mathrm{f}\) :
-
Parameter defined in Eq. (18)
- \(Q_\mathrm{fr}\) :
-
Heat source ratio in the fluid phase, \(Q_\mathrm{f2}/Q_\mathrm{f1}\)
- \(\hat{{Q}}_\mathrm{s}\) :
-
Parameter defined in Eq. (18)
- \(Q_\mathrm{sr}\) :
-
Heat source ratio in the solid phase, \(Q_\mathrm{s2}/Q_\mathrm{s1}\)
- Ra :
-
Internal thermal Rayleigh number, \(\frac{(\rho c)_\mathrm{f} \rho _0 g\beta K_1 H^{3}Q_\mathrm{f1} }{2\mu k_\mathrm{f1}^{2}}\)
- \(Ra_\mathrm{C}\) :
-
Internal solutal Rayleigh number, \(\frac{(\rho c)_\mathrm{f} \rho _0 g\beta _\mathrm{C} K_1 H^{3}Q_\mathrm{C1} }{2\mu k_\mathrm{f1} D_\mathrm{C}}\)
- \({Ra}_\mathrm{eff}\) :
-
Effective combined Rayleigh number defined in Eq. (63)
- t :
-
Dimensionless time,\(\frac{k_\mathrm{f1} }{(\rho c)_\mathrm{f} H^{2}}t^{*}\)
- \(t^{*}\) :
-
Time
- T :
-
Dimensionless temperature, \(\frac{(\rho c)_\mathrm{f} \rho _0 g\beta K_1 H}{\mu k_\mathrm{f1} }(T^{*} -T_0)\)
- \(T^{*}\) :
-
Temperature
- \(T_{0}\) :
-
Temperature at each of the upper and lower boundaries
- (u, v, w):
-
Dimensionless velocity components, \(\frac{(\rho c)_\mathrm{f} H}{k_\mathrm{f1} }(u^{*},v^{*},w^{*})\)
- \(\mathbf{u}^{*}\) :
-
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; \(z^{*}\) is the vertically upward coordinate
- \(\alpha \) :
-
Modified thermal diffusivity ratio, \(\frac{(\rho c)_\mathrm{s1} }{(\rho c)_\mathrm{f1} }\frac{k_\mathrm{f1} }{k_\mathrm{s1}}\)
- \(\beta \) :
-
Thermal expansion coefficient of the fluid
- \(\beta _\mathrm{C}\) :
-
Solutal expansion coefficient of the fluid
- \(\gamma \) :
-
Modified thermal conductivity ratio, \(\frac{\phi _1 k_\mathrm{f1} }{(1-\phi _1)k_\mathrm{s1}}\)
- \(\delta \) :
-
Dimensionless layer depth ratio (interface position)
- \(\hat{{\delta }}\) :
-
Parameter defined in Eq. (18)
- \(\delta _\mathrm{r}\) :
-
Inverse solid fraction ratio, \(\frac{1-\phi _1 }{1-\phi _2}\)
- \(\hat{{\varepsilon }}\) :
-
Parameter defined in Eq. (18)
- \(\varepsilon _\mathrm{r}\) :
-
Solid heat capacity ratio, \(\frac{(\rho c)_\mathrm{s2} }{(\rho c)_\mathrm{s1}}\)
- \(\mu \) :
-
Viscosity of the fluid
- \(\rho _{0}\) :
-
Fluid density at temperature \(T_{0}\)
- \(\rho _\mathrm{f}\) :
-
Fluid density
- \((\rho c)_\mathrm{f}\) :
-
Heat capacity of the fluid
- \((\rho c)_\mathrm{s}\) :
-
Heat capacity of the solid
- \(\phi \) :
-
Porosity
- \(\hat{{\phi }}\) :
-
Parameter defined in Eq. (18)
- \(\phi _\mathrm{r}\) :
-
Porosity ratio, \(\phi _{2}/\phi _{1}\)
- \(\mathrm{B}\) :
-
Basic state
- \(\mathrm{f}\) :
-
Fluid phase
- \(\mathrm{r}\) :
-
Relative quantity
- \(\mathrm{s}\) :
-
Solid phase
- 1:
-
The region \(0\le z^{*}<\delta H\)
- 2:
-
The region \(\delta H\le z^{*}\le H\)
- \('\) :
-
Perturbation variable
- *:
-
Dimensional variable
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Acknowledgments
A.V.K. gratefully acknowledges the support of the Alexander von Humboldt Foundation through the Humboldt Research Award.
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Kuznetsov, A.V., Nield, D.A., Barletta, A. et al. Local Thermal Non-equilibrium and Heterogeneity Effects on the Onset of Double-Diffusive Convection in an Internally Heated and Soluted Porous Medium. Transp Porous Med 109, 393–409 (2015). https://doi.org/10.1007/s11242-015-0525-6
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DOI: https://doi.org/10.1007/s11242-015-0525-6