Onset of Buoyancy-Driven Convection in a Fluid-Saturated Porous Medium Bounded by a Long Cylinder
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Onset of Buoyancy-Driven Convection in a Fluid-Saturated Porous Medium Bounded by a Long Cylinder Min Chan Kim
Received: 8 October 2012 / Accepted: 21 January 2013 / Published online: 7 February 2013 © Springer Science+Business Media Dordrecht 2013
Abstract A theoretical analysis of convective instability driven by buoyancy forces under the transient concentration fields is conducted in an initially quiescent, liquid-saturated, cylindrical porous column. 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 principle of exchange of stabilities, the stability equations are derived in self-similar boundary-layer coordinate. The present predictions suggest the critical R D , and the onset time and corresponding wavenumber for a given R D . The onset time becomes smaller with increasing R D and follows the asymptotic relation derived in the infinite horizontal porous layer. Keywords Buoyancy-driven convection · Porous medium · Cylindrical geometry · Linear stability analysis
Nomenclature a∗ C c De g K P R (r, θ, Z ) (¯r , θ, z) RD
√ Modified wave number, αl,m τ Concentration (M) Dimensionless concentration Effective diffusivity (m2 /s) Gravitation acceleration (m/s2 ) Permeability of porous media (m2 ) Pressure (Pa) Radius of cylinder (m) Cylindrical coordinates Dimensionless cylindrical coordinates Darcy–Rayleigh number, R D = gβ K C R/(ε De ν)
M. C. Kim (B) Department of Chemical Engineering, Jeju National University, Jeju 690-756, Republic of Korea e-mail: [email protected]
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t U W w
Time (s) Velocity vector (m/s) Vertical velocity (m/s) Dimensionless vertical velocity
Greek Letters αl,m β ε ζ μ ν τ ρ
Wave number Volumetric expansion coefficient (M−1 ) Porosity of porous media Dimensionless similarity variable, z/τ 1/2 Viscosity (Pa s) Kinematic viscosity (m2 /s) Dimensionless time Density (kg/m3 )
Subscripts 0 1 c
Basic quantities Perturbation quantities Critical conditions
Superscript *
Transformed quantities
1 Introduction The buoyancy-driven convection is well-known natural phenomena, and has attracted many researchers’ interests due to a wide variety of engineering applications, such as geothermal reservoirs, agricultural product storage system, packed-bed catalytic reactors, the pollutant transport in underground and the heat removal of nuclear power plants. Recently the phenomena of natural convection in porous media have been investigated in connection with the enhanced carbon dioxide dissolution into the saline water confined within the geologically stable formations (Ennis-King et al. 2005; Xu et al. 2006; Riaz et al. 2006; Hassanzadeh et al. 2006), and the novel-enhanced oil recovery (EOR) method where diffusion–convection may be employed as drive to force heavy, viscous oil to move (Rashidi et al. 2000). The dissolution process can be enhanced by the motion driven by the density gradient. Therefore, the onset condition of the buoyanc
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