Abstract
In this paper, we consider the effect of mechanical vibration on the onset of convection in porous media. The porous medium is saturated either by a pure fluid or by a binary mixture. The importance of a transport model on stability diagrams is presented and discussed. The stability threshold for the Darcy–Brinkman case in the Ra Tc -R and k c -R diagrams is presented (where Ra Tc , k c and R are the critical Rayleigh number, the critical wave number and the vibration parameters, respectively). It is shown that there is a significant deviation from the Darcy model. In the thermo-solutal case with the Soret effect, the influence of vibration on the reduction of multi-cellular convection is emphasized. A new analytical relation for obtaining the threshold of mono-cellular convection is derived. This relation shows how the separation factor Ψ is related to the controlling parameters of the problem, Ψ = f (R, ε*, Le), when the wave number k → 0. The importance of vibrational parameter definition is highlighted and it is shown how, by using a proper definition for vibrational parameter, we may obtain compact relationship. It is also shown how this result may be used to increase component separation.
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Abbreviations
- a*:
-
Effective thermal diffusivity, m2 s−1
- b :
-
Vibration amplitude, m
- C i :
-
Initial mass fraction
- D*:
-
Mass diffusion coefficient
- D T :
-
Thermodiffusion coefficient
- Da :
-
Darcy number
- e :
-
The direction of vibration
- g :
-
Gravitational acceleration, m s−2
- H :
-
Height, m
- j :
-
Unit vector in y direction
- k :
-
Wave number
- K :
-
Permeability, m2
- Le :
-
Lewis number (a/D)*
- P :
-
Pressure, N m−2
- R :
-
Vibration parameter
- Ra :
-
Rayleigh number
- Ra v :
-
Vibrational Rayleigh number
- T :
-
Temperature, K
- t :
-
Dimensional time
- V :
-
Velocity, m s−1
- W :
-
Solenoidal vector
- α :
-
Direction of vibration
- β C :
-
Coefficient of mass expansion
- β T :
-
Coefficient of thermal expansion
- γ :
-
The ratio of the Brinkman effective viscosity to the fluid viscosity (μ e /μ f )
- ε :
-
Porosity
- ε*:
-
Normalized porosity
- λ*:
-
Effective thermal conductivity
- ν :
-
Kinematic viscosity, m2 s−1
- ρ :
-
Density, kg m−3
- (ρc)*:
-
Volumic heat capacity of medium
- τ :
-
Vibration period
- ψ :
-
Separation factor
- ω :
-
Dimensional pulsation
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Razi, Y.P., Mojtabi, A. & Charrier-Mojtabi, M.C. A Summary of New Predictive High Frequency Thermo-Vibrational Models in Porous Media. Transp Porous Med 77, 207–228 (2009). https://doi.org/10.1007/s11242-008-9332-7
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DOI: https://doi.org/10.1007/s11242-008-9332-7