View of ANALYTICAL STUDY ON THERMOHALINE CONVECTIVE INSTABILITY IN A MICROPOLAR FERROFLUID

ANALYTICAL STUDY ON THERMOHALINE CONVECTIVE INSTABILITY IN A

MICROPOLAR FERROFLUID

Nirmala P Ratchagar a_{, Seyalmurugan}

A,a_{Department of Mathematics, Annamalai University, Chidambaram, Tamilnadu - 608 002, India}

b_{Department of Mathematics, Jayagovind Harigopal Agarwal Agarsen College, Madhavaram, Chennai, Tamilnadu - 600 060,} India

Article History: Received: 11 January 2021; Accepted: 27 February 2021; Published online: 5 April 2021 Abstract: The present investigation is on linear analytical study of thermohaline convective instability in micropolar ferrofluid using perturbation technique. The fluid layer is heated from below and salted from above. The theory of linear stability is used to reduce the non-linear effects on governing equations and normal mode analysis is taken to study. The critical magnetic thermal Rayleigh number NSC is obtained for sufficient large value of M1 and an oscillatory instability is determined. The parameters N1 _and N5'_{are analyzed for stabilizing behavior and} N3'_,  _{and M3 give the destabilizing behavior. The results} are depicted graphically.

Key words:

Thermohaline convection, Micropolar ferrofluid, Salinity Rayleigh number, Perturbation technique, linear stability analysis

1. Introduction

Ferrofluids are colloidal suspension of fine magnetic mono domain nano-particles in non - conducting liquids. Such types of ferrofluids have wonderful applications in science and technology. Generally, the ferrofluids are used for cancer treatment in the biomedicine field. An excellent introduction and reviews of this extremely interesting monograph has been given by Rosensweig [1]. In his monograph, the fascinating information is introduced on magnetization. The convection in ferromagnetic fluid is analyzed in various aspects by Chandrasekhar [2]. Finlayson [3] has been investigated the convection in ferrofluid in single component fluid with uniform magnetic fluid. This investigation is extended to porous medium by Vaidyanathan et al. [4]. In non-presence of buoyancy effects, the thermoconvective instability in ferrofluid is given by Lalas and Carmi [5]. The micropolar fluids respond to spin inertia and micro-rotational motions. It can support couple stress and distributed body couples. Eringen [6] introduced the micropolar fluids theory. This theory has been developed by Eringen [7] on thermal effect. An excellent reviews and applications of this fluids theory can be obtained in by Ariman et al. [8] and Eringen [9]. Later, Ahmadi [10] employed firstly the energy method on convective instability of micropolar fluid with use of stability analysis. Pérez-Garcia and Rubi [11] analyzed the micropolar fluids with the effects of overstable motions. Narasimma Murty [12] examined the instability of the Bénard convection in a micropolar fluid using linear stability analysis.

In the effect of porous media, the double-diffusive convection is of greatest interest in mechanical and chemical engineering. In some special case, sodium chloride and temperature field are involving and this is often called as thermohaline convection. Thermohaline convection in a ferrofluid has been analyzed by Vaidyanathan et al. [13] with two-component fluid. The presence of porous medium on ferrothermohaline convection has been given by Vaidyanathan et al. [14].

The theoretical investigation of a micropolar ferromagnetic two component fluid in non-presence of Darcy porous effect has been undertaken by Sunil et al. [15]. The Soret effect is investigated on two component ferrofluid by Vaidyanathan et al. [16] and this is continued to large and small porous effect by Sekar et al. [17, 18]. Reena and Rana [19] have been analyzed the thermosolutal convective instability of micropolar rotating fluids in a porous effect. They used the Darcy model. Chand [20] studied the porous effect on triple-diffusive convective instability in micropolar ferromagnetic fluid.

In present investigation, our intension is to consider salinity gradient on magnetization and magnetic potential equation and thermal convection problem in micropolar fluid of Eringen extend to the thermohaline convection in micropolar ferrofluid. Also, an effect of salinity gradient and how micropolar parameters affect the stability in micropolar ferromagnetic fluid heated from below and salted from above. The stationary and oscillatory instabilities are studied.

Fig.1 Geometrical configuration

Here we consider, an infinite horizontal micropolar ferrofluid layer heated from below and salted from above. The fluid layer is of thickness d and the fluid is considered as an electrically non-conducting incompressible one. The temperature and salinity at the bottom and top surfaces z d/ 2 are

T

0



(

DT

) / 2

_and

S

0

(

DS

) / 2,

respectively and



t

( |



dT dz

/

|)

_and



s

( |



dS dz

/

|)

_{are maintained. The governing equations are}

The continuity equation is

. 0

 q 1

The momentum and internal angular momentum equations are





 



 



2 0 . p . 2 t



_   _   



  



 

 

    q q g HB ω q (2)











 

 



2 0I . 2 2 0 ' ' . ' t



_   _ 



  



 

 

    



  q ω q ω M H ω ω (3)

The temperature equation is





2 0 , 0 1 , , . . . v H o s s v H v H DT T D C C T K T T T Dt t T Dt        _{ }  _{   }           _ _{ }  _{ } _ _{ } _     M M H H ω (4)

The mass flux equation is





2

0 / t . S KS S



   q   (5)

We can assume the magnetization using Maxwell’s equation for non-conducting fluids [16-18] is = M H T S( , , ) /H.

M H _{The linearized magnetic equation in term of} H₀, T_{a and} S_{a is}

0

0 (H ) ( a) 2( a)

M M  H K TT K SS (6)

The density equation of state is 0[1 t(T Ta) S(S Sa)]

      (7)

where q- velocity of fluid,



0 _{- mean density of the clean fluid, p- pressure,}



_{- density of the fluid, g -}

gravitational field, H-magnetic field, B-magnetic induction,



-coupling viscosity,



-microrotation,



-shear viscosity coefficient, I-moment of inertia, M-magnetization,



'

-bulk spin viscosity,



'

-shear spin viscosity,

,

v H

C

-effective heat capacity at constant volume,

C

s_{-specific heat solid material,}



0_{-viscosity of the fluid}

when the applied magnetic field is absent,

K

1_{-thermal diffusivity, T-temperature,}



_{-micropolar heat}

conduction coefficient, S-solute concentration,

K

s_{-concentration diffusivity,}

H

0_{-uniform magnetic field,}

T

a -average temperature,

S

a_{-average salinity,}



t_{-thermal expansion coefficient and}



s_{-analogous solvent} coefficient.

2 0 0 2 0 0 0 0 0 0 ( ) , ( ) [1 ], (0, 0, 0), 1 1 ( ) , (0, 0, 0), 1 1 , , ( ) and . S b S S b S b b ext b b S b t t t t t K z K z z M z z z K z K z z H T T z S S z p p z M H H   _{     }           _   _            _   _              M k q q H k ω ω (8)

where subscript b − the basic state and ˆk− unit vector vertical direction.

A small thermal disturbance is made on the system. Let us take the perturbed components of M and H be 1

'

2

'

0 3

'

[

M M

,

,

M z

( )



M

]

_and

[

H H

1

'

,

2

'

,

H z

0

( )



H

3

'

],

_{respectively. The perturbed quantities are}

( )

',

',

( )

',

(z)

',

( )

',

',

',

(z)

,

b b b b b b b b

z

p

p z

p

S

S

S

z

 



T

T































H

H

H

ω ω

ω

M

M

M

q = q + q

(9)

the superscript ' denotes perturbed state.

The perturbed density equation can be calculated as

0

' ( t sS')

     (10)

1.

normal mode analysis method

We undertake the perturbation quantities by use of normal modes are

( , , , ) ( , ) exp[ ] ( , , , ) ( , ) exp[ ] ( , , , ) ( , ) exp[ ] ( , , , ) ( , ) exp[ ] x y x y x y x y w x y z t w z t i k x i k y x y z t z t i k x i k y x y z t z t i k x i k y S x y z t S z t i k x i k y             (11)

In Eq. (2), one can get the kth _{component is}





2 2 2 2 0 2 0 2 2 2 0 2 2 0 ₂ 0 0 0 0 0 0 2 0 0 2 2 2 2 2 2 2 ' 0 2 0 0 0 0 0 2 0 3 2 ' 1 1 1 ' 2 1 t t S t S S t S K KK KK k w K k k k S K k k t z z z K k S g k g k S k z z                     _{       }               _  _  _ _       _{  } _{          }         _ _    _  _     _ _    2 2 0 k w             _    (12)

Internal angular Eq. (3) can be manipulated as 2 2 3 2 2 0I 2 ₂ k0 w 2 3 ' ₂ k0 3 t z z



_  



_  _   



_   _          ' ' ' ₍₁₃₎

Eq. (4) can be calculated as

2 2 2 2 0 0 0 2 0 1 0 0 1 0 2 3 1 1 t S t t K T K K T C K T K k C w t t z z                        _   _  _{      }       _{   }              ' (14)

The Salinity equation is



2 2 2



0

( / t S) _SwK_S ( / z )k S' (15)

Using Vaidyanathan et al. [14], one gets

2 2 0 0 2 2 0 (1 ) 1 M k K K S 0 H z z z            _{ }       _{ } (16)





2 2 1/ 2 1/ 2 1/ 2 1 1 1 5 1/ 2 1/ 2 1/ 2 4 4 1 5 5 2 2 * 2 2 2 1 3 ( ) * * (1 ) * * * (1 ) * * * 2 ( ) ( ) * S S D a w a M R D a M R T a R M M D t M a R M S a R M M T a R S M N D a D a w        _  _{   }  _    _               _  ' (17)









'* 3 2 2 '* 2 2 ' * 3 1 3 3 ' 2 * 2 I D a w N D a N t  _{ }   _    _     ' ₍₁₈₎ ' 2 2 1/ 2 1/ 2 * 2 2 2 5 5 3 * ( *) ( ) * (1 ) * * * r r T P M P D D a T a M M M R w aN R t t     _ _{      }  _{ }    ' ' (19) 2 2 1/ 2 5 6 * ( ) * *, * r S M S P D a S aR w t  M    _{  }    _{ } (20) 1/2 2 2 5 3 * * * * 0, S M R D M a DT DS R







     _{ }    (21)

where the dimensionless quantities are

1/ 2 1/ 2 1 1 0 2 2 0 , 0 , 1/ 2 2 2 3 '* 3 0 0 3 1 2 1 0 , 0 0 0 , 3 (1 ) * , * , * , * , * , , , * * , , , , , (1 ) (1 ) 1 v H t v H t S S t v H S t v H K aR K aR wd t z w t T z a k d D C d d z d C K d K aR K K T S S d M M N C d g C M M   _{  }                                 _ _ _ _            _ _             0 0 0 2 2 ' 4 5 6 0 , 3 2 0 1 1 4 4 0 , 0 , ' ' 5 2 2 2 1 1 1 1 2 / ' , , , , , , (1 ) (1 ) , ' , , , , s s S S v H s t v H s s v H t t r r S S H K K K K M M M C N g K K K d C gd C gd I N I P C P C R R K K K K C d d    _{ }                _  _                        _{  }          (22)

2.

linear stability analysis

The stationary and oscillatory instabilities have been studied using linear theory. The boundary conditions are

2 *

3

* * * * * 0

w D w D



S   T 

at z* 1 / 2. (23)

The exact solutions satisfying above Eq. (23) are

* * *

1 2 3

* 4 * *

4 3 5

*

cos

*,

*

cos

*,

*

cos

*

*

cos

*,

*

sin

*,

cos

*

t t t t t

w

X e

z

T

X e

z

S

X e

z

X

D

X e

z

z

X e

z

    



































 

_

(24)

where

X

1

,

X

2

,

X

3

,

X

4 _and

X

5 _{are constants. Eqs. (17)–(21) can be mathematical manipulated using Eq. (23)} as





2 2 2 2 1/ 2 1 1 5 1 2 1 1/ 2 1/ 2 2 2 4 4 5 3 5 1 4 1 5 2 ( ) (1 )( ) 1 (1 ) (1 ) _S (1 ) 2( ) 0, a N a X aR M M X a M M M R X a M R M X a N X          _    _    _          _{ } (25) 2 2 2 2 1 1 1 3 5 2(



a N X) 4N (



a N) I'



X 0,   _   ' _  (26) 1/ 2 2 2 1/ 2 2 2 5 1 2 2 4 5 5 (1 ) ( _r ) _r 0, a M M M R X  P  a X PM X aN R' X  (27) 1/ 2 2 2 6 S 1 ( ) r 3 0, aM R X _



a

 

 P X_  (28) 1/ 2 2 1/ 2 2 1 1/ 2 2 2 2 ( 5/ 6 ) 3 ( 3) 5 0, S S R  X R  M M X R  a M X      (29)

To evaluate the Eigen function, determination of the co-efficient of X1, X2, X3, X4_and X5 _{in Eqs.} (25)–(29) is equal to zero. Using the analyses Vaidyanathan et al. [13, 14], Eqs. (25)–(29) have been adopted to get 4 3 2 1 2 3 4 5 0 T



T



T



T



T  (30) where 2 1 1 2 1 1 ' ' ' ' ' ( ' ), ' (1 ) ( ') r r r r r T P Ix T P I x P P x P I 2 2 2 2 2 2 3 4 8 6 1 8 2 1 1 6 7 2 6 4 9 3 8 ' '2 ' ' ' ' ' ' (1 )( ') '( ) ' ( ) r r r r r r r T a x P I R x x P x P P I x P I a x x x x x P a M P I R x x x x



          2 2 2 2 2 2 4 1 4 4 7 6 5 8 6 2 5 1 6 1 6 8 2 2 3 2 2 2 1 1 7 8 7 1 6 3 8 6 4 9 ' ' ' ' ' ' ' ' ' ( ) ( ) ( ')( ) (1 ) ( ')( ) r r r r r r T a x x I R a P x R x x N x x P a x N R x x x x x P P I a x R x x x x P P x x I a M x x a M x x R

 







               2 2 2 2 2 3 5 1 4 7 4 6 5 8 6 2 1 6 7 2 1 2 7 6 1 4 9 1 3 9 '

(

(

)

(

(

)

(

)))

(

_s

)

T

x

a x x R

a x x N R

x x a x R

x x

x a x R

x

a x M

x x x R

x x x R

 

















2 2 1 2 4 3 4 4 5 4 1 5 2 2 2 5 1 6 1 5 7 1 1 3 8 3 9 5 6 '

,

1

,

1

(

/

),

(1

),

2

,

,

4

,

,

/

x

a

x

x

x

M

M

M

x

M

M

x

N

x

x x

x

N

x N

x

a M

x

M

M











 

 





















4.1 The case of stationary instability

For steady state, we have σ = 0 at the marginal stability. Then the Eq. (30) leads to get Eigen value Rsc for which solution exists. Using the analyses [14]-[15], the critical magnetic Rayleigh number Rsc has been obtained using

sc Nr R Dr  (31) where

















2 2 3 2 2 2 2 1 2 2 ' 1 1 3 1 1 4 4 5 1 3 6 ( ) 4 ( ) 1 4 (1 ) 4 ( ) _s Nr  a N   a N' N  N a M M M N   a N M R























2 2 2 ' 1 5 1 3 1 5 2 2 2 2 1 2 2 1 5 1 3 5 1 5 2 2 3 1 (1 ) 4 ( ) 2 (1 ) 4 ( ) 1 2 ( ) ( ) Dr a M M N a N N N a M M N a N M N N a a M





_

_

_



               ' ' '

When M1 is very large, one can gets Nsc (= M1Rsc ).

sc Nr N Dr  (32) where



















2 2 2 ' 5 1 3 1 5 2 2 2 2 1 2 2 5 1 3 5 1 5 2 2 3 (1 ) 4 ( ) 2 (1 ) 4 ( ) 1 2 ( ) ( ) Dr a M N a N N N a M N a N M N N a a M





_

_

_



              ' ' '

Here a is denoted as critical wave number ac. Analysis of the classical results is given below: Assuming

'

3 1

1, N 1,N 0,

   _and N₅' 0 _{in Eq. (32), one get}









2 2 3 2 1 1 4 4 5 6 2 2 1 2 2 5 5 3

(

)

(1

)

(1

) 1

1

/ (

)

sc s

a

a

M

M M

M

R

N

a

M

M

a M











  



















(33)

which is an expression for Nsc of Vaidyanathan et al. [13]. Moreover, if

1

4, 6, , s 0

M M  R  _{in Eq. (33), it gives N}

Takingi1 (10) _{in Eq. (30), it leads to Roc has been derived using}



6 4 2



1 2 1 ( 1 5 2 3) 1 ( 2( 2 6) 3 4) 1 5( 2 6) / oc R  T Y  T Y T Y   Y Y Y Y Y  Y Y Y Dr where 2 2 2 1 6 4 9 3 8 4 2 8 2 2 3 2 2 1 8 6 1 8 1 8 7 2 2 2 2 2 2 2 3 1 4 4 7 6 5 3 6 2 5 1 6 8 2 2 6 4 9 7 1 4 ' ' ' ' ' ' ' '2 ' ' ' ' ' ' ' '( ) ' ', (1 )( ') ' , ' ( ) ( ') ( '), r r r r r r r r r r r r r Y a M P I x x x x a x P I a x x P I Y x P P I x x P x x P I P x x x Y a I x x a P x x x N a x x P x N x x x P a x P I a M x x P x x I Y                         





2 3 2 8 6 1 1 7 8 7 1 6 3 8 2 2 2 2 2 2 5 1 4 7 4 6 5 2 6 8 5 2 6 7 8 6 1 4 7 9 4 2 2 2 6 1 7 8 6 7 8 1 6 8 2 2 2 2 2 2 2 1 5 1 3 1 2 2 1 3 ' ' ' ' ' ' ' ( ') (1 ) ( ') , ( ) , , , ( 4 ) / 2 r r r r s r x x P P I x x x x P P x x I a M x x R Y x a x x a x x N a x x x N a x x x x a M x x x x Y x x x a M P x x x x x Dr Y Y Y A A A A                                1 1 2 2 1 3 2 2 5 4 2 3 4 5 3 2 6 , , , and ( ) A A T Y T Y A T Y Y Y A Y Y Y Y Y

3.

Discussion of Results

In this investigation, thermohaline convection in micropolar ferrofluid layer is studied. The fluid layer is heated from below and salted from above and the convective system is subjected to a transverse uniform magnetic field. The thermal perturbations and linear stability analysis are used in the study. Here we consider the free boundary conditions. The magnetic numbers M1 and M2 are considered the values 1000 and 0, respectively. M3 is taken from 5 to 25 (Vaidyanthan et al. [14]) and



ranges from 0.05 to 0.11 (Vaidyanthan et al. [14]) and Rs taken from -500 to 500. The magnetization parameters M4 and M6 are taken to be 0.1 and M5 = 0.5 (Vaidyanthan et al. [16]). Further, N1,

' 3

N _and ' 5

N _{are taken to be non-negative values which is presented by}

Eringen [21] and he assumed the clausius-Duhem inequality. Pr is taken as 0.01.

The variation of Nsc with the coupling parameter N1 is depicted in Fig. 2 (a) and (b). It is observed from the Fig. 2 (a) that the convective system gives stabilizing behavior, when increasing values of M3 and RS. Due to the increasing value of N1 from 0 to 1 and RS from -500 to 500 on the system, Nsc gets the highest values and the system has more stabilizing effect. But, an increasing of M3 from 5 to 25, the system shows the stabilizing effect and it is less pronounced. M3 analyzed for destabilizing behavior always [13-14, 16-17]. But, introducing of N1 on M3, the system gets stabilizing effect. Fig. 2 (b) represents the plot of Nsc versus N1 for different



. This figure shows that N1 has the stabilizing behavior for increasing value of



.

This is because, the greater the mass and heat transports and more buoyancy energy, which contributes to thermal instability. Also, it is shown from the Fig. 2 (c) that the increase in N1 stabilizes the system for increasing of



and RS. Also, in the presence of RS = 500, ac is close to zero. In this moment, the system has an equilibrium state.

Figs. 3 (a) and (b) display the variation of Nsc versus spin diffusion parameter

N

3

'

for increasing of RS and M3 and , respectively. In Fig. 3 (a), we observe that Nsc decreases with increasing of

N

3

',

which leads to destabilize the system. Moreover, when RS = 500, Nsc gets zero value. Therefore, the system has a null effect. From Fig. 3 (b), it is seen that as

N

3

'

_{increases from 2 to 8, there is a decrease in Nsc indicating destabilization} for different



. Fig. 3 (c) shows the variation of ac versus

N

3

'

for various , RS and M3. When



increases from 0.05 to 0.1, RS increases from – 500 to 500 and M3 increases from 5 to 25, there is a decrease in ac. It is clear that there is a destabilization on the system which is not much pronounced and when RS = 500, there is an oscillation in ac.

Figs. 4 (a) and (b) show the variation of Nsc versus micropolar heat conduction parameter

N

5

'

for different M3,



and. It is clear from the Fig. 4 (a) that

N

5

'

leads to an increase in Nsc. Therefore,

N

5

'

has a stabilizing effect. It is very clear from the Fig. 4 (b) that increase in

N

5

',

_{it is stabilizing behavior for various RS.} Fig. 4 (c) represents the critical wave number ac versus the

N

5

'

for various physical parameter



, RS and M3. In this figure

N

5

'

_{shows a stabilizing behavior. In such situation also the system ha no effect when RS = 500.}

The increase in non-buoyancy magnetization parameter M3 is obtained to cause large destabilization, because both thermal and magnetic mechanisms favor destabilization. This can be studied from Figs. 5 (a) and (b) in which the increase in M3 and , decrease in Nsc and ac, respectively.

From Fig. 6 (a), it is seen that an increase in RS, decrease in Nsc. An increase of RS would means that the system is salted from above. Also, when RS = 500 and



(=0.05, 0.07, 0.09), the Nsc gets small values. But for the value



= 0.11, suddenly Nsc gets highest value. In this moment, the convective system gets stabilizing effect. Fig. 6 (b) shows the variation of ac versus RS for different



. When RS increases from -500 to 500, there is a decrease in ac promoting instability. When the highest value of RS (=500) the system tends to the same effect. That is, the system converges to the small values. But, when the highest values of



(=0.11), the system has an equilibrium position.

4.

Conclusion

In the present analysis, the results of a theoretical study on thermohaline convection in a micropolar ferrofluid are considered with free boundary conditions. We conclude that the effect of non-buoyancy magnetization M3, salinity effect RS, spin diffusion parameter

N

3

'

have destabilizing behavior and the effect of coupling parameter N1 and the micropolar heat conduction parameter

N

5

'

have a stabilizing effect due to the microrotation on the onset of convection.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 10000 20000 30000 40000 50000 60000 RS500 RS100 RS RS00 RS500 M 3 = 5 M₃ = 10 M 3 = 15 M₃ = 20 M 3 = 25 NSC N1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 5000 10000 15000 20000 25000 30000 35000 40000     NSC N₁

Fig. 2 (a). Variation of Nsc versus N1 for different M3 and RS, N3'2,N5'0.2 and 0.05

Fig. 2(b). Variation of Nsc versus N1 for different ,  N3'2,N5'0.2, _{M3 = 5 and RS = -500.} 0.2 0.3 0.4 0.5 0.6 0.7 0.8 5 6 7 8 9 10 11     R_S500 R_S00 R_S R_S00 R_S 500 ac N 1

2 3 4 5 6 7 8 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 NC N'₃ M3 = 5 M3 = 10 M3 = 15 M3 = 20 M3 = 25 RS500 RS100 RS RS00 R S500 2 3 4 5 6 7 8 6000 8000 10000 12000 14000 16000 18000     NSC N'3

Fig. 3 (a). Variation of Nsc versus N3' for different M3 and RS, N10.2,N5'0.2 and  0.05.

Fig. 3 (b). Variation of Nsc versus N3' for different ,  N10.2,N5'0.2, _{M3 = 5 and RS = -500.} 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9     N'₃ a c RS500 RS100 R_S R_S00 R_S500 M₃ = 5 M₃ = 10 M₃ = 15 M₃ = 20 M₃ = 25

Fig. 3 (c) – Variation of ac versus N3' for different RS, M3 and , N10.2,N5'0.2 and M3 = 5.

0.0 0.2 0.4 0.6 0.8 1.0 0 20000 40000     M₃ = 5 M₃ = 10 M 3 = 15 M₃ = 20 M₃ = 25 NSC N' 5 0.0 0.2 0.4 0.6 0.8 1.0 20000 40000 RS = -500 R_S = -100 R_S = 0 R_S = 100 RS = 500 N' 5 NSC 0 2 4 6 8 10

Fig. 4 (a). Variation of Nsc versusN5'for different M3 and , N10.2,N3'2 _{and RS = -500.}

Fig. 4 (b). Variation of Nsc versus N5' for different RS, 0.05, N10.2,N3'2, M3 = 5 and RS = -500.

0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 M 3 = 5 M 3 = 10 M 3 = 15 M3 = 20 M3 = 25 R S500 R S100 R S R S00 R S500     ac N' 3

Fig. 4 (c). Variation of ac versus N5' for different RS, M3 and , N10.2,N5'0.2and M3 = 5.

5 10 15 20 25 2000 4000 6000 8000 10000 12000 14000 16000 18000     NSC M₃ 0 2 4 6 8 10 5 10 15 20 25 4 5 6 7 8 9 ac M 3    

Fig. 5 (a). Variation of Nsc versus M3 for different ,

1 0.2, 3' 2,

N  N  N₅'0.2

and RS = -500.

Fig. 5 (b). Variation of ac versus M3 for different ,

1 0.2, 3' 2, 5' 0.2 N  N  N  and RS = -500. 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 R_S = -500 R_S = -100 R_S = 0 R_S = 100 R_S = 500 NSC  -600 -400 -200 0 200 400 600 0 2 4 6 8 10     a c R_S

Fig. 6 (a). Variation of Nsc versus for different RS,

1 0.2, 3' 2,

N  N  N₅'0.2

and M3 = 5.

Fig. 6 (b). Variation of ac versus RS for different

,

ACKNOWLEDGEMNT

The author S. Seyalmurugan is grateful to Dr. M. Mohana Krishnnan, Principal, Jayagovind Harigopal Agarwal Agarsen College, Madhavaram, Chennai for his constant encouragement and thank to Dr.A. R. Vijayalakshmi, Department of Applied Mathematics, Sri Venkateswara College of Engineering, Sriperambudur, Chennai, for her support.

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