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Thermal convection problem of micropolar fluid subjected to hall current
Applied Mathematical Modelling 34 (2010) 508–519
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Thermal convection problem of micropolar fluid subjected to hall current Neela Rani a, S.K. Tomar b,* a b
Department of Mathematics, MCMDAV College, Chandigarh 160 036, India Department of Mathematics, Panjab University, Chandigarh 160 014, India
a r t i c l e
i n f o
Article history: Received 3 June 2008 Received in revised form 6 May 2009 Accepted 1 June 2009 Available online 6 June 2009
Keywords: Micropolar Convection Rayleigh number Hall currents Wavenumber Magnetic
a b s t r a c t Thermal instability of a micropolar fluid layer heated from below in the presence of hall currents is investigated. Using the appropriate boundary conditions on the boundary surfaces of the fluid layer, the frequency equation is derived and then critical Rayleigh number is determined. It is found that hall current parameter has destabilizing effect on the system. For specific values of parameters, oscillatory convection in observed in the system. The behavior of Rayleigh number with wavenumber is also computed for different values of various parameters. The results of some earlier workers have been reduced as a special case from the present problem. Ó 2009 Published by Elsevier Inc.
1. Introduction The theory of micropolar fluids was developed by Eringen [1–3], in which the fluid elements possess rotatory motion, in addition to usual translatory motion. The translatory motion is characterized by the velocity vector, while the rotatory motion is characterized by the gyration vector. Physically speaking, a micropolar fluid may be thought of the fluids containing dumb-bell shaped molecules, e.g., polymeric fluids, fluid suspension, blood, muddy fluids like crude oils etc. These fluids differ from the classical fluids in the sense that they support couple stresses due to rotatory motion, in addition to force stress due to translatory motion and they are asymmetric in nature. The former is absent in the classical fluids. The subject of fluid stability is not only of academic interest but has many practical application, e.g. oil extraction from reservoirs, chemical engineering, geophysics etc. The instability of a fluid layer heated from below or above produce a fixed temperature difference and is known as Rayleigh–Benard convection problem in the literature. A detailed account of thermal convection in a horizontal thin layer of Newtonian fluid heated from below, under varying assumption of hydrodynamics and hydromagnetics has been nicely given in the book by Chandrasekhar [4]. Linear and non-linear instability problems of micropolar fluids have been studied by Perez-Garcia and Rubi [5], Siddeshwar and Pranesh [6,7], Ahmadi [8], Sharma and Kumar [9–12], Datta and Sastry [13], Sharma and Gupta [14], Ezzat and Othman [15], Abraham [16], Sunil et al. [17], Sharma and Kumar [18,19], Rani and Tomar [20], Othman and Zaki [21], Dragomirescu [22] including several others. In the presence of strong electric field, the electric conductivity is affected by the magnetic field. Consequently, the conductivity parallel to the electric field is reduced. Hence, the current is reduced in the direction normal to both electric and magnetic field. This phenomena in the literature is known as ‘Hall Effect’. The effect of hall current on thermal instability has also been studied by several authors, e.g., Raghavachar and Gothandaraman [23], Sharma and Gupta [24], Raptis and Ram
* Corresponding author. Tel.: +91 172 2534523; fax: +91 172 2541132. E-mail addresses: [email protected] (N. Rani), [email protected] (S.K. Tomar). 0307-904X/$ - see front matter Ó 2009 Published by Elsevier Inc. doi:10.1016/j.apm.2009.06.007
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[25], Gupta [26], Sharma and Kishore [27], Sharma et. al. [28], Sunil and Singh [29], Sunil et al. [30]. In the present problem, we have studied the thermal instability of electrically conducting incompressible micropolar fluid heated from below in the presence of uniform magnetic field and under the influence of hall currents. Frequency equation is derived and effect of various parameters is studied on the plot of Rayleigh number with wavenumber. It is found that the hall current has destabilizing effect on the system. The existence of negative values of the Rayleigh number is also noticed for some specific values of the certain parameters. 2. Formulations and basic equations In reference to the Cartesian coordinates ðx; y; zÞ, let x- and y- axes be horizontal and z- axis is directing vertically upward. Consider a horizontal layer of an incompressible electrically conducting micropolar fluid having thickness d. The region S occupied by the micropolar fluid layer L is defined as S ¼ f0 6 z 6 d; 1 < x; y < 1g. Let the system be permeated by a uniform external magnetic field H ¼ ð0; 0; HÞ and the effect of hall current is taken into account. Let the fluid be heated from below and the convection does not setin until a specific temperature gradient is attained between the lower and upper limiting surfaces. This is a well known Rayleigh–Benard instability problem in micropolar fluid. Within the framework of Boussinesq approximation, the basic equations of an incompressible heat conducting fluid, in the absence of external couple density and heat sources are
r v ¼ 0;
ð1Þ
dv 1 q0 ¼ rp þ ðl þ jÞr2 v þ jr P qg ebz þ ðr HÞ H; 4p dt dP q0 j ¼ ð þ b0 Þrðr PÞ þ cr2 P þ jr v 2jP; dt dT q0 cv ¼ kT r2 T þ d1 ðr PÞ rT; dt
ð2Þ ð3Þ ð4Þ
q ¼ q0 ½1 aðT T 0 Þ:
ð5Þ
where v ; P; T; q; p; g; ebz ; j; cv ; kT ; d1 and l denote velocity, spin, temperature, density, pressure, gravitational effect, unit vector along z- axis, micro-inertia, specific heat at constant volume, thermal conductivity, coefficient giving account of coupling between the spin flux and heat flux, coefficient of viscosity respectively; ; b0 ; c and j are micropolar constants of viscosity. q0 and T 0 are reference density and reference temperature, respectively at the lower boundary and a is the coefficient of thermal expansion. The well known Maxwell’s equations are
@H 1 r ððr HÞ HÞ; ¼ r ðv HÞ þ gr2 H @t 4pNe r H ¼ 0;
ð6Þ ð7Þ
where g is electrical resistivity, N is the number density and e is the charge of an electron. The magnetic permeability is taken as unity. In the above equations, the operator dtd stands for convective derivative given by
d @ þ v r: dt @t We now study the stability of the system wherein we shall give small perturbations to the initial state of rest and see the reaction of the disturbances on the system. The initial state is given by
v ¼ 0;
P ¼ 0;
p ¼ pðzÞ;
q ¼ qðzÞ; T ¼ TðzÞ:
Let u ¼ ðux ; uy ; uz Þ; C; dp; dq; h and h ¼ ðhx ; hy ; hz Þ denote respectively the perturbations in velocity v , spin P, pressure p, density q, temperature T and magnetic field H. Eqs. (1)–(7) yield the following perturbed equations
r u ¼ 0;
ð8Þ
du 1 q0 ¼ rdp þ ðl þ jÞr2 u þ jðr CÞ þ g aq0 h^ez þ ððr hÞ HÞ; dt 4p dC q0 j ¼ ð þ b0 Þrðr CÞ þ cr2 C þ jðr uÞ 2jC; dt dh d1 d1 ðr CÞz þ ðr CÞ rh þ buz ; ¼ jT r2 h b dt q0 c v q0 c v @h 1 r ððr hÞ HÞ; ¼ r ðu HÞ þ gr2 h @t 4pNe r h ¼ 0; dT dz
where b ¼ is the uniform adverse temperature gradient and corresponds to z-component. Using the notations
jT ¼
kT q0 cv
ð9Þ ð10Þ ð11Þ ð12Þ ð13Þ
is the thermal diffusivity. The suffix z in Eq. (11)
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p¼
ljT 2
d
jT
2
z ¼ z d;
h ¼ bdh ; u ¼ u ; d rffiffiffiffiffiffiffiffiffi jT ljT h; C ¼ 2 C ; h ¼ 2 d d
t ¼ lq0 d t ; p ;
ð14Þ
and then dropping the asterisk for convenience, the non-dimensional forms of Eqs. (8)–(13) are
r u ¼ 0;
ð15Þ
du 1 ððr hÞ HÞ; ¼ rdp þ ð1 þ KÞr2 u þ Kðr CÞ þ Rh^ez þ dt 4p j dC ¼ c1 rðr CÞ c0 r ðr CÞ þ Kððr uÞ 2CÞ; dt dh p1 ¼ r2 h þ uz þ d½rh ðr CÞ ðr CÞz ; dt @h 1 1 ¼ r ðu HÞ þ r2 h r ððr hÞ HÞ; @t p2 4pNe r h ¼ 0;
ð16Þ ð17Þ ð18Þ ð19Þ ð20Þ
where the new dimensionless coefficients are
j ¼ j ; 2 d p1 ¼
d ¼
d1 2
q0 c v d
;
c1 ¼
þ b0 þ c ; ld2
4
R¼
g abq0 d
ljT
;
l l c j ; p2 ¼ ; c0 ¼ 2 ; K ¼ : q0 jT q0 g l ld
Here, the coefficients, namely, R; p1 and p2 correspond to Rayleigh, Prandtl and magnetic numbers respectively, while the d correspond to spin diffusion, coupling between vorticity and spin effects and coupling between spin coefficients c0 , K and flux and heat flux, respectively. To obtain the analytical solutions of these equations, we shall assume that both the bounding surfaces are free from stresses. In view of (14), the boundary surfaces given by z ¼ 0 and z ¼ d after dropping asterisk, reduce to z ¼ 0 and z ¼ 1. Now, the appropriate boundary conditions on both the boundary surfaces z ¼ 0 and at z ¼ 1 are given by (see Ref. [4]):
uz ¼ h ¼ ðr CÞz ¼
@ 2 uz ¼ 0: @z2
ð21Þ
3. Linear stability Applying Curl operator twice to Eq. (16) and once to Eqs. (16), (17) and (19) under the assumption of linear theory, the zcomponent of resulting equations are given by (see Ref. [4, p. 20])
@ @2h @2h þ ðr2 uz Þ ¼ R @t @x2 @y2
! þ ð1 þ KÞr4 uz þ K r2 Xz þ
@fz H @nz ¼ ð1 þ KÞr2 fz þ ; 4p @z @t j @ Xz ¼ c0 r2 Xz K r2 uz þ 2Xz ; @t @nz @f 1 H @ 2 r hz ; ¼ H z þ r2 nz þ 4pNe @z @t @z p2
H @ 2 r hz ; 4p @z
ð22Þ ð23Þ ð24Þ ð25Þ
where fz ¼ ðr uÞz and nz ¼ ðr hÞz are the z-components of vorticity and current density respectively, while Xz ¼ ðr CÞz . The z-component of Eq. (19) and the linearized form of Eq. (18) are given by
@hz 1 @uz H @nz ¼ r2 hz þ H þ ; p2 @t @z 4pNe @z @h ¼ r2 h þ uz dXz : p1 @t
ð26Þ ð27Þ
Now, we assume that: (i) the medium adjoining the fluid is electrically non-conducting, and (ii) the normal component of vorticity vanishes at the boundary surfaces. Due to these assumptions, we shall have following restrictions on the free boundary surfaces,
nz ¼
@hz ¼ 0; @z
@fz ¼ 0: @z
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Thus the complete set of boundary conditions at the stress free isothermal boundary surfaces z ¼ 0 and z ¼ 1 are given by (see Ref. [4, p. 22])
uz ¼ nz ¼ Xz ¼ h ¼
@hz @fz @ 2 uz ¼ 0: ¼ ¼ @z @z @z2
ð28Þ
Using normal mode method, the solutions of Eqs. (22)–(27) are taken as
ðuz ; Xz ; fz ; nz ; hz ; hÞ ¼ fU; G; Z; X; B; HgðzÞ expfıðkx x þ ky yÞ þ rtg;
ð29Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where r is, in general, a complex constant and denotes the stability parameter and k ¼ kx þ ky is the wavenumber. Plugging (29) into Eqs. (22)–(27), we obtain
H 2 2 2 2 ðD2 k Þ½r ð1 þ KÞðD2 k ÞU ¼ Rk H þ ðD2 k Þ KG þ DB ; 4p H 2 ½r ð1 þ KÞðD2 k ÞZ ¼ DX; 4p 2 2 2 ½‘r þ 2A ðD k ÞG ¼ AðD2 k ÞU; 1 H 2 r ðD2 k2 Þ X ¼ HDZ þ ðD2 k ÞDB; p2 4pNe 1 H r ðD2 k2 Þ B ¼ HDU þ DX; p2 4pNe h i 2 p1 r ðD2 k Þ H ¼ U dG;
ð30Þ ð31Þ ð32Þ ð33Þ ð34Þ ð35Þ
d where ‘ ¼ jA=K; A ¼ K=c0 and D dz . Owing to relation (29), the above mentioned boundary conditions given in (28) transform into:
U ¼ X ¼ G ¼ H ¼ DZ ¼ DB ¼ D2 U ¼ 0;
at z ¼ 0 and z ¼ 1:
ð36Þ
Using these boundary conditions into Eqs. (30)–(35), we obtain
D2 G ¼ D2 H ¼ D2 X ¼ D3 B ¼ 0:
ð37Þ
Differentiating Eq. (30) twice with respect to z and using Eq. (37), it can be shown that D4 U ¼ 0. It is easy to show from Eqs. (30)–(35) and boundary conditions (36) and (37) that all even derivatives of U vanish on the boundaries. Therefore, the proper solution for U characterizing the lowest mode is (see Ref. [4])
U ¼ U 0 sin pz;
ð38Þ
where U 0 is a constant quantity. Eliminating G; Z; X; B and H from Eqs. (30)–(35), and using Eq. (38) we obtain
"
2 ) # H H2 p b ð‘r þ 2A þ b dAbÞ b þ rþ 4Ne p2 4 " ( 2 ) 2 # b H H2 p b ðp1 r þ bÞð‘r þ 2A þ bÞðr þ bð1 þ KÞÞ ¼ b fr þ bð1 þ KÞg rþ b þ rþ p2 4Ne p2 4 " ( 2 ) 2 # b H H2 p b 2 ðp1 r þ bÞ KAb fr þ bð1 þ KÞg rþ b þ rþ p2 4Ne p2 4 " # H2 pb b H2 p ; þ ðr þ bð1 þ KÞÞ þ ðrp1 þ bÞð‘r þ 2A þ bÞ r þ 4 p2 4 2
Rk
(
fr þ bð1 þ KÞg
rþ
b p2
2
ð39Þ
2
where b ¼ p2 þ k . This is the dispersion relation for electrically conducting incompressible micropolar fluid in the presence of magnetic field and hall currents.
4. Overstability motions Let us write, the complex quantity r as r ¼ rR þ irI , where rR ; rI 2 R are the real and imaginary parts of r. The marginal state of stationary convection can be achieved when r ¼ 0. This means that at stationary convection rR ¼ rI ¼ 0 and the principal of exchange of stability is valid. For overstability motion, which corresponds to the case when r–0 and rR ¼ 0. This means r ¼ ırI . Therefore, to determine the state at which the convection sets in as overstability motion, we separate the right hand side of dispersion relation (39) into real and imaginary parts by putting r ¼ ırI . We can obtain R ¼ X þ ırI Y, d and rI and explicit expressions of these functions are as under where X and Y are real valued functions of p1 ; p2 ; k; H; A; K;
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X¼
C 1 C 3 þ r2I C 2 C 4 2 k C 21 þ C 22
and Y ¼
C1 C4 C2C3 ; 2 C 21 þ C 22
ð40Þ
k
where
"
2
2
2
2b b ‘ 2b ‘ 4b 2 ð1 dAÞ þ 2 þ ð1 þ KÞ þ b ð1 þ KÞð1 dAÞ þ A p2 p2 p2 p2 # 2 4 3 2 H p‘ H b 2b þ b‘ þ 2Abð1 þ KÞ þ ð1 þ KÞð1 dAÞ 2 þ Að1 þ KÞ 2 4 4Ne p2 p2 2 2 H 2 pb H 2 pb H 2 þ ð1 dAÞ þ 2A ð2A þ b dAbÞð1 þ KÞb ; 4p2 4Ne 4p2 " 3 2 3 3 2 2b 2Ab ‘b b þ Að1 þ KÞ 4b C 2 ¼ rI ð1 dAÞð1 þ KÞ þ 2 þ ð1 þ KÞ 2 þ ð1 dAÞ 2 p2 p2 p2 p2 p2 # 2 2 2 H pb H p b‘ H 2 þ ð1 dAÞ þ 2A fð1 þ KÞb ‘ þ ð2A þ b dAbÞbg þ 4 p2 4Ne 4 2‘b r3I ð1 dAÞb þ þ 2A þ ‘bð1 þ KÞ ; p2 " 3 b ‘ p 2p 2 3 C 3 ¼ p1 ‘br6I þ r4I 2ð2A þ bÞp1 b ð1 þ KÞ þ 2 þ 1 þ 2b ‘ 1 þ 1 ð1 þ KÞ p2 p2 p2 # 2 2 H p‘ p1 H 3 2 2 2 2 þ 2p1 b p1 b ‘ þ p1 b ‘ð1 þ KÞ þ ð2A þ bÞb 1 þ 2 KAp1 b 4 4Ne p2 " 4 p 2b 2p p H 2 p‘ 4 3 r2I ð2A þ bÞ 2 þ 1 ð1 þ KÞ þ b ð2A þ bÞ 1 þ 1 ð1 þ KÞ2 þ 2b ð1 þ KÞ 1 þ 1 4 p2 p2 p2 p2 5 5 3 2 2 2b ‘ 2b ‘ H p p b ‘ 2H p þ 2ð2A þ bÞð1 þ KÞp1 b þ ð1 þ KÞ 2 þ þ ð1 þ KÞ2 2 þ 1 p2 4 4 p2 p2 p2 !2 ( ( ) 2 4 4 p H p b H2 p p b p 2 4 þ ð2A þ bÞ 2b 1 þ 1 KA 2þ 1 þ b 1 þ 2 1 ð1 þ KÞ þ 2 þ p1 b‘ p2 4 4 p2 p2 p2 p2 ( )) 2 2n 2 H H 2 H p 3 4 þ p1 b b 2ð2A þ bÞð1 þ KÞp1 b þ ð1 þ KÞ2 p1 b ‘ 4Ne 4Ne 4 )# " 6 4 b b H2 p 4 3 þ 2ð1 þ KÞb ‘ þ ð2A þ bÞb þ ð2A þ bÞð1 þ KÞ2 2 þ 2ð2A þ bÞð1 þ KÞ p2 4 p2 3 !2 ( ) 6 4 2 b H2 p b H2 p H 2 5 5 þ ð2A þ bÞ b fð2A þ b KAÞð1 þ KÞb g ; KA ð1 þ KÞ 2 þ 4Ne 4 p2 4 p2 " 3 p1 2 p b 2 5 3 b ‘ rI ð2A þ bÞ 2 þ 1 C 4 ¼ rI ð2A þ bÞp1 b þ 2p1 ‘b ð1 þ KÞ þ 1 þ 2 p2 p2 p2 4 p b ‘ p p 3 4 þ 2b ð2A þ bÞ 1 þ 2 1 ð1 þ KÞ þ b ‘ 1 þ 2 1 ð1 þ KÞ2 þ 2 2 þ 1 ð1 þ KÞ p2 p2 p2 p2 4 I
C 1 ¼ ‘r r
2 I
4
H2 p H2 p b ‘ 2 3 þ 2p1 b ‘ð1 þ KÞ þ 2 þ p1 b ð2A þ bÞð1 þ KÞ2 4 4 p2
2 p p 2 H p 3 3 þ 2b ‘ 1 þ 1 KA b 1 þ 2 1 þ p1 b ð1 þ KÞ 4 p2 p2 # 2 n o H 2 3 ð2A þ bÞp1 b þ ð2p1 ð1 þ KÞ þ 1Þb ‘ 4Ne " 5 6 3 4 b b ‘ H2 p b H2 p b ‘ þ rI 2ð2A þ bÞð1 þ KÞ 2 þ ð1 þ KÞ2 2 þ 2ð2A þ bÞ þ 2ð1 þ KÞ 4 p2 4 p2 p2 p2 !2 5 2 p1 b p1 H p H2 p 3 2 2 þ ð2A þ bÞð1 þ KÞ 2 þ þ 2b ð2A þ bÞð1 þ KÞ 1 þ þb ‘ p2 p2 p2 4 4 þ 2p1 bð2A þ bÞ
N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519
513
!2 ( ) 5 5 H2 p p1 b b p1 H2 p 3 þ p1 bð2A þ bÞ KA ð1 þ KÞ 2 þ þ þb 1þ 4 p2 p2 p22 p2 4 # 2 H 4 2 b f2ð2A þ bÞð1 þ KÞ þ ð1 þ KÞ fb‘ þ ð2A þ bÞp1 g KAf1 þ p1 ð1 þ KÞgg : 4Ne For stationary convection
R¼
C3 2
k C1
rI ¼ rR ¼ 0, we have ð41Þ
;
and for overstability case, i.e., when
R¼
r
C 1 C 3 þ 2I C 2 C 4 2 k C 21 þ C 22
;
rR ¼ 0 and rI –0, we have
and C 1 C 4 C 2 C 3 ¼ 0:
ð42Þ
For given values of parameters p1 ; p2 ; l; ; A; K; d and H, one can determine the Rayleigh number R from the equations given by (41) and (42) for the relevant case. 5. Special cases (i) For stationary convection, (i.e., when rI ¼ 0), Eq. (41) in the absence of coupling between spin and heat flux (i.e., when d ¼ 0Þ and in the presence of hall currents, reduces to
R¼
C3 2
k C1
ð43Þ
;
where
" C 1 ¼ ð2A þ bÞ ð1 þ KÞ
2 # 3 b H2 pb H 2 ; þ ð1 þ KÞb 4p2 4Ne p22 6
4
2 b H2 p b 2 H p C 3 ¼ ð1 þ KÞ ð2A þ bÞ 2 þ 2ð2A þ bÞð1 þ KÞ þ ð2A þ bÞb 4 p 4 p2 2 ( 4 ) 2 6 2 b H p b H 5 KA : þ ð1 þ KÞ 2 ð2A þ b KAÞð1 þ KÞb 4Ne p2 4 p2
!2
2
(ii) In the absence of hall current and magnetic field, (i.e., when H ¼ 0) and in the presence of coupling between spin and heat flux, i.e. d–0, the Eq. (43) reduces to
1000 900
Rayleigh Number (R)
800 700 600 500 400 300 200 100 0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 1a. Effect of d on Rayleigh Number (R) versus wavenumber (k).
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N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519 3
R¼
fð2A þ bÞð1 þ KÞ KAgb ; 2 k fð2A þ bÞ dAbg
ð44Þ
A result derived by Datta and Sastry [13] with for the corresponding problem. (iii) In the absence of hall current and magnetic field, i.e. H ¼ 0 and in the absence of coupling between spin and heat flux, i.e., d–0, the Eq. (43) reduces to
R¼
1þ
3 ðA þ bÞK b ; 2A þ b k2
ð45Þ
(iv) For a Newtonian fluid, (i.e., when d ¼ K ¼ H ¼ A ¼ 0), Eq. (43) reduces to
R¼
b
3
2
k
ð46Þ
;
which agrees with the classical result of Chandrasekhar [4] for the relevant problem. On comparing Eqs. (46) and (45), we note that the micropolar fluid heated from below is more stable than a viscous newtonian fluid for all types of boundary conditions discussed in this paper.
1000 900
Rayleigh Number (R)
800 700 600 500 400 300 200 100 0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 1b. Effect of magnetic field (H) on Rayleigh Number (R) versus wavenumber (k).
500
Rayleigh Number (R)
400
300
200
100
0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 1c. Effect of hall parameter (M) on Rayleigh Number (R) versus wavenumber (k).
N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519
515
6. Numerical results and discussion The problems of thermal convection of a fluid layer heated from below or above have great practical application in the fields such as extraction of oils/gas from oil fields and fluid flow in chemical engineering etc. In the field of oil extractions from oil fields, most of the oil companies today are looking for the best efficient and most economic techniques to get maximum oil contents. In nature, some crude oils or similar fluids may be best approximated to micropolar fluids. The convection problems provide us most reliable and accurate value of critical Rayleigh number of fluids. This value helps a lot to get the idea of most suitable temperature to be given, in order to get the maximum amount of oil extraction. To seek the impact of present investigation on the real world problems as mentioned above, we shall study the effect of different parameters on the transition of micropolar fluid from stable to unstable motion. This transition is given by the critical Rayleigh number in the convection problems. We shall compute the Rayleigh number R from (41) and (42)1 for stationary and overstability motions, respectively. To the best of authors’ knowledge, the numerical values corresponding to various parameters of micropolar fluids are not available in the literature, therefore, as an illustration these values are taken hypothetically. However, if these values are replaced by real data of a particular micropolar fluid, then we are sure enough that the results corresponding to a realistic convection problem may be obtained. The values taken are as:
p1 ¼ p2 ¼ d ¼ K ¼ 1:0;
A ¼ 0:1;
‘ ¼ 0:5;
H ¼ 1:5;
1500 1400 1300 1200
Rayleigh Number (R)
1100 1000 900 800 700 600 500 400 300 200 100 0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 2a. Effect of d on Rayleigh Number (R) versus wavenumber (k).
1500 1400 1300 1200
Rayleigh Number (R)
1100 1000 900 800 700 600 500 400 300 200 100 0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 2b. Effect of magnetic field (H) on Rayleigh Number (R) versus wavenumber (k).
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N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519
2 and M ¼ 0:2 wherever not mentioned. Here, M ¼ 4pHNe corresponds to hall effect, H corresponds to magnetic effect and the parameters d; K; A and ‘ are responsible for micropolar effects. The variation of Rayleigh number (R) is computed numerically for different values of wavenumber (k) for the cases of stationary and overstability motions. For stationary motion case, the value of stability parameter rI is taken to be zero and that for overstability motion, it is taken to be unity. The results obtained are then plotted as the variation of Rayleigh number with the wavenumber. Figs. 1a–1c depict the variation of Rayleigh number R with wavenumber k for stationary convection at different values of parameters d; H and M, while Figs. 2a–2c depict that of corresponding to overstability convection. In Fig. 1a, we notice that the parameter d has strong effect on critical Rayleigh number. It can be seen that the critical Rayleigh number goes high for large value of parameter d for stationary convection and consequently making the system more stable. On the other hand, we observe from Fig. 2a that the effect of corresponding values of parameter d on critical Rayleigh number for overstationary convection is comparatively low than that for stationary convection. Thus, we can conclude that this parameter d has more stabilizing effect in stationary convection as compared to that of in overstationary convection. It can be noticed from Figs. 1b and 2b that the effect of magnetic parameter (H) on critical Rayleigh number in stationary and overstationary cases is almost similar and that it increases with increase of (H). The effect of hall current parameter (M) on critical Rayleigh number, in case of stationary motion, can be clearly seen through Fig. 1c, which is similar but comparatively smaller than that of magnetic parameter (H). This means that with increase of hall current and magnetic field, the system becomes more stable. Significant as well as interesting effect of hall current parameter (M) on critical Rayleigh number is noticed in case of overstationary
Rayleigh Number (R)
200
100
0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 2c. Effect of Hall current (M) on Rayleigh Number (R) versus wavenumber (k).
40000
Rayleigh Number (R)
30000
20000
10000
0
-10000 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 3a. Effect of micropolar parameter (A) on Rayleigh number (R) versus wavenumber (k).
N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519
517
convection depicted in Fig. 2c. It is found that in the range 0 < k < 6, the critical Rayleigh number increases for the values of parameter (M) ranging between 0 and 3, after which the critical Rayleigh number decreases and may take negative values for large values of M. Moreover, beyond the above specified range of wavenumber k, the values of Rayleigh number increases with increase of (M). This means that the effect of hall current parameter M is significant enough in the range 0 < k < 6 and makes the system stable when 0 < M < 3 and unstable when M > 3. Perhaps the reason for this is that the general character of parameter M is to make the system unstable, which becomes dominant in the presence of other parameters only when M takes values greater than 3. From Fig. 3a, we see that the Rayleigh number is positive for smaller values of parameter (A) namely at A = 0.1 and 1.0, but when A = 2.0, the curve corresponding to Rayleigh number exhibits two branches. This is not surprising and similar phenomena has been encountered earlier by Datta and Sastry [13] in the relevant problem. Datta and Sastry [13] investigated the thermal instability of a horizontal layer of micropolar fluid heated from below. They found that the plot of Rayleigh number versus wavenumber has two branches separating the zones of stability and observed that instability is due to the existence of negative values of Rayleigh number. This behavior may be called oscillatory convection. As the parameter (A) takes values 3 and higher the Rayleigh number becomes negative, which indicates the instability in the system. A similar behavior of parameter d is noticed on Rayleigh number in stationary convection and depicted in Fig. 3b. Fig. 3c indicates the effect of micropolar parameter (K) in case of oscillatory convection. We see that as the parameter K takes higher values, the positive branch of the curve corresponding to Rayleigh number goes up and negative branch of the curve goes down. For sufficiently 40000
Rayleigh Number (R)
30000
20000
10000
0
-10000 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 3b. Effect of d on Rayleigh Number (R) versus wavenumber (k). (A=.
70000 60000
Rayleigh Number (R)
50000 40000 30000 20000 10000 0 -10000 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 3c. Effect of micropolar parameter (K) on Rayleigh number (R) versus wavenumber (k).
518
N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519 80000 70000
Rayleigh Number (R)
60000 50000 40000 30000 20000 10000 0 -10000 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 3d. Effect of Hall parameter (M) on Rayleigh number (R) versus wavenumber (k).
higher values of parameter (K), the negative branch of the curve becomes asymptotic to the value zero. Fig. 3d indicates the behavior of oscillatory convection at different values of hall current parameter (M). We note that the Rayleigh number remains positive for the values of parameter M ranging between 1 < M < 4, while at M = 4 and above, the curve bifurcates into positive and negative branches. It is found that the position of discontinuity of the branches is an increasing function of hall current parameter M. 7. Conclusion Thermal convection of a micropolar fluid layer heated from below in the presence of hall currents has been analyzed. The effects of micropolar and hall current parameters on critical Rayleigh number have been studied. It is concluded that (i) The critical Rayleigh number of overstability convection is always found to be less than the critical Rayleigh number for stationary convection. (ii) After crossing certain values of different parameters, oscillatory convection is observed. (iii) Increase of micropolar parameter ð dÞ and magnetic parameter (H) results in increase of critical Rayleigh number for stationary as well as overstationary convection. (iv) For stationary convection, an increase in hall current parameter (M) results in increase of critical Rayleigh number, however for overstationary convection, the critical Rayleigh number first increases with increase of hall current parameter (M) up to certain limit, and then decreases with further increase of parameter (M). (v) The parameters ð dÞ, (K) and (M) are found to be responsible for setting instability, beyond certain fixed values of them.
Acknowledgement One of the authors (NR) is thankful to University Grants Commission, New Delhi for providing financial assistance through Scheme No. F.6-2(25)/2008(MRP/NRCB). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1) (1966) 1–18. A.C. Eringen, Theory of thermo-microfluids, J. Math. Anal. Appl. 38 (2) (1972) 480–496. A.C. Eringen, Microcontinuum Field Theories II: Fluent Media, Springer-Verlag, New York, 2001. S. Chandrasekhar, Hydrodynamic and Hydromagnetic stability, Dover Publications, New York, 1981. C. Perez-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. P.G. Siddheshwar, S. Pranesh, Effect of non-uniform basic temperature gradient on Rayleigh–Bernard convection in micropolar fluid, Int. J. Eng. Sci. 36 (1998) 1183–1196. P.G. Siddheshwar, S. Pranesh, Magnetoconvection in a micropolar fluid, Int. J. Eng. Sci. 36 (10) (1998) 1173–1181. G. Ahmadi, Stability of micropolar fluid layer heated from below, Int. J. Eng. Sci. 14 (1976) 81–89. R.C. Sharma, P. Kumar, On micropolar fluids heated from below in Hydromagnetics, J. Non-Equilib. Thermodyn. 20 (1995) 150–159. R.C. Sharma, P. Kumar, Effect of rotation on thermal convection in micropolar fluids, Int. J. Eng. Sci. 32 (3) (1994) 545–551. R.C. Sharma, P. Kumar, On micropolar fluids heated from below in hydromagnetics in porous medium, Czech. J. Phys. 47 (6) (1997) 637–647. V. Sharma, S. Kumar, Thermosolutal convection of micropolar fluids in hydromagnetics, Czech. J. Phys. 50 (10) (2000) 1133–1146.
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A.B. Datta, V.U.K. Sastry, Thermal instability of a horizontal layer of micropolar fluid heated from below, Int. J. Eng. Sci. 14 (1976) 631–637. R.C. Sharma, U. Gupta, Thermal convection in micropolar fluids in porous medium, Int. J. Eng. Sci. 33 (13) (1995) 1887–1892. M.A. Ezzat, M.I.A. Othman, Thermal instability in a rotating micropolar fluid layer subject to an electric field, Int. J. Eng. Sci. 38 (16) (2000) 1851–1867. A. Abraham, Rayleigh–Benard convection in a micropolar ferromagnetic fluid, Int. J. Eng. Sci. 40 (4) (2002) 449–460. Sunil, A. Sharma, P.K. Bharti, R.G. Shandil, Effect of rotation on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium, Int. J. Eng. Sci. 44 (11–12) (2006) 683–698. R.C. Sharma, Pardeep Kumar, On micropolar fluids heated from below in hydromagnetics in porous medium, Czec. J. Phys. 47 (6) (1997) 637–647. R.C. Sharma, P. Kumar, Effect of rotation on thermal convection in micropolar fluids in porous medium, Ind. J. Pure Appl. Math. 29 (1) (1998) 95–104. N. Rani, S.K. Tomar, Convection of micropolar heated fluid layer with rotation in hydromagnetics, Int. J. Fluid Mech. Res. 32 (6) (2005) 675–690. M.I.A. Othman, S.A. Zaki, Thermal relaxation effect on magnetohydrodynamic instability in a rotating micropolar fluid layer heated from below, Acta Mech. 170 (2004) 187–197. I.F. Dragomirescu, Rayleigh number in a stability problem for a micropolar fluid, Turk. J. Math. 31 (2007) 123–137. M.R. Raghavachar, V.S. Gothandaraman, Hydromagnetic convection in a rotating fluid layer in the presence of Hall current, Geophys. Astro. Fluid Dyn. 45 (1988) 199–211. R.C. Sharma, U. Gupta, Thermal instability of compressible fluids with hall currents and suspended particles in porous medium, Int. J. Eng. Sci. 31 (7) (1993) 1053–1060. A. Raptis, P.C. Ram, Role of rotation and hall currents on free convection and mass transfer flow through a porous medium, Int. Commun. Heat Mass Trans. 11 (4) (1984) 385–397. A.S. Gupta, Hall effects on thermal instability, Rev. Roum. Math. Pure Appl. 12 (1967) 665–677. V. Sharma, K. Kishore, Hall effect on thermosolutal instability of Rivlin–Ericksen fluid with varying gravity field in porous medium, Ind. J. Pure Appl. Math. 32 (11) (2001) 1643–1657. R.C. Sharma, Sunil, S. Chand, Hall effect on thermal instability of Rivlin–Ericksen fluid, Ind. J. Pure Appl. Math. 31 (1) (2000) 49–59. Sunil, P. Singh, Thermal instability of a porous medium with relaxation and inertia in the presence of Hall effects, Arch. Appl. Mech. 70 (8–9) (2000) 649–658. Sunil, Y.D. Sharma, P.K. Bharti, R.C. Sharma, Thermosolutal instability of compressible Rivlin–Ericksen fluid with Hall currents, Int. J. Appl. Mech. Eng. 10 (2) (2005) 329–343.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Thermal convection problem of micropolar fluid subjected to hall current Neela Rani a, S.K. Tomar b,* a b
Department of Mathematics, MCMDAV College, Chandigarh 160 036, India Department of Mathematics, Panjab University, Chandigarh 160 014, India
a r t i c l e
i n f o
Article history: Received 3 June 2008 Received in revised form 6 May 2009 Accepted 1 June 2009 Available online 6 June 2009
Keywords: Micropolar Convection Rayleigh number Hall currents Wavenumber Magnetic
a b s t r a c t Thermal instability of a micropolar fluid layer heated from below in the presence of hall currents is investigated. Using the appropriate boundary conditions on the boundary surfaces of the fluid layer, the frequency equation is derived and then critical Rayleigh number is determined. It is found that hall current parameter has destabilizing effect on the system. For specific values of parameters, oscillatory convection in observed in the system. The behavior of Rayleigh number with wavenumber is also computed for different values of various parameters. The results of some earlier workers have been reduced as a special case from the present problem. Ó 2009 Published by Elsevier Inc.
1. Introduction The theory of micropolar fluids was developed by Eringen [1–3], in which the fluid elements possess rotatory motion, in addition to usual translatory motion. The translatory motion is characterized by the velocity vector, while the rotatory motion is characterized by the gyration vector. Physically speaking, a micropolar fluid may be thought of the fluids containing dumb-bell shaped molecules, e.g., polymeric fluids, fluid suspension, blood, muddy fluids like crude oils etc. These fluids differ from the classical fluids in the sense that they support couple stresses due to rotatory motion, in addition to force stress due to translatory motion and they are asymmetric in nature. The former is absent in the classical fluids. The subject of fluid stability is not only of academic interest but has many practical application, e.g. oil extraction from reservoirs, chemical engineering, geophysics etc. The instability of a fluid layer heated from below or above produce a fixed temperature difference and is known as Rayleigh–Benard convection problem in the literature. A detailed account of thermal convection in a horizontal thin layer of Newtonian fluid heated from below, under varying assumption of hydrodynamics and hydromagnetics has been nicely given in the book by Chandrasekhar [4]. Linear and non-linear instability problems of micropolar fluids have been studied by Perez-Garcia and Rubi [5], Siddeshwar and Pranesh [6,7], Ahmadi [8], Sharma and Kumar [9–12], Datta and Sastry [13], Sharma and Gupta [14], Ezzat and Othman [15], Abraham [16], Sunil et al. [17], Sharma and Kumar [18,19], Rani and Tomar [20], Othman and Zaki [21], Dragomirescu [22] including several others. In the presence of strong electric field, the electric conductivity is affected by the magnetic field. Consequently, the conductivity parallel to the electric field is reduced. Hence, the current is reduced in the direction normal to both electric and magnetic field. This phenomena in the literature is known as ‘Hall Effect’. The effect of hall current on thermal instability has also been studied by several authors, e.g., Raghavachar and Gothandaraman [23], Sharma and Gupta [24], Raptis and Ram
* Corresponding author. Tel.: +91 172 2534523; fax: +91 172 2541132. E-mail addresses: [email protected] (N. Rani), [email protected] (S.K. Tomar). 0307-904X/$ - see front matter Ó 2009 Published by Elsevier Inc. doi:10.1016/j.apm.2009.06.007
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509
[25], Gupta [26], Sharma and Kishore [27], Sharma et. al. [28], Sunil and Singh [29], Sunil et al. [30]. In the present problem, we have studied the thermal instability of electrically conducting incompressible micropolar fluid heated from below in the presence of uniform magnetic field and under the influence of hall currents. Frequency equation is derived and effect of various parameters is studied on the plot of Rayleigh number with wavenumber. It is found that the hall current has destabilizing effect on the system. The existence of negative values of the Rayleigh number is also noticed for some specific values of the certain parameters. 2. Formulations and basic equations In reference to the Cartesian coordinates ðx; y; zÞ, let x- and y- axes be horizontal and z- axis is directing vertically upward. Consider a horizontal layer of an incompressible electrically conducting micropolar fluid having thickness d. The region S occupied by the micropolar fluid layer L is defined as S ¼ f0 6 z 6 d; 1 < x; y < 1g. Let the system be permeated by a uniform external magnetic field H ¼ ð0; 0; HÞ and the effect of hall current is taken into account. Let the fluid be heated from below and the convection does not setin until a specific temperature gradient is attained between the lower and upper limiting surfaces. This is a well known Rayleigh–Benard instability problem in micropolar fluid. Within the framework of Boussinesq approximation, the basic equations of an incompressible heat conducting fluid, in the absence of external couple density and heat sources are
r v ¼ 0;
ð1Þ
dv 1 q0 ¼ rp þ ðl þ jÞr2 v þ jr P qg ebz þ ðr HÞ H; 4p dt dP q0 j ¼ ð þ b0 Þrðr PÞ þ cr2 P þ jr v 2jP; dt dT q0 cv ¼ kT r2 T þ d1 ðr PÞ rT; dt
ð2Þ ð3Þ ð4Þ
q ¼ q0 ½1 aðT T 0 Þ:
ð5Þ
where v ; P; T; q; p; g; ebz ; j; cv ; kT ; d1 and l denote velocity, spin, temperature, density, pressure, gravitational effect, unit vector along z- axis, micro-inertia, specific heat at constant volume, thermal conductivity, coefficient giving account of coupling between the spin flux and heat flux, coefficient of viscosity respectively; ; b0 ; c and j are micropolar constants of viscosity. q0 and T 0 are reference density and reference temperature, respectively at the lower boundary and a is the coefficient of thermal expansion. The well known Maxwell’s equations are
@H 1 r ððr HÞ HÞ; ¼ r ðv HÞ þ gr2 H @t 4pNe r H ¼ 0;
ð6Þ ð7Þ
where g is electrical resistivity, N is the number density and e is the charge of an electron. The magnetic permeability is taken as unity. In the above equations, the operator dtd stands for convective derivative given by
d @ þ v r: dt @t We now study the stability of the system wherein we shall give small perturbations to the initial state of rest and see the reaction of the disturbances on the system. The initial state is given by
v ¼ 0;
P ¼ 0;
p ¼ pðzÞ;
q ¼ qðzÞ; T ¼ TðzÞ:
Let u ¼ ðux ; uy ; uz Þ; C; dp; dq; h and h ¼ ðhx ; hy ; hz Þ denote respectively the perturbations in velocity v , spin P, pressure p, density q, temperature T and magnetic field H. Eqs. (1)–(7) yield the following perturbed equations
r u ¼ 0;
ð8Þ
du 1 q0 ¼ rdp þ ðl þ jÞr2 u þ jðr CÞ þ g aq0 h^ez þ ððr hÞ HÞ; dt 4p dC q0 j ¼ ð þ b0 Þrðr CÞ þ cr2 C þ jðr uÞ 2jC; dt dh d1 d1 ðr CÞz þ ðr CÞ rh þ buz ; ¼ jT r2 h b dt q0 c v q0 c v @h 1 r ððr hÞ HÞ; ¼ r ðu HÞ þ gr2 h @t 4pNe r h ¼ 0; dT dz
where b ¼ is the uniform adverse temperature gradient and corresponds to z-component. Using the notations
jT ¼
kT q0 cv
ð9Þ ð10Þ ð11Þ ð12Þ ð13Þ
is the thermal diffusivity. The suffix z in Eq. (11)
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N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519
p¼
ljT 2
d
jT
2
z ¼ z d;
h ¼ bdh ; u ¼ u ; d rffiffiffiffiffiffiffiffiffi jT ljT h; C ¼ 2 C ; h ¼ 2 d d
t ¼ lq0 d t ; p ;
ð14Þ
and then dropping the asterisk for convenience, the non-dimensional forms of Eqs. (8)–(13) are
r u ¼ 0;
ð15Þ
du 1 ððr hÞ HÞ; ¼ rdp þ ð1 þ KÞr2 u þ Kðr CÞ þ Rh^ez þ dt 4p j dC ¼ c1 rðr CÞ c0 r ðr CÞ þ Kððr uÞ 2CÞ; dt dh p1 ¼ r2 h þ uz þ d½rh ðr CÞ ðr CÞz ; dt @h 1 1 ¼ r ðu HÞ þ r2 h r ððr hÞ HÞ; @t p2 4pNe r h ¼ 0;
ð16Þ ð17Þ ð18Þ ð19Þ ð20Þ
where the new dimensionless coefficients are
j ¼ j ; 2 d p1 ¼
d ¼
d1 2
q0 c v d
;
c1 ¼
þ b0 þ c ; ld2
4
R¼
g abq0 d
ljT
;
l l c j ; p2 ¼ ; c0 ¼ 2 ; K ¼ : q0 jT q0 g l ld
Here, the coefficients, namely, R; p1 and p2 correspond to Rayleigh, Prandtl and magnetic numbers respectively, while the d correspond to spin diffusion, coupling between vorticity and spin effects and coupling between spin coefficients c0 , K and flux and heat flux, respectively. To obtain the analytical solutions of these equations, we shall assume that both the bounding surfaces are free from stresses. In view of (14), the boundary surfaces given by z ¼ 0 and z ¼ d after dropping asterisk, reduce to z ¼ 0 and z ¼ 1. Now, the appropriate boundary conditions on both the boundary surfaces z ¼ 0 and at z ¼ 1 are given by (see Ref. [4]):
uz ¼ h ¼ ðr CÞz ¼
@ 2 uz ¼ 0: @z2
ð21Þ
3. Linear stability Applying Curl operator twice to Eq. (16) and once to Eqs. (16), (17) and (19) under the assumption of linear theory, the zcomponent of resulting equations are given by (see Ref. [4, p. 20])
@ @2h @2h þ ðr2 uz Þ ¼ R @t @x2 @y2
! þ ð1 þ KÞr4 uz þ K r2 Xz þ
@fz H @nz ¼ ð1 þ KÞr2 fz þ ; 4p @z @t j @ Xz ¼ c0 r2 Xz K r2 uz þ 2Xz ; @t @nz @f 1 H @ 2 r hz ; ¼ H z þ r2 nz þ 4pNe @z @t @z p2
H @ 2 r hz ; 4p @z
ð22Þ ð23Þ ð24Þ ð25Þ
where fz ¼ ðr uÞz and nz ¼ ðr hÞz are the z-components of vorticity and current density respectively, while Xz ¼ ðr CÞz . The z-component of Eq. (19) and the linearized form of Eq. (18) are given by
@hz 1 @uz H @nz ¼ r2 hz þ H þ ; p2 @t @z 4pNe @z @h ¼ r2 h þ uz dXz : p1 @t
ð26Þ ð27Þ
Now, we assume that: (i) the medium adjoining the fluid is electrically non-conducting, and (ii) the normal component of vorticity vanishes at the boundary surfaces. Due to these assumptions, we shall have following restrictions on the free boundary surfaces,
nz ¼
@hz ¼ 0; @z
@fz ¼ 0: @z
N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519
511
Thus the complete set of boundary conditions at the stress free isothermal boundary surfaces z ¼ 0 and z ¼ 1 are given by (see Ref. [4, p. 22])
uz ¼ nz ¼ Xz ¼ h ¼
@hz @fz @ 2 uz ¼ 0: ¼ ¼ @z @z @z2
ð28Þ
Using normal mode method, the solutions of Eqs. (22)–(27) are taken as
ðuz ; Xz ; fz ; nz ; hz ; hÞ ¼ fU; G; Z; X; B; HgðzÞ expfıðkx x þ ky yÞ þ rtg;
ð29Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where r is, in general, a complex constant and denotes the stability parameter and k ¼ kx þ ky is the wavenumber. Plugging (29) into Eqs. (22)–(27), we obtain
H 2 2 2 2 ðD2 k Þ½r ð1 þ KÞðD2 k ÞU ¼ Rk H þ ðD2 k Þ KG þ DB ; 4p H 2 ½r ð1 þ KÞðD2 k ÞZ ¼ DX; 4p 2 2 2 ½‘r þ 2A ðD k ÞG ¼ AðD2 k ÞU; 1 H 2 r ðD2 k2 Þ X ¼ HDZ þ ðD2 k ÞDB; p2 4pNe 1 H r ðD2 k2 Þ B ¼ HDU þ DX; p2 4pNe h i 2 p1 r ðD2 k Þ H ¼ U dG;
ð30Þ ð31Þ ð32Þ ð33Þ ð34Þ ð35Þ
d where ‘ ¼ jA=K; A ¼ K=c0 and D dz . Owing to relation (29), the above mentioned boundary conditions given in (28) transform into:
U ¼ X ¼ G ¼ H ¼ DZ ¼ DB ¼ D2 U ¼ 0;
at z ¼ 0 and z ¼ 1:
ð36Þ
Using these boundary conditions into Eqs. (30)–(35), we obtain
D2 G ¼ D2 H ¼ D2 X ¼ D3 B ¼ 0:
ð37Þ
Differentiating Eq. (30) twice with respect to z and using Eq. (37), it can be shown that D4 U ¼ 0. It is easy to show from Eqs. (30)–(35) and boundary conditions (36) and (37) that all even derivatives of U vanish on the boundaries. Therefore, the proper solution for U characterizing the lowest mode is (see Ref. [4])
U ¼ U 0 sin pz;
ð38Þ
where U 0 is a constant quantity. Eliminating G; Z; X; B and H from Eqs. (30)–(35), and using Eq. (38) we obtain
"
2 ) # H H2 p b ð‘r þ 2A þ b dAbÞ b þ rþ 4Ne p2 4 " ( 2 ) 2 # b H H2 p b ðp1 r þ bÞð‘r þ 2A þ bÞðr þ bð1 þ KÞÞ ¼ b fr þ bð1 þ KÞg rþ b þ rþ p2 4Ne p2 4 " ( 2 ) 2 # b H H2 p b 2 ðp1 r þ bÞ KAb fr þ bð1 þ KÞg rþ b þ rþ p2 4Ne p2 4 " # H2 pb b H2 p ; þ ðr þ bð1 þ KÞÞ þ ðrp1 þ bÞð‘r þ 2A þ bÞ r þ 4 p2 4 2
Rk
(
fr þ bð1 þ KÞg
rþ
b p2
2
ð39Þ
2
where b ¼ p2 þ k . This is the dispersion relation for electrically conducting incompressible micropolar fluid in the presence of magnetic field and hall currents.
4. Overstability motions Let us write, the complex quantity r as r ¼ rR þ irI , where rR ; rI 2 R are the real and imaginary parts of r. The marginal state of stationary convection can be achieved when r ¼ 0. This means that at stationary convection rR ¼ rI ¼ 0 and the principal of exchange of stability is valid. For overstability motion, which corresponds to the case when r–0 and rR ¼ 0. This means r ¼ ırI . Therefore, to determine the state at which the convection sets in as overstability motion, we separate the right hand side of dispersion relation (39) into real and imaginary parts by putting r ¼ ırI . We can obtain R ¼ X þ ırI Y, d and rI and explicit expressions of these functions are as under where X and Y are real valued functions of p1 ; p2 ; k; H; A; K;
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X¼
C 1 C 3 þ r2I C 2 C 4 2 k C 21 þ C 22
and Y ¼
C1 C4 C2C3 ; 2 C 21 þ C 22
ð40Þ
k
where
"
2
2
2
2b b ‘ 2b ‘ 4b 2 ð1 dAÞ þ 2 þ ð1 þ KÞ þ b ð1 þ KÞð1 dAÞ þ A p2 p2 p2 p2 # 2 4 3 2 H p‘ H b 2b þ b‘ þ 2Abð1 þ KÞ þ ð1 þ KÞð1 dAÞ 2 þ Að1 þ KÞ 2 4 4Ne p2 p2 2 2 H 2 pb H 2 pb H 2 þ ð1 dAÞ þ 2A ð2A þ b dAbÞð1 þ KÞb ; 4p2 4Ne 4p2 " 3 2 3 3 2 2b 2Ab ‘b b þ Að1 þ KÞ 4b C 2 ¼ rI ð1 dAÞð1 þ KÞ þ 2 þ ð1 þ KÞ 2 þ ð1 dAÞ 2 p2 p2 p2 p2 p2 # 2 2 2 H pb H p b‘ H 2 þ ð1 dAÞ þ 2A fð1 þ KÞb ‘ þ ð2A þ b dAbÞbg þ 4 p2 4Ne 4 2‘b r3I ð1 dAÞb þ þ 2A þ ‘bð1 þ KÞ ; p2 " 3 b ‘ p 2p 2 3 C 3 ¼ p1 ‘br6I þ r4I 2ð2A þ bÞp1 b ð1 þ KÞ þ 2 þ 1 þ 2b ‘ 1 þ 1 ð1 þ KÞ p2 p2 p2 # 2 2 H p‘ p1 H 3 2 2 2 2 þ 2p1 b p1 b ‘ þ p1 b ‘ð1 þ KÞ þ ð2A þ bÞb 1 þ 2 KAp1 b 4 4Ne p2 " 4 p 2b 2p p H 2 p‘ 4 3 r2I ð2A þ bÞ 2 þ 1 ð1 þ KÞ þ b ð2A þ bÞ 1 þ 1 ð1 þ KÞ2 þ 2b ð1 þ KÞ 1 þ 1 4 p2 p2 p2 p2 5 5 3 2 2 2b ‘ 2b ‘ H p p b ‘ 2H p þ 2ð2A þ bÞð1 þ KÞp1 b þ ð1 þ KÞ 2 þ þ ð1 þ KÞ2 2 þ 1 p2 4 4 p2 p2 p2 !2 ( ( ) 2 4 4 p H p b H2 p p b p 2 4 þ ð2A þ bÞ 2b 1 þ 1 KA 2þ 1 þ b 1 þ 2 1 ð1 þ KÞ þ 2 þ p1 b‘ p2 4 4 p2 p2 p2 p2 ( )) 2 2n 2 H H 2 H p 3 4 þ p1 b b 2ð2A þ bÞð1 þ KÞp1 b þ ð1 þ KÞ2 p1 b ‘ 4Ne 4Ne 4 )# " 6 4 b b H2 p 4 3 þ 2ð1 þ KÞb ‘ þ ð2A þ bÞb þ ð2A þ bÞð1 þ KÞ2 2 þ 2ð2A þ bÞð1 þ KÞ p2 4 p2 3 !2 ( ) 6 4 2 b H2 p b H2 p H 2 5 5 þ ð2A þ bÞ b fð2A þ b KAÞð1 þ KÞb g ; KA ð1 þ KÞ 2 þ 4Ne 4 p2 4 p2 " 3 p1 2 p b 2 5 3 b ‘ rI ð2A þ bÞ 2 þ 1 C 4 ¼ rI ð2A þ bÞp1 b þ 2p1 ‘b ð1 þ KÞ þ 1 þ 2 p2 p2 p2 4 p b ‘ p p 3 4 þ 2b ð2A þ bÞ 1 þ 2 1 ð1 þ KÞ þ b ‘ 1 þ 2 1 ð1 þ KÞ2 þ 2 2 þ 1 ð1 þ KÞ p2 p2 p2 p2 4 I
C 1 ¼ ‘r r
2 I
4
H2 p H2 p b ‘ 2 3 þ 2p1 b ‘ð1 þ KÞ þ 2 þ p1 b ð2A þ bÞð1 þ KÞ2 4 4 p2
2 p p 2 H p 3 3 þ 2b ‘ 1 þ 1 KA b 1 þ 2 1 þ p1 b ð1 þ KÞ 4 p2 p2 # 2 n o H 2 3 ð2A þ bÞp1 b þ ð2p1 ð1 þ KÞ þ 1Þb ‘ 4Ne " 5 6 3 4 b b ‘ H2 p b H2 p b ‘ þ rI 2ð2A þ bÞð1 þ KÞ 2 þ ð1 þ KÞ2 2 þ 2ð2A þ bÞ þ 2ð1 þ KÞ 4 p2 4 p2 p2 p2 !2 5 2 p1 b p1 H p H2 p 3 2 2 þ ð2A þ bÞð1 þ KÞ 2 þ þ 2b ð2A þ bÞð1 þ KÞ 1 þ þb ‘ p2 p2 p2 4 4 þ 2p1 bð2A þ bÞ
N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519
513
!2 ( ) 5 5 H2 p p1 b b p1 H2 p 3 þ p1 bð2A þ bÞ KA ð1 þ KÞ 2 þ þ þb 1þ 4 p2 p2 p22 p2 4 # 2 H 4 2 b f2ð2A þ bÞð1 þ KÞ þ ð1 þ KÞ fb‘ þ ð2A þ bÞp1 g KAf1 þ p1 ð1 þ KÞgg : 4Ne For stationary convection
R¼
C3 2
k C1
rI ¼ rR ¼ 0, we have ð41Þ
;
and for overstability case, i.e., when
R¼
r
C 1 C 3 þ 2I C 2 C 4 2 k C 21 þ C 22
;
rR ¼ 0 and rI –0, we have
and C 1 C 4 C 2 C 3 ¼ 0:
ð42Þ
For given values of parameters p1 ; p2 ; l; ; A; K; d and H, one can determine the Rayleigh number R from the equations given by (41) and (42) for the relevant case. 5. Special cases (i) For stationary convection, (i.e., when rI ¼ 0), Eq. (41) in the absence of coupling between spin and heat flux (i.e., when d ¼ 0Þ and in the presence of hall currents, reduces to
R¼
C3 2
k C1
ð43Þ
;
where
" C 1 ¼ ð2A þ bÞ ð1 þ KÞ
2 # 3 b H2 pb H 2 ; þ ð1 þ KÞb 4p2 4Ne p22 6
4
2 b H2 p b 2 H p C 3 ¼ ð1 þ KÞ ð2A þ bÞ 2 þ 2ð2A þ bÞð1 þ KÞ þ ð2A þ bÞb 4 p 4 p2 2 ( 4 ) 2 6 2 b H p b H 5 KA : þ ð1 þ KÞ 2 ð2A þ b KAÞð1 þ KÞb 4Ne p2 4 p2
!2
2
(ii) In the absence of hall current and magnetic field, (i.e., when H ¼ 0) and in the presence of coupling between spin and heat flux, i.e. d–0, the Eq. (43) reduces to
1000 900
Rayleigh Number (R)
800 700 600 500 400 300 200 100 0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 1a. Effect of d on Rayleigh Number (R) versus wavenumber (k).
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N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519 3
R¼
fð2A þ bÞð1 þ KÞ KAgb ; 2 k fð2A þ bÞ dAbg
ð44Þ
A result derived by Datta and Sastry [13] with for the corresponding problem. (iii) In the absence of hall current and magnetic field, i.e. H ¼ 0 and in the absence of coupling between spin and heat flux, i.e., d–0, the Eq. (43) reduces to
R¼
1þ
3 ðA þ bÞK b ; 2A þ b k2
ð45Þ
(iv) For a Newtonian fluid, (i.e., when d ¼ K ¼ H ¼ A ¼ 0), Eq. (43) reduces to
R¼
b
3
2
k
ð46Þ
;
which agrees with the classical result of Chandrasekhar [4] for the relevant problem. On comparing Eqs. (46) and (45), we note that the micropolar fluid heated from below is more stable than a viscous newtonian fluid for all types of boundary conditions discussed in this paper.
1000 900
Rayleigh Number (R)
800 700 600 500 400 300 200 100 0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 1b. Effect of magnetic field (H) on Rayleigh Number (R) versus wavenumber (k).
500
Rayleigh Number (R)
400
300
200
100
0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 1c. Effect of hall parameter (M) on Rayleigh Number (R) versus wavenumber (k).
N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519
515
6. Numerical results and discussion The problems of thermal convection of a fluid layer heated from below or above have great practical application in the fields such as extraction of oils/gas from oil fields and fluid flow in chemical engineering etc. In the field of oil extractions from oil fields, most of the oil companies today are looking for the best efficient and most economic techniques to get maximum oil contents. In nature, some crude oils or similar fluids may be best approximated to micropolar fluids. The convection problems provide us most reliable and accurate value of critical Rayleigh number of fluids. This value helps a lot to get the idea of most suitable temperature to be given, in order to get the maximum amount of oil extraction. To seek the impact of present investigation on the real world problems as mentioned above, we shall study the effect of different parameters on the transition of micropolar fluid from stable to unstable motion. This transition is given by the critical Rayleigh number in the convection problems. We shall compute the Rayleigh number R from (41) and (42)1 for stationary and overstability motions, respectively. To the best of authors’ knowledge, the numerical values corresponding to various parameters of micropolar fluids are not available in the literature, therefore, as an illustration these values are taken hypothetically. However, if these values are replaced by real data of a particular micropolar fluid, then we are sure enough that the results corresponding to a realistic convection problem may be obtained. The values taken are as:
p1 ¼ p2 ¼ d ¼ K ¼ 1:0;
A ¼ 0:1;
‘ ¼ 0:5;
H ¼ 1:5;
1500 1400 1300 1200
Rayleigh Number (R)
1100 1000 900 800 700 600 500 400 300 200 100 0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 2a. Effect of d on Rayleigh Number (R) versus wavenumber (k).
1500 1400 1300 1200
Rayleigh Number (R)
1100 1000 900 800 700 600 500 400 300 200 100 0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 2b. Effect of magnetic field (H) on Rayleigh Number (R) versus wavenumber (k).
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N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519
2 and M ¼ 0:2 wherever not mentioned. Here, M ¼ 4pHNe corresponds to hall effect, H corresponds to magnetic effect and the parameters d; K; A and ‘ are responsible for micropolar effects. The variation of Rayleigh number (R) is computed numerically for different values of wavenumber (k) for the cases of stationary and overstability motions. For stationary motion case, the value of stability parameter rI is taken to be zero and that for overstability motion, it is taken to be unity. The results obtained are then plotted as the variation of Rayleigh number with the wavenumber. Figs. 1a–1c depict the variation of Rayleigh number R with wavenumber k for stationary convection at different values of parameters d; H and M, while Figs. 2a–2c depict that of corresponding to overstability convection. In Fig. 1a, we notice that the parameter d has strong effect on critical Rayleigh number. It can be seen that the critical Rayleigh number goes high for large value of parameter d for stationary convection and consequently making the system more stable. On the other hand, we observe from Fig. 2a that the effect of corresponding values of parameter d on critical Rayleigh number for overstationary convection is comparatively low than that for stationary convection. Thus, we can conclude that this parameter d has more stabilizing effect in stationary convection as compared to that of in overstationary convection. It can be noticed from Figs. 1b and 2b that the effect of magnetic parameter (H) on critical Rayleigh number in stationary and overstationary cases is almost similar and that it increases with increase of (H). The effect of hall current parameter (M) on critical Rayleigh number, in case of stationary motion, can be clearly seen through Fig. 1c, which is similar but comparatively smaller than that of magnetic parameter (H). This means that with increase of hall current and magnetic field, the system becomes more stable. Significant as well as interesting effect of hall current parameter (M) on critical Rayleigh number is noticed in case of overstationary
Rayleigh Number (R)
200
100
0 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 2c. Effect of Hall current (M) on Rayleigh Number (R) versus wavenumber (k).
40000
Rayleigh Number (R)
30000
20000
10000
0
-10000 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 3a. Effect of micropolar parameter (A) on Rayleigh number (R) versus wavenumber (k).
N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519
517
convection depicted in Fig. 2c. It is found that in the range 0 < k < 6, the critical Rayleigh number increases for the values of parameter (M) ranging between 0 and 3, after which the critical Rayleigh number decreases and may take negative values for large values of M. Moreover, beyond the above specified range of wavenumber k, the values of Rayleigh number increases with increase of (M). This means that the effect of hall current parameter M is significant enough in the range 0 < k < 6 and makes the system stable when 0 < M < 3 and unstable when M > 3. Perhaps the reason for this is that the general character of parameter M is to make the system unstable, which becomes dominant in the presence of other parameters only when M takes values greater than 3. From Fig. 3a, we see that the Rayleigh number is positive for smaller values of parameter (A) namely at A = 0.1 and 1.0, but when A = 2.0, the curve corresponding to Rayleigh number exhibits two branches. This is not surprising and similar phenomena has been encountered earlier by Datta and Sastry [13] in the relevant problem. Datta and Sastry [13] investigated the thermal instability of a horizontal layer of micropolar fluid heated from below. They found that the plot of Rayleigh number versus wavenumber has two branches separating the zones of stability and observed that instability is due to the existence of negative values of Rayleigh number. This behavior may be called oscillatory convection. As the parameter (A) takes values 3 and higher the Rayleigh number becomes negative, which indicates the instability in the system. A similar behavior of parameter d is noticed on Rayleigh number in stationary convection and depicted in Fig. 3b. Fig. 3c indicates the effect of micropolar parameter (K) in case of oscillatory convection. We see that as the parameter K takes higher values, the positive branch of the curve corresponding to Rayleigh number goes up and negative branch of the curve goes down. For sufficiently 40000
Rayleigh Number (R)
30000
20000
10000
0
-10000 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 3b. Effect of d on Rayleigh Number (R) versus wavenumber (k). (A=.
70000 60000
Rayleigh Number (R)
50000 40000 30000 20000 10000 0 -10000 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 3c. Effect of micropolar parameter (K) on Rayleigh number (R) versus wavenumber (k).
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N. Rani, S.K. Tomar / Applied Mathematical Modelling 34 (2010) 508–519 80000 70000
Rayleigh Number (R)
60000 50000 40000 30000 20000 10000 0 -10000 0
1
2
3
4
5
6
7
8
9
10
Wavenumber (k) Fig. 3d. Effect of Hall parameter (M) on Rayleigh number (R) versus wavenumber (k).
higher values of parameter (K), the negative branch of the curve becomes asymptotic to the value zero. Fig. 3d indicates the behavior of oscillatory convection at different values of hall current parameter (M). We note that the Rayleigh number remains positive for the values of parameter M ranging between 1 < M < 4, while at M = 4 and above, the curve bifurcates into positive and negative branches. It is found that the position of discontinuity of the branches is an increasing function of hall current parameter M. 7. Conclusion Thermal convection of a micropolar fluid layer heated from below in the presence of hall currents has been analyzed. The effects of micropolar and hall current parameters on critical Rayleigh number have been studied. It is concluded that (i) The critical Rayleigh number of overstability convection is always found to be less than the critical Rayleigh number for stationary convection. (ii) After crossing certain values of different parameters, oscillatory convection is observed. (iii) Increase of micropolar parameter ð dÞ and magnetic parameter (H) results in increase of critical Rayleigh number for stationary as well as overstationary convection. (iv) For stationary convection, an increase in hall current parameter (M) results in increase of critical Rayleigh number, however for overstationary convection, the critical Rayleigh number first increases with increase of hall current parameter (M) up to certain limit, and then decreases with further increase of parameter (M). (v) The parameters ð dÞ, (K) and (M) are found to be responsible for setting instability, beyond certain fixed values of them.
Acknowledgement One of the authors (NR) is thankful to University Grants Commission, New Delhi for providing financial assistance through Scheme No. F.6-2(25)/2008(MRP/NRCB). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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