Transport in Porous Media

, Volume 85, Issue 3, pp 941–951 | Cite as

The Onset of Double-Diffusive Nanofluid Convection in a Layer of a Saturated Porous Medium



The paper develops a theory of double-diffusive nanofluid convection in porous media. This theory is applied to investigating the onset of nanofluid convection in a horizontal layer of a porous medium saturated by a nanofluid for the case when the base fluid of the nanofluid is itself a binary fluid such as salty water. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis, while the Darcy model is used for the porous medium. In addition the thermal energy equations include regular diffusion and cross-diffusion terms. Both non-oscillatory and oscillatory cases are investigated by using Galerkin method; the stability boundaries for these cases are approximated by simple and useful analytical expressions.


Nanofluid convection Porous media Brownian motion Thermophoresis Natural convection Horizontal layer 



Solute concentration


Brownian diffusion coefficient (m2/s)


Thermophoretic diffusion coefficient (m2/s)


Dimensional layer depth (m)


Thermal conductivity of the nanofluid (W/m K)


Permeability (m2)


Thermo-solutal Lewis number, defined by Eq. (30)


Thermo-nanofluid Lewis number, defined by Eq. (23)


Modified diffusivity ratio, defined by Eq. (28)


Modified particle-density increment, defined by Eq. (29)


Soret parameter, defined by Eq. (32)


Dufour parameter, defined by Eq. (31)


Pressure (Pa)


Dimensionless pressure, \({p^{\ast}K/\mu \alpha_{\rm m}}\)


Thermal Rayleigh-Darcy number, defined by Eq. (24)


Basic-density Rayleigh number, defined by Eq. (26)


Nanoparticle Rayleigh number, defined by Eq. (27)


Solutal Rayleigh number, defined by Eq. (25)


Time (s)


Dimensionless time, \({t^{\ast}\alpha_{\rm m} /H^{2}}\)


Nanofluid temperature (K)


Dimensionless temperature, \({\frac{T^\ast-T^\ast_c}{T^\ast_h -T^\ast_c }}\)

\({T^*_{\rm c}}\)

Temperature at the upper wall (K)

\({T^*_{\rm h}}\)

Temperature at the lower wall (K)

(u, v, w)

Dimensionless velocity components, \({(u^\ast,v^\ast,w^\ast)H/\alpha_{\rm m}}\) (m/s)


Nanofluid velocity (m/s)

(x, y, z)

Dimensionless Cartesian coordinates, (x*, y*, z*)/H; z is the vertically-upward coordinate

(x*, y*, z*)

Cartesian coordinates (m)



Effective thermal diffusivity of the porous medium (m/s2)


Solutal volumetric coefficient


Thermal volumetric coefficient (K−1)




Viscosity of the fluid (N s/m2)


Fluid density (kg/m3)


Nanoparticle mass density (kg/m3)


Thermal capacity ratio


Nanoparticle volume fraction


Relative nanoparticle volume fraction, \({\frac{\phi^\ast-\phi^\ast_0}{\phi^\ast_1 -\phi^\ast_0}}\)



Dimensional variable

Perturbation variable



Basic solution






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  1. Abbassi H., Aghanajafi C.: Evaluation of heat transfer augmentation in a nanofluid-cooled microchannel heat sink. J. Fusion Energy 25, 187–196 (2006)CrossRefGoogle Scholar
  2. Anoop K.B., Kabelac S., Sundararajan T., Das S.K.: Rheological and flow characteristics of nanofluids: influence of electroviscous effects and particle agglomeration. J. Appl. Phys. 106, 034909 (2009)CrossRefGoogle Scholar
  3. Buongiorno J.: Convective transport in nanofluids. ASME J. Heat Transf. 128, 240–250 (2006)CrossRefGoogle Scholar
  4. Buongiorno, J., Hu, L.-W.: Nanofluid coolants for advanced nuclear power plants. Paper no. 5705, Proceedings of ICAPP ‘05, Seoul, May 15–19 (2005)Google Scholar
  5. Choi, S.: Enhancing thermal conductivity of fluids with nanoparticles. ASME FED-Vol. 231/ MD-Vol. 66, 99–105 (1995)Google Scholar
  6. Das S.K., Choi S.U.S.: A review of heat transfer in nanofluids. Adv. Heat Transf. 41, 81–197 (2009)Google Scholar
  7. Das S.K., Choi S.U.S., Yu W., Pradeep T.: Nanofluids: Science and Technology. Wiley, Hoboken, NY (2008)Google Scholar
  8. Feng Y., Yu B., Feng K., Xu P., Zou M.: Thermal conductivity of nanofluids and size distribution of nanoparticles by monte carlo simulations. J. Nanoparticle Res. 10, 1319–1328 (2008)CrossRefGoogle Scholar
  9. Ganguly S., Sikdar S., Basu S.: Experimental investigation of the effective electrical conductivity of aluminum oxide nanofluids. Powder Technol. 196, 326–330 (2009)CrossRefGoogle Scholar
  10. Ghazvini M., Akhavan-Behabadi M.A., Esmaeili M.: The effect of viscous dissipation on laminar nanofluid flow in a microchannel heat sink. IME J. Mech. Eng. Sci. 223, 2697–2706 (2009)CrossRefGoogle Scholar
  11. Ghazvini M., Shokouhmand H.: Investigation of a nanofluid-cooled microchannel heat sink using fin and porous media approaches. Energy Convers. Manage. 50, 2373–2380 (2009)CrossRefGoogle Scholar
  12. Hwang K.S., Jang S.P., Choi S.U.S.: Flow and convective heat transfer characteristics of water-based Al2O3 nanofluids in fully developed laminar flow regime. Int. J. Heat Mass Transf. 52, 193–199 (2009)CrossRefGoogle Scholar
  13. Jain S., Patel H.E., Das S.K.: Brownian dynamic simulation for the prediction of effective thermal conductivity of nanofluid. J. Nanoparticle Res. 11, 767–773 (2009)CrossRefGoogle Scholar
  14. Kim S.Y., Koo J.M., Kuznetsov A.V.: Effect of anisotropy in permeability and effective thermal conductivity on thermal performance of an aluminum foam heat sink. Numer. Heat Transf. A 40, 21–36 (2001)Google Scholar
  15. Kim S.Y., Kuznetsov A.V.: Optimization of pin-fin heat sinks using anisotropic local thermal nonequilibrium porous model in a jet impinging channel. Numer. Heat Transf. A 44, 771–787 (2003)CrossRefGoogle Scholar
  16. Kuznetsov A.V., Nield D.A.: Thermal instability in a porous medium saturated by a nanofluid: Brinkman model. Transp. Porous Med. 81, 409–422 (2010a)CrossRefGoogle Scholar
  17. 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 (2010b)CrossRefGoogle Scholar
  18. Kuznetsov, A.V., Nield, D.A.: The effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid: Brinkman model. J. Porous Med 14, to appear (2011)Google Scholar
  19. Lee S., Choi S.U.S., Li S., Eastman J.A.: Measuring thermal conductivity of fluids containing oxide nanoparticles. ASME J. Heat Transf. 121, 280–289 (1999)CrossRefGoogle Scholar
  20. 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)Google Scholar
  21. Merabia S., Shenogin S., Joly L., Keblinski P., Barrat J.: Heat transfer from nanoparticles: a corresponding state analysis. Proc. Nat. Acad. Sci. 106, 15113–15118 (2009)CrossRefGoogle Scholar
  22. Nelson I.C., Banerjee D., Ponnappan R.: Flow loop experiments using polyalphaolefin nanofluids. J. Thermophys. Heat Transf. 23, 752–761 (2009)CrossRefGoogle Scholar
  23. Nield D.A.: Onset of thermohaline convection in a porous medium. Water Resour. Res. 4, 533–560 (1968)CrossRefGoogle Scholar
  24. Nield D.A., Kuznetsov A.V.: Thermal instability in a porous medium layer saturated by a nanofluid. Int. J. Heat Mass Transf. 52, 5796–5801 (2009)CrossRefGoogle Scholar
  25. Nield, D.A., Kuznetsov, A.V.: The onset of double-diffusive convection in a nanofluid layer. ASME J. Heat Transf. submitted (2010)Google Scholar
  26. Pearlstein A.J., Harris R.M., Terrones G.: The onset of convective instability in a triple diffusive layer. J. Fluid Mech. 202, 443–465 (1989)CrossRefGoogle Scholar
  27. Rea U., McKrell T., Hu L., Buongiorno J.: Laminar convective heat transfer and viscous pressure loss of alumina-water and zirconia-water nanofluids. Int. J. Heat Mass Transf. 52, 2042–2048 (2009)CrossRefGoogle Scholar
  28. Salloum M., Ma R.H., Weeks D., Zhu L.: Controlling nanoparticle delivery in magnetic nanoparticle hyperthermia for cancer treatment: experimental study in agarose gel. Int. J. Hyperthermia 24, 337–345 (2008a)CrossRefGoogle Scholar
  29. Salloum M., Ma R., Zhu L.: An in-vivo experimental study of temperature elevations in animal tissue during magnetic nanoparticle hyperthermia. Int. J. Hyperthermia 24, 589–601 (2008b)CrossRefGoogle Scholar
  30. Salloum M., Ma R., Zhu L.: Enhancement in treatment planning for magnetic nanoparticle hyperthermia: optimization of the heat absorption pattern. Int. J. Hyperthermia 25, 309–321 (2009)CrossRefGoogle Scholar
  31. Tsai T., Chein R.: Performance analysis of nanofluid-cooled microchannel heat sinks. Int. J. Heat Fluid Flow 28, 1013–1026 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringNorth Carolina State UniversityRaleighUSA
  2. 2.Department of Engineering ScienceUniversity of AucklandAucklandNew Zealand

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