The onset of convection in a horizontal nanofluid layer of finite depth: A revised model

International Journal of Heat and Mass Transfer 77 (2014) 915–918

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Technical Note

The onset of convection in a horizontal nanofluid layer of finite depth: A revised model D.A. Nield a, A.V. Kuznetsov b,⇑ a b

Department of Engineering Science, University of Auckland, Private Bag 92019, Auckland 1142, 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 17 April 2014 Received in revised form 8 June 2014 Accepted 9 June 2014

Keywords: Nanofluid Brownian motion Thermophoresis Natural convection Horizontal layer

a b s t r a c t This paper presents a revised linear stability analysis for the onset of natural convection in a horizontal nanofluid layer. The employed model incorporates the effects of Brownian motion and thermophoresis. It is now assumed that the value of the temperature can be imposed on the boundaries, but the nanoparticle fraction adjusts so that the nanoparticle flux is zero on the boundaries. It is shown that, with the new boundary conditions, oscillatory convection can no longer occur. The pertinent dimensionless nanofluid parameters have been rescaled. The effect of the nanoparticles on non-oscillatory convection is destabilizing. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction

2. Analysis

An extension to the classical Rayleigh–Bénard problem (the onset of convection in a horizontal fluid layer, uniformly heated from below) to the case of a nanofluid (a suspension of particles with diameters tens or hundreds nanometres) was made by Nield and Kuznetsov [1]. This study was based on a model of Buongiorno [2] that incorporates the effects of Brownian motion and thermophoresis. In this paper it was assumed that it was possible to control the value of the nanoparticle fraction at the boundary in the same way as the temperature there can be controlled, but no indication was given of how this could be done in practice. A more physically realistic assumption is to assume that nanoparticle flux at the boundary is zero. This change has some significant consequences. The scaling for dimensionless parameters needs to be changed. The basic solution for the nanoparticle fraction is also affected. Importantly, there are no longer two opposing buoyancy agencies affecting the instability, and as a result the possibility of oscillatory instability is removed. This is a companion paper to one on the corresponding problem in a porous medium [3].

The analysis here follows closely that in [1] and so it is abbreviated as much as possible. We select a coordinate frame in which the z-axis is aligned vertically upwards as shown in Fig. 1. We consider a horizontal layer of nanofluid confined between the planes z⁄ = 0 and z⁄ = H. Asterisks are used to distinguish dimensional variables. Each boundary wall is assumed to be perfectly thermally conducting. The temperatures at the lower and upper wall are taken to be T h and T c , the former being the greater, and the latter is chosen as the reference temperature. In this revision we treat just the case of rigid boundaries, at which the normal component of the particle flux is taken to be zero. We employed the Oberbeck–Boussinesq approximation. 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 c . The conservation equations take the form

⇑ Corresponding author. Tel.: +1 (919) 515 5292; fax: +1 (919) 515 7968. E-mail addresses: [email protected] (D.A. Nield), [email protected] (A.V. Kuznetsov). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.06.020 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

r  v  ¼ 0;

ð1Þ

   h @v   ¼  r p þ lr2 v  þ / qp þ ð1  / Þ qf þ v  r v  @t n oi  qf ð1  bðT   T c ÞÞ g;

ð2Þ

   h @T   ¼ kr2 T  þ ðqcÞp DB r /  r T  þ v  r T @t i þ ðDT =T c Þr T   r T  ;

ð3Þ

ðqcÞf

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Nomenclature cf cp DB DT H k Le NA NB

nanofluid specific heat at constant pressure (J/kg K) specific heat of the nanoparticle material (J/kg K) Brownian diffusion coefficient (m2/s) thermophoretic diffusion coefficient (m2 /s) dimensional layer depth (m) thermal conductivity of the nanofluid (W/m K) Lewis number, defined by Eq. (15a) modified diffusivity ratio, defined by Eq. (15c) modified particle-density increment, defined by Eq. (15c) pressure (Pa) dimensionless pressure, p⁄H2/laf Prandtl number, defined by Eq. (15a) thermal Rayleigh number, defined by Eq. (15a) basic-density Rayleigh number, defined by Eq. (15b) concentration Rayleigh number, defined by Eq. (15b) dimensionless growth factor time (s) dimensionless time, t⁄af/H2 nanofluid temperature (K)   T T dimensionless temperature, T  T c

p⁄ p Pr Ra Rm Rn s t⁄ t T⁄ T

h

c

T c temperature at the upper wall (K) T h temperature at the lower wall (K) (u, v, w) dimensionless velocity components, (u⁄, (m/s)

v⁄, w⁄)H/af

@/ þ v   r / ¼ DB r2 / þ ðDT =T c Þr2 T  : @t

ð4Þ

We write v⁄ = (u⁄, v⁄, w⁄). It will be noted that in writing down Eqs. (2) and (3) we have assumed that the spatial variation of l and k is negligible. Unless otherwise stated properties such as conductivity are those of the nanofluid as a whole. We assume that the temperature is held constant and the normal component of the nanoparticle flux is zero at the boundaries. Thus the boundary conditions are now 

@w ¼ 0; @z

w ¼ 0;



T  ¼ T h ;

DB



@/ DT @T þ ¼ 0 at z ¼ 0; @z T c @z ð5Þ

w ¼ 0;

@w ¼ 0; @z

T  ¼ T c ;

DB

@/ DT @T  þ ¼ 0 at z ¼ H: @z T c @z

ð6Þ

v (x, y, z)

nanofluid velocity (m/s) dimensionless Cartesian coordinates, (x⁄, y⁄, z⁄)/H; z is the vertically-upward coordinate (x⁄, y⁄, z⁄) Cartesian coordinates (m) Greek symbols a dimensionless horizontal wave number af thermal diffusivity of the nanofluid (m/s2) l viscosity of the nanofluid (N s/m2) q base fluid density (kg/m3) qp nanoparticle mass density (kg/m3) /⁄ nanoparticle volume fraction / / / relative nanoparticle volume fraction, / 0 0 /0 reference value of the nanoparticle volume fraction x dimensionless frequency (a real number) Superscripts ⁄ dimensional variable 0 perturbation variable Subscripts b basic solution 0 reference value

where

af ¼

k ; ðqcP Þf

and the subscript 0 denotes a reference value. Then Eqs. (1)–(6) take the form:

r  v ¼ 0;

t ¼ t af =H2 ;

ðu; v ; wÞ ¼ ðu ; v  ; w ÞH=af ; 

/  /¼ /0

/0



;

T  T¼  Th 

T c T c

p ¼ p H2 =laf ;

;

ð10Þ

@T NB N N þ v  rT ¼ r2 T þ r/  rT þ A B rT  rT; @t Le Le

ð11Þ

@/ 1 NA 2 þ v  r/ ¼ r2 / þ r T; @t Le Le

ð12Þ

w ¼ 0;

@w ¼ 0; @z

T ¼ 1;

@/ @T þ NA ¼ 0 at z ¼ 0; @z @z

ð13Þ

w ¼ 0;

@w ¼ 0; @z

T ¼ 0;

@/ @T þ NA ¼ 0 at z ¼ 1: @z @z

ð14Þ

ð7Þ

In Eqs. (9)–(14)

Pr ¼

qf 0 gbH3 ðT h  T c Þ af l ; Le ¼ ; Ra ¼ ; qaf DB laf

Rm ¼

NA ¼

Fig. 1. Definition sketch.

ð9Þ

  1 @v ^z þ RaT e ^z  Rn/e ^z ; þ v  rv ¼ rp þ r2 v  Rme Pr @t

We introduce dimensionless variables as follows. We define

ðx; y; zÞ ¼ ðx ; y ; z Þ=H;

ð8Þ

½qp /0 þ qf 0 ð1  /0 ÞgH3

laf

DT ðT h  T c Þ ; DB T c /0

NB ¼

;

Rn ¼

ðqcÞp  / : ðqcÞf 0

ðqp  qf 0 Þ/0 gH3 ; laf ;

ð15aÞ

ð15bÞ

ð15cÞ

The parameter Le is a Lewis number and Ra is the familiar thermal Rayleigh number. The parameters Rm and Rn may be regarded as a

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basic-density Rayleigh number and a concentration Rayleigh number, respectively. The parameter NA is a modified diffusivity ratio while NB is a modified particle-density increment. Eq. (10) has been linearized by the neglect of a term proportional to the product of / and T. 2.1. Basic solution We seek a time-independent quiescent solution of Eqs. (9)–(14) with temperature and nanoparticle volume fraction varying in the z-direction only, that is a solution of the form v = 0, p = pb(z), T = Tb(z), / = /b(z). Eqs. (10)–(12) reduce to

0¼

dpb  Rm þ RaT b  Rn/b ; dz

2

d Tb 2

dz

þ

 2 NB d/b dT b NA NB dT b þ ¼ 0; Le dz dz Le dz

2

d /b 2

dz

ð16Þ

d Tb 2

dz

¼ 0:

ð18Þ

Using the boundary conditions (13) and (14), Eq. (18) may be integrated to give

d/b

dT b þ NA ¼ 0; 2 dz dz

2

ð20Þ

T b ¼ 1  z:

ð21Þ

Eq. (19) then integrates, when Eqs. (13) and (14) are applied, to

/b ¼ /0 þ NA z:

ð22Þ

2.2. Perturbation solution We now superimpose perturbations on the basic solution. We write 0

0

p ¼ pb þ p ;

T ¼ Tb þ T ;

0

/ ¼ /b þ / ;

ð23Þ

substitute in Eqs. (9)–(14), and linearize by neglecting products of primed quantities. The following equations are obtained when Eqs. (21) and (22) are used.

r  v 0 ¼ 0;

ð24Þ

1 @ v0 ^z  Rn/0 e ^z ; ¼ rp0 þ r2 v 0 þ RaT 0 e Pr @t

ð25Þ

  @T 0 NB @T 0 @/0 2NA NB @T 0   w0 ¼ r2 T 0 þ NA  ; @t Le @z @z Le @z

ð26Þ

@/ 1 NA 2 0 þ w0 ¼ r2 /0 þ r T; Le @t Le

0

w ¼ 0;

@w0 ¼ 0; @z @w0 ¼ 0; @z

ð31Þ

 2 s  2 ðD2  a2 Þ  D  a2 W  Ra a2 H þ Rn a2 U ¼ 0; Pr   NB 2NA NB NB W þ D2  Dþ D  a2  s H þ DU ¼ 0; Le Le Le W

  NA 2 1 2 ðD  a2 ÞH  ðD  a2 Þ  s U ¼ 0; Le Le

ð32Þ

ð33Þ

ð34Þ

T 0 ¼ 0; 0

T ¼ 0;

ð27Þ @/0 @T 0 þ NA ¼ 0 at z ¼ 0; @z @z @/0 @T 0 þ NA ¼0 @z @z

DW ¼ 0;

H ¼ 0;

and at

z ¼ 1;

DU þ NA DH ¼ 0 ð35Þ

where 1=2 d 2 and a ¼ ðl þ m2 Þ : dz

ð36Þ

Thus a is a dimensionless horizontal wave number and s is a dimensionless growth factor. 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. Further, since oscillatory instability is ruled out, we can put s = 0. We now employ a Galerkin-type weighted residuals method to obtain an approximate solution to the system of Eqs. (32)–(35). We choose as trial functions (satisfying the boundary conditions) Wp, Hp, Up; p = 1, 2, 3, . . . , and write

W¼

N X Ap W p ;

H¼

p¼1

N X Bp Hp ; p¼1

U¼

N X C p Up ;

ð37Þ

p¼1

substitute into Eqs. (32)–(34), and make the expressions on the left-hand sides of those equations (the residuals) orthogonal to the trial functions, thereby obtaining a system of 3N linear algebraic equations in the 3N unknowns Ap, Bp, Cp; 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. 3. Results and discussion

0

w0 ¼ 0;

 w0 ; T 0 ; /0 ¼ ½WðzÞ; HðzÞ; UðzÞ expðst þ ilx þ imyÞ;

D

The solution of Eq. (20) satisfying Eqs. (13) and (14) is

v¼v;



at z ¼ 0

¼ 0:

0

In Eq. (30), r2H is the two-dimensional Laplacian operator in the horizontal plane. The differential Eqs. (30), (26), (27) and the boundary conditions (28) and (29) constitute a linear boundaryvalue problem that can be solved using the method of normal modes. We write

W ¼ 0;

2

dz

ð30Þ

ð19Þ

and substitution of this into Eq. (17) gives

d Tb

1 @ 2 0 r w  r4 w0 ¼ Rar2H T 0  Rnr2H /0 : Pr @t

and substitute into the differential equations to obtain

ð17Þ

2

þ NA

^z  curl curl and use the idenWe can then operate on Eq. (25) with e tity curl curl  grad div  r2 together with Eq. (24). As a result of this procedure, the six unknowns u0 , v0 , w0 , p0 , T0 , /0 are reduced to three, related by

We confine our analysis to a one-term Galerkin approximation. Appropriate trial functions satisfying the boundary conditions, which are now

W ¼ 0; ð28Þ

DW ¼ 0;

H ¼ 0;

DU þ NA DH ¼ 0

at z ¼ 0 and at z ¼ 1;

ð38Þ

are

at

z ¼ 1:

ð29Þ

W 1 ¼ z2 ð1  zÞ2 ;

H1 ¼ zð1  zÞ;

U1 ¼ NA zð1  zÞ:

ð39Þ

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With this choice of trial functions, the eigenvalue equation takes the form

Ra þ ðNA þ LeÞRn ¼

28ð504 þ 24a2 þ a4 Þð10 þ a2 Þ : 27a2

ð40Þ

The right-hand side of this equation takes a minimum when

a = 3.12 and its minimum value is 1750. Hence the onset of non-oscillatory instability is characterized by

Ra þ ðNA þ LeÞRn ¼ 1750:

ð41Þ

The value 1750 obtained using the one-term Galerkin approximation is about 3% greater than the well-known exact value 1707.762 for the critical Rayleigh number for the classical Rayleigh-Bénard problem. We observe that the parameter NB does not appear to this order of approximation. (Terms in NB drop out because of an orthogonality property of the first-order trial functions and their first derivatives. This means that, at this order of approximation, the average contribution of the nanoparticle flux to the thermal energy equation is zero.) We also observe that with the present analysis we had no need to limit the magnitude of the Lewis number. Formerly, in order to make analytical progress, we had to assume that Le was large. 4. Conclusions A revised analytical study has been made, using linear instability theory, of the onset of convection in a horizontal layer of a nanofluid, employing a model that incorporates the effects of

Brownian motion and thermophoresis. Eq. (41) is formally the same as Eq. (78) of [1]. Thus the conclusions in [1] concerning non-oscillatory instability are unchanged by the change of boundary conditions. These conclusions are that the parameter NB is not significant at first order, and the effect of the nanofluid Rayleigh number Rn is destabilizing by an amount that depends on the Lewis number and the modified diffusion ratio NA defined by Eq. (15c). However, it should be noted that Rn now involves a different scaling (a typical nanofluid fraction rather than the difference between two fractions). A major difference is that the sign of Rn cannot be negative, and hence oscillatory convection is now ruled out, in contrast to the conclusion in [1]. Conflict of interest None declared. Acknowledgment AVK gratefully acknowledges the support of the Alexander von Humboldt Foundation though the Humboldt Research Award. References [1] D.A. Nield, A.V. Kuznetsov, The onset of convection in a horizontal nanofluid layer of finite depth, Eur. J. Mech. – B/Fluids 29 (2010) 217–223. [2] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [3] D.A. Nield, A.V. Kuznetsov, Thermal instability in a porous medium layer saturated by a nanofluid: a revised model, Int. J. Heat Mass Transfer 68 (2014) 211–214.