- Email: [email protected]
Onset of Rayleigh–Bénard MHD convection in a micropolar fluid
International Journal of Heat and Mass Transfer 55 (2012) 1164–1169
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Onset of Rayleigh–Bénard MHD convection in a micropolar fluid Z. Alloui a,⇑, P. Vasseur a,b a b
École Polytechnique, Université de Montréal, C.P. 6079, Succ. «Centre Ville», Montréal, Québec, Canada H3C 3A7 Laboratoire des Technologies Innovantes, Université Jules Vernes d’Amiens, rue des Facultés le Bailly, 800025 Amiens Cedex, France
a r t i c l e
i n f o
Article history: Received 11 May 2011 Received in revised form 23 September 2011 Accepted 23 September 2011 Available online 13 November 2011
Keywords: Micropolar fluids Rayleigh–Bénard Magnetic field
a b s t r a c t The present work is concerned with the effect of a uniform magnetic field on the onset of convection in an electrically conducting micropolar fluid. A flat fluid layer bounded by horizontal rigid boundaries, subjected to thermal boundary conditions of the Neumann type, is considered. The parallel flow approximation is used to predict analytically the critical Rayleigh number for the onset of convection. The onset of motion is found to depend on the Hartmann number Ha, materials parameters K, B, k, and the micro -rotation boundary condition n. A linear stability analysis is carried out to study numerically the onset of convection. The predictions of the analytical model are found to be in good agreement with the numerical solution. The above results are also compared with those obtained numerically for the case of a system subject to Dirichlet thermal boundary conditions. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The determination of the criterion for the onset of Rayleigh– Bénard convection, in a horizontal layer of micropolar fluids, has been investigated by several investigators. In general, these studies are based on the theory proposed by Eringen [1,2]. Micropolar fluids are of interest since they are able to describe satisfactory the behaviour of colloidal solutions, suspension solutions, liquid crystals, animal bloods, paper production, metal extraction, etc. Rama Rao [3] and Ahmadi [4] were among the first to study the onset of convection of a heat conducting micropolar fluid layer confined between two horizontal rigid boundaries and heated from below. The critical Rayleigh number was predicted in terms of the governing parameters of the problem. The possibility of overstable motions of micropolar fluids heated from below has been investigated by Pérez-Garcia and Rubi [5]. It was found that such motions are possible only for fluids with a very large coupling parameter between the spin flux and the heat flux. The instability of rotating micropolar fluids has been considered by Sastry and Ramamohan [6]. The linear stability theory was used by Narasimha Murty [7] to investigate the instability of the Bénard convection in a layer of micropolar fluid. Critical Rayleigh numbers were obtained for various hydrodynamic boundary conditions and initial temperature profiles. Natural convection in a shallow cavity filled with a micropolar fluid has been studied analytically and numerically by Alloui and Vasseur [8]. Neumann boundary conditions for temperature are applied to the horizontal walls of the enclosure. The ⇑ Corresponding author. E-mail address: [email protected] (Z. Alloui). URL: http://www.meca.polymtl.ca/convection (Z. Alloui). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.09.054
critical Rayleigh number for the onset of motion was predicted explicitly. Also, the analytical model yields the effects of the characteristic parameters of micropolar fluids on the convective heat transfer. The effect of a non-uniform basic temperature gradient on the onset of Marangoni convection in a horizontal micropolar fluid layer was considered by Melviana et al. [9]. The influence of various parameters on the onset of convection is discussed. The effect of non-uniform basic temperature gradients on the onset of Bénard–Marangoni convection in a micropolar fluid was performed by Idris et al. [10]. It was found that the presence of micron-sized suspended particles delays the onset of convection. The onset of Bénard–Marangoni convection, within a horizontal layer of micropolar fluid with rigid–rigid, rigid-free or free–free boundaries, was investigated by Alloui and Vasseur [11]. Both analytical and numerical results are reported by these authors. Explicit expressions for the critical Rayleigh and Marangoni numbers are obtained in terms of material parameters of the fluid and micro-rotation boundary conditions. The effect of a magnetic field on the onset of convection in a horizontal micropolar fluid layer heated from the bottom has also been investigated by several authors. The extension of micropolar flows to include magnetohydrodynamic effects is of interest in regard to various engineering applications such as in the design of the cooling systems for nuclear reactors, MHD electrical power generation, shock tubes, pump, flow meters, etc. The effects of throughflow and magnetic field on the onset of Bénard convection in a horizontal layer of micropolar fluid confined between two rigid, isothermal and micro-rotation free, boundaries have been studied by Narasimha Murty [12]. The critical Rayleigh number was predicted on the basis of a single-term Galerking technique. Rayleigh Bénard convection in a micropolar ferromagnetic fluid
1165
Z. Alloui, P. Vasseur / International Journal of Heat and Mass Transfer 55 (2012) 1164–1169
Nomenclature Da⁄ g Ha j k K n N Pr R Rc t u v
magnetic Darcy number, Ha2 gravitational acceleration Hartmann number, B0 H0 (r0 /q0m)1/2 micro inertia per unit mass thermal conductivity vortex viscosity parameter, j/l dimensionless micro-gyration parameter dimensionless microrotation, N0 H0 2/a Prandtl number, m/a magnetic Rayleigh number, RaDa⁄ critical magnetic Rayleigh number dimensionless time, t0 a/H0 2 dimensionless velocity in x direction, u0 H0 /a dimensionless velocity in y direction, v0 H0 /a
has been investigated analytically by Abraham [13] for a layer with free–free, isothermal, spin-vanishing magnetic boundaries. It was demonstrated that the micropolar ferromagnetic fluid layer heated from below is more stable as compared to the Newtonian ferromagnetic fluid. The effect of a magnetic field on the onset of Marangoni convection in micropolar fluid has been considered by Mahmud et al. [14–16]. The presence of the magnetic field was found to always have a stability effect of increasing the critical Marangoni number. The aim of the present paper is to study the effect of a magnetic field on the onset of convection in an electrically conducting micropolar fluid layer, bounded by solid boundaries, and heated from below. A linear stability analysis is conducted to predict numerically the critical Rayleigh number for the onset of motion from the rest state. In the case of a layer subjected to Neumann thermal boundary conditions, an analytical solution, which yields explicitly the critical Rayleigh number in terms of the governing parameters of the problem, is proposed. A good agreement between the two approaches is found.
2. Mathematical formulation The configuration considered in this study is a horizontal shallow layer, of thickness H0 and length L0 , filled with a micropolar fluid with an electrical conductivity r0 . The center of the layer is taken as the origin of coordinates x0 and y0 which are along it and across it, respectively. The two vertical walls of the layer are assumed adiabatic while Dirichlet or Neumann boundary conditions are applied for temperature on the horizontal walls. A magnetic field ~ B0 is applied upward parallel to gravity. The magnetic Reynolds number is assumed to be small, so that the induced magnetic field can be neglected in comparison with the applied magnetic field. Also, as discussed for instance by Garandet et al. [17], for a two-dimensional situation the electric field vanishes everywhere since there is always an electrically insulating boundary around the enclosure. The thermophysical properties of the fluid at a reference T 00 are assumed to be constant, except for the density in the buoyancy term, such that the micropolar fluid is assumed to satisfy the Boussinesq approximation. The density variation with temperature is described by the state equation q ¼ q0 ½1 b0T ðT 0 T 00 Þ where q0 is the fluid mixture density at temperature T 0 ¼ T 00 and b0T is the thermal expansion coefficient, respectively. The governing equations, which describe the system behaviour are conservation of momentum, microrotation and energy, are given below in terms of the stream function W as (see for instance [18,19]):
Greek symbols l dynamic viscosity m kinematic viscosity of fluid, l/q k material parameter, c/(lj) q density of fluid c spin gradient viscosity j vortex viscosity Subscripts 0 reference state c refers to critical conditions Superscript 0 refers to dimensional variable
" # 2 @ r2 W @T 2 2 2 2 2@ W þ JðW; r WÞ ¼ Pr ð1 þ KÞr ðr WÞ þ K r N Ra Ha 2 @t @x dy
ð1Þ
h i @N þ JðW; NÞ ¼ Pr kr2 N BKð2N þ r2 WÞ @t
ð2Þ
@T þ JðW; TÞ ¼ r2 T @t
ð3Þ
where, as usual, in order to satisfy the continuity equation, the stream function W is defined such that u = o W/ o y, v = o W/ o x and J(f, g) = o f/ o y o g/ o x o f/ o x o g/ o y. The above equations have been nondimensionalized by scaling length by H’, stream function by the thermal diffusivity a, microrotation by a/H’2 and time by H’2/a. Also, we introduce the reduced temperature T = (T0 T00 )/DT0 . The appropriate boundary conditions applied on the walls of the layer are:
A x¼ ; 2
W¼
@W @v ¼ 0; N ¼ n ; @x @x
1 @W @u y¼ ; W¼ ¼ 0; N ¼ n 2 @y @y
e
@T ¼0 @x
@T eþ1 ð1 eÞT ¼ @y 2
ð4Þ
ð5Þ
The parameter e is set equal to zero for Dirichlet boundary conditions and to 1 for Neumann ones. In the above equations n is a constant (0 6 n 6 1). The case n = 0 represents concentrate particle flows in which the microelements close to the wall are unable to rotate (Jena and Mathur [20]). The case n = 0.5 represents weak concentration and corresponds to the vanishing of antisymmetric part of the stress tensor (Ahmadi [21]). Finally, it was postulated by Peddieson [22] that the case n = 1 is applicable to the modeling of turbulent boundary layer flows. From the above equations it is observed that the present problem is governed by eight parameters, namely the thermal Rayleigh number Ra ¼ gb0T DT 0 H03 =am, Hartmann number Ha = B0 H0 (r0 /qom)1/2, aspect ratio of the layer A = L0 H0 , Prandtl number Pr = m/a, materials parameter k ¼ c=ðljÞ, vortex viscosity parameterK = j/l, microinertia parameter B = H0 2/j and micro-gyration parameter n. 3. Linear stability analysis In this section the onset of motion is investigated on the basis of the linear stability theory. To do so, the stability to small perturbations from the quiescent state (WB = NB = 0 and TB = 1 – y) of the physical situation described by Eqs. (1)–(3) is examined now. It
1166
Z. Alloui, P. Vasseur / International Journal of Heat and Mass Transfer 55 (2012) 1164–1169
is convenient to rewrite the governing equations using ^ ¼ W WB , g ^ ¼ N N B and ^ w h ¼ T T B . As usual, the perturbed solution is assumed to have the following functional form:
9 ptþiax ^ x; yÞ ¼ wðyÞe ~ wðt; > = g^ ðt; x; yÞ ¼ g~ ðyÞeptþiax > ^hðt; x; yÞ ¼ ~hðyÞeptþiax ;
105
8
ac
K=0
6
10
4 0
ð6Þ
50
100
(1)
Ha
(2)
^
Ψ
~ g ~ and ~ where w, h describe the vertical perturbation profiles, p is the growth rate of the perturbation and a the real wave number. Introducing Eq. (6) into Eqs. (1)–(3) and neglecting second higher-order nonlinear terms yields the following linear system:
K = 10
Ha = 10
Ra
104
Ha = 0
K=0
(2)
h i ~ ~ þ KðD2 a2 Þg ~ iaRa~h Ha2 D2 w Pr ð1 þ KÞðD2 a2 ÞðD2 a2 Þw
Ha = 10
~ ¼ pðD2 a2 Þw
ε=1 ε=0 Chandrasekhar [23]
(1)
Ha = 0
103
ð7Þ
720
n o ~ ¼ pg ~ BK½2g ~ þ ðD2 a2 Þw ~ Pr kðD2 a2 Þg
ð8Þ
~ ¼ p~h ðD2 a2 Þ~h iaw
0
2
4
a
6
8
ð9Þ Fig. 1. Marginal stability curve for B = 1, k ¼ 1 and n = 0.
The boundary conditions, corresponding to Eq. (5), are:
y ¼ 1=2;
~ ¼ Dw ~ ¼ 0; w
~ g~ ¼ nD2 w; eD~h þ ð1 eÞ~h ¼ 0 ð10Þ
The perturbed state Eqs. (7)–(9) with the boundary conditions (10) may be written in a compact matrix form as:
L1 ðaÞY ¼ pL2 ðaÞY
ð11Þ
~ g ~; ~ where Y ¼ ½w; h is a three-component vector of the perturbation and L1(a) and L2(a) are two linear differential operators that depend on the control parameters Ra, Ha, Pr, K, B, k and n. The set of Eq. (11) is solved using a finite differences scheme. The system is discretized using a fourth-order scheme in the domain between y = 1/2 and y = 1/2. For M computational points, the resulting discrete system has 3M eigenvalues that can be found using a standard IMSL subroutine such as DGVCCG. The validation of the present stability analysis was made for the case of a horizontal fluid layer, with free upper and lower boundaries, heated isothermally from the bottom and under the influence of a uniform magnetic field. Typical results are presented in Table 1. For the case of a pure fluid (K = 0) the present numerical solution is found to agree very well with the analytical solution reported in the past by Chandrasekhar [23]. For the case of a micropolar fluid (K > 0, B ¼ k ¼ 1 and n = 0), the agreement is also found to be excellent with the analytical solution of Sharma and Kumar [24]. Fig. 1 illustrates the variation of the wave number a with the Rayleigh number Ra for B = 1, k ¼ 1, n = 0, Ha = 0, 10 and K = 0, 10. The results obtained for the case of Dirichlet thermal boundary conditions (e = 0), depicted by dashed lines, indicate that as expected the minimum Rayleigh number, i.e. the critical Rayleigh number, Rac, occurs at a given critical wave number, ac, which depends upon the other governing parameters. For a Newtonian fluid (K = 0), the analytical results reported by Chandrasekhar [23],
depicted by squares, are found to be in good agreement with the present numerical procedure. Also depicted on the graph are the effects of Ha and K on ac. For a given value of K, an increase of Ha results in an increase of the critical wave number (as illustrated by the streamlines of points 1 and 2). An opposite trend is obtained upon increasing K for a given value of Ha. On the other hand, it is observed that the results obtained for the case of e = 1, depicted by solid lines, are considerably different. Thus, Fig. 1 indicates that, in the case of a layer heated by a constant heat flux, the minimum Rayleigh number Rac, occurs at a zero wave number (ac = 0) and this, independently of the governing parameters of the problem. This point is well known and has been discussed in literature by many authors; see for instance Nield [25]. The analytical procedure, developed in the following section, relies on this fact. 4. Analytical solution In this section an exact solution, for the prediction of the critical Rayleigh number, is proposed for the case of a layer heated and cooled by constant heat fluxes (e = 1). The procedure to predict analytically the onset of motion relies on the parallel flow approximation which has been described in the past in details (see for instance Vasseur et al. [27]). Thus, it is assumed that, in the limit of a shallow cavity (A 1), W(x, y) W(y), N(x, y) N(y) and T(x, y) Cx + h(y), where C is unknown constant temperature gradient in x-direction. Using the above approximations, it is found that Eqs. (1)–(3) reduce to the following system of ordinary differential equations: 4
ð1 þ KÞ
d W 4
dy
2
þK
d N dy
2
2
¼ RaC þ Ha2
d W
ð12Þ
2
dy
Table 1 Validation of the numerical code for free–free BC and e = 0. Sharma and Kumar [24]
Chandrasekhar [23]
Present results
Ha
K
Rac
ac
Rac
ac
Rac
ac
0 0 0 10 100
0 1 10 0 0
657.51 1275.82 5333.80 – –
2.221 2.207 2.142 – –
657.51 – – 2653.71 119832
2.233 – – 3.702 8.588
657.46 1275.72 5333.37 2653.63 119831
2.221 2.207 2.142 3.701 8.606
Z. Alloui, P. Vasseur / International Journal of Heat and Mass Transfer 55 (2012) 1164–1169
"
2
k
d N dy
2
2
d h dy
2
2
¼ BK 2N þ
d W
# ð13Þ
2
dy
dW C dy
¼
ð14Þ
As discussed recently by Alloui and Vasseur [11] the general solution of the above equations, satisfying the boundary conditions Eq. (5), has the following form:
uðyÞ ¼ RaCu ðyÞ
ð15Þ
WðyÞ ¼ RaC W ðyÞ
ð16Þ
NðyÞ ¼ RaCN ðyÞ
ð17Þ
hðyÞ ¼ RaC 2 h ðyÞ y ⁄
⁄
ð18Þ ⁄
⁄
where W (y), u (y), N (y) and h (y) are functions that depend upon the boundary conditions on the velocity component u, imposed on the horizontal walls of the fluid layer and on the Hartmann number Ha. In the above equations the constant C is determined by imposing an energy flux condition in the end regions of the system. This yields:
C¼
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 A1 RaÞ=A2 Ra
ð19Þ
where the constantsA1 and A2 depend upon W⁄, u⁄, N⁄ and h⁄ (see Alloui and Vasseur [11]). Eq. (19) leads us to the critical value for existence of convection in the layer. The supercritical Rayleigh number Rac for the onset of motion from the rest state, corresponding to C = 0 in Eq. (19), is given by:
Rac ¼
Ha2 2
1=12 þ 1=Ha cothðHa=2Þ=2Ha
1167
ð21Þ
It is interesting to mention that the above expression is similar to that reported by Vasseur et al. [26], while studying the onset of convection in a horizontal porous layer, modeled according to the Brinkman extended Darcy’s equation. As matter of fact, as discussed by Vasseur et al. [27], there is an analogy between the bulk resistance exerted by the magnetic drag in a fluid layer and the bulk resistance exerted by the solid matrix in a fluid saturated porous layer. The retarding effect of the magnetic field is equivalent to that of a solid matrix in a porous medium having a permeability K⁄ = H0 2/ Ha2 = lB0 r0 . Thus, the effect of the Hartmann number Ha, and more exactly the parameter Da⁄ = 1/Ha2, in a fluid layer is similar to that of the classical Darcy number in a Brinkman layer. This analogy is, however, restricted to parallel flows, i.e. flows with only one velocity component. In the following discussion the parameter Da⁄ will be referred as to the magnetic Darcy number and R = Ra/Ha2 as the magnetic Rayleigh number. In the high Hartmann number limit Ha >> 1 (i.e. Da⁄ << 1), Eq. (21) yields Rc = 12 (i.e. Rac = 12Ha2) as predicted by Nield [25] while studying the onset of motion in a Darcy layer heated from the bottom by a constant heat flux. Similarly, Rac = 720 in the limit of a pure fluid layer, as predicted by Sparrow et al. [28]. 4.2. The micropolar fluid in the absence of a magnetic fluid For Ha = 0, which corresponds to the case of the onset of convection in a micropolar fluid layer, it is readily found that Eq. (20) reduces to the following expression:
Rac ¼
1 ½sinhðX0 Þ X coshðX0 ÞC 1 C 2 =240 C 3 =12
ð22Þ
2
Rac ¼
Ha 1=12 þ k1 k2
ð20Þ
C 1 ¼ 2½BKð1 2nÞ þ 6keX0 ½12Bð2 þ KÞX2 ða1 eX þ a2 Þ1
where
C 2 ¼ 1=ð2 þ KÞ; C 3 ¼ 2X2 C 1 sinhðX0 Þ C 2 =12
2C 2 X01 1 1 K1 ¼ e ½ 2 sinhðX01 Þ coshðX01 Þ 2X1 C0 X1
K2 ¼
where
a1 ¼ nKðX þ 2Þ XðK þ 1Þ K; a2 ¼ nKðX 2Þ XðK þ 1Þ þ K
X ¼ fBKð2 þ KÞ½kð1 þ KÞ1 g1=2 ;
2C 1 X02 1 1 e ½ 2 sinhðX02 Þ coshðX02 Þ 2X2 C0 X2
1 2
X0 ¼ X
X1 ¼ fðK 2 B þ kHa2 þ 2KB A0 Þ½2kð1 þ KÞ1 g1=2
The above results are in agreement with those reported recently by Alloui and Vasseur [11]. In the limit K ? 0, it is found that:
X2 ¼ fðK 2 B þ kHa2 þ 2KB þ A0 Þ½2kð1 þ KÞ1 g1=2 1 2
1 2
X01 ¼ X1 ; X02 ¼ X2 2
Rac ¼ 720ð1 þ KÞ 2 2
2 1=2
A0 ¼ f½K þ 4ð1 þ KÞK B ½ð3K þ 2ÞKB þ A1 kHa g A1 ¼ ð3K þ 2ÞKB kHa2 B1 ¼ ðA1 A0 Þ=ð4K 2 BÞ; B2 ¼ ðA1 þ A0 Þ=ð4K 2 BÞ C 0 ¼ ðD1 D2 ÞðB1 nÞX1 ðD1 þ D2 ÞðB2 nÞX2
C 1 ¼ ½ðn B1 ÞX1 2n þ 1eX1 þ ðn B1 ÞX1 þ 2n 1 C 2 ¼ ½ðn B2 ÞX2 þ 2n 1eX2 þ ðn B2 ÞX2 2n þ 1 D1 ¼ eX1 eX2 ; D2 ¼ eX1 X2 1 The above expression can be considerably simplified upon considering the following limits: 4.1. The Newtonian fluid under the influence of a magnetic field For K = 0, which corresponds to the case of the onset of MHD convection in a Newtonian fluid layer, it is readily found that Eq. (20) reduces to:
120 2 20 BK ð1 þ 7nÞ þ 2 B2 K 3 ð1 þ 9nÞ þ OðK 7=2 Þ 7k 7k
ð23Þ
The effect of the magnetic Darcy number, Da⁄, and the micropolar parameters K, B, k and n on the critical magnetic Rayleigh number, Rc is illustrated in Figs. 2–5. In these graphs, the dashed lines are the numerical predictions of the linear stability theory for e = 0 and the solid lines correspond to e = 1. Also depicted in these graphs is the influence of Ha and the micropolar parameters on the critical wave number ac for the case of e = 0. On the other hand the squared symbols the prediction of the analytical solution, Eq. (20). A bird eye view on these graphs indicates that for e = 1, as discussed above, the critical magnetic Rayleigh number Rc ? 12 as the value of Da⁄ ? 0 (i.e. Ha ? 1) and this, independently of the value of the micropolar parameters. This behaviour is only true in the case of a unidirectional flow and such a limit is not reached for e = 0, for which the flow at the onset of motion is multicellular. The curves presented for the micropolar layer heated isothermally from the bottom are presented here for comparison since this situation has already be investigated in the past, a least for the case of a layer with free boundaries by a few authors (see for instance [4], [5],
1168
Z. Alloui, P. Vasseur / International Journal of Heat and Mass Transfer 55 (2012) 1164–1169
ε=1 ε=0
Linear stability 10
3
Eq. (20) Eq. (22)
Eq. (20) 10000
3.4
3.5
10
3.4
3.3
Ha = 10
Rc
10
Rc
3 0 3.1
0
5
3.2
5
3.2
2
a c 3.3
8
ac
10
ε=1 ε=0
Linear Stability
20
K
3.1
Ha = 0 2
4
6
8
10
B
30
B = 0.1
5000
1
K = 10
10
K=0 10
1
0
10-5
10-4
10-3
10-2
10-1
100
0
0.2
0.4
Da
Fig. 2. Variation of critical Rayleigh number Rc with parameter Da⁄ for B = 1, k ¼ 1; n ¼ 0 and K = 0, 10.
Fig. 4. Variation of critical Rayleigh number Rc with parameter Da⁄ for K = 10, k ¼ 1, n = 0 and B = 0.1, 1, 10.
1500
ε=1 ε=0
Linear Stability
Linear stability
0.6
Da *
*
ε=1 ε=0
Eq. (20) 10000
3.4
Eq. (20) 1000
3.3
Ha = 10
ac 3.2
5 0
3.5
Rc
3.1
Rc
0
ac 3
500 0
4
6
8
10
λ
λ = 0.1
5000
0.5 n=1 10
2
1
1
20
10
Ha
0.5 n=0 0
0
10-3
10-2
10-1
*
Da
0
0.2
0.4
0.6
Da*
Fig. 3. Variation of critical Rayleigh number Rc with parameter Da⁄ for K = 10, B = 1, k ¼ 1, and n = 0, 0.5, 1.
Fig. 5. Variation of critical Rayleigh number Rc with parameter Da⁄ for K = 10, B = 1, n = 0 and k ¼ 0:1; 1; 10.
[24]). Thus the following discussion will be restricted to the case of a layer heated from below by a constant heat flux. Fig. 2 exemplifies the effect of the vortex viscosity parameter, Kon Rc for B = 1, k ¼ 1 and n = 0. The limit Da⁄ ? 1 corresponds to a micropolar fluid in the absence of a Hartmann force for which the only resistance to the flow motion is due to the viscous effects. According to Eq. (23), for the parameters considered here, the critical Rayleigh number is given by:
Da⁄ 106, i.e. for a Hartmann number of the order of Ha 103. For a given value of Da⁄, an increase of the vortex viscosity parameter K, i.e. of the total viscosity of the fluid, results naturally in an increase of the critical Rayleigh number for the onset of motion. Fig. 3 illustrates the influence of the micro-gyration parameter n on Rc for K = 10, B = 1 and k ¼ 1. For a given value of Da⁄ it is found that the critical Rayleigh increases upon decreasing n. This follows from the fact that reducing n implies an increase of the concentration of the microelements. As a result, the onset of motion is delayed since a greater part of the energy of the system is required in developing gyrational velocities of the fluid. The effects of B and k on Rc are depicted on Figs. 4 and 5, respectively. The influence of these parameters on the critical Rayleigh number is weaker than that of K and n. For a given value of Da⁄ it is observed that Rc decreases upon increasing B, for which the micro-inertia density is decreased. Similarly, Fig. 5 indicates that a decrease in k, which
Rac ¼ 720ð1 þ KÞ
120 2 20 3 K þ K þ OðK 7=2 Þ 7 7
ð24Þ
This regime is reached asymptotically at about Da⁄ 101. Upon increasing the magnetic field Ha, i.e. decreasing the value of Da⁄, the retarding effects of the magnetic drag adds to the viscous effect such that a higher Rac (lower Rc) is required to destabilize the fluid layer. The limit Da⁄ ? 0, for which Rc ? 12, is reached at about
Z. Alloui, P. Vasseur / International Journal of Heat and Mass Transfer 55 (2012) 1164–1169
reduces the strength of the couple stress of the fluid, destabilizes the system. Thus, the onset of motion occurs at a lower Rc. 5. Conclusions In this study, the onset of convection in a micropolar fluid layer heated from the bottom, and subject to a vertical magnetic field, is studied analytically on the basis of the parallel flow approximation. Closed form expressions are obtained for the critical Rayleigh number in terms of the governing parameters of the problem. Numerical results, obtained on the basis of the linear stability theory, are also obtained and found to be in good agreement with the analytical solution. Furthermore, the numerical procedure yields also results for the case of a layer heated isothermally from the bottom. It is demonstrated that the magnetic field and the micropolar parameters have a considerable impact on the onset of motion. In particular, Rc depends considerably upon the vortex viscosity parameter K and micro-gyration parameter n. References [1] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 1–18. [2] A.C. Eringen, Theory of thermomicrofluids, J. Math. Anal. Appl. 38 (1972) 480– 496. [3] K.V. Rama Rao, Onset of instability in a heat conducting micropolar fluid layer, Acta Mech. 32 (1974) 79–84. [4] G. Ahmadi, Stability of a micropolar layer heated from below, Int. J. Eng. Sci. 14 (1976) 81–89. [5] C. Pérez-Garcia, J.M. Rubi, On the possibility of overstable motions of micropolar fluids heated from below, Int. J. Eng. Sci. 20 7 (1982) 873–878. [6] V.U.K. Sastry, V. Ramamohan, Numerical study of thermal instability of a rotating micropolar fluid layer, Int. J. Eng. Sci. 21 (1983) 449–461. [7] Y. Narasimha Murty, Analysis of non-uniform temperature profiles on Bénard convection in micropolar fluids, Appl. Math. Comput. 134 (2003) 473–486. [8] Z. Alloui, P. Vasseur, Natural convection in a shallow cavity filled with a micropolar fluid, Int. J. Heat Mass Transfer 53 (2010) 2750–2759. [9] J.F. Melviana, M.A. Norihan, N.S. Mohd, M.N. Roslinda, Effect of non-uniform temperature gradient on Marangoni convection in a micropolar fluid, Eur. J. Sci. Res. 4 (2009) 612–620. [10] R. Idris, H. Othman, I. Hashim, On effect of non-uniform basic temperature gradient on Bénard–Marangoni convection in micropolar fluid, Int. Com. Heat Mass Transfer 36 (2009) 255–258.
1169
[11] Z. Alloui, P. Vasseur, Onset of Bénard–Marangoni convection in a micropolar fluid, Int. J. Heat Mass Transfer doi:10.1016/i.jheatmasstransfer.2011.02.0629. [12] Y. Narasimha Murty, Effect of throughflow and magnetic field on Bénard convection in micropolar fluid, Acta Mech. 150 (2001) 11–21. [13] A. Abraham, Rayleigh–Benard convection in a micropolar ferromagnetic fluid, Int. J. Eng. Sci. 40 (2002) 449–460. [14] M.N. Mahmud, R. Idris, I. Hashim, Effect of a magnetic field on the onset of Marangoni convection in a micropolar fluid, in: Proceedings World Acadey Sciences, Engineering and Technology 38 (2009) (pp. 866–868). [15] M.N. Mahmud, R. Idris, I. Hashim, Effects of a magnetic field and nonlinear temperature profile on Marangoni convection in a micropolar fluid, Diff. Eq. Nonlinear (2009). Art. no. 748794. [16] M.N. Mahmud, Z. Mustafa, I. Hashim, Effect of control on the onset of Bénard– Marangoni convection in a micropolar fluid, Int. Comm. Heat Mass Transfer 37 (2011) 1335–1339. [17] J.P. Garandet, T. Alboussiere, R. Moreau, Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field, Int. J. Heat Mass Transfer 35 (1992) 741–748. [18] M. Zadravec, M. Hribersek, L. Skerget, Natural convection of micropolar fluid in enclosure with boundary element method, Eng. Anal. Bound. Element. 33 (2009) 485–492. [19] S. Alchaar, P. Vasseur, E. Bilgen, Natural convection heat transfer in a rectangular enclosure with a transverse magnetic field, J. Heat Transfer 117 (1995) 668–673. [20] S.K. Jena, M.N. Mathur, Similarity solutions for laminar free convection flow of a thermomicropolar fluid past a nonisothermal vertical flate, Int. J. Eng. Sci. 19 (1981) 1431–1439. [21] G. Ahmadi, Self-similar solutions of incompressible micropolar boundary layer flow over a semi-infinite plate, Int. J. Eng. Sci. 14 (1976) 639–646. [22] J. Peddieson, An application of the micropolar fluid model to the calculation of a turbulent shear flow, Int. J. Eng. Sci. 10 (1972) 23–32. [23] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (1981) New York Dover. [24] V. Sharma, S. Kumar, Thermosolutal convection of micropolar fluids in hydromagnetics, Czecholovak J. Physics 50 (2000) 1133–1146. [25] D.A. Nield, The thermohaline Rayleigh–Jeffreys problem, J. Fluid Mech. 29 (1967) 545–558. [26] P. Vasseur, C.H. Wang, M. Sen, The Brinkman model for natural convection in a shallow porous cavity with uniform heat flux, Numer. Heat Transfer 15 (1989) 221–242. [27] P. Vasseur, M. Hasnaoui, E. Bilgen, L. Robillard, Natural convection in an inclined fluid layer with a transverse magnetic field: analogy with a porous medium, J. Heat Transfer 117 (1995) 121128. [28] E.M. Sparrow, R.J. Goldstein, V.K. Jonsson, Thermal instability in a horizontal fluid layer: effect of boundary conditions and nonlinear temperature profile, J. Fluid Mech. 18 (1964) 513–528.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Onset of Rayleigh–Bénard MHD convection in a micropolar fluid Z. Alloui a,⇑, P. Vasseur a,b a b
École Polytechnique, Université de Montréal, C.P. 6079, Succ. «Centre Ville», Montréal, Québec, Canada H3C 3A7 Laboratoire des Technologies Innovantes, Université Jules Vernes d’Amiens, rue des Facultés le Bailly, 800025 Amiens Cedex, France
a r t i c l e
i n f o
Article history: Received 11 May 2011 Received in revised form 23 September 2011 Accepted 23 September 2011 Available online 13 November 2011
Keywords: Micropolar fluids Rayleigh–Bénard Magnetic field
a b s t r a c t The present work is concerned with the effect of a uniform magnetic field on the onset of convection in an electrically conducting micropolar fluid. A flat fluid layer bounded by horizontal rigid boundaries, subjected to thermal boundary conditions of the Neumann type, is considered. The parallel flow approximation is used to predict analytically the critical Rayleigh number for the onset of convection. The onset of motion is found to depend on the Hartmann number Ha, materials parameters K, B, k, and the micro -rotation boundary condition n. A linear stability analysis is carried out to study numerically the onset of convection. The predictions of the analytical model are found to be in good agreement with the numerical solution. The above results are also compared with those obtained numerically for the case of a system subject to Dirichlet thermal boundary conditions. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The determination of the criterion for the onset of Rayleigh– Bénard convection, in a horizontal layer of micropolar fluids, has been investigated by several investigators. In general, these studies are based on the theory proposed by Eringen [1,2]. Micropolar fluids are of interest since they are able to describe satisfactory the behaviour of colloidal solutions, suspension solutions, liquid crystals, animal bloods, paper production, metal extraction, etc. Rama Rao [3] and Ahmadi [4] were among the first to study the onset of convection of a heat conducting micropolar fluid layer confined between two horizontal rigid boundaries and heated from below. The critical Rayleigh number was predicted in terms of the governing parameters of the problem. The possibility of overstable motions of micropolar fluids heated from below has been investigated by Pérez-Garcia and Rubi [5]. It was found that such motions are possible only for fluids with a very large coupling parameter between the spin flux and the heat flux. The instability of rotating micropolar fluids has been considered by Sastry and Ramamohan [6]. The linear stability theory was used by Narasimha Murty [7] to investigate the instability of the Bénard convection in a layer of micropolar fluid. Critical Rayleigh numbers were obtained for various hydrodynamic boundary conditions and initial temperature profiles. Natural convection in a shallow cavity filled with a micropolar fluid has been studied analytically and numerically by Alloui and Vasseur [8]. Neumann boundary conditions for temperature are applied to the horizontal walls of the enclosure. The ⇑ Corresponding author. E-mail address: [email protected] (Z. Alloui). URL: http://www.meca.polymtl.ca/convection (Z. Alloui). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.09.054
critical Rayleigh number for the onset of motion was predicted explicitly. Also, the analytical model yields the effects of the characteristic parameters of micropolar fluids on the convective heat transfer. The effect of a non-uniform basic temperature gradient on the onset of Marangoni convection in a horizontal micropolar fluid layer was considered by Melviana et al. [9]. The influence of various parameters on the onset of convection is discussed. The effect of non-uniform basic temperature gradients on the onset of Bénard–Marangoni convection in a micropolar fluid was performed by Idris et al. [10]. It was found that the presence of micron-sized suspended particles delays the onset of convection. The onset of Bénard–Marangoni convection, within a horizontal layer of micropolar fluid with rigid–rigid, rigid-free or free–free boundaries, was investigated by Alloui and Vasseur [11]. Both analytical and numerical results are reported by these authors. Explicit expressions for the critical Rayleigh and Marangoni numbers are obtained in terms of material parameters of the fluid and micro-rotation boundary conditions. The effect of a magnetic field on the onset of convection in a horizontal micropolar fluid layer heated from the bottom has also been investigated by several authors. The extension of micropolar flows to include magnetohydrodynamic effects is of interest in regard to various engineering applications such as in the design of the cooling systems for nuclear reactors, MHD electrical power generation, shock tubes, pump, flow meters, etc. The effects of throughflow and magnetic field on the onset of Bénard convection in a horizontal layer of micropolar fluid confined between two rigid, isothermal and micro-rotation free, boundaries have been studied by Narasimha Murty [12]. The critical Rayleigh number was predicted on the basis of a single-term Galerking technique. Rayleigh Bénard convection in a micropolar ferromagnetic fluid
1165
Z. Alloui, P. Vasseur / International Journal of Heat and Mass Transfer 55 (2012) 1164–1169
Nomenclature Da⁄ g Ha j k K n N Pr R Rc t u v
magnetic Darcy number, Ha2 gravitational acceleration Hartmann number, B0 H0 (r0 /q0m)1/2 micro inertia per unit mass thermal conductivity vortex viscosity parameter, j/l dimensionless micro-gyration parameter dimensionless microrotation, N0 H0 2/a Prandtl number, m/a magnetic Rayleigh number, RaDa⁄ critical magnetic Rayleigh number dimensionless time, t0 a/H0 2 dimensionless velocity in x direction, u0 H0 /a dimensionless velocity in y direction, v0 H0 /a
has been investigated analytically by Abraham [13] for a layer with free–free, isothermal, spin-vanishing magnetic boundaries. It was demonstrated that the micropolar ferromagnetic fluid layer heated from below is more stable as compared to the Newtonian ferromagnetic fluid. The effect of a magnetic field on the onset of Marangoni convection in micropolar fluid has been considered by Mahmud et al. [14–16]. The presence of the magnetic field was found to always have a stability effect of increasing the critical Marangoni number. The aim of the present paper is to study the effect of a magnetic field on the onset of convection in an electrically conducting micropolar fluid layer, bounded by solid boundaries, and heated from below. A linear stability analysis is conducted to predict numerically the critical Rayleigh number for the onset of motion from the rest state. In the case of a layer subjected to Neumann thermal boundary conditions, an analytical solution, which yields explicitly the critical Rayleigh number in terms of the governing parameters of the problem, is proposed. A good agreement between the two approaches is found.
2. Mathematical formulation The configuration considered in this study is a horizontal shallow layer, of thickness H0 and length L0 , filled with a micropolar fluid with an electrical conductivity r0 . The center of the layer is taken as the origin of coordinates x0 and y0 which are along it and across it, respectively. The two vertical walls of the layer are assumed adiabatic while Dirichlet or Neumann boundary conditions are applied for temperature on the horizontal walls. A magnetic field ~ B0 is applied upward parallel to gravity. The magnetic Reynolds number is assumed to be small, so that the induced magnetic field can be neglected in comparison with the applied magnetic field. Also, as discussed for instance by Garandet et al. [17], for a two-dimensional situation the electric field vanishes everywhere since there is always an electrically insulating boundary around the enclosure. The thermophysical properties of the fluid at a reference T 00 are assumed to be constant, except for the density in the buoyancy term, such that the micropolar fluid is assumed to satisfy the Boussinesq approximation. The density variation with temperature is described by the state equation q ¼ q0 ½1 b0T ðT 0 T 00 Þ where q0 is the fluid mixture density at temperature T 0 ¼ T 00 and b0T is the thermal expansion coefficient, respectively. The governing equations, which describe the system behaviour are conservation of momentum, microrotation and energy, are given below in terms of the stream function W as (see for instance [18,19]):
Greek symbols l dynamic viscosity m kinematic viscosity of fluid, l/q k material parameter, c/(lj) q density of fluid c spin gradient viscosity j vortex viscosity Subscripts 0 reference state c refers to critical conditions Superscript 0 refers to dimensional variable
" # 2 @ r2 W @T 2 2 2 2 2@ W þ JðW; r WÞ ¼ Pr ð1 þ KÞr ðr WÞ þ K r N Ra Ha 2 @t @x dy
ð1Þ
h i @N þ JðW; NÞ ¼ Pr kr2 N BKð2N þ r2 WÞ @t
ð2Þ
@T þ JðW; TÞ ¼ r2 T @t
ð3Þ
where, as usual, in order to satisfy the continuity equation, the stream function W is defined such that u = o W/ o y, v = o W/ o x and J(f, g) = o f/ o y o g/ o x o f/ o x o g/ o y. The above equations have been nondimensionalized by scaling length by H’, stream function by the thermal diffusivity a, microrotation by a/H’2 and time by H’2/a. Also, we introduce the reduced temperature T = (T0 T00 )/DT0 . The appropriate boundary conditions applied on the walls of the layer are:
A x¼ ; 2
W¼
@W @v ¼ 0; N ¼ n ; @x @x
1 @W @u y¼ ; W¼ ¼ 0; N ¼ n 2 @y @y
e
@T ¼0 @x
@T eþ1 ð1 eÞT ¼ @y 2
ð4Þ
ð5Þ
The parameter e is set equal to zero for Dirichlet boundary conditions and to 1 for Neumann ones. In the above equations n is a constant (0 6 n 6 1). The case n = 0 represents concentrate particle flows in which the microelements close to the wall are unable to rotate (Jena and Mathur [20]). The case n = 0.5 represents weak concentration and corresponds to the vanishing of antisymmetric part of the stress tensor (Ahmadi [21]). Finally, it was postulated by Peddieson [22] that the case n = 1 is applicable to the modeling of turbulent boundary layer flows. From the above equations it is observed that the present problem is governed by eight parameters, namely the thermal Rayleigh number Ra ¼ gb0T DT 0 H03 =am, Hartmann number Ha = B0 H0 (r0 /qom)1/2, aspect ratio of the layer A = L0 H0 , Prandtl number Pr = m/a, materials parameter k ¼ c=ðljÞ, vortex viscosity parameterK = j/l, microinertia parameter B = H0 2/j and micro-gyration parameter n. 3. Linear stability analysis In this section the onset of motion is investigated on the basis of the linear stability theory. To do so, the stability to small perturbations from the quiescent state (WB = NB = 0 and TB = 1 – y) of the physical situation described by Eqs. (1)–(3) is examined now. It
1166
Z. Alloui, P. Vasseur / International Journal of Heat and Mass Transfer 55 (2012) 1164–1169
is convenient to rewrite the governing equations using ^ ¼ W WB , g ^ ¼ N N B and ^ w h ¼ T T B . As usual, the perturbed solution is assumed to have the following functional form:
9 ptþiax ^ x; yÞ ¼ wðyÞe ~ wðt; > = g^ ðt; x; yÞ ¼ g~ ðyÞeptþiax > ^hðt; x; yÞ ¼ ~hðyÞeptþiax ;
105
8
ac
K=0
6
10
4 0
ð6Þ
50
100
(1)
Ha
(2)
^
Ψ
~ g ~ and ~ where w, h describe the vertical perturbation profiles, p is the growth rate of the perturbation and a the real wave number. Introducing Eq. (6) into Eqs. (1)–(3) and neglecting second higher-order nonlinear terms yields the following linear system:
K = 10
Ha = 10
Ra
104
Ha = 0
K=0
(2)
h i ~ ~ þ KðD2 a2 Þg ~ iaRa~h Ha2 D2 w Pr ð1 þ KÞðD2 a2 ÞðD2 a2 Þw
Ha = 10
~ ¼ pðD2 a2 Þw
ε=1 ε=0 Chandrasekhar [23]
(1)
Ha = 0
103
ð7Þ
720
n o ~ ¼ pg ~ BK½2g ~ þ ðD2 a2 Þw ~ Pr kðD2 a2 Þg
ð8Þ
~ ¼ p~h ðD2 a2 Þ~h iaw
0
2
4
a
6
8
ð9Þ Fig. 1. Marginal stability curve for B = 1, k ¼ 1 and n = 0.
The boundary conditions, corresponding to Eq. (5), are:
y ¼ 1=2;
~ ¼ Dw ~ ¼ 0; w
~ g~ ¼ nD2 w; eD~h þ ð1 eÞ~h ¼ 0 ð10Þ
The perturbed state Eqs. (7)–(9) with the boundary conditions (10) may be written in a compact matrix form as:
L1 ðaÞY ¼ pL2 ðaÞY
ð11Þ
~ g ~; ~ where Y ¼ ½w; h is a three-component vector of the perturbation and L1(a) and L2(a) are two linear differential operators that depend on the control parameters Ra, Ha, Pr, K, B, k and n. The set of Eq. (11) is solved using a finite differences scheme. The system is discretized using a fourth-order scheme in the domain between y = 1/2 and y = 1/2. For M computational points, the resulting discrete system has 3M eigenvalues that can be found using a standard IMSL subroutine such as DGVCCG. The validation of the present stability analysis was made for the case of a horizontal fluid layer, with free upper and lower boundaries, heated isothermally from the bottom and under the influence of a uniform magnetic field. Typical results are presented in Table 1. For the case of a pure fluid (K = 0) the present numerical solution is found to agree very well with the analytical solution reported in the past by Chandrasekhar [23]. For the case of a micropolar fluid (K > 0, B ¼ k ¼ 1 and n = 0), the agreement is also found to be excellent with the analytical solution of Sharma and Kumar [24]. Fig. 1 illustrates the variation of the wave number a with the Rayleigh number Ra for B = 1, k ¼ 1, n = 0, Ha = 0, 10 and K = 0, 10. The results obtained for the case of Dirichlet thermal boundary conditions (e = 0), depicted by dashed lines, indicate that as expected the minimum Rayleigh number, i.e. the critical Rayleigh number, Rac, occurs at a given critical wave number, ac, which depends upon the other governing parameters. For a Newtonian fluid (K = 0), the analytical results reported by Chandrasekhar [23],
depicted by squares, are found to be in good agreement with the present numerical procedure. Also depicted on the graph are the effects of Ha and K on ac. For a given value of K, an increase of Ha results in an increase of the critical wave number (as illustrated by the streamlines of points 1 and 2). An opposite trend is obtained upon increasing K for a given value of Ha. On the other hand, it is observed that the results obtained for the case of e = 1, depicted by solid lines, are considerably different. Thus, Fig. 1 indicates that, in the case of a layer heated by a constant heat flux, the minimum Rayleigh number Rac, occurs at a zero wave number (ac = 0) and this, independently of the governing parameters of the problem. This point is well known and has been discussed in literature by many authors; see for instance Nield [25]. The analytical procedure, developed in the following section, relies on this fact. 4. Analytical solution In this section an exact solution, for the prediction of the critical Rayleigh number, is proposed for the case of a layer heated and cooled by constant heat fluxes (e = 1). The procedure to predict analytically the onset of motion relies on the parallel flow approximation which has been described in the past in details (see for instance Vasseur et al. [27]). Thus, it is assumed that, in the limit of a shallow cavity (A 1), W(x, y) W(y), N(x, y) N(y) and T(x, y) Cx + h(y), where C is unknown constant temperature gradient in x-direction. Using the above approximations, it is found that Eqs. (1)–(3) reduce to the following system of ordinary differential equations: 4
ð1 þ KÞ
d W 4
dy
2
þK
d N dy
2
2
¼ RaC þ Ha2
d W
ð12Þ
2
dy
Table 1 Validation of the numerical code for free–free BC and e = 0. Sharma and Kumar [24]
Chandrasekhar [23]
Present results
Ha
K
Rac
ac
Rac
ac
Rac
ac
0 0 0 10 100
0 1 10 0 0
657.51 1275.82 5333.80 – –
2.221 2.207 2.142 – –
657.51 – – 2653.71 119832
2.233 – – 3.702 8.588
657.46 1275.72 5333.37 2653.63 119831
2.221 2.207 2.142 3.701 8.606
Z. Alloui, P. Vasseur / International Journal of Heat and Mass Transfer 55 (2012) 1164–1169
"
2
k
d N dy
2
2
d h dy
2
2
¼ BK 2N þ
d W
# ð13Þ
2
dy
dW C dy
¼
ð14Þ
As discussed recently by Alloui and Vasseur [11] the general solution of the above equations, satisfying the boundary conditions Eq. (5), has the following form:
uðyÞ ¼ RaCu ðyÞ
ð15Þ
WðyÞ ¼ RaC W ðyÞ
ð16Þ
NðyÞ ¼ RaCN ðyÞ
ð17Þ
hðyÞ ¼ RaC 2 h ðyÞ y ⁄
⁄
ð18Þ ⁄
⁄
where W (y), u (y), N (y) and h (y) are functions that depend upon the boundary conditions on the velocity component u, imposed on the horizontal walls of the fluid layer and on the Hartmann number Ha. In the above equations the constant C is determined by imposing an energy flux condition in the end regions of the system. This yields:
C¼
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 A1 RaÞ=A2 Ra
ð19Þ
where the constantsA1 and A2 depend upon W⁄, u⁄, N⁄ and h⁄ (see Alloui and Vasseur [11]). Eq. (19) leads us to the critical value for existence of convection in the layer. The supercritical Rayleigh number Rac for the onset of motion from the rest state, corresponding to C = 0 in Eq. (19), is given by:
Rac ¼
Ha2 2
1=12 þ 1=Ha cothðHa=2Þ=2Ha
1167
ð21Þ
It is interesting to mention that the above expression is similar to that reported by Vasseur et al. [26], while studying the onset of convection in a horizontal porous layer, modeled according to the Brinkman extended Darcy’s equation. As matter of fact, as discussed by Vasseur et al. [27], there is an analogy between the bulk resistance exerted by the magnetic drag in a fluid layer and the bulk resistance exerted by the solid matrix in a fluid saturated porous layer. The retarding effect of the magnetic field is equivalent to that of a solid matrix in a porous medium having a permeability K⁄ = H0 2/ Ha2 = lB0 r0 . Thus, the effect of the Hartmann number Ha, and more exactly the parameter Da⁄ = 1/Ha2, in a fluid layer is similar to that of the classical Darcy number in a Brinkman layer. This analogy is, however, restricted to parallel flows, i.e. flows with only one velocity component. In the following discussion the parameter Da⁄ will be referred as to the magnetic Darcy number and R = Ra/Ha2 as the magnetic Rayleigh number. In the high Hartmann number limit Ha >> 1 (i.e. Da⁄ << 1), Eq. (21) yields Rc = 12 (i.e. Rac = 12Ha2) as predicted by Nield [25] while studying the onset of motion in a Darcy layer heated from the bottom by a constant heat flux. Similarly, Rac = 720 in the limit of a pure fluid layer, as predicted by Sparrow et al. [28]. 4.2. The micropolar fluid in the absence of a magnetic fluid For Ha = 0, which corresponds to the case of the onset of convection in a micropolar fluid layer, it is readily found that Eq. (20) reduces to the following expression:
Rac ¼
1 ½sinhðX0 Þ X coshðX0 ÞC 1 C 2 =240 C 3 =12
ð22Þ
2
Rac ¼
Ha 1=12 þ k1 k2
ð20Þ
C 1 ¼ 2½BKð1 2nÞ þ 6keX0 ½12Bð2 þ KÞX2 ða1 eX þ a2 Þ1
where
C 2 ¼ 1=ð2 þ KÞ; C 3 ¼ 2X2 C 1 sinhðX0 Þ C 2 =12
2C 2 X01 1 1 K1 ¼ e ½ 2 sinhðX01 Þ coshðX01 Þ 2X1 C0 X1
K2 ¼
where
a1 ¼ nKðX þ 2Þ XðK þ 1Þ K; a2 ¼ nKðX 2Þ XðK þ 1Þ þ K
X ¼ fBKð2 þ KÞ½kð1 þ KÞ1 g1=2 ;
2C 1 X02 1 1 e ½ 2 sinhðX02 Þ coshðX02 Þ 2X2 C0 X2
1 2
X0 ¼ X
X1 ¼ fðK 2 B þ kHa2 þ 2KB A0 Þ½2kð1 þ KÞ1 g1=2
The above results are in agreement with those reported recently by Alloui and Vasseur [11]. In the limit K ? 0, it is found that:
X2 ¼ fðK 2 B þ kHa2 þ 2KB þ A0 Þ½2kð1 þ KÞ1 g1=2 1 2
1 2
X01 ¼ X1 ; X02 ¼ X2 2
Rac ¼ 720ð1 þ KÞ 2 2
2 1=2
A0 ¼ f½K þ 4ð1 þ KÞK B ½ð3K þ 2ÞKB þ A1 kHa g A1 ¼ ð3K þ 2ÞKB kHa2 B1 ¼ ðA1 A0 Þ=ð4K 2 BÞ; B2 ¼ ðA1 þ A0 Þ=ð4K 2 BÞ C 0 ¼ ðD1 D2 ÞðB1 nÞX1 ðD1 þ D2 ÞðB2 nÞX2
C 1 ¼ ½ðn B1 ÞX1 2n þ 1eX1 þ ðn B1 ÞX1 þ 2n 1 C 2 ¼ ½ðn B2 ÞX2 þ 2n 1eX2 þ ðn B2 ÞX2 2n þ 1 D1 ¼ eX1 eX2 ; D2 ¼ eX1 X2 1 The above expression can be considerably simplified upon considering the following limits: 4.1. The Newtonian fluid under the influence of a magnetic field For K = 0, which corresponds to the case of the onset of MHD convection in a Newtonian fluid layer, it is readily found that Eq. (20) reduces to:
120 2 20 BK ð1 þ 7nÞ þ 2 B2 K 3 ð1 þ 9nÞ þ OðK 7=2 Þ 7k 7k
ð23Þ
The effect of the magnetic Darcy number, Da⁄, and the micropolar parameters K, B, k and n on the critical magnetic Rayleigh number, Rc is illustrated in Figs. 2–5. In these graphs, the dashed lines are the numerical predictions of the linear stability theory for e = 0 and the solid lines correspond to e = 1. Also depicted in these graphs is the influence of Ha and the micropolar parameters on the critical wave number ac for the case of e = 0. On the other hand the squared symbols the prediction of the analytical solution, Eq. (20). A bird eye view on these graphs indicates that for e = 1, as discussed above, the critical magnetic Rayleigh number Rc ? 12 as the value of Da⁄ ? 0 (i.e. Ha ? 1) and this, independently of the value of the micropolar parameters. This behaviour is only true in the case of a unidirectional flow and such a limit is not reached for e = 0, for which the flow at the onset of motion is multicellular. The curves presented for the micropolar layer heated isothermally from the bottom are presented here for comparison since this situation has already be investigated in the past, a least for the case of a layer with free boundaries by a few authors (see for instance [4], [5],
1168
Z. Alloui, P. Vasseur / International Journal of Heat and Mass Transfer 55 (2012) 1164–1169
ε=1 ε=0
Linear stability 10
3
Eq. (20) Eq. (22)
Eq. (20) 10000
3.4
3.5
10
3.4
3.3
Ha = 10
Rc
10
Rc
3 0 3.1
0
5
3.2
5
3.2
2
a c 3.3
8
ac
10
ε=1 ε=0
Linear Stability
20
K
3.1
Ha = 0 2
4
6
8
10
B
30
B = 0.1
5000
1
K = 10
10
K=0 10
1
0
10-5
10-4
10-3
10-2
10-1
100
0
0.2
0.4
Da
Fig. 2. Variation of critical Rayleigh number Rc with parameter Da⁄ for B = 1, k ¼ 1; n ¼ 0 and K = 0, 10.
Fig. 4. Variation of critical Rayleigh number Rc with parameter Da⁄ for K = 10, k ¼ 1, n = 0 and B = 0.1, 1, 10.
1500
ε=1 ε=0
Linear Stability
Linear stability
0.6
Da *
*
ε=1 ε=0
Eq. (20) 10000
3.4
Eq. (20) 1000
3.3
Ha = 10
ac 3.2
5 0
3.5
Rc
3.1
Rc
0
ac 3
500 0
4
6
8
10
λ
λ = 0.1
5000
0.5 n=1 10
2
1
1
20
10
Ha
0.5 n=0 0
0
10-3
10-2
10-1
*
Da
0
0.2
0.4
0.6
Da*
Fig. 3. Variation of critical Rayleigh number Rc with parameter Da⁄ for K = 10, B = 1, k ¼ 1, and n = 0, 0.5, 1.
Fig. 5. Variation of critical Rayleigh number Rc with parameter Da⁄ for K = 10, B = 1, n = 0 and k ¼ 0:1; 1; 10.
[24]). Thus the following discussion will be restricted to the case of a layer heated from below by a constant heat flux. Fig. 2 exemplifies the effect of the vortex viscosity parameter, Kon Rc for B = 1, k ¼ 1 and n = 0. The limit Da⁄ ? 1 corresponds to a micropolar fluid in the absence of a Hartmann force for which the only resistance to the flow motion is due to the viscous effects. According to Eq. (23), for the parameters considered here, the critical Rayleigh number is given by:
Da⁄ 106, i.e. for a Hartmann number of the order of Ha 103. For a given value of Da⁄, an increase of the vortex viscosity parameter K, i.e. of the total viscosity of the fluid, results naturally in an increase of the critical Rayleigh number for the onset of motion. Fig. 3 illustrates the influence of the micro-gyration parameter n on Rc for K = 10, B = 1 and k ¼ 1. For a given value of Da⁄ it is found that the critical Rayleigh increases upon decreasing n. This follows from the fact that reducing n implies an increase of the concentration of the microelements. As a result, the onset of motion is delayed since a greater part of the energy of the system is required in developing gyrational velocities of the fluid. The effects of B and k on Rc are depicted on Figs. 4 and 5, respectively. The influence of these parameters on the critical Rayleigh number is weaker than that of K and n. For a given value of Da⁄ it is observed that Rc decreases upon increasing B, for which the micro-inertia density is decreased. Similarly, Fig. 5 indicates that a decrease in k, which
Rac ¼ 720ð1 þ KÞ
120 2 20 3 K þ K þ OðK 7=2 Þ 7 7
ð24Þ
This regime is reached asymptotically at about Da⁄ 101. Upon increasing the magnetic field Ha, i.e. decreasing the value of Da⁄, the retarding effects of the magnetic drag adds to the viscous effect such that a higher Rac (lower Rc) is required to destabilize the fluid layer. The limit Da⁄ ? 0, for which Rc ? 12, is reached at about
Z. Alloui, P. Vasseur / International Journal of Heat and Mass Transfer 55 (2012) 1164–1169
reduces the strength of the couple stress of the fluid, destabilizes the system. Thus, the onset of motion occurs at a lower Rc. 5. Conclusions In this study, the onset of convection in a micropolar fluid layer heated from the bottom, and subject to a vertical magnetic field, is studied analytically on the basis of the parallel flow approximation. Closed form expressions are obtained for the critical Rayleigh number in terms of the governing parameters of the problem. Numerical results, obtained on the basis of the linear stability theory, are also obtained and found to be in good agreement with the analytical solution. Furthermore, the numerical procedure yields also results for the case of a layer heated isothermally from the bottom. It is demonstrated that the magnetic field and the micropolar parameters have a considerable impact on the onset of motion. In particular, Rc depends considerably upon the vortex viscosity parameter K and micro-gyration parameter n. References [1] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 1–18. [2] A.C. Eringen, Theory of thermomicrofluids, J. Math. Anal. Appl. 38 (1972) 480– 496. [3] K.V. Rama Rao, Onset of instability in a heat conducting micropolar fluid layer, Acta Mech. 32 (1974) 79–84. [4] G. Ahmadi, Stability of a micropolar layer heated from below, Int. J. Eng. Sci. 14 (1976) 81–89. [5] C. Pérez-Garcia, J.M. Rubi, On the possibility of overstable motions of micropolar fluids heated from below, Int. J. Eng. Sci. 20 7 (1982) 873–878. [6] V.U.K. Sastry, V. Ramamohan, Numerical study of thermal instability of a rotating micropolar fluid layer, Int. J. Eng. Sci. 21 (1983) 449–461. [7] Y. Narasimha Murty, Analysis of non-uniform temperature profiles on Bénard convection in micropolar fluids, Appl. Math. Comput. 134 (2003) 473–486. [8] Z. Alloui, P. Vasseur, Natural convection in a shallow cavity filled with a micropolar fluid, Int. J. Heat Mass Transfer 53 (2010) 2750–2759. [9] J.F. Melviana, M.A. Norihan, N.S. Mohd, M.N. Roslinda, Effect of non-uniform temperature gradient on Marangoni convection in a micropolar fluid, Eur. J. Sci. Res. 4 (2009) 612–620. [10] R. Idris, H. Othman, I. Hashim, On effect of non-uniform basic temperature gradient on Bénard–Marangoni convection in micropolar fluid, Int. Com. Heat Mass Transfer 36 (2009) 255–258.
1169
[11] Z. Alloui, P. Vasseur, Onset of Bénard–Marangoni convection in a micropolar fluid, Int. J. Heat Mass Transfer doi:10.1016/i.jheatmasstransfer.2011.02.0629. [12] Y. Narasimha Murty, Effect of throughflow and magnetic field on Bénard convection in micropolar fluid, Acta Mech. 150 (2001) 11–21. [13] A. Abraham, Rayleigh–Benard convection in a micropolar ferromagnetic fluid, Int. J. Eng. Sci. 40 (2002) 449–460. [14] M.N. Mahmud, R. Idris, I. Hashim, Effect of a magnetic field on the onset of Marangoni convection in a micropolar fluid, in: Proceedings World Acadey Sciences, Engineering and Technology 38 (2009) (pp. 866–868). [15] M.N. Mahmud, R. Idris, I. Hashim, Effects of a magnetic field and nonlinear temperature profile on Marangoni convection in a micropolar fluid, Diff. Eq. Nonlinear (2009). Art. no. 748794. [16] M.N. Mahmud, Z. Mustafa, I. Hashim, Effect of control on the onset of Bénard– Marangoni convection in a micropolar fluid, Int. Comm. Heat Mass Transfer 37 (2011) 1335–1339. [17] J.P. Garandet, T. Alboussiere, R. Moreau, Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field, Int. J. Heat Mass Transfer 35 (1992) 741–748. [18] M. Zadravec, M. Hribersek, L. Skerget, Natural convection of micropolar fluid in enclosure with boundary element method, Eng. Anal. Bound. Element. 33 (2009) 485–492. [19] S. Alchaar, P. Vasseur, E. Bilgen, Natural convection heat transfer in a rectangular enclosure with a transverse magnetic field, J. Heat Transfer 117 (1995) 668–673. [20] S.K. Jena, M.N. Mathur, Similarity solutions for laminar free convection flow of a thermomicropolar fluid past a nonisothermal vertical flate, Int. J. Eng. Sci. 19 (1981) 1431–1439. [21] G. Ahmadi, Self-similar solutions of incompressible micropolar boundary layer flow over a semi-infinite plate, Int. J. Eng. Sci. 14 (1976) 639–646. [22] J. Peddieson, An application of the micropolar fluid model to the calculation of a turbulent shear flow, Int. J. Eng. Sci. 10 (1972) 23–32. [23] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (1981) New York Dover. [24] V. Sharma, S. Kumar, Thermosolutal convection of micropolar fluids in hydromagnetics, Czecholovak J. Physics 50 (2000) 1133–1146. [25] D.A. Nield, The thermohaline Rayleigh–Jeffreys problem, J. Fluid Mech. 29 (1967) 545–558. [26] P. Vasseur, C.H. Wang, M. Sen, The Brinkman model for natural convection in a shallow porous cavity with uniform heat flux, Numer. Heat Transfer 15 (1989) 221–242. [27] P. Vasseur, M. Hasnaoui, E. Bilgen, L. Robillard, Natural convection in an inclined fluid layer with a transverse magnetic field: analogy with a porous medium, J. Heat Transfer 117 (1995) 121128. [28] E.M. Sparrow, R.J. Goldstein, V.K. Jonsson, Thermal instability in a horizontal fluid layer: effect of boundary conditions and nonlinear temperature profile, J. Fluid Mech. 18 (1964) 513–528.