Transp Porous Med (2014) 102:137–138DOI 10.1007/s11242-014-0269-8

A Note on the Paper of Nield and Kuznetsov

Jaimala · Reema Singh

Received: 29 October 2013 / Accepted: 3 January 2014 / Published online: 14 January 2014© Springer Science+Business Media Dordrecht 2014

Nield and Kuznetsov (2011) discussed the effect of vertical throughflow on the onset ofconvection in a horizontal layer of a porous medium saturated by a nanofluid. Using linearstability theory based on normal mode technique, the expression of Rayleigh number hasbeen derived for non-oscillatory and oscillatory modes. The dependence of Rayleigh number(Ra) on thermophoresis and Brownian motion parameters NA and NB has been investigatedby them in the absence and presence of through flow.

The following are the main conclusions in their paper:

1. In the absence of throughflow (Q = 0), the critical Rayleigh number (Ra) for non-oscillatory modes depends upon the thermophoresis parameter NA but for oscillatorymodes it neither depends upon the thermophoresis parameter NA nor Brownian motionparameter NB. The frequency does depend on the thermophoresis parameter NA.

2. In the presence of throughflow (Q �= 0), the critical Rayleigh number (Ra) for non-oscillatory modes depends on both the thermophoresis and Brownian motion parametersNA and NB whereas for oscillatory modes, it is independent of both of them.

On further critical examination of their work, we found that the conclusion about criticalRayleigh number for oscillatory modes in the presence of throughflow is not correct.

The erroneous conclusion in the paper of Nield and Kuznetsov (2011) is due to the fact thatone term in their Eq. (65) obtained for oscillatory convection in the presence of throughflowis missing. The correct form of Eq. (65) is

Ra(1 + KT )

σ+ Rn

(1 − Kφ)

ε= 4π2

[1

Le+ 1

σ

]− 2

N1

σ,

which includes the effect of NA and NB both through the parameter N1 given by

N1 = π

Le

[NB Jφ + 2NA NB JT

].

Jaimala · R. Singh (B)Department of Mathematics, Ch. Charan Singh University, Meerut 250004, UP, Indiae-mail: reemamalik28@gmail.com; reemaveerbhan@gmail.com

Jaimalae-mail: jaimalccsu@gmail.com

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138 Jaimala, R. Singh

Moreover, the effect of NA on Rayleigh number (Ra) also enters through Kφ , where theexpression for Kφ is written as

Kφ =Q2

[Le2

ε2 (4π2 + Q2)]

(4π2 + Q2)

[4π2 +

(LeQ

ε

)2] + 4π2 NA

( Leε

+ 1)

(4π2 + Q2)

[4π2 +

(LeQ

ε

)2] = K ′

φ + K ′′φ.

Here the contribution K ′φ to K ′′

φ is due to the presence of thermophoresis parameter NA.The conclusion about the frequency that it depends both on the thermophoresis and Brown-

ian motion parameters NA and NB is correct, though there is some misprint in Eq. (66) andthe correct form of this equation should read as

Leω2

σα2 = 4π2[

1 − N1

2π2 + NA N2

2π2

]− Ra

[1 + KT − (1 − Kφ)

LeN2

2π2ε

]

−Rn

[(1 + KT )NA + (1 − Kφ)

Le

ε

(1 − N1

2π2

)].

Our study is important in view of the fact that the critical Rayleigh number (Ra) for oscilla-tory modes in the presence of throughflow (Q �= 0) depends both on the thermophoresis andBrownian motion parameters NA and NB, contrary to the conclusion of Nield and Kuznetsov(2011).

Reference

Nield, D.A., Kuznetsov, A.V.: The effect of vertical throughflow on thermal instability in a porous mediumlayer saturated by a nanofluid. Transp. Porous Med. 87, 765–775 (2011)

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