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The onset of double-diffusive convection in a nanofluid layer
International Journal of Heat and Fluid Flow 32 (2011) 771–776
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International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
The onset of double-diffusive convection in a nanofluid layer D.A. Nield a, A.V. Kuznetsov b,⇑ a b
Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland, New Zealand Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC 27695-7910, USA
a r t i c l e
i n f o
Article history: Received 20 August 2010 Received in revised form 14 February 2011 Accepted 31 March 2011 Available online 6 May 2011
a b s t r a c t The onset of double-diffusive convection in a horizontal layer of a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. In addition the thermal energy equations include regular diffusion and cross-diffusion terms. The stability boundaries for both non-oscillatory and oscillatory cases have been approximated by simple analytical expressions. Physical significance of the obtained results is discussed. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The term ‘‘nanofluid’’ refers to a liquid containing a suspension of metallic or non-metallic nanometer-sized solid particles and fibres (nanoparticles). The term was suggested by Choi (1995). The characteristic feature of nanofluids is thermal conductivity enhancement, a phenomenon observed by Masuda et al. (1993). A significant body of work addresses physical mechanisms of thermal conductivity enhancement in nanofluids (utilizing both classical and non-classical approaches), see for example Eapen et al. (2010), Fan and Wang (2010a, 2010b), Gao et al. (2009), Wang and Wei (2009), Wang and Fan (2010). There is a variety of potential engineering applications of nanofluids, including advanced nuclear systems (Buongiorno and Hu, 2005). 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). On the particular topic of convective transport in nanofluids, a comprehensive survey was made by Buongiorno (2006) who, after considering alternative agencies, proposed a model incorporating the effects of Brownian diffusion and the thermophoresis. This model was applied to the Rayleigh-Bénard problem (the onset of convection in a horizontal layer uniformly heated from below) by Nield and Kuznetsov (2010a, 2010b). Both Brownian diffusion and thermophoresis give rise to cross-diffusion terms that are in some ways analogous to the familiar Soret and Dufour crossdiffusion terms that arise with a binary fluid. This aspect of transport in nanofluids has been discussed by Kim et al. (2006, 2007) and by Savino and Paterna (2008). In the present paper a fresh approach to cross-diffusion in nanofluids is made, with the aim of making a detailed comparison with regular cross-diffusion effects and the cross-diffusion effects peculiar to nanofluids, and at the same time investigating the ⇑ Corresponding author. E-mail address: [email protected] (A.V. Kuznetsov). 0142-727X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2011.03.010
interaction between these effects when the base fluid of the nanofluid is itself a binary fluid such as salty water. The outcome is that we investigating a sort of triple-diffusion problem involving heat, the nanoparticles and the solute. The analysis is an extension of that presented by Nield and Kuznetsov (2010a). 2. Conservation equations for a nanofluid The conservation equations as formulated by Buongiorno (2006) are now extended as follows. It is assumed that no nanoparticle agglomeration occurs and that the nanoparticle suspension remains stable. The continuity equation for the nanofluid is
r v ¼ 0:
ð1Þ
Here v is the nanofluid velocity. In the presence of thermophoresis the conservation equation for the nanoparticles, in the absence of chemical reactions, takes the form
@/ rT þ v r/ ¼ r DB r/ þ DT ; @t T
ð2Þ
where / is nanoparticle volume fraction, T is the temperature, DB is the Brownian diffusion coefficient, and DT is the thermophoretic diffusion coefficient. It is assumed that the solute does not affect the transport of the nanoparticles. If one introduces a buoyancy force and adopts the Boussinesq approximation, then the momentum equation can be written as
@v þ v rv ¼ rp þ lr2 v þ qg; @t
q
ð3Þ
where q is the overall density of the nanofluid, which we now assume to be given by
q ¼ /qp þ ð1 /Þq0 ½1 bT ðT T 0 Þ bC ðC C 0 Þ;
ð4Þ
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Nomenclature C DB DCT DS DT DTc H k Le Ln NA NB NCT NTC p⁄ p Ra Rm Rn Rs t⁄ t T⁄ T T⁄c T⁄h
solute concentration Brownian diffusion coefficient (m2/s) diffusivity of Soret type (m2/s) solutal diffusivity (m2/s) thermophoretic diffusion coefficient (m2/s) diffusivity of Dufour type (m2/s) dimensional layer depth (m) thermal conductivity of the nanofluid (W/m K) thermo-solutal Lewis number, defined by Eq. (32) thermo-nanofluid Lewis number, defined by Eq. (25) modified diffusivity ratio, defined by Eq. (30) modified particle-density increment, defined by Eq. (31) Soret parameter, defined by Eq. (34) Dufour parameter, defined by Eq. (33) pressure (Pa) dimensionless pressure, p⁄H2/laf thermal Rayleigh number, defined by Eq. (26) basic-density Rayleigh number, defined by Eq. (28) nanoparticle Rayleigh number, defined by Eq. (29) solutal Rayleigh number, defined by Eq. (27) time (s) dimensionless time, t⁄af/H2 nanofluid temperature (K) T T dimensionless temperature, T T c c h temperature at the upper wall (K) temperature at the lower wall (K)
where qp is the particle density, q0 is a reference density for the fluid, bT is the thermal volumetric expansion coefficient and bC is the analogous solutal coefficient (something that is negative if, as is usual, the solute is more dense that the solvent). The thermal energy equation for a nanofluid can be written as
qc
@T rT rT þ v rT ¼ kr2 T þ qp cp DB r/ rT þ DT @t T þ qcDTC r2 C;
ð5Þ
where c is the fluid specific heat (at constant pressure), k is the nanofluid thermal conductivity, cp is the specific heat of the material constituting the nanoparticles, and DTC is a diffusivity of Dufour type. Eq. (5) differs from that used in Nield and Kuznetsov (2010a) by the addition of the last term. To this we add a conservation equation for the solute of the form
@C þ v rC ¼ DS r2 C þ DCT r2 T; @t
ð6Þ
(u, v, w) v (x, y, z)
dimensionless velocity components, (u⁄, v⁄, w⁄)H/af nanofluid velocity (m/s) dimensionless Cartesian coordinates, (x⁄, y⁄, z⁄)/H; z is the vertically-upward coordinate (x⁄, y⁄, z⁄) Cartesian coordinates (m) Greek
af bC bT k1, k2
l q qp /⁄ /
1
bC DCT ; bT DS
Subscripts b basic solution f fluid p particle
asterisk has not been needed because all the variables were dimensional). Each boundary wall is assumed to be perfectly thermally conducting. The temperatures at the lower and upper boundary are taken to be T 0 þ DT and T 0 and the corresponding solute concentrations are taken as C 0 þ DC and C 0 . The Oberbeck–Boussinesq approximation is employed. Homogeneity of the fluid is assumed. In the linear theory being applied here the temperature change in the fluid is assumed to be small in comparison with T 0 and likewise the concentration change in the fluid is assumed to be small in comparison with C 0 . The conservation equations take the form
r v ¼ 0;
q0
is employed in lieu of DCT.) It has been assumed that the nanoparticles do not affect the transport of the solute.
3. Application to the Rayleigh-Bénard problem We select a coordinate frame in which the z-axis is aligned vertically-upwards. We consider a horizontal layer of fluid confined between the planes z⁄ = 0 and z⁄ = H. From now on asterisks are used to denote dimensional variables (previously an
@ v þ v r v @t
ð8Þ
h ¼ r p þ lr2 v þ / qp þ ð1 / Þq0 ½1 bT ðT T 0 Þ bC ðC C 0 Þ g;
qc ð7Þ
0
Superscripts ⁄ dimensional variable 0 perturbation variable
where DS is the solutal diffusivity and DCT is a diffusivity of Soret type. (Commonly a Soret parameter S defined by
S¼
thermal diffusivity of the fluid (m2/s) solutal volumetric coefficient thermal volumetric coefficient (K1) parameters introduced in Eqs. (13) and (14) viscosity of the fluid (N s/m2) fluid density (kg/m3) nanoparticle mass density (kg/m3) nanoparticle volume fraction / / relative nanoparticle volume fraction, / /0
ð9Þ
@T þ v r T @t
¼ kr2 T þ ðqcÞp DB r / r T þ ðDT =T c Þr T r T þ qcDTC r2 C ;
ð10Þ
@C þ v r C ¼ DS r2 C þ DCT r2 T ; @t
ð11Þ
@/ þ v r / ¼ DB r2 / þ ðDT =T c Þr2 T : @t
ð12Þ
We write v⁄ = (u⁄, v⁄, w⁄).
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D.A. Nield, A.V. Kuznetsov / International Journal of Heat and Fluid Flow 32 (2011) 771–776
We assume that the temperature, the solutal concentration and the volumetric fraction of the nanoparticles are constant on the boundaries. Thus the boundary conditions are
@w @ 2 w þ k H ¼ 0; T ¼ T 0 þ DT ; 1 @z @z2 C ¼ C 0 þ DC ; / ¼ /0 at z ¼ 0;
w ¼ 0;
@w @ 2 w k2 H 2 ¼ 0; @z @z C ¼ C 0 ; / ¼ /1 at z ¼ H: w ¼ 0;
ðx; y; zÞ ¼ ðx ; y ; z Þ=H; p ¼ p H2 =laf ;
ð14Þ
2
t ¼ t af =H ;
k
T T 0 T¼ ; DT
C¼
C C 0 ; DC
ð16Þ
Then Eqs. (8)–(14) take the form:
r v ¼ 0;
ð17Þ
1 @v ^z þ RaT e ^z þ v rv ¼ rp þ r2 v Rme Pr @t ^z Rn/e ^z ; þ ðRs=LeÞC e
ð18Þ
@T NB N N þ v rT ¼ r2 T þ r/ rT þ A B rT rT þ NTC r2 C; @t Ln Ln
ð19Þ
@C 1 þ v rC ¼ r2 C þ NCT r2 T; @t Le
ð20Þ
@/ 1 2 N þ v r/ ¼ r / þ A r2 T; @t Ln Ln
ð21Þ
w ¼ 0;
T ¼ 1;
C ¼ 1;
/ ¼ 0 at z ¼ 0; ð22Þ
@w @2w k2 2 ¼ 0; @z @z
w ¼ 0;
ð27Þ
½qp /0 þ qð1 /0 ÞgH3
laf
Rn ¼
ðqp qÞð/1 /0 ÞgH3
laf
C ¼ 0;
DT DT DB T c ð/1
;
ð30Þ
NB ¼
ðqcÞp ð/ /0 Þ; ðqcÞf 1
ð31Þ
af
Le ¼
DS
af
ð33Þ
NCT ¼
DCT DT : af DC
ð34Þ
The parameter Ln is a nanofluid Lewis number, while Le is the familiar Lewis number (the ratio of the Schmidt number to the Prandtl number Pr), Ra is the familiar thermal Rayleigh number, and Rs is the familiar solutal Rayleigh number. (In Nield and Kuznetsov (2010a) the symbol Le was used for the parameter now denoted by Ln.) The new parameters Rm and Rn may be regarded as a basicdensity Rayleigh number and a nanoparticle concentration Rayleigh number, respectively. The parameter NA is a modified diffusivity ratio and is somewhat similar to the Soret parameter that arises in cross-diffusion phenomena in solutions, while NB is a modified particle-density increment. In the spirit of the Oberbeck–Boussinesq approximation, Eq. (18) has been linearized by the neglect of terms, one proportional to the product of / and T and another proportional to the product of / and C. This assumption is likely to be valid in the case of small thermal and solutal gradients in a dilute suspension of nanoparticles. We now also have as additional parameters the regular Soret parameter NCT and the regular Dufour parameter NTC. 3.1. Basic solution We seek a time-independent quiescent solution of Eqs. (17)– (23) with temperature, concentration and nanoparticle volume fraction varying in the z-direction only, that is a solution of the form
p ¼ pb ðzÞ;
DB
T ¼ T b ðzÞ;
C ¼ C b ðzÞ;
/ ¼ /b ðzÞ:
ð35Þ
Then Eqs. (18)–(21) reduce to
0¼
dpb Rm þ RaT b þ ðRs=LeÞC b Rn/b ; dz
2
d Tb
þ
2 2 NB d/b dT b NA NB dT b d Cb þ NTC ¼ 0; þ 2 Ln dz dz Ln dz dz
ð24Þ
dz
ð25Þ
1 d Cb d Tb þ N CT ¼ 0; 2 Le dz2 dz
2
2
;
ð32Þ
;
DTC DC ; af DT
Here
Ln ¼
/0 Þ
NTC ¼
/ ¼ 1 at z ¼ 1: ð23Þ
l Pr ¼ ; qaf
ð29Þ
;
NA ¼
v ¼ 0; T ¼ 0;
ð28Þ
;
:
@w @2w þ k1 2 ¼ 0; @z @z
qgbC H3 DC ; lDS
ðu; v ; wÞ ¼ ðu ; v ; w ÞH=af ;
where
qc
Rs ¼
T ¼ T 0 ;
ð15Þ
af ¼
ð26Þ
Rm ¼
/ /0 /¼ ; /1 /0
qgbT H3 DT ; laf
ð13Þ
The parameters k1 and k2 each take the value 0 for the case of a rigid boundary and 1 for a free boundary. We recognize that our choice of boundary conditions imposed on /⁄ is somewhat arbitrary. It could be argued that zero particle flux on the boundaries is more realistic physically, but then one is faced with the problem that it appears that no steady-state solution for the basic conduction equations is then possible (we have tried to find one and met a contradiction) so that in order to make analytical progress it is necessary freeze the basic profile for /⁄, and at that stage our choice of boundary conditions is seen to be quite realistic. We introduce dimensionless variables as follows. We define
Ra ¼
ð36Þ
ð37Þ
2
ð38Þ
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D.A. Nield, A.V. Kuznetsov / International Journal of Heat and Fluid Flow 32 (2011) 771–776
2
d /b 2
dz
2
þ NA
d Tb 2
dz
¼ 0:
ð39Þ
ðw0 ; T 0 ; C 0 ; /0 Þ ¼ ½WðzÞ; HðzÞ; RðzÞ; UðzÞ expðst þ ilx þ imyÞ;
According toBuongiorno (2006), for most nanofluids investi gated so far Ln= /1 /0 is large, of order 105–106, and since the nanoparticle fraction decrement /1 /0 is typically no smaller than 103 this means so that Ln is large, of order 102–103, while NA is no greater than about 10. Using this approximation, the basic solution is found to be
T b ¼ 1 z;
We write
C b ¼ 1 z;
/b ¼ z:
ð40a; b; cÞ
and substitute into the differential equations to obtain
s ðD2 a2 Þ2 ðD2 a2 Þ W Raa2 H ðRs=LeÞa2 R þ Rna2 U ¼ 0; Pr ð51Þ
NB 2NA NB NB W þ D2 þ D D a2 s H DU þ NTC D2 a2 R ¼ 0: Ln Ln Ln ð52Þ
3.2. Perturbation solution We now superimpose perturbations on the basic solution. We write
v ¼ v0 ;
T ¼ Tb þ T0;
p ¼ pb þ p0 ;
C ¼ C b þ C 0 ; / ¼ /b þ /0 ;
substitute in Eqs. (17)–(23), and linearize by neglecting products of primed quantities. The following equations are obtained when Eq. ((40a)–(c)) are used.
r v ¼ 0; 0
1 2 ðD a2 sÞR þ NCT D2 a2 H ¼ 0; Le NA 2 1 2 ðD a2 Þ s U ¼ 0; ðD a2 ÞH W Ln Ln
ð53Þ
ð54Þ
W ¼ 0; DW þ k1 D2 W ¼ 0; H ¼ 0; R ¼ 0; U ¼ 0 at z ¼ 0;
ð55Þ
ð42Þ
1 @v0 ^z þ ðRs=LeÞC 0 e ^z Rn/0 e ^z ; ¼ rp0 þ r2 v 0 þ RaT 0 e Pr @t
ð45Þ
@/0 1 2 0 NA 2 0 r / þ r T; þ w0 ¼ Ln @t Ln
ð46Þ
C 0 ¼ 0;
/0 ¼ 0 at z ¼ 0: ð47Þ
0
2
0
@w @ w ¼ 0; k2 @z @z2
C 0 ¼ 0;
D
d dz
2
and a ¼ ðl þ m2 Þ1=2 :
ð48Þ It will be noted that the parameter Rm is not involved in these and subsequent equations. It is just a measure of the basic static pressure gradient. For the case of a regular binary fluid (not a nanofluid) the parameters Rn, NA and NB are zero, the second term on the lefthand side in Eq. (46) is absent because d/b/dz = 0 and then Eq. (46) is satisfied trivially. The remaining equations are reduced to the familiar equations for the double-diffusive Rayleigh-Bénard problem. The seven unknowns u0 , v0 , w0 , p0 , T0 , C0 , /0 can be reduced to four ^z curl curl and using the identity by operating on Eq. (43) with e curl curl grad div r2 together with Eq. (42). The result is
ð57Þ
Thus a is a dimensionless horizontal wavenumber. For neutral stability the real part of s is zero. Hence we now write s = ix, where x is real and is a dimensionless frequency. We now employ a Galerkin-type weighted residuals method to obtain an approximate solution to the system of Eqs. (51)–(56). We choose as trial functions (satisfying the boundary conditions) Wp, Hp, Rp, Up; p = 1, 2, 3, . . ., and write
W¼
/0 ¼ 0 at z ¼ 1:
1 @ 2 0 r w r4 w0 ¼ Rar2H T 0 þ ðRs=LeÞr2H C 0 Rnr2H /0 : Pr @t
U ¼ 0 at z ¼ 1: ð56Þ
N X
Ap W p ;
p¼1
T 0 ¼ 0;
R ¼ 0;
where
@C 0 1 w0 ¼ r2 C 0 þ NCT r2 T 0 ; Le @t
T 0 ¼ 0;
H ¼ 0;
ð43Þ
ð44Þ
@w0 @ 2 w0 ¼ 0; þ k1 @z @z2
DW k2 D2 W ¼ 0;
W ¼ 0;
@T 0 NB @T 0 @/0 2NA NB @T 0 w0 ¼ r2 T 0 þ þ NTC r2 C 0 ; @t Ln @z @z Ln @z
w0 ¼ 0;
(In the corresponding equation in Nield and Kuznetsov (2010a) the sign of NB should be changed but no consequent changes are needed.)
Wþ ð41Þ
w0 ¼ 0;
ð50Þ
U¼
N X
H¼
N X
Bp Hp ;
p¼1
R¼
N X
C p Rp ;
p¼1
Dp Up ;
ð58Þ
p¼1
substitute into Eqs. (51)–(54), and make the expressions on the lefthand sides of those equations (the residuals) orthogonal to the trial functions, thereby obtaining a system of 4N linear algebraic equations in the 4N unknowns Ap, Bp, Cp, Dp; p = 1, 2, . . . , N. The vanishing of the determinant of coefficients produces the eigenvalue equation for the system. One can regard Ra as the eigenvalue. Thus Ra is found in terms of the other parameters. 4. Results and discussion 4.1. Case A: Free-free boundaries For this case the boundary conditions are
ð49Þ
Here r2H is the two-dimensional Laplacian operator in the horizontal plane. The differential Eqs. (49), (44)–(46) and the boundary conditions (47) and (48) constitute a linear boundary-value problem that can be solved using the method of normal modes.
W ¼ 0; D2 W ¼ 0; H ¼ 0; R ¼ 0; U ¼ 0 at z ¼ 0 and at z ¼ 1; ð59Þ and the trial functions can be chosen as
W p ¼ Hp ¼ Rp ¼ Up ¼ sin ppz;
p ¼ 1; 2; 3; . . .
ð60Þ
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D.A. Nield, A.V. Kuznetsov / International Journal of Heat and Fluid Flow 32 (2011) 771–776
The non-oscillatory stability boundary (s = 0) is given approximately, using the one-term Galerkin approximation, by
W p ¼ z2 ð1 zÞ½2 þ p 2zp
ð1 LeNTC ÞðRa þ NA RnÞ þ ð1 LeN CT NTC ÞLnRn þ ð1 NCT ÞRs ¼ ð1 LeNCT NTC ÞR0 ;
ð61Þ
where
p
R0 ¼ 27p4 =4 ¼ 657:5 and with ac ¼ p= 2 ¼ 2:22:
ð62Þ
In the case of double diffusion in a regular fluid, when NTC, NCT and NA are all zero, Eq. (61) reduces to Ra + Rs = R0, as expected. (See, for example, Nield, 1967.) The non-oscillatory boundary does not depend on the value of Pr. It is noteworthy that the particle increment parameter NB does not appear in the eigenvalue equation at this level of approximation, and the diffusivity ratio NA appears only in association with the nanoparticle Rayleigh number Rn. The case of oscillatory in stability is much more complicated. Indeed, it is well known that the linear stability problem for triply diffusive systems is complex (Pearlstein et al., 1989). Numerical values of Rac can be calculated readily using the Mathematica software package, but it appears that even a moderately simple analytical solution can be obtained only in the case Pr ? 1, so we now make this assumption. The oscillatory stability boundary (s = ix, where x is real), is found by splitting the eigenvalue equation into its real and imaginary parts. When the common factor is cancelled from the latter it gives x2 as a linear expression in Ra, while the former gives Ra as a bilinear expression in x2. It is the smallest positive root that has physical significance. The solution for the case on general Ln is too lengthy to present here so we make a further approximation. In view of the fact that Ln is large for a typical nanofluid we consider just the case where Ln ? 1. With the wave p number a approximated by p/ 2, one then obtains from the real and imaginary parts of the eigenvalue equation the following two formulas involving x2
3x2 ðRa þ Rn þ Rs 2R0 Þ ¼ ð1 LeNCT NTC ÞR0 Rn;
ð63Þ
ð1 LeNTC ÞRa þ 2Rn þ ð1 NCT ÞRs ¼ ð1 LeN CT NTC ÞR0 3x2 :
ð64Þ
Eliminating x between these two equations and selecting the smaller root we get 2ð1 LeNTC ÞRa ¼ ½3 Leð2 þ NCT ÞN TC R0 ð3 LeNTC ÞRn ð2 NCT 8 91=2 2 2 2 > < ½1 LeNTC ð2 NCT Þ R0 þ ½ð1 þ LeNTC ÞRn þ ðLeNTC NCT ÞRs > = LeN TC ÞRs 2½1 LeNTC ð1 þ 3NCT Þ þ Le2 N2TC ð2 þ NCT ÞR0 Rn : > > : ; 2ðNCT LeNTC Þ½1 LeNTC ð2 NCT ÞR0 Rs ð65Þ
For the case, where the Soret and Dufour parameters NCT and NTC are negligible, Eq. (65) reduces to
Ra ¼ R0 Rs Rn:
ð66Þ
On the other hand, Eq. (61) then reduces to
Ra ¼ R0 Rs ðNA þ LnÞRn:
ð67Þ
Comparing these two expressions for Ra, we see that if Rn is positive then Ra is minimized by a non-oscillatory mode. This result is as expected, because for oscillations to occur two of the buoyancy forces have to be in opposite directions. 4.2. Case B: Rigid-free boundaries For this case the boundary conditions are
W ¼ 0; W ¼ 0;
DW ¼ 0; 2
D W ¼ 0;
H ¼ 0; H ¼ 0;
R ¼ 0; R ¼ 0;
U ¼ 0 at z ¼ 0; U ¼ 0 at z ¼ 1:
Appropriate trial functions are
ð68Þ
Hp ¼ Rp ¼ Up ¼ zp ð1 zÞ;
p ¼ 1; 2; 3; . . .
ð69Þ
The non-oscillatory stability boundary is given approximately by Eq. (61) while the oscillatory stability boundary is given by Eq. (65), where now
R0 ¼ 1140 and with ac ¼ 2:68:
ð70Þ
The value 1140 is about 4% greater than the well known exact value 1100.65 that applies for a regular fluid. 4.3. Rigid-rigid boundaries For this case the boundary conditions are
W ¼ 0;
DW ¼ 0;
H ¼ 0;
R ¼ 0;
U¼0
at z ¼ 0 and z ¼ 1:
ð71Þ
Appropriate trial functions are
W p ¼ z1þp ð1 zÞ2 ;
Hp ¼ Rp ¼ Up ¼ zp ð1 zÞ;
p ¼ 1; 2; 3; . . .
ð72Þ
The non-oscillatory stability boundary is given approximately by Eq. (61) while the oscillatory stability boundary is given by Eq. (65), where now
R0 ¼ 1750 and with ac ¼ 3:12:
ð73Þ
The value 1750 is about 3% greater than the well known exact value 1707.762 that applies for a regular fluid. Thus the change in boundary conditions leads to a change in the values of R0 and ac but that is all. 5. Conclusions The onset of double-diffusive convection in a horizontal layer of a nanofluid has been studied analytically using linear instability theory. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. In addition the thermal energy equations include regular diffusion and cross-diffusion terms. Both non-oscillatory and oscillatory stability have been considered. The stability boundaries have been approximated using a one-term Galerkin approximation, something that produces an upper bound on the critical Rayleigh number that is about 5% higher than the true value. The results are encapsulated in Eqs. (61) and (65) for non-oscillatory and oscillatory stability, respectively. These equations relate thermal, solutal and nanofluid Rayleigh numbers Ra, Rs, and Rn to thermo-solutal and thermo-nanoparticle Lewis numbers Le and Ln, the Soret and Dufour parameters NCT and NTC and the nanoparticle parameters NA and NB. (At the order of approximation used NB does not appear.) The analytical results for oscillatory stability are limited to the case of large Prandtl number and large nanoparticle Lewis number. References Buongiorno, J., Hu, L.-W., 2005. Nanofluid coolants for advanced nuclear power plants. In: Proceedings of ICAPP ’05, May 15–19, Paper no. 5705, Seoul. Buongiorno, J., 2006. Convective transport in nanofluids. ASME J. Heat Transfer 128, 240–250. Choi, S., 1995. Enhancing thermal conductivity of fluids with nanoparticles. In: Siginer, D.A., Wang, H.P. (Eds.), Developments and Applications of NonNewtonian Flows, ASME FED-vol. 231/MD-vol. 66. ASME, New York, pp. 99– 105. Das, S.K., Choi, S.U.S., 2009. A review of heat transfer in nanofluids. Adv. Heat Transfer 41, 81–197.
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Das, S.K., Choi, S.U.S., Yu, W., Pradeep, T., 2008. Nanofluids: Science and Technology. Wiley, Hoboken, NY. Eapen, J., Rusconi, R., Piazza, R., Yip, S., 2010. The classical nature of thermal conduction in nanofluids. ASME J. Heat Transfer 132, 102402. Fan, J., Wang, L., 2010a. Effective thermal conductivity of nanofluids: the effects of microstructure. J. Phys. D – Appl. Phys. 43, 165501. Fan, J., Wang, L., 2010b. Microstructural effects on macroscale thermal properties in nanofluids. Nano 5, 117–125. Gao, J.W., Zheng, R.T., Ohtani, H., Zhu, D.S., Chen, G., 2009. Experimental investigation of heat conduction mechanisms in nanofluids. Clue Cluster. Nano Lett. 9, 4128–4132. Kim, J., Choi, C.K., Kang, Y.T., Kim, M.G., 2006. Effects of thermodiffusion and nanoparticles on convective instabilities in binary nanofluids. Nanoscale Microscale Thermophys. Eng. 10, 29–39. Kim, J., Kang, Y.T., Choi, C.K., 2007. Soret and Dufour effects on convective instabilities in binary nanofluids for absorption application. Int. J. Refrig. 30, 323–328.
Masuda, H., Ebata, A., Teramae, K., Hishinuma, N., 1993. Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 7, 227–233. Nield, D.A., 1967. The thermohaline Rayleigh–Jeffreys problem. J. Fluid Mech. 29, 545–558. Nield, D.A., Kuznetsov, A.V., 2010a. The onset of convection in a nanofluid layer of finite depth. Eur. J. Mech. B/Fluids 29, 217–223. Nield, D.A., Kuznetsov, A.V., 2010b. The effect of local thermal non-equilibrium on the onset of convection in a nanofluid. ASME J. Heat Transfer 132, 052405. Pearlstein, A.J., Harris, R.M., Terrones, G., 1989. The onset of convective instability in a triple diffusive layer. J. Fluid Mech. 202, 443–465. Savino, R., Paterna, D., 2008. Thermodiffusion in nanofluids under different gravity conditions. Phys. Fluids 20, 017101. Wang, L., Fan, J., 2010. Nanofluids Research: key issues. Nanoscale Res. Lett. 5, 1241–1252. Wang, L., Wei, X., 2009. Heat conduction in nanofluids. Chaos Solitons Fract. 39, 2211–2215.
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International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
The onset of double-diffusive convection in a nanofluid layer D.A. Nield a, A.V. Kuznetsov b,⇑ a b
Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland, New Zealand Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC 27695-7910, USA
a r t i c l e
i n f o
Article history: Received 20 August 2010 Received in revised form 14 February 2011 Accepted 31 March 2011 Available online 6 May 2011
a b s t r a c t The onset of double-diffusive convection in a horizontal layer of a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. In addition the thermal energy equations include regular diffusion and cross-diffusion terms. The stability boundaries for both non-oscillatory and oscillatory cases have been approximated by simple analytical expressions. Physical significance of the obtained results is discussed. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The term ‘‘nanofluid’’ refers to a liquid containing a suspension of metallic or non-metallic nanometer-sized solid particles and fibres (nanoparticles). The term was suggested by Choi (1995). The characteristic feature of nanofluids is thermal conductivity enhancement, a phenomenon observed by Masuda et al. (1993). A significant body of work addresses physical mechanisms of thermal conductivity enhancement in nanofluids (utilizing both classical and non-classical approaches), see for example Eapen et al. (2010), Fan and Wang (2010a, 2010b), Gao et al. (2009), Wang and Wei (2009), Wang and Fan (2010). There is a variety of potential engineering applications of nanofluids, including advanced nuclear systems (Buongiorno and Hu, 2005). 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). On the particular topic of convective transport in nanofluids, a comprehensive survey was made by Buongiorno (2006) who, after considering alternative agencies, proposed a model incorporating the effects of Brownian diffusion and the thermophoresis. This model was applied to the Rayleigh-Bénard problem (the onset of convection in a horizontal layer uniformly heated from below) by Nield and Kuznetsov (2010a, 2010b). Both Brownian diffusion and thermophoresis give rise to cross-diffusion terms that are in some ways analogous to the familiar Soret and Dufour crossdiffusion terms that arise with a binary fluid. This aspect of transport in nanofluids has been discussed by Kim et al. (2006, 2007) and by Savino and Paterna (2008). In the present paper a fresh approach to cross-diffusion in nanofluids is made, with the aim of making a detailed comparison with regular cross-diffusion effects and the cross-diffusion effects peculiar to nanofluids, and at the same time investigating the ⇑ Corresponding author. E-mail address: [email protected] (A.V. Kuznetsov). 0142-727X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2011.03.010
interaction between these effects when the base fluid of the nanofluid is itself a binary fluid such as salty water. The outcome is that we investigating a sort of triple-diffusion problem involving heat, the nanoparticles and the solute. The analysis is an extension of that presented by Nield and Kuznetsov (2010a). 2. Conservation equations for a nanofluid The conservation equations as formulated by Buongiorno (2006) are now extended as follows. It is assumed that no nanoparticle agglomeration occurs and that the nanoparticle suspension remains stable. The continuity equation for the nanofluid is
r v ¼ 0:
ð1Þ
Here v is the nanofluid velocity. In the presence of thermophoresis the conservation equation for the nanoparticles, in the absence of chemical reactions, takes the form
@/ rT þ v r/ ¼ r DB r/ þ DT ; @t T
ð2Þ
where / is nanoparticle volume fraction, T is the temperature, DB is the Brownian diffusion coefficient, and DT is the thermophoretic diffusion coefficient. It is assumed that the solute does not affect the transport of the nanoparticles. If one introduces a buoyancy force and adopts the Boussinesq approximation, then the momentum equation can be written as
@v þ v rv ¼ rp þ lr2 v þ qg; @t
q
ð3Þ
where q is the overall density of the nanofluid, which we now assume to be given by
q ¼ /qp þ ð1 /Þq0 ½1 bT ðT T 0 Þ bC ðC C 0 Þ;
ð4Þ
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D.A. Nield, A.V. Kuznetsov / International Journal of Heat and Fluid Flow 32 (2011) 771–776
Nomenclature C DB DCT DS DT DTc H k Le Ln NA NB NCT NTC p⁄ p Ra Rm Rn Rs t⁄ t T⁄ T T⁄c T⁄h
solute concentration Brownian diffusion coefficient (m2/s) diffusivity of Soret type (m2/s) solutal diffusivity (m2/s) thermophoretic diffusion coefficient (m2/s) diffusivity of Dufour type (m2/s) dimensional layer depth (m) thermal conductivity of the nanofluid (W/m K) thermo-solutal Lewis number, defined by Eq. (32) thermo-nanofluid Lewis number, defined by Eq. (25) modified diffusivity ratio, defined by Eq. (30) modified particle-density increment, defined by Eq. (31) Soret parameter, defined by Eq. (34) Dufour parameter, defined by Eq. (33) pressure (Pa) dimensionless pressure, p⁄H2/laf thermal Rayleigh number, defined by Eq. (26) basic-density Rayleigh number, defined by Eq. (28) nanoparticle Rayleigh number, defined by Eq. (29) solutal Rayleigh number, defined by Eq. (27) time (s) dimensionless time, t⁄af/H2 nanofluid temperature (K) T T dimensionless temperature, T T c c h temperature at the upper wall (K) temperature at the lower wall (K)
where qp is the particle density, q0 is a reference density for the fluid, bT is the thermal volumetric expansion coefficient and bC is the analogous solutal coefficient (something that is negative if, as is usual, the solute is more dense that the solvent). The thermal energy equation for a nanofluid can be written as
qc
@T rT rT þ v rT ¼ kr2 T þ qp cp DB r/ rT þ DT @t T þ qcDTC r2 C;
ð5Þ
where c is the fluid specific heat (at constant pressure), k is the nanofluid thermal conductivity, cp is the specific heat of the material constituting the nanoparticles, and DTC is a diffusivity of Dufour type. Eq. (5) differs from that used in Nield and Kuznetsov (2010a) by the addition of the last term. To this we add a conservation equation for the solute of the form
@C þ v rC ¼ DS r2 C þ DCT r2 T; @t
ð6Þ
(u, v, w) v (x, y, z)
dimensionless velocity components, (u⁄, v⁄, w⁄)H/af nanofluid velocity (m/s) dimensionless Cartesian coordinates, (x⁄, y⁄, z⁄)/H; z is the vertically-upward coordinate (x⁄, y⁄, z⁄) Cartesian coordinates (m) Greek
af bC bT k1, k2
l q qp /⁄ /
1
bC DCT ; bT DS
Subscripts b basic solution f fluid p particle
asterisk has not been needed because all the variables were dimensional). Each boundary wall is assumed to be perfectly thermally conducting. The temperatures at the lower and upper boundary are taken to be T 0 þ DT and T 0 and the corresponding solute concentrations are taken as C 0 þ DC and C 0 . The Oberbeck–Boussinesq approximation is employed. Homogeneity of the fluid is assumed. In the linear theory being applied here the temperature change in the fluid is assumed to be small in comparison with T 0 and likewise the concentration change in the fluid is assumed to be small in comparison with C 0 . The conservation equations take the form
r v ¼ 0;
q0
is employed in lieu of DCT.) It has been assumed that the nanoparticles do not affect the transport of the solute.
3. Application to the Rayleigh-Bénard problem We select a coordinate frame in which the z-axis is aligned vertically-upwards. We consider a horizontal layer of fluid confined between the planes z⁄ = 0 and z⁄ = H. From now on asterisks are used to denote dimensional variables (previously an
@ v þ v r v @t
ð8Þ
h ¼ r p þ lr2 v þ / qp þ ð1 / Þq0 ½1 bT ðT T 0 Þ bC ðC C 0 Þ g;
qc ð7Þ
0
Superscripts ⁄ dimensional variable 0 perturbation variable
where DS is the solutal diffusivity and DCT is a diffusivity of Soret type. (Commonly a Soret parameter S defined by
S¼
thermal diffusivity of the fluid (m2/s) solutal volumetric coefficient thermal volumetric coefficient (K1) parameters introduced in Eqs. (13) and (14) viscosity of the fluid (N s/m2) fluid density (kg/m3) nanoparticle mass density (kg/m3) nanoparticle volume fraction / / relative nanoparticle volume fraction, / /0
ð9Þ
@T þ v r T @t
¼ kr2 T þ ðqcÞp DB r / r T þ ðDT =T c Þr T r T þ qcDTC r2 C ;
ð10Þ
@C þ v r C ¼ DS r2 C þ DCT r2 T ; @t
ð11Þ
@/ þ v r / ¼ DB r2 / þ ðDT =T c Þr2 T : @t
ð12Þ
We write v⁄ = (u⁄, v⁄, w⁄).
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D.A. Nield, A.V. Kuznetsov / International Journal of Heat and Fluid Flow 32 (2011) 771–776
We assume that the temperature, the solutal concentration and the volumetric fraction of the nanoparticles are constant on the boundaries. Thus the boundary conditions are
@w @ 2 w þ k H ¼ 0; T ¼ T 0 þ DT ; 1 @z @z2 C ¼ C 0 þ DC ; / ¼ /0 at z ¼ 0;
w ¼ 0;
@w @ 2 w k2 H 2 ¼ 0; @z @z C ¼ C 0 ; / ¼ /1 at z ¼ H: w ¼ 0;
ðx; y; zÞ ¼ ðx ; y ; z Þ=H; p ¼ p H2 =laf ;
ð14Þ
2
t ¼ t af =H ;
k
T T 0 T¼ ; DT
C¼
C C 0 ; DC
ð16Þ
Then Eqs. (8)–(14) take the form:
r v ¼ 0;
ð17Þ
1 @v ^z þ RaT e ^z þ v rv ¼ rp þ r2 v Rme Pr @t ^z Rn/e ^z ; þ ðRs=LeÞC e
ð18Þ
@T NB N N þ v rT ¼ r2 T þ r/ rT þ A B rT rT þ NTC r2 C; @t Ln Ln
ð19Þ
@C 1 þ v rC ¼ r2 C þ NCT r2 T; @t Le
ð20Þ
@/ 1 2 N þ v r/ ¼ r / þ A r2 T; @t Ln Ln
ð21Þ
w ¼ 0;
T ¼ 1;
C ¼ 1;
/ ¼ 0 at z ¼ 0; ð22Þ
@w @2w k2 2 ¼ 0; @z @z
w ¼ 0;
ð27Þ
½qp /0 þ qð1 /0 ÞgH3
laf
Rn ¼
ðqp qÞð/1 /0 ÞgH3
laf
C ¼ 0;
DT DT DB T c ð/1
;
ð30Þ
NB ¼
ðqcÞp ð/ /0 Þ; ðqcÞf 1
ð31Þ
af
Le ¼
DS
af
ð33Þ
NCT ¼
DCT DT : af DC
ð34Þ
The parameter Ln is a nanofluid Lewis number, while Le is the familiar Lewis number (the ratio of the Schmidt number to the Prandtl number Pr), Ra is the familiar thermal Rayleigh number, and Rs is the familiar solutal Rayleigh number. (In Nield and Kuznetsov (2010a) the symbol Le was used for the parameter now denoted by Ln.) The new parameters Rm and Rn may be regarded as a basicdensity Rayleigh number and a nanoparticle concentration Rayleigh number, respectively. The parameter NA is a modified diffusivity ratio and is somewhat similar to the Soret parameter that arises in cross-diffusion phenomena in solutions, while NB is a modified particle-density increment. In the spirit of the Oberbeck–Boussinesq approximation, Eq. (18) has been linearized by the neglect of terms, one proportional to the product of / and T and another proportional to the product of / and C. This assumption is likely to be valid in the case of small thermal and solutal gradients in a dilute suspension of nanoparticles. We now also have as additional parameters the regular Soret parameter NCT and the regular Dufour parameter NTC. 3.1. Basic solution We seek a time-independent quiescent solution of Eqs. (17)– (23) with temperature, concentration and nanoparticle volume fraction varying in the z-direction only, that is a solution of the form
p ¼ pb ðzÞ;
DB
T ¼ T b ðzÞ;
C ¼ C b ðzÞ;
/ ¼ /b ðzÞ:
ð35Þ
Then Eqs. (18)–(21) reduce to
0¼
dpb Rm þ RaT b þ ðRs=LeÞC b Rn/b ; dz
2
d Tb
þ
2 2 NB d/b dT b NA NB dT b d Cb þ NTC ¼ 0; þ 2 Ln dz dz Ln dz dz
ð24Þ
dz
ð25Þ
1 d Cb d Tb þ N CT ¼ 0; 2 Le dz2 dz
2
2
;
ð32Þ
;
DTC DC ; af DT
Here
Ln ¼
/0 Þ
NTC ¼
/ ¼ 1 at z ¼ 1: ð23Þ
l Pr ¼ ; qaf
ð29Þ
;
NA ¼
v ¼ 0; T ¼ 0;
ð28Þ
;
:
@w @2w þ k1 2 ¼ 0; @z @z
qgbC H3 DC ; lDS
ðu; v ; wÞ ¼ ðu ; v ; w ÞH=af ;
where
qc
Rs ¼
T ¼ T 0 ;
ð15Þ
af ¼
ð26Þ
Rm ¼
/ /0 /¼ ; /1 /0
qgbT H3 DT ; laf
ð13Þ
The parameters k1 and k2 each take the value 0 for the case of a rigid boundary and 1 for a free boundary. We recognize that our choice of boundary conditions imposed on /⁄ is somewhat arbitrary. It could be argued that zero particle flux on the boundaries is more realistic physically, but then one is faced with the problem that it appears that no steady-state solution for the basic conduction equations is then possible (we have tried to find one and met a contradiction) so that in order to make analytical progress it is necessary freeze the basic profile for /⁄, and at that stage our choice of boundary conditions is seen to be quite realistic. We introduce dimensionless variables as follows. We define
Ra ¼
ð36Þ
ð37Þ
2
ð38Þ
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D.A. Nield, A.V. Kuznetsov / International Journal of Heat and Fluid Flow 32 (2011) 771–776
2
d /b 2
dz
2
þ NA
d Tb 2
dz
¼ 0:
ð39Þ
ðw0 ; T 0 ; C 0 ; /0 Þ ¼ ½WðzÞ; HðzÞ; RðzÞ; UðzÞ expðst þ ilx þ imyÞ;
According toBuongiorno (2006), for most nanofluids investi gated so far Ln= /1 /0 is large, of order 105–106, and since the nanoparticle fraction decrement /1 /0 is typically no smaller than 103 this means so that Ln is large, of order 102–103, while NA is no greater than about 10. Using this approximation, the basic solution is found to be
T b ¼ 1 z;
We write
C b ¼ 1 z;
/b ¼ z:
ð40a; b; cÞ
and substitute into the differential equations to obtain
s ðD2 a2 Þ2 ðD2 a2 Þ W Raa2 H ðRs=LeÞa2 R þ Rna2 U ¼ 0; Pr ð51Þ
NB 2NA NB NB W þ D2 þ D D a2 s H DU þ NTC D2 a2 R ¼ 0: Ln Ln Ln ð52Þ
3.2. Perturbation solution We now superimpose perturbations on the basic solution. We write
v ¼ v0 ;
T ¼ Tb þ T0;
p ¼ pb þ p0 ;
C ¼ C b þ C 0 ; / ¼ /b þ /0 ;
substitute in Eqs. (17)–(23), and linearize by neglecting products of primed quantities. The following equations are obtained when Eq. ((40a)–(c)) are used.
r v ¼ 0; 0
1 2 ðD a2 sÞR þ NCT D2 a2 H ¼ 0; Le NA 2 1 2 ðD a2 Þ s U ¼ 0; ðD a2 ÞH W Ln Ln
ð53Þ
ð54Þ
W ¼ 0; DW þ k1 D2 W ¼ 0; H ¼ 0; R ¼ 0; U ¼ 0 at z ¼ 0;
ð55Þ
ð42Þ
1 @v0 ^z þ ðRs=LeÞC 0 e ^z Rn/0 e ^z ; ¼ rp0 þ r2 v 0 þ RaT 0 e Pr @t
ð45Þ
@/0 1 2 0 NA 2 0 r / þ r T; þ w0 ¼ Ln @t Ln
ð46Þ
C 0 ¼ 0;
/0 ¼ 0 at z ¼ 0: ð47Þ
0
2
0
@w @ w ¼ 0; k2 @z @z2
C 0 ¼ 0;
D
d dz
2
and a ¼ ðl þ m2 Þ1=2 :
ð48Þ It will be noted that the parameter Rm is not involved in these and subsequent equations. It is just a measure of the basic static pressure gradient. For the case of a regular binary fluid (not a nanofluid) the parameters Rn, NA and NB are zero, the second term on the lefthand side in Eq. (46) is absent because d/b/dz = 0 and then Eq. (46) is satisfied trivially. The remaining equations are reduced to the familiar equations for the double-diffusive Rayleigh-Bénard problem. The seven unknowns u0 , v0 , w0 , p0 , T0 , C0 , /0 can be reduced to four ^z curl curl and using the identity by operating on Eq. (43) with e curl curl grad div r2 together with Eq. (42). The result is
ð57Þ
Thus a is a dimensionless horizontal wavenumber. For neutral stability the real part of s is zero. Hence we now write s = ix, where x is real and is a dimensionless frequency. We now employ a Galerkin-type weighted residuals method to obtain an approximate solution to the system of Eqs. (51)–(56). We choose as trial functions (satisfying the boundary conditions) Wp, Hp, Rp, Up; p = 1, 2, 3, . . ., and write
W¼
/0 ¼ 0 at z ¼ 1:
1 @ 2 0 r w r4 w0 ¼ Rar2H T 0 þ ðRs=LeÞr2H C 0 Rnr2H /0 : Pr @t
U ¼ 0 at z ¼ 1: ð56Þ
N X
Ap W p ;
p¼1
T 0 ¼ 0;
R ¼ 0;
where
@C 0 1 w0 ¼ r2 C 0 þ NCT r2 T 0 ; Le @t
T 0 ¼ 0;
H ¼ 0;
ð43Þ
ð44Þ
@w0 @ 2 w0 ¼ 0; þ k1 @z @z2
DW k2 D2 W ¼ 0;
W ¼ 0;
@T 0 NB @T 0 @/0 2NA NB @T 0 w0 ¼ r2 T 0 þ þ NTC r2 C 0 ; @t Ln @z @z Ln @z
w0 ¼ 0;
(In the corresponding equation in Nield and Kuznetsov (2010a) the sign of NB should be changed but no consequent changes are needed.)
Wþ ð41Þ
w0 ¼ 0;
ð50Þ
U¼
N X
H¼
N X
Bp Hp ;
p¼1
R¼
N X
C p Rp ;
p¼1
Dp Up ;
ð58Þ
p¼1
substitute into Eqs. (51)–(54), and make the expressions on the lefthand sides of those equations (the residuals) orthogonal to the trial functions, thereby obtaining a system of 4N linear algebraic equations in the 4N unknowns Ap, Bp, Cp, Dp; p = 1, 2, . . . , N. The vanishing of the determinant of coefficients produces the eigenvalue equation for the system. One can regard Ra as the eigenvalue. Thus Ra is found in terms of the other parameters. 4. Results and discussion 4.1. Case A: Free-free boundaries For this case the boundary conditions are
ð49Þ
Here r2H is the two-dimensional Laplacian operator in the horizontal plane. The differential Eqs. (49), (44)–(46) and the boundary conditions (47) and (48) constitute a linear boundary-value problem that can be solved using the method of normal modes.
W ¼ 0; D2 W ¼ 0; H ¼ 0; R ¼ 0; U ¼ 0 at z ¼ 0 and at z ¼ 1; ð59Þ and the trial functions can be chosen as
W p ¼ Hp ¼ Rp ¼ Up ¼ sin ppz;
p ¼ 1; 2; 3; . . .
ð60Þ
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D.A. Nield, A.V. Kuznetsov / International Journal of Heat and Fluid Flow 32 (2011) 771–776
The non-oscillatory stability boundary (s = 0) is given approximately, using the one-term Galerkin approximation, by
W p ¼ z2 ð1 zÞ½2 þ p 2zp
ð1 LeNTC ÞðRa þ NA RnÞ þ ð1 LeN CT NTC ÞLnRn þ ð1 NCT ÞRs ¼ ð1 LeNCT NTC ÞR0 ;
ð61Þ
where
p
R0 ¼ 27p4 =4 ¼ 657:5 and with ac ¼ p= 2 ¼ 2:22:
ð62Þ
In the case of double diffusion in a regular fluid, when NTC, NCT and NA are all zero, Eq. (61) reduces to Ra + Rs = R0, as expected. (See, for example, Nield, 1967.) The non-oscillatory boundary does not depend on the value of Pr. It is noteworthy that the particle increment parameter NB does not appear in the eigenvalue equation at this level of approximation, and the diffusivity ratio NA appears only in association with the nanoparticle Rayleigh number Rn. The case of oscillatory in stability is much more complicated. Indeed, it is well known that the linear stability problem for triply diffusive systems is complex (Pearlstein et al., 1989). Numerical values of Rac can be calculated readily using the Mathematica software package, but it appears that even a moderately simple analytical solution can be obtained only in the case Pr ? 1, so we now make this assumption. The oscillatory stability boundary (s = ix, where x is real), is found by splitting the eigenvalue equation into its real and imaginary parts. When the common factor is cancelled from the latter it gives x2 as a linear expression in Ra, while the former gives Ra as a bilinear expression in x2. It is the smallest positive root that has physical significance. The solution for the case on general Ln is too lengthy to present here so we make a further approximation. In view of the fact that Ln is large for a typical nanofluid we consider just the case where Ln ? 1. With the wave p number a approximated by p/ 2, one then obtains from the real and imaginary parts of the eigenvalue equation the following two formulas involving x2
3x2 ðRa þ Rn þ Rs 2R0 Þ ¼ ð1 LeNCT NTC ÞR0 Rn;
ð63Þ
ð1 LeNTC ÞRa þ 2Rn þ ð1 NCT ÞRs ¼ ð1 LeN CT NTC ÞR0 3x2 :
ð64Þ
Eliminating x between these two equations and selecting the smaller root we get 2ð1 LeNTC ÞRa ¼ ½3 Leð2 þ NCT ÞN TC R0 ð3 LeNTC ÞRn ð2 NCT 8 91=2 2 2 2 > < ½1 LeNTC ð2 NCT Þ R0 þ ½ð1 þ LeNTC ÞRn þ ðLeNTC NCT ÞRs > = LeN TC ÞRs 2½1 LeNTC ð1 þ 3NCT Þ þ Le2 N2TC ð2 þ NCT ÞR0 Rn : > > : ; 2ðNCT LeNTC Þ½1 LeNTC ð2 NCT ÞR0 Rs ð65Þ
For the case, where the Soret and Dufour parameters NCT and NTC are negligible, Eq. (65) reduces to
Ra ¼ R0 Rs Rn:
ð66Þ
On the other hand, Eq. (61) then reduces to
Ra ¼ R0 Rs ðNA þ LnÞRn:
ð67Þ
Comparing these two expressions for Ra, we see that if Rn is positive then Ra is minimized by a non-oscillatory mode. This result is as expected, because for oscillations to occur two of the buoyancy forces have to be in opposite directions. 4.2. Case B: Rigid-free boundaries For this case the boundary conditions are
W ¼ 0; W ¼ 0;
DW ¼ 0; 2
D W ¼ 0;
H ¼ 0; H ¼ 0;
R ¼ 0; R ¼ 0;
U ¼ 0 at z ¼ 0; U ¼ 0 at z ¼ 1:
Appropriate trial functions are
ð68Þ
Hp ¼ Rp ¼ Up ¼ zp ð1 zÞ;
p ¼ 1; 2; 3; . . .
ð69Þ
The non-oscillatory stability boundary is given approximately by Eq. (61) while the oscillatory stability boundary is given by Eq. (65), where now
R0 ¼ 1140 and with ac ¼ 2:68:
ð70Þ
The value 1140 is about 4% greater than the well known exact value 1100.65 that applies for a regular fluid. 4.3. Rigid-rigid boundaries For this case the boundary conditions are
W ¼ 0;
DW ¼ 0;
H ¼ 0;
R ¼ 0;
U¼0
at z ¼ 0 and z ¼ 1:
ð71Þ
Appropriate trial functions are
W p ¼ z1þp ð1 zÞ2 ;
Hp ¼ Rp ¼ Up ¼ zp ð1 zÞ;
p ¼ 1; 2; 3; . . .
ð72Þ
The non-oscillatory stability boundary is given approximately by Eq. (61) while the oscillatory stability boundary is given by Eq. (65), where now
R0 ¼ 1750 and with ac ¼ 3:12:
ð73Þ
The value 1750 is about 3% greater than the well known exact value 1707.762 that applies for a regular fluid. Thus the change in boundary conditions leads to a change in the values of R0 and ac but that is all. 5. Conclusions The onset of double-diffusive convection in a horizontal layer of a nanofluid has been studied analytically using linear instability theory. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. In addition the thermal energy equations include regular diffusion and cross-diffusion terms. Both non-oscillatory and oscillatory stability have been considered. The stability boundaries have been approximated using a one-term Galerkin approximation, something that produces an upper bound on the critical Rayleigh number that is about 5% higher than the true value. The results are encapsulated in Eqs. (61) and (65) for non-oscillatory and oscillatory stability, respectively. These equations relate thermal, solutal and nanofluid Rayleigh numbers Ra, Rs, and Rn to thermo-solutal and thermo-nanoparticle Lewis numbers Le and Ln, the Soret and Dufour parameters NCT and NTC and the nanoparticle parameters NA and NB. (At the order of approximation used NB does not appear.) The analytical results for oscillatory stability are limited to the case of large Prandtl number and large nanoparticle Lewis number. References Buongiorno, J., Hu, L.-W., 2005. Nanofluid coolants for advanced nuclear power plants. In: Proceedings of ICAPP ’05, May 15–19, Paper no. 5705, Seoul. Buongiorno, J., 2006. Convective transport in nanofluids. ASME J. Heat Transfer 128, 240–250. Choi, S., 1995. Enhancing thermal conductivity of fluids with nanoparticles. In: Siginer, D.A., Wang, H.P. (Eds.), Developments and Applications of NonNewtonian Flows, ASME FED-vol. 231/MD-vol. 66. ASME, New York, pp. 99– 105. Das, S.K., Choi, S.U.S., 2009. A review of heat transfer in nanofluids. Adv. Heat Transfer 41, 81–197.
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Das, S.K., Choi, S.U.S., Yu, W., Pradeep, T., 2008. Nanofluids: Science and Technology. Wiley, Hoboken, NY. Eapen, J., Rusconi, R., Piazza, R., Yip, S., 2010. The classical nature of thermal conduction in nanofluids. ASME J. Heat Transfer 132, 102402. Fan, J., Wang, L., 2010a. Effective thermal conductivity of nanofluids: the effects of microstructure. J. Phys. D – Appl. Phys. 43, 165501. Fan, J., Wang, L., 2010b. Microstructural effects on macroscale thermal properties in nanofluids. Nano 5, 117–125. Gao, J.W., Zheng, R.T., Ohtani, H., Zhu, D.S., Chen, G., 2009. Experimental investigation of heat conduction mechanisms in nanofluids. Clue Cluster. Nano Lett. 9, 4128–4132. Kim, J., Choi, C.K., Kang, Y.T., Kim, M.G., 2006. Effects of thermodiffusion and nanoparticles on convective instabilities in binary nanofluids. Nanoscale Microscale Thermophys. Eng. 10, 29–39. Kim, J., Kang, Y.T., Choi, C.K., 2007. Soret and Dufour effects on convective instabilities in binary nanofluids for absorption application. Int. J. Refrig. 30, 323–328.
Masuda, H., Ebata, A., Teramae, K., Hishinuma, N., 1993. Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 7, 227–233. Nield, D.A., 1967. The thermohaline Rayleigh–Jeffreys problem. J. Fluid Mech. 29, 545–558. Nield, D.A., Kuznetsov, A.V., 2010a. The onset of convection in a nanofluid layer of finite depth. Eur. J. Mech. B/Fluids 29, 217–223. Nield, D.A., Kuznetsov, A.V., 2010b. The effect of local thermal non-equilibrium on the onset of convection in a nanofluid. ASME J. Heat Transfer 132, 052405. Pearlstein, A.J., Harris, R.M., Terrones, G., 1989. The onset of convective instability in a triple diffusive layer. J. Fluid Mech. 202, 443–465. Savino, R., Paterna, D., 2008. Thermodiffusion in nanofluids under different gravity conditions. Phys. Fluids 20, 017101. Wang, L., Fan, J., 2010. Nanofluids Research: key issues. Nanoscale Res. Lett. 5, 1241–1252. Wang, L., Wei, X., 2009. Heat conduction in nanofluids. Chaos Solitons Fract. 39, 2211–2215.