Abstract
The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is analytically studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Darcy model is employed. The effect of local thermal non-equilibrium among the particle, fluid, and solid-matrix phases is investigated using a three-temperature model. The analysis reveals that in some circumstances the effect of LTNE can be significant, but for a typical dilute nanofluid (with large Lewis number and with small particle-to-fluid heat capacity ratio) the effect is small.
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Abbreviations
- D B :
-
Brownian diffusion coefficient
- D T :
-
Thermophoretic diffusion coefficient
- g :
-
Gravitational acceleration
- g :
-
Gravitational acceleration vector
- H :
-
Dimensional layer depth
- k :
-
Thermal conductivity
- K :
-
Permeability of the porous medium
- Le :
-
Lewis number, defined by Eq. 20
- N A :
-
Modified thermophoresis to Brownian-motion diffusivity ratio, defined by Eq. 24
- N B :
-
Modified particle-density increment, defined by Eq. 25
- N HP :
-
Nield number for the fluid/particle interface, defined by Eq. 26a
- N HS :
-
Nield number for the fluid/solid-matrix interface, defined by Eq. 26b
- p*:
-
Pressure
- p :
-
Dimensionless pressure, p* K/μ α f
- Ra :
-
Thermal Rayleigh–Darcy number, defined by Eq. 21
- Rm :
-
Basic-density Rayleigh number, defined by Eq. 22
- Rn :
-
Concentration Rayleigh number, defined by Eq. 23
- t*:
-
Time
- t :
-
Dimensionless time, t*α f/H 2
- T*:
-
Temperature
- T :
-
Dimensionless temperature, \({\frac{T^*-T^*_{\rm c}}{T^*_{\rm h}-T^*_{\rm c}}}\)
- \({T^*_{\rm c}}\) :
-
Temperature at the upper wall
- \({T^*_{\rm h}}\) :
-
Temperature at the lower wall
- (u, v, w):
-
Dimensionless Darcy velocity components, (u*, v*, w*)H/α f
- v :
-
Dimensionless Darcy velocity, \({H{\bf v}^*_{\rm D}/\alpha_{\rm f}}\)
- \({{\bf v}^*_{\rm D}}\) :
-
Dimensional 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
- α f :
-
Thermal diffusivity of the fluid, \({\frac{k_{\rm f}}{(\rho c)_{\rm f}}}\)
- β :
-
Volumetric expansion coefficient of the fluid
- γ P :
-
Modified thermal capacity ratio defined by Eq. 27a
- γ S :
-
Modified thermal capacity ratio defined by Eq. 27b
- \({\varepsilon}\) :
-
Porosity
- \({\varepsilon_{\rm P}}\) :
-
Modified thermal diffusivity ratio defined by Eq. 28a
- \({\varepsilon_{\rm S}}\) :
-
Modified thermal diffusivity ratio defined by Eq. 28b
- μ :
-
Viscosity of the fluid
- ρ f :
-
Fluid density
- ρ p :
-
Nanoparticle mass density
- (ρ c)f :
-
Heat capacity of the fluid
- (ρ c)p :
-
Heat capacity of the particle material
- (ρ c)s :
-
Heat capacity of the solid-matrix material
- \({\phi^*}\) :
-
Nanoparticle volume fraction
- \({\phi}\) :
-
Relative nanoparticle volume fraction, \({\frac{\phi^*-\phi^*_0}{\phi^*_1-\phi^*_0}}\)
- *:
-
Dimensional variable
- ′:
-
Perturbation variable
- b:
-
Basic solution
- f:
-
Fluid phase
- p:
-
Particle phase
- s:
-
Solid-matrix phase
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Kuznetsov, A.V., Nield, D.A. Effect of Local Thermal Non-equilibrium on the Onset of Convection in a Porous Medium Layer Saturated by a Nanofluid. Transp Porous Med 83, 425–436 (2010). https://doi.org/10.1007/s11242-009-9452-8
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DOI: https://doi.org/10.1007/s11242-009-9452-8