J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012)
Upper Limits to the Complex Growth Rate in Couple-stress Fluid in the Presence of Magnetic Field AJAIB S. BANYAL1 and MONIKA KHANNA2 1
Department of Mathematics, Govt. College Nadaun, Dist. Hamirpur-177033, H. P., India. 2 Department of Mathematics, Govt. College Dehri, Dist. Kangra-176022, H. P., India. . (Received on : April 2, 2012) ABSTRACT The thermal instability of a couple-stress fluid acted upon by uniform vertical magnetic field and heated from below is investigated. Following the linearized stability theory and normal mode analysis, the paper through mathematical analysis of the governing equations of couple-stress fluid convection with a uniform vertical magnetic field, for the case of free and perfectly conducting boundaries shows that the complex growth rate σ of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle
R 2 σ 〈 1 p1π 2 3π 2 F + p 2 in the right half of a complex number,
2
,
σ -plane, where R is the Rayleigh
p1 is the thermal Prandtl number, p 2 is the magnetic
Prandtl number and F is the couple-stress parameter, which prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in the couple-stress fluid heated from below in the presence of uniform vertical magnetic field. The result is important since the exact solutions of the problem investigated in closed form, are not obtainable. Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
238
Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012) Keywords: Thermal convection; Couple-Stress Fluid; Magnetic field; Complex growth rate; Chandrasekhar number. MSC 2000 No.: 76A05, 76E06, 76E15; 76E07; 76U05.
1. INTRODUCTION The thermal instability of a fluid layer with maintained adverse temperature gradient by heating the underside plays an important role in Geophysics, interiors of the Earth, Oceanography and Atmospheric Physics etc. A detailed account of the theoretical and experimental study of the onset of BĂŠnard Convection in Newtonian fluids, under varying assumptions of hydrodynamics and hydromagnetics, has been given by Chandrasekhar4. The use of Boussinesq approximation has been made throughout, which states that the density changes are disregarded in all other terms in the equation of motion except the external force term. Sharma et al.8 has considered the effect of suspended particles on the onset of BĂŠnard convection in hydromagnetics. The fluid has been considered to be Newtonian in all above studies. With the growing importance of non-Newtonian fluids in modern technology and industries, the investigations on such fluids are desirable. Stokes12 proposed and postulated the theory of couple-stress fluid. One of the applications of couple-stress fluid is its use to the study of the mechanism of lubrication of synovial joints, which has become the object of scientific research. According to the theory of Stokes12, couple-stresses are found to appear in noticeable magnitude in fluids having very large molecules. Since the long chain hylauronic acid molecules are found as additives in synovial fluid, Walicki
and Walicka13 modeled synovial fluid as couple-stress fluid in human joints. Sharma and Thakur9 have studied the thermal convection in couple-stress fluid in porous medium in hydromagnetics. An electrically conducting couple-stress fluid heated from below in porous medium in the presence of uniform horizontal magnetic field has been studied by Sharma and Sharma10. Sharma and Sharma11 and Kumar and Kumar5 have studied the effect of dust particles, magnetic field and rotation on couple-stress fluid heated from below and for the case of stationary convection, found that dust particles have destabilizing effect on the system, where as the rotation is found to have stabilizing effect on the system, however couple-stress and magnetic field are found to have both stabilizing and destabilizing effects under certain conditions. Banerjee et al.2 gave a new scheme for combining the governing equations of thermohaline convection which is shown to lead to bounds for the complex growth rate of the arbitrary oscillatory perturbations, neutral or unstable for all combinations of dynamically rigid or free boundaries. However no such result existed for nonNewtonian fluid configurations, in general and for couple-stress fluid configurations, in particular. Banyal3 have characterized the non-oscillatory motions in couple-stress fluid. Keeping in mind the importance of nonNewtonian fluids, the present paper is an
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
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Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012)
attempt to prescribe the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude, in a layer of incompressible couple-stress fluid heated from below in the presence of uniform vertical magnetic field opposite to force field of gravity, when the bounding surfaces of infinite horizontal extension, at the top and bottom of the fluid are free and the region outside the fluid is perfectly conducting. The result is important since the exact solutions of the problem investigated in closed form, are not obtainable. 2. FORMULATION OF THE PROBLEM AND PERTURBATION EQUATIONS Considered an infinite, horizontal, incompressible couple-stress fluid layer, of thickness d, heated from below so that, the temperature and density at the bottom surface z = 0 are T0 , ρ 0 respectively and at the upper surface z = d are Td , ρ d and that a uniform adverse temperature gradient
β =
dT dz
is maintained. The fluid is acted
→
∇. q = 0 , →
→ dH → → 2 = H .∇ . q + η∇ H dt
→
→
denote respectively the density, pressure, temperature and velocity of the fluid. Then the momentum balance, mass balance equations of the couple-stress fluid (Stokes12, Chandrasekhar4 and Scanlon and Segel6) are →
→ → → δρ ∂ q 1 + q .∇ q = − ∇ p + g 1 + ρ ρ 0 ∂t 0 → → → µ e µ' + ν − ∇ 2 ∇ 2 q + ∇ × H × H ρ 4 πρ 0 0
(3)
→
∇. H = 0 .
(4)
The equation of state
ρ = ρ 0 [1 − α (T − T0 )],
(5)
Where the suffix zero refer to the values at the reference level z = 0. Here η stands for electrical resistivity. Let cv denote the heat capacity of the fluid at constant volume. Assuming that the particles and the fluid are in thermal equilibrium, the equation of heat conduction gives → ∂ → + q .∇ T = q ∇ 2T , ∂t
ρ 0 cv Or
∂T → + q .∇ T = κ∇ 2T , ∂t
(6)
The kinematic viscosityν , couple-
stress viscosity µ , thermal diffusivity κ , and coefficient of thermal expansion α are all assumed to be constants. The basic motionless solution is '
→
, (1)
,
And
upon by a uniform vertical magnetic field
H (0,0, H ) . Let ρ , p, T and q (u, v, w)
(2)
→
q = (0,0,0 ) , T = T0 − βz , H = (0,0, H )
and ρ = ρ 0 (1 + αβz ) .
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Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012)
Assume small perturbations around basic solution and let δρ , δp , θ ,
the
→
→
q (u, v, w) ,
(
and h = hx , h y , hz
)
denote
respectively the perturbations in density, pressure p, temperature T, couple-stress fluid velocity (0,0,0) and magnetic field respectively. The change in density δρ caused mainly by the perturbation θ in temperature is given by
δρ = −αρ 0θ .
(8)
Then the linearized hydromagnetic perturbation equations are of the couplestress fluid becomes →
→ µ ' 2 ∇ δp − g αθ + ν − ∇ ρ0 ∂t ρ0 → → → ∇ 2 q + ∇ × h × H ∂ q
=−
,
(9)
→
∇. q = 0 ,
(10)
∂θ = β w + κ∇ 2θ , ∂t
(11)
q κ= ρ 0 cv
(12)
(13)
(15)
∂2 ∂2 ∂2 ∇ = 2 + 2 + 2 ∂x ∂y ∂z 2
where
.
Analyzing the disturbances into normal modes, we assume that the Perturbation quantities are of the form
[w,θ , hz ] = [W (z ), Θ(z ), K (z )]
(
)
Exp ik x x + ik y y + nt ,
(16)
Where k x , k y are the wave numbers along
(
x
and 2
),
1 2 2
y-directions
respectively
is the resultant wave
number and n is the growth rate which is, in general, a complex constant. Using (16), equations (14), (15) and (11), on using (10) and (13), in non-dimensional form, become
(D .
,(14)
Together with (11),
k = kx + k y
→
Where
µ H ∂ − e ∇ 2 hz 4πρ ∂ z 0
∂w ∂ 2 − η ∇ h = H , z ∂t ∂z
the
→
∇. h = 0 ,
2 2 ∂ ∇ 2w − gα ∂ θ + ∂ θ ∂x 2 ∂t ∂y 2
3. NORMAL MODE ANALYSIS
1
→ dh → → = H .∇ . q + η∇ 2 h , dt
Within the framework of Boussinesq approximation, equations (9) and (12), on using equations (10) and (13), gives
2
=−
)[
(
) − (D − a )]W D (D − a )K
−a2 σ + F D 2 − a2 gα d 2a 2Θ
ν
µ Hd + e 4πρ 0ν
2
2
2
2
2
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Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012)
(D
2
(D
2
)
Hd DW , − a 2 − p 2σ K = − η
)
− a 2 − p1σ Θ = −
(18)
D=
nd 2
ν
, p1 =
βd 2 W, κ
(19)
ν ν µ' , , p2 = , F = κ η ρ 0 d 2ν
d and D⊕ = dD and dropping (⊕ ) dz
for convenience. Here
p1 =
thermal prandtl number,
ν , is the κ ν p 2 = , is η
magnetic prandtl number and F is the couple-stress parameter. Applying the transformations, W = W⊕ ,
βd 2 Θ= Θ ⊕ and κ
Hd K = K ⊕ η
in
equations (17), (18) and (19) and dropping (⊕ ) for convenience, in non-dimensional form becomes, D 2
2 − a 2 σ + F D 2 − a 2 − D 2 − a 2
W = − Ra 2 Θ + QD D 2 − a 2 K
(D
2
(D
2
)
− a − p 2σ K = − DW , 2
)
− a 2 − p1σ Θ = −W ,
Where R =
gαβd 4
κν
,
the thermal
Q=
Where a = kd , σ =
is
, (20)
(22)
Rayleigh
number
and
2
is
the
Chandrasekhar
number. Since both the boundaries are free and the region outside the fluid is perfectly conducting and are maintained at constant temperature, the perturbations in the temperature are zero at the boundaries. The appropriate boundary conditions with respect to which equations (20), (21) and (22) must be solved are W = D 2W = 0, Θ = 0 and K = 0 at z = 0 and z = 1.
(23)
Equations (20)-(22), along with boundary conditions (23), poses an eigenvalue problem for σ and we wish to Characterize σ i when σ r ≥ 0 . We prove the following theorem: Theorem: If R 〉 0 , F 〉 0, Q 〉 0 σ r ≥ 0 and
σ i ≠ 0 then the necessary condition for the
existence of non-trivial solution (W , Θ, K ) of equations (20), (21) and (22) together with boundary conditions (23) is that
σ〈 (21)
µe H d , 4πρ 0νη 2
241
R . 1 p1π 2 3π 2 F + p 2
Proof: Multiplying equation (20) by W ∗ (the complex conjugate of W) throughout and integrating the resulting equation over the vertical range of z, we get
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Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012)
σ ∫ W ∗ (D 2 − a 2 )Wdz +F ∫ W ∗ (D 2 − a 2 ) Wdz − ∫ W ∗ (D 2 −a 2 ) Wdz = − Ra 2 ∫ W ∗ Θdz + Q ∫ W ∗ D(D 2 − a 2 )Kdz , 1
0
1
3
0
2
0
1
1
0
0
(24) Taking complex conjugate on both sides of equation (22), we get
(D
2
− a 2 − p1σ ∗ )Θ∗ = −W ∗ ,
(25)
Therefore, using (25), we get 1
(
1
)
∗ 2 2 ∗ ∗ ∫ W Θdz = − ∫ Θ D − a − p1σ Θ dz , 0
(26)
0
Also taking complex conjugate on both sides of equation (21), we get
[D
2
]
− a 2 − p 2σ ∗ K ∗ = − DW ∗ ,
(27)
Therefore, using (27), we get
(
1
)
1
∗ 2 2 ∫ W D D − a Kdz = − ∫ DW 0
(D
∗
2
)
∫ K (D 1
− a 2 Kdz =
0
2
)(
− a 2 D 2 − a 2 − p 2σ
∗
)K
∗
dz , (28)
0
Substituting (26) and (28) in the right hand side of equation (24), we get
σ ∫ W (D − a )Wdz +F ∫ W (D − a 1
1
∗
2
∗
2
0
2
) Wdz − ∫ W (D 1
2 3
0
(
1
∗
2
)
2
−a 2 Wdz
0
)
(
1
)(
)
= Ra 2 ∫ Θ D 2 − a 2 − p1σ ∗ Θ ∗ dz + Q ∫ K D 2 − a 2 D 2 − a 2 − p 2σ ∗ K ∗ dz , 0
(29)
0
Integrating the terms on both sides of equation (29) for an appropriate number of times by making use of the appropriate boundary conditions (23), along with (21), we get 1
1
0
2 2 2 2 2 2 σ ∫ DW + a2 W dz +F∫ D3W + 3a2 D2W + 3a4 DW + a6 W dz + ∫ D2W + 2a2 DW + a4 W dz 1
0
1
{
2
2
0
2
}
= Ra 2 ∫ DΘ + a 2 Θ + p1σ ∗ Θ dz 2
2
2
0
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243
Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012) 1
− Q ∫ D 2 K 0
2
2
+ 2 a 2 DK
2
+ a4 K
dz − Qp σ 2
∗
∫ ( DK 1
2
+ a2 K
2
)dz ,
(30)
0
And equating the real and imaginary parts on both sides of equation (30), and cancelling σ i (≠ 0) throughout from imaginary part, we get 1
0
2
2
1
0
σ r ∫ DW + a 2 W dz +F ∫ D3W
{
1
2
+ 3a 2 D 2W
2
+ 3a 4 DW
2
1 2 + a 6 W dz + ∫ D 2W 0
2
+ 2a 2 DW
2
2 + a 4 W dz
}
= Ra 2 ∫ DΘ + a 2 Θ dz 2
2
0
(
)
(
)
1 1 1 1 2 2 2 2 2 2 2 2 − Q∫ D 2 K + 2a 2 DK + a 4 K dz − Qp2σ r ∫ DK + a 2 K dz + σ r Ra 2 p1 ∫ Θ dz − Qp2 ∫ DK + a 2 K dz 0 0 0 0
(31)
and
∫ {DW 1
2
+ a2 W
2
}dz = −Ra
1 2
1
(
p1 ∫ Θ dz + Qp 2 ∫ DK + a 2 K
0
2
0
2
2
)dz ,
(32)
0
Equation (32) implies that, 1
1
(
Ra 2 p1 ∫ Θ dz − Qp 2 ∫ DK + a 2 K 2
0
2
2
)dz ,
(33)
0
is negative definite and also, 1
{
Q ∫ DK + a 2 K 2
2
}dz ≥ p1
1
∫ DW
2
dz ,
34)
2 0
0
We first note that since W and Z satisfy W (0) = 0 = W (1) and K (0) = 0 = K (1) in addition to satisfying to governing equations and hence we have from the Rayleigh-Ritz inequality7 1
∫ 0
1
DW dz ≥ π 2 ∫ W dz , 2
2
0
Further, for W (0) = 0 = W (1) , Banerjee et al.1 have shown that Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
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Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012)
1
∫DW 2
2
dz ≥ π
1 2
0
2
∫ DW
dz ,
(36)
0
Further, multiplying equation (22) and its complex conjugate (25), and integrating by parts each term on right hand side of the resulting equation for an appropriate number of times and making use of boundary conditions on Θ namely Θ(0) = 0 = (1) along with (22), we get
∫ (D 1
2
)
1
2
(
)
− a 2 Θ dz + 2 p1σ r ∫ DΘ + a 2 Θ dz + p1 σ
0
2
2
2
1 2
0
1
∫ Θ dz = ∫ W dz , 2
0
2
(37)
0
since σ r ≥ 0 , σ i ≠ 0 therefore the equation (37) gives,
∫ (D 1
2
−a
2
)Θ
2
1 2
dz 〈 ∫ W dz ,
0
(38)
0
And 1
∫Θ
2
dz 〈
0
1
1 p1 σ 2
2
∫W
2
dz ,
(39)
0
It is easily seen upon using the boundary conditions (23) that
∫ ( DΘ 1
0
2
)
1 2 + a 2 Θ dz = Real part of − ∫ Θ ∗ D 2 − a 2 Θdz 0
(
1
(
)
(
)
(
)
)
≤ ∫ Θ ∗ D 2 − a 2 Θdz , 0 1
≤ ∫ Θ ∗ D 2 − a 2 Θ dz , 0 1
≤ ∫ Θ ∗ D 2 − a 2 Θ dz , 0 1
(
)
= ∫ Θ D 2 − a 2 Θ dz , 0
Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
245
Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012) 1 2
1 2
2 2 ≤ ∫ Θ dz ∫ D 2 − a 2 Θ dz , 0 0 1
1
(
)
(40)
(Utilizing Cauchy-Schwartz-inequality) Upon utilizing the inequality (38) and (39), inequality (40) gives
∫ ( DΘ 1
2
)
1
+ a 2 Θ dz ≤ 2
p1 σ
0
1
∫W
2
dz ,
(41)
0
Now R 〉 0, Q 〉0 and σ r ≥ 0 , thus upon utilizing (33) and the inequalities (34)-( (36) and (41), the equation (31) gives, π2 I 1 + a 2 3π 4 F + p2 where 1
{
I 1 = σ r ∫ DW
1 R 2 − ∫ W dz 〈0 , p σ 0 1
}dz +F ∫ {D W 1
2
+ a2 W
2
3
0
2
(42)
2
+ a6 W
0
1
}dz + ∫ D W 1
+ 3a 4 DW
2
2
0
1
(
2 2 2 Q ∫ D 2 K + a 2 DK dz + Qp 2σ r ∫ DK + a 2 K 0 0
2
2
2 2 + 2a 2 Dw + a 4 W dz +
)dz ,
is positive definite. And therefore , we must have
σ〈
R
,
(43)
1 p1π 2 3π 2 F + p 2
Hence, if
σ r ≥ 0 and σ i ≠ 0 , then σ 〈
R
.
1 p1π 2 3π 2 F + p 2
And this completes the proof of the theorem. Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)
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Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012)
4. CONCLUSIONS
5. REFERENCES
The inequality (43) for σ r ≥ 0 and
σ i ≠ 0 , can be written as
2
σr2
R 2 , + σ i 〈 1 p1π 2 3π 2 F + p 2
The essential content of the theorem, from the point of view of linear stability theory is that for the configuration of couple-stress fluid of infinite horizontal extension heated form below, having top and bottom bounding surfaces free and the region outside the fluid is perfectly conducting, in the presence of uniform vertical magnetic field parallel to the force field of gravity, the complex growth rate of an arbitrary oscillatory motions of growing amplitude, lies inside the semi-circle in the right half of the σ r σ i - plane whose centre is at the origin and radius is equal to R , 1 p1π 2 3π 2 F + p 2
where R is the thermal Rayleigh number, p1 is the thermal Prandtl number, p 2 is magnetic Prandtl number and F is couple-stress parameter. The result important since the exact solutions of problem investigated in closed form, are obtainable.
the the is the not
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Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012)
medium, Indian J. Phys, Vol. 75B, pp.59-61 (2001). 11. Sharma, R.C. and Sharma, M. Effect of suspended particles on couple-stress fluid heated from below in the presence of rotation and magnetic field, Indian J. pure. Appl. Math., Vol. 35(8), pp. 973-989 (2004).
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12. Stokes, V. K. Couple-stress in fluids, Phys. Fluids, Vol. 9, pp.1709-15 (1966). 13. Walicki, E. and Walicka, A. Inertial effect in the squeeze film of couplestress fluids in biological bearings, Int. J. Appl. Mech. Engg., Vol. 4, pp. 363-73 (1999).
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