Upper Limits to the Complex Growth Rate in Couple-stress Fluid in the Presence of Magnetic Field

Upper Limits to the Complex Growth Rate in Couple-stress Fluid in the Presence of Magnetic Field

AJAIB S. BANYAL¹ and MONIKA KHANNA²

1Department of Mathematics,

Govt. College Nadaun, Dist. Hamirpur-177033, H. P., India.

2Department 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

2

2 2 2 1 2

3 1



























 +

〈

F p p

R π π

σ ^,

in the right half of a complex σ -plane, where R is the Rayleigh number, p₁ is the thermal Prandtl number, p₂ 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.

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 Bénard 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 Bénard 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 non- Newtonian 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 non- Newtonian fluids, the present paper is an

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,ρ₀ respectively and at the upper surface z = d are T_d,ρ_d and that a uniform adverse temperature gradient







= dz

β dT is maintained. The fluid is acted

upon by a uniform vertical magnetic field

(

^H

)

H 0,0,

→

. Letρ, p, T and ^q→

(

^u,v,w

)

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

→

→

→

→

→

→ →

×











∇ × +

∇













∇

− +











 +

+

∇

−

 =











 ∇

∂ +

∂

H H q

g p q

t q q

e 0 2

2 0

'

0 0

4 1 1 .

πρ µ ρµ

ν

ρ δρ ρ

, (1)

0

. =

∇

^→

q

, (2)

→

→

→ →

∇ +

 

 



 ∇

= H q H

dt H

d

₂

.

. η

_{, (3)}

And

0

. =

∇ H

^→ . (4) The equation of state

( )

[

0

]

0 1− T−T

=ρ α

ρ ^{, (5)}

Where the suffix zero refer to the values at the reference level z = 0. Here η stands for electrical resistivity.

Let c_v 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

T q T t q

c_v ²

0 .  = ∇



 



 + ∇

∂

∂ ^{→ →}

ρ ^,

Or

T T

t q

T 2

.  = ∇



 



 ∇

∂ +

∂ ^→ κ ^{, (6)}

The kinematic viscosityν^{, couple-}

stress viscosityµ^', thermal diffusivityκ ^, and coefficient of thermal expansionα ^are

all assumed to be constants.

The basic motionless solution is

(

^0,0,0

)

→ =

q , T =T₀ −βz^{, H→ =}

(

^0,0,H

)

and ^{ρ =ρ}0

(

^1+αβz

)

. (7)

240 Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012) Assume small perturbations around

the basic solution and letδρ^,δp^,θ^,

(

^{u v w}

)

q , ,

→

, and^→h=

(

hx,hy,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

θ αρ

δρ =− ₀ . (8)

Then the linearized hydromagnetic perturbation equations are of the couple- stress fluid becomes

→

→

→

→ →

×











∇× +

∇











 − ∇

+

−

∇

−

∂ =

∂

H h q

g t p

q

2

2 0

'

0

1

ρ ν µ αθ ρ δ

, (9)

0

. =

∇

^→

q

, (10)

θ κ θ = β + ∇

2

∂

∂ w

t

^{, (11)}

→

→

→ →

∇ +

 

 



 ∇

= H q h

dt h

d

₂

.

. η

_{, (12)}

0

. =

∇

^→

h

, (13)

Where

c

v

q ρ

0

κ =

.

Within the framework of Boussinesq approximation, equations (9) and (12), on using equations (10) and (13), gives

















∂ ∇

− ∂













∂ + ∂

∂

− ∂

∂ ∇

∂

z

e h

z H y

x g w t

2 2 0

2 2 2 2

4µπρ θ

α θ ,(14)

h

z

t  

 

∇

∂ −

∂ η

2

z H w

∂

= ∂

, (15)

Together with (11),

where ₂

2 2

2 2

2 2

z y

x ∂

+ ∂

∂ + ∂

∂

= ∂

∇

.

3. NORMAL MODE ANALYSIS 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

(

^ikxx+ikyy+nt

)

, (16) Where k ,_x k_y are the wave numbers along the x and y-directions respectively

(

^{kx2 ky2}

)

²¹

k = + , 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 a}

)

^K

Hd D a

d g

W a D a D F a

D

e 2 2

0 2

2

2 2 2

2 2 2

2

4 −

Θ +

−

=

−

−

− +

−

ν πρ µ ν

α

σ (17)

(

^{D a p}

)

^{K Hd}^DW







−

=

−

− ² ₂σ η

2 , (18)

(

^{D a p}1^σ

)

^β_κ^d2W

2

2 − − Θ=− , (19)

Where

ν ρµ η

ν κ

ν

σ ν ₂

0 ' 2

1 2

, ,

,

, nd p p F d

kd

a= = = = = ,

dz

D= d andD_⊕ =dD and dropping

( )

⊕ for convenience. Here ₁ ,

κ

=ν

p is the

thermal prandtl number, ₂ , η

=ν

p is

magnetic prandtl number and F is the couple-stress parameter.

Applying the transformations,W =W_⊕, Θ⊕

= Θ κ

β^d2

and  _⊕







=Hd K

K η ⁱⁿ

equations (17), (18) and (19) and dropping

( )

⊕ for convenience, in non-dimensional form becomes,

K a D QD Ra W

a D a

D F a

D



 



 −

+ Θ

−

=











 



 



 −

−



 



 −

+



 



 −

2 2 2

2 2 2

2 2 2

2 σ

, (20)

(

^{D2 −a2 − p2σ}

)

^{K=−DW , (21)}

(

^{D −a − p}1^σ

)

^{Θ = −W}

2

2 , (22)

Where

κν αβ^d4 R= g ^,

is the thermal Rayleigh number and

νη πρ µ

0 2 2

4 d

Q= ^eH ^{, 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 = D²W = 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 σr ≥0 and

≠0

σi 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







 +

〈

2 2 2 1

3 1 F p p

R π π

σ .

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

242 Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012)

( ) ∫ ( ) ∫ ( ) ∫ ∫ ( )

∫

^{∗ − + ∗ − − ∗ − =− ∗Θ + 1 ∗ −}

0

2 2 1

0 2 2

2 2 1

0 1

0

2 3 2 0

2

2 a Wdz F W D a Wdz W D a Wdz Ra W dz Q W DD a Kdz

D

σ W ,

(24)

Taking complex conjugate on both sides of equation (22), we get

(

^{D2 −a2 − p1σ∗}

)

^{Θ∗ =−W∗,} (25)

Therefore, using (25), we get

( )

∫

^{∗Θ =−}

∫

^{Θ − − ∗ Θ∗}

1

0

1

0

1 2

2 a p dz

D dz

W σ , (26)

Also taking complex conjugate on both sides of equation (21), we get

[

^{D −a − p}2^σ∗

]

^{K∗=−DW∗}

2

2 , (27)

Therefore, using (27), we get

( ) ( ) ( )( )

∫

^{1 ∗ − = −}

∫

^{∗ − =}

∫

^{− − − ∗ ∗}

0

1

0

1

0

2 2 2 2 2 2

2 2

2 a Kdz DW D a Kdz K D a D a p K dz

D D

W σ , (28)

Substituting (26) and (28) in the right hand side of equation (24), we get

(

^{D a}

)

^{Wdz F W}

(

^{D a}

)

^{Wdz W}

(

^{D a}

)

^Wdz

W ^{2 2 2}

1

0 1

0

2 3 2 1

0

2

2 − +

∫

− −

∫

−

∫

^{∗ ∗ ∗}

σ

( ) ∫ ( )( )

∫

^{Θ − − ∗Θ∗ + − − − ∗ ∗}

= ¹

0

2 2 2 2 2 1

0

1 2 2

2 D a p dz Q K D a D a p K dz

Ra σ σ , (29)

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

0

4 2 2 2

2 2 1

0

6 2 4 2

2 2 2 2

1 3 0

2 2

2 a W dz F DW 3a DW 3a DW a W dz DW 2a DW a W dz

σ DW

{

^{D a p}

}

^dz

Ra

∫

^{Θ + Θ + Θ}

= ^{1 ∗}

0

2 1 2 2

2 2 σ

( )

∫

^_^{ + + }_^{ −}

∫

⁺

− ^∗

1

0

1

0

2 2 2 2

4 2 2 2

2 2

2a DK a K dz Qp DK a K dz

K D

Q σ , (30)

And equating the real and imaginary parts on both sides of equation (30), and cancelling )

(≠0

σi throughout from imaginary part, we get

∫ 









 + +

+

∫ 









 + + +

∫ +







 + ¹

0

4 2 2 2

2 2 1

0

6 2 4 2

2 2 2 2

1 3 0

2 2

2 a W dz F DW 3a D W 3a DW a W dz D W 2a DW a W dz

r DW σ

{

^{D a}

}

^dz

Ra

∫

^{Θ + Θ}

=

1

0

2 2 2 2

( ) ( )

^



 



 Θ − +

+ +

−



 



 + +

− ∫ ∫ ∫¹ ∫

0

1

0

2 2 2 2 2 1 2 1

0

1

0

2 2 2 2

4 2 2 2

2 2

2a DK a K dz Qp DK a K dz Ra p dz Qp DK a K dz

K D

Q σ_r σ_r

(31)

and

{ } ∫ ∫ ( )

∫

^{+ =− 1 Θ + +}

0

1

0

2 2 2 2

2 1

0

1 2 2

2 2

dz K a DK Qp dz p

Ra dz W a

DW , (32)

Equation (32) implies that,

( )

∫

^{1 Θ −}

∫

⁺

0

1

0

2 2 2 2

2 1

2p dz Qp DK a K dz

Ra , (33)

is negative definite and also,

{ }

∫

^{+ ≥}

∫

1

0

1

0 2

2 2 2

2 1

dz p DW

dz K a DK

Q , 34)

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 2 2 1

0

2dz W dz

DW π , (35)

Further, for W(0)=0=W(1), Banerjee et al.¹ have shown that

244 Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012)

∫

∫

^≥

1

0 2 2 1

0 2 2

dz DW dz

W

D π ^, (36)

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

( ) ^∫ ( ) ^{∫ ∫}

∫

^{− Θ + Θ + Θ + 1 Θ =}

0

1

0 2 2

2 2 1 1

0

2 2 2 1

1

0

2 2

2 a dz 2p D a dz p dz W dz

D σ_r σ , (37)

since σr ≥0,σ_i ≠0 therefore the equation (37) gives,

( ) ∫

∫

^{− Θ 〈}

1

0 2 1

0

2 2

2 a dz W dz

D , (38)

And

∫

∫

^{Θ 〈 1}

0 2 2 2

1 1

0

2 1

dz W p

dz σ ^, (39)

It is easily seen upon using the boundary conditions (23) that

(

^{Θ + Θ}

)

⁼

∫

¹

0

2 2

2 a dz

D Real part of

( )







−

∫

¹Θ^∗ − Θ

0

2

2 a dz

D

( )

∫

^{Θ − Θ}

≤ ^{1 ∗}

0

2

2 a dz

D ,

(

^{D a}

)

^dz

∫

^{Θ − Θ}

≤^{1 ∗}

0

2

2 ,

(

^{D a}

)

^dz

∫

^{Θ − Θ}

≤^{1 ∗}

0

2

2 ,

(

^{D a}

)

^dz

∫

^{Θ − Θ}

= ¹

0

2

2 ,

( )

²

1 1

0

2 2 2 2

1 1

0 2







 − Θ







 Θ

≤

∫

^dz

∫

^{D a dz ,} (40)

(Utilizing Cauchy-Schwartz-inequality) Upon utilizing the inequality (38) and (39), inequality (40) gives

( ) ∫

∫

^{Θ + Θ ≤}

1

0 2

1 1

0

2 2

2 1

dz p W

dz a

D σ , (41)

Now R 〉 0, Q〉0 and σ_r ≥0, thus upon utilizing (33) and the inequalities (34)-( (36) and (41), the equation (31) gives,

0 3

1

0 2

1 2 2 4 2

1 〈











 −



 



 +

+^{a F} _{p p}^R

∫

^{W dz}

I π π σ , (42)

where

{ } { }

 ^ ⁺



 + +

+ +

+ +

+

=

∫ ∫ ∫

1

0

4 2 2 2

2 2 1

0

6 2 4 2

3 2 1

0

2 2 2

1 DW a W dz F DW 3a DW a W dz D W 2a Dw a W dz

I σr

( )

∫

∫

^_^{ + }_^{ + +}

1

0

2 2 2 2

1

0

2 2 2 2

dz K a DK Qp

dz DK a K D

Q σ_r ,

is positive definite.

And therefore , we must have



 



 +

〈

2 2 2 1

3 1 F p p

R π π

σ , (43)

Hence, if

≥0

σr ^and σ_i ≠⁰, then







 +

〈

2 2 2 1

3 1 F p p

R π π

σ .

And this completes the proof of the theorem.

246 Ajaib S. Banyal, et al., J. Comp. & Math. Sci. Vol.3 (2), 237-247 (2012) 4. CONCLUSIONS

The inequality (43) for σ_r ≥0 ^and

≠0

σi , can be written as

2

2 2 2 1 2 2

3 1



























 +

+ 〈

F p p

R

i r

π π σ

σ ,

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_rσi - plane whose centre is at the origin and radius is equal to



























 +

2 2 2 1

3 1 F p p

R π π

,

where R is the thermal Rayleigh number, p₁ is the thermal Prandtl number, p₂ is the magnetic Prandtl number and F is the couple-stress parameter. The result is important since the exact solutions of the problem investigated in closed form, are not obtainable.

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