Abstract
A non-autonomous complex Ginzburg-Landau equation (CGLE) for the finite amplitude of convection is derived, and a method is presented here to determine the amplitude of this convection with a weakly nonlinear thermal instability for an oscillatory mode under throughflow and gravity modulation. Only infinitesimal disturbances are considered. The disturbances in velocity, temperature, and solutal fields are treated by a perturbation expansion in powers of the amplitude of the applied gravity field. Throughflow can stabilize or destabilize the system for stress free and isothermal boundary conditions. The Nusselt and Sherwood numbers are obtained numerically to present the results of heat and mass transfer. It is found that throughflow and gravity modulation can be used alternately to heat and mass transfer. Further, oscillatory flow, rather than stationary flow, enhances heat and mass transfer.
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Abbreviations
- A :
-
amplitude of convection
- δ :
-
amplitude of gravity modulation
- g :
-
acceleration due to gravity
- R 0 :
-
critical Rayleigh number
- d :
-
depth of fluid layer
- q :
-
fluid velocity
- (x, z):
-
horizontal and vertical coordinates
- Nu :
-
Nusselt number
- Pe :
-
Pćclet number, \(Pe = \frac{{{w_0}d}}{{{\kappa _T}}}\)
- p :
-
reduced pressure
- Sh :
-
Sherwood number
- ΔS :
-
solutal difference across porous media
- Rs :
-
solutal Rayleigh number, \(Rs = \frac{{{\beta _\user1{S}}g\Delta SdK}}{{v{\kappa _\user1{S}}}}\)
- T :
-
temperature
- ΔT :
-
temperature difference across porous media
- Ra :
-
thermal Rayleigh number, \(Ra = \frac{{{\alpha _\user1{S}}g\Delta TdK}}{{v{\kappa _\user1{T}}}}\)
- t :
-
time
- a :
-
wavenumber
- αT :
-
coefficient of thermal expansion
- Γ:
-
diffusivity ratio, \(\Gamma = \frac{{{\kappa _\user1{S}}}}{{{\kappa _\user1{T}}}}\)
- ω :
-
dimensionless oscillatory frequency
- μ :
-
dynamic viscosity of fluid
- κT :
-
effective thermal diffusivity
- ρ :
-
fluid density
- Ω:
-
frequency of modulation
- γ :
-
heat capacity ratio
- ν :
-
kinematic viscosity, \(v = \frac{\mu }{{{\rho _0}}}\)
- χ :
-
perturbation parameter
- ε :
-
porosity of porous media
- s :
-
slow time
- β S :
-
solutal expansion coefficient
- ψ :
-
stream function
- \(\hat k\) :
-
vertical unit vector
- ∇2 :
-
\(\frac{{{\partial ^2}}}{{\partial {x^2}}} + \frac{{{\partial ^2}}}{{\partial {y^2}}} + \frac{{{\partial ^2}}}{{\partial {z^2}}}\)
- b:
-
basic state
- c:
-
critical
- 0:
-
reference value
- *:
-
dimensionless quantity
- ′:
-
perturbed quantity
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Kiran, P. Throughflow and g-jitter effects on binary fluid saturated porous medium. Appl. Math. Mech.-Engl. Ed. 36, 1285–1304 (2015). https://doi.org/10.1007/s10483-015-1984-9
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DOI: https://doi.org/10.1007/s10483-015-1984-9