Abstract
The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Brinkman model is employed. Three cases of free–free, rigid–rigid, and rigid–free boundaries are considered. The analysis reveals that for a typical nanofluid (with large Lewis number), the prime effect of the nanofluids is via a buoyancy effect coupled with the conservation of nanoparticles, whereas the contribution of nanoparticles to the thermal energy equation is a second-order effect. It is found that the critical thermal Rayleigh number can be reduced or increased by a substantial amount, depending on whether the basic nanoparticle distribution is top-heavy or bottom-heavy, by the presence of the nanoparticles. Oscillatory instability is possible in the case of a bottom-heavy nanoparticle distribution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- D B :
-
Brownian diffusion coefficient
- D T :
-
thermophoretic diffusion coefficient
- Da :
-
Darcy number, defined by Eq. 15b
- g :
-
Gravitational acceleration
- g :
-
Gravitational acceleration vector
- H :
-
Dimensional layer depth
- k m :
-
Effective thermal conductivity of the porous medium
- K :
-
Permeability of the porous medium
- Le :
-
Lewis number, defined by Eq. 15c
- N A :
-
Modified diffusivity ratio, defined by Eq. 19
- N B :
-
Modified particle-density increment, defined by Eq. 20
- p * :
-
Pressure
- p :
-
Dimensionless pressure, p * K/μα m
- Pr :
-
Prandtl number, defined by Eq. 15a
- Ra :
-
Thermal Rayleigh–Darcy number, defined by Eq. 16
- Rm :
-
Basic-density Rayleigh number, defined by Eq. 17
- Rn :
-
Concentration Rayleigh number, defined by Eq. 18
- t * :
-
Time
- t :
-
Dimensionless time, t * α m/σH 2
- T * :
-
Temperature
- T :
-
Dimensionless temperature, \({\frac{T^\ast-T^\ast_{\rm c}}{T^\ast_{\rm h} -T^\ast_{\rm c}}}\)
- \({T^*_{\rm c}}\) :
-
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 m
- v :
-
Dimensionless Darcy velocity, \({H{\bf v}^\ast_{\rm D} /\alpha _{\rm m} }\)
- \({{\bf v}^\ast_{\rm D} }\) :
-
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 m :
-
Thermal diffusivity of the porous medium, \({\frac{k_{\rm m}}{(\rho c)_{\rm f}}}\)
- β :
-
Volumetric expansion coefficient of the fluid
- ε :
-
Porosity
- λ i :
-
Parameter that takes value 0 for the case of a rigid boundary and ∞ for a free boundary, i = 1, 2
- μ :
-
Viscosity of the fluid
- \({\tilde {\mu}}\) :
-
Effective viscosity of the porous medium
- ρ f :
-
Fluid density
- ρ p :
-
Nanoparticle mass density
- (ρc)f :
-
Heat capacity of the fluid
- (ρc)m :
-
Effective heat capacity of the porous medium
- (ρc)p :
-
Effective heat capacity of the nanoparticle material
- σ :
-
Heat capacity ratio, defined by Eq. 8
- \({\phi*}\) :
-
Nanoparticle volume fraction
- \({\phi }\) :
-
Relative nanoparticle volume fraction, \({\frac{\phi^ \ast-\phi ^\ast_0}{\phi ^\ast_1 -\phi^ \ast_0}}\)
- * :
-
Dimensional variable
- ′:
-
Perturbation variable
- b:
-
basic solution
References
Buongiorno J.: Convective transport in nanofluids. ASME J. Heat Transf. 128, 240–250 (2006). doi:10.1115/1.2150834
Buongiorno, J., Hu, W.: Nanofluid coolants for advanced nuclear power plants, Paper no. 5705. In: Proceedings of ICAPP ‘05, Seoul (2005)
Choi, S.: Enhancing thermal conductivity of fluids with nanoparticles. In: Siginer D.A., Wang, H.P. (eds.) Developments and Applications of Non-Newtonian Flows, ASME FED- Vol. 231/ MD-Vol. 66, New York, pp. 99–105 (1995)
Kuznetsov A.V., Avramenko A.A.: Effect of small particles on the stability of bioconvection in a suspension of gyrotactic microorganisms in a layer of finite length. Int. Comm. Heat Mass Transf. 31, 1–10 (2004). doi:10.1016/S0735-1933(03)00196-9
Masuda H., Ebata A., Teramae K., Hishinuma N.: Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 7, 227–233 (1993)
Nield D.A., Bejan A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)
Nield, D.A., Kuznetsov, A.V.: The onset of convection in a nanofluid layer. ASME J. Heat Transf. (2009a, submitted)
Nield, D.A., Kuznetsov, A.V.: Thermal instability in a porous layer saturated by a nanofluid. Int. J. Heat Mass Transf. (2009b, submitted)
Platten J.K., Legros J.C.: Convection in Liquids, p. 372. Springer, New York (1984)
Tzou D.Y.: Instability of nanofluids in natural convection. ASME J. Heat Transf. 130, 072401 (2008a)
Tzou D.Y.: Thermal instability of nanofluids in natural convection. Int. J. Heat Mass Transf. 51, 2967–2979 (2008b). doi:10.1016/j.ijheatmasstransfer.2007.09.014
Vadasz, P.: Nanofluids suspensions: possible explanations for the apparent enhanced effective thermal conductivity, ASME paper # HT2005-72258. In: Proceedings of 2005 ASME Summer Heat Transfer Conference, San Francisco, CA, 17–22 July 2005
Vadasz P.: Heat conduction in nanofluid suspensions. ASME J. Heat Transf. 128, 465–477 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kuznetsov, A.V., Nield, D.A. Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: Brinkman Model. Transp Porous Med 81, 409–422 (2010). https://doi.org/10.1007/s11242-009-9413-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-009-9413-2