Transp Porous Med (2011) 87:121–123 DOI 10.1007/s11242-010-9671-z

A Note on the Onset of Convection in a Layer of a Porous Medium Saturated by a Non-Newtonian Nanofluid of Power-Law Type D. A. Nield

Received: 15 August 2010 / Accepted: 28 September 2010 / Published online: 12 October 2010 © Springer Science+Business Media B.V. 2010

Abstract It is pointed out that the Horton–Rogers–Lapwood problem becomes singular when a Newtonian fluid is replaced by a standard power-law fluid. It is shown how this singularity can be removed. When this is done, the nanofluid effects due to thermophoresis and Brownian motion become independent of the power-law index. Keywords

Onset of convection · Porous medium · Power-law fluid · Nanofluid

1 Introduction and Discussion Convection in a porous medium saturated by a non-Newtonian fluid is of interest in several fields of application such as oil recovery, food processing, the spread of contaminants in the environment, and in various processes in the chemical and materials industry. It is often convenient to model the non-Newtonian behavior by use of a power-law fluid, in which the drag force is assumed to vary with the nth power of the velocity. (In a fluid clear of solid material, the shear stress is assumed to vary as the nth power of the velocity shear, but integration over a representative elementary volume leads to an expression involving the same power of the velocity itself.) Since the pioneering work of Chen and Chen (1988) many other articles on convection in a porous medium saturated by a power-law fluid have been published, but as far as the author is aware they are all concerned with boundary-layer flows along a vertical plate or with an enclosure heated from the side. None of them have been concerned with the onset of convection in a horizontal layer heated from below (the Horton–Rogers–Lapwood (HRL) problem). All the studies of the HRL problem with a non-Newtonian fluid have dealt with a fluid of special type (such as a micropolar or viscoelastic fluid); see the articles surveyed in Sect. 6.23 of Nield and Bejan (2006). Where a power-law fluid and a horizontal layer are

D. A. Nield (B) Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand e-mail: [email protected]

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involved the studies have been not on the onset of convection but rather, as in Amari et al. (1994), on situations in which the convection is well advanced. This dearth of articles on the HRL problem for a power-law fluid came to the author’s attention when investigating the case of a nanofluid (a fluid with a suspension of nanoparticles, that is particles with a diameter of the order of 10 or 100 nm). Convection in nanofluids is a subject of intense current interest. The general topic of heat transfer in nanofluids has been surveyed in a review article by Das and Choi (2009) and a book by Das et al. (2008). In particular, studies of the HRL problem with nanofluids were made by Nield and Kuznetsov (2009) and Kuznetsov and Nield (2010a,b,c, 2011) in which the effects of thermophoresis and Brownian motion were included. In each of these articles, the nanofluid itself was modeled as a Newtonian fluid. The present article is concerned with what happens when the Newtonian fluid is replaced by a power-law fluid. (For completeness it should be noted that Straughan (2010) has developed the theory of Green and Naghdi to describe the convection of nanofluids. The theory involves vorticity and spin of vorticity. This is non-Newtonian behavior of a very special kind.) However, first the case of a regular power-law fluid must be considered. When one attempts to apply the usual linear stability theory for the HRL problem, with a perturbation amplitude ε, to the case of a power-law fluid one strikes immediately a severe mathematical problem, unless the power-law index n has the value 1 (so that one has simple Darcy drag). The drag term is of order ε n , and unless n = 1 this is incommensurate with the other terms in the perturbation equation (which are of orders ε, ε 2 , . . . ,). It is of no use to abandon the perturbation scheme and perform calculations and extrapolate to ε = 0 in order to determine the critical Rayleigh number, because it is notorious that this numerical methodology is deceptively inaccurate in this situation (see, for example, the refutation by Lage et al. (1992)) of claims by other researchers, who have relied on standard computation, that the critical Rayleigh number depends on the Prandtl number). Clearly, a fresh approach is needed. n where c is a constant and u is The author suggests that the drag term, of the form   cu , n−1 . For large values of u, the the Darcy velocity, be replaced by the expression cu 1 + δu usual power law with index n is retained, and the empirical fit between the new law and the old law is good when n > 1. For small values of u, the drag is linear in u, and the usual linear perturbation procedure can be followed. If one adopts this scheme then one concludes that the critical Rayleigh number is independent of the power-law index. This applies whether or not the fluid is a nanofluid or not. Thus, all the results of Nield and Kuznetsov (2009) and Kuznetsov and Nield (2010a,b,c, 2011) are independent of the index n when the nanofluid is a power-law fluid.

References Amari, B., Vasseuer, P., Bilgen, E.: Natural convection of non-Newtonian fluids in a horizontal porous layer. Wärme-Stoffübertrag. 29, 185–193 (1994) Chen, H.T., Chen, C.K.: Free convection flow of non-Newtonian fluids along a vertical plate embedded in a porous medium. ASME J. Heat Transf. 110, 257–260 (1988) Das, S.K., Choi, S.U.S.: A review of heat transfer in nanofluids. Adv. Heat Transf. 41, 81–197 (2009) Das, S.K., Choi, S.U.S., Yu, W., Pradeep, T.: Nanofluids: Science and Technology. Wiley, Hoboken, NY (2008) Kuznetsov, A.V., Nield, D.A.: Thermal instability in a porous medium saturated by a nanofluid: Brinkman model. Transp. Porous Media 81, 409–422 (2010a) Kuznetsov, A.V., Nield, D.A.: The effect of local thermal equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid. Transp. Porous Media, to appear (2010b)

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Kuznetsov, A.V., Nield, D.A.: The onset of double-diffusive convection in a layer of a saturated porous medium. porous medium saturated by a nanofluid: Brinkman model. Transp. Porous Media, to appear (2010c) 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 Media 14, to appear (2011) Lage, J.L., Bejan, A., Georgiadis, J.: The Prandtl number effect near the onset of Bénard convection in a porous medium. Int. J. Heat Fluid Flow 13, 408–411 (1992) Nield, D.A., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006) 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) Poulikakos, D., Spatz, T.L.: Non-Newtonian natural convection at a melting front in a permeable solid matrix. Int. Commun. Heat Mass Transf. 15, 593–603 (1988) Rastogi, S.K., Poulikakos, D.: Double-diffusion from a vertical surface in a porous region saturated with a non-Newtonian fluid. Int. J. Heat Mass Transf. 38, 935–946 (1995) Straughan, B.: Green-Naghdi fluid with non-thermal equilibrium effects. Proc. Roy. Soc. A 466, 2021– 2032 (2010)

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