Abstract
A theoretical analysis of convective instability driven by buoyancy forces under the transient concentration fields is conducted in an initially quiescent, liquid-saturated, and anisotropic cylindrical porous layer supported by a gas phase. Darcy’s law and Boussinesq approximation are used to explain the characteristics of fluid motion, and linear stability theory is employed to predict the onset of buoyancy-driven motion. Under the quasi-steady-state approximation, the stability equations are derived in a similar boundary layer coordinate and solved by the numerical shooting method. The critical \(Ra_D\) is determined as a function of the anisotropy ratio. Also, the onset time and corresponding wavelength are obtained for the various anisotropic ratios. The onset time becomes smaller with increasing \(Ra_D\) and follows the asymptotic relation derived in the infinite horizontal porous layer. Anisotropy effect makes the system more stable by suppressing the vertical velocity.
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Abbreviations
- \(C\) :
-
Concentration (M)
- \(c\) :
-
Dimensionless concentration
- \(D_e\) :
-
Effective diffusivity \((\hbox {m}^{2}/\hbox {s})\)
- \(g\) :
-
Gravitation acceleration \((\hbox {m}/\hbox {s}^{2})\)
- \(K\) :
-
Permeability of porous media \((\hbox {m}^{2})\)
- \(k^{*}\) :
-
Modified wavenumber, \(\alpha _{l,m} \sqrt{\tau }\)
- \(P\) :
-
Pressure (Pa)
- \(R\) :
-
Radius of cylinder (m)
- \((r,\theta ,Z)\) :
-
Cylindrical coordinates
- \((\overline{r},\theta ,z)\) :
-
Dimensionless cylindrical coordinates
- \(Ra_D\) :
-
Darcy–Rayleigh number, \(Ra_D={g\beta KC_i R}/{\left( {\varepsilon D_e \nu } \right) }\)
- \(t\) :
-
Time (s)
- \(\mathbf{U}\) :
-
Velocity vector (m/s)
- \(W\) :
-
Vertical velocity (m/s)
- \(w\) :
-
Dimensionless vertical velocity
- \(\alpha _{l,m}\) :
-
Wavenumber
- \(\beta \) :
-
Volumetric expansion coefficient \((\hbox {M}^{-1})\)
- \(\gamma \) :
-
Anisotropy ratio
- \(\varepsilon \) :
-
Porosity of porous media
- \(\zeta \) :
-
Dimensionless similarity variable, \(z/\tau ^{1/2}\)
- \(\mu \) :
-
Viscosity (Pa s)
- \(\nu \) :
-
Kinematic viscosity \((\hbox {m}^{2}/\hbox {s})\)
- \(\tau \) :
-
Dimensionless time
- \(\rho \) :
-
Density \((\hbox {kg/m}^{3})\)
- H:
-
Horizontal quantities
- V:
-
Vertical quantities
- 0:
-
Basic quantities
- 1:
-
Perturbation quantities
- c:
-
Critical conditions
- *:
-
Transformed quantities
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Acknowledgments
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2038983).
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Kim, M.C. Onset of Buoyancy-Driven Convection in a Liquid-Saturated Cylindrical Anisotropic Porous Layer Supported by a Gas Phase. Transp Porous Med 102, 31–42 (2014). https://doi.org/10.1007/s11242-013-0259-2
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DOI: https://doi.org/10.1007/s11242-013-0259-2