Skip to main content
SpringerLink
Log in

The Effect of Vertical Throughflow on Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

The effect of vertical throughflow on 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. The dependences of the critical Rayleigh number for the non-oscillatory and oscillatory modes of instability on the thermophoresis and Brownian motion parameters for the cases with and without throughflow are investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

c :

Nanofluid specific heat at constant pressure

c p :

Specific heat of the nanoparticle material

D B :

Brownian diffusion coefficient

D T :

Thermophoretic diffusion coefficient

F :

Parameter defined by Eq. 52

g :

Gravitational acceleration

H :

Dimensional layer depth

J T :

Parameter defined by Eq. 46

\({J_{\phi}}\) :

Parameter defined by Eq. 48

k :

Thermal conductivity of the nanofluid

k m :

Effective thermal conductivity of the porous medium

k p :

Thermal conductivity of the nanoparticle material

K :

Permeability of the porous medium

K T :

Parameter defined by Eq. 47

\({K_{\phi}}\) :

Parameter defined by Eq. 49

Le :

Lewis number, defined by Eq. 16

N A :

Modified diffusivity ratio, defined by Eq. 20

N B :

Modified particle-density increment, defined by Eq. 21

N 1 :

Parameter defined by Eq. 50

N 2 :

Parameter defined by Eq. 51

p*:

Pressure

p :

Dimensionless pressure, p*K/μα m

Q :

Péclet number defined by Eq. 15

Ra :

Thermal Rayleigh–Darcy number, defined by Eq. 17

Rm:

Basic-density Rayleigh number, defined by Eq. 18

Rn:

Concentration Rayleigh number, defined by Eq. 19

t*:

Time

t :

Dimensionless time, t*α m/σ H 2

T*:

Nanofluid temperature

T :

Dimensionless temperature, \({\frac{T^{\ast}-T^{\ast}_{\rm c}}{T^\ast_{\rm h} -T^\ast_{\rm c}}}\)

\({T^{\ast}_{\rm c}}\) :

Temperature at the upper wall

\({T^{\ast}_{\rm h}}\) :

Temperature at the lower wall

(u, v, w):

Dimensionless Darcy velocity components, (u*, v*, w*)H/α m

v :

Dimensionless Darcy velocity

v*:

Dimensional Darcy velocity, (u*, v*, w*)

V*:

Vertical throughflow velocity

(x, y, z):

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

(x*, y*, z*):

Cartesian coordinates

α :

Dimensionless horizontal wavenumber

α m :

Thermal diffusivity of the porous medium, \({\frac{k_{\rm m}}{(\rho c_{\rm p} )_{\rm f}}}\)

ε :

Porosity of the porous medium

λ:

Parameter defined by Eq. 27

μ :

Viscosity of the fluid

ρ :

Fluid density

ρ p :

Nanoparticle mass density

(ρc)f :

Heat capacity of the fluid

(ρc)m :

Effective heat capacity of the porous medium

σ :

Parameter defined by Eq. 8

\({\phi^{\ast}}\) :

Nanoparticle volume fraction

\({\phi_{0}^{\ast}}\) :

Nanoparticle volume fraction at the lower wall

\({\phi_{1}^{\ast}}\) :

Nanoparticle volume fraction at the upper wall

\({\phi}\) :

Relative nanoparticle volume fraction, \({\frac{\phi^{\ast}-\phi^{\ast}_0}{\phi^{\ast}_1 -\phi^{\ast}_0}}\)

ω :

Dimensionless frequency

*:

Dimensional variable

′:

Perturbation variable

b:

Basic solution

References

  • Buongiorno J.: Convective transport in nanofluids. ASME J. Heat Transfer 128, 240–250 (2006)

    Article  Google Scholar 

  • Buongiorno, J., Hu, W.: Nanofluid coolants for advanced nuclear power plants, Paper no. 5705. Proceedings of ICAPP ’05 Seoul, 15–19 (2005)

  • Buongiorno J., Hu L., Kim S.J., Hannink R., Truong B., Forrest E.: Nanofluids for enhanced economics and safety of nuclear reactors: An evaluation of the potential features, issues, and research gaps. Nucl. Technol. 162, 80–91 (2008)

    Google Scholar 

  • Choi, S.: Enhancing thermal conductivity of fluids with nanoparticle. In: Siginer, D.A., Wang, H.P. (eds.) Developments and Applications of Non-Newtonian Flows, ASME FED-vol. 231/ MD-vol. 66, pp. 99–105 (1995)

  • Choi S.U.S.: Nanofluids: from vision to reality through research. ASME J. Heat Transfer 131, 033106 (2009)

    Article  Google Scholar 

  • Choi S.U.S., Zhang Z.G., Yu W., Lockwood F.E., Grulke E.A.: Anomalous thermal conductivity enhancement in nanotube suspensions. Appl. Phys. Lett. 79, 2252–2254 (2001)

    Article  Google Scholar 

  • Choi S.U.S., Zhang Z., Keblinski P.: Nanofluids. In: Nalwa, H. (ed.) Encyclopedia of Nanoscience and Nanotechnology, pp. 757–773. American Scientific Publishers, New York (2004)

    Google Scholar 

  • Das S., Choi S.U.S., Yu W., Pradeep T.: Nanofluids Science and Technology. Wiley, Hoboken (2008)

    Google Scholar 

  • Eastman J.A., Choi S.U.S., Li S., Yu W., Thompson L.J.: Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl. Phys. Lett. 78, 718–720 (2001)

    Article  Google Scholar 

  • Kim S.J., McKrell T., Buongiorno J., Hu L.: Enhancement of flow boiling critical heat flux (CHF) in alumina/water nanofluids. Adv. Sci. Lett. 2, 100–102 (2009)

    Google Scholar 

  • Kleinstreuer C., Li J., Koo J.: Microfluidics of nano-drug delivery. Int. J. Heat Mass Transf. 51, 5590–5597 (2008)

    Article  Google Scholar 

  • Kuznetsov A.V., Nield D.A.: Thermal instability in a porous medium layer saturated by a nanofluid: Brinkman model. Transp. Porous Media 81, 409–422 (2010)

    Article  Google Scholar 

  • Lee S., Choi S.U.S., Li S., Eastman J.A.: Measuring thermal conductivity of fluids containing oxide nanoparticles. ASME J. Heat Transfer 121, 280–289 (1999)

    Article  Google Scholar 

  • 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 

  • Nield D.A.: Convective instability in porous media with throughflow. AIChE J. 33, 1222–1224 (1987)

    Article  Google Scholar 

  • Nield D.A., Bejan A.: Convection in Porous Media, 3rd ed. Springer, New York (2006)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • Nield D.A., Kuznetsov A.V.: The effect of local thermal nonequilibrium on the onset of convection in a nanofluid. ASME J. Heat Transfer 132, 052405 (2010)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Wu C., Cho T.J., Xu J., Lee D., Yang B., Zachariah M.R.: Effect of nanoparticle clustering on the effective thermal conductivity of concentrated silica colloids. Phys. Rev. E 81, 011406 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    D. A. Nield

  2. Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC, 27695-7910, USA

    A. V. Kuznetsov

Authors
  1. D. A. Nield
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Kuznetsov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nield, D.A., Kuznetsov, A.V. The Effect of Vertical Throughflow on Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid. Transp Porous Med 87, 765–775 (2011). https://doi.org/10.1007/s11242-011-9717-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-011-9717-x

Keywords

Search

Navigation