Effect of inclined magnetic field on unsteady free ... - CiteSeerX

International Journal of Applied Mathematics and Modeling ©KINDI PUBLICATIONS www.kindipublication.com

Vol.1, No. 1, 16-33 May, 2013 ISSN: 2336-0054

Effect of inclined magnetic field on unsteady free convection flow of a dusty viscous fluid between two infinite flat plates filled by a porous medium N.Sandeep Assistant Professor, Department of mathematics, School of Engineering and Technology, Jain University, Banglore, India. Dr.V.Sugunamma Associate Professor,Department of mathematics,S.V.U College of Sciences,Tirupati,India. Abstract: In this paper we analyses the laminar convective flow of a dusty viscous fluid through a porous medium of non-conducting walls in the presence of inclined magnetic field with volume fraction, heat source and considering porous parameter. Governing equations are solved by perturbative technique and we analyze the effect of velocity of fluid and particle phase, temperature of the dusty fluid at the inclined magnetic field with various parameters as t (time), M (Magnetic parameter), Pr(Prandtl number) ,Gr (Grashof number), S(Heat source parameter), θ(Inclined magnetic field angle), ε3(Porous parameter), Φ(Volume fraction of dusty particals), p(Pressure gradient).From these we observed that increase in inclined magnetic field angle causes the decrease of velocity of the fluid and fluid temperature decreases by increase in heat source parameter.

Key Words: Convective flow, inclined angle, Porous medium, Heat source, MHD, Dusty fluid. 1.1 INTRODUCTION The concept of heat transfer of a dusty fluid has a wide range of applications in air conditioning ,refrigeration, pumps, accelerators, nuclear reactors space heating, power generation, chemical processing, , filtration and geothermal systems etc. The good example of heat transfer is the radiator in a car, in which the hot radiator fluid is cooled by the flow of air over the radiator 16


Vol.1, No. 1, 16-33 May, 2013 ISSN: 2336-0054

surface. Keeping above facts many authors attracted in this field of study of heat and mass transfer through dusty fluids. Ramamurthy [1] analyses the free convection effects on the Stoke’s problems for an infinite vertical plate in a dusty fluid and the boundary layer region for convection in a rectangular cavity was discussed by Gill [2]. Attia [3] has investigated an unsteady MHD Couette flow and heat transfer of dusty fluid with variable physical properties. Mishra et al. [4] have studied the transient conduction and radiation heat transfer with temperature dependent thermal conductivity on two dimensional flows. Chakraborty [5] studied MHD flow and heat transfer of a dusty viscoelastic stratified fluid down an inclined channel in porous medium under variable viscosity. Nag and Jana [6] have discussed unsteady couette flow of a dusty gas between two infinite parallel plates, when one plate is kept fixed and the other plate moves in its own plane by using Laplace Transform technique. Here they found that the dust velocity in the case of accelerated start of the plate is less than the fluid velocity. Ghosh [7] studied the hydromagnetic flow of a dusty viscoelastic Maxwell fluid through a rectangular channel for an arbitrary pressure gradient. Unsteady hydromagnetic flow and heat transfer from a non isothermal stretching sheet immersed in a porous medium was discussed by Chamkha [8].Viscous heating of high Prandtl number fluids with temperature dependent viscosity. Eckert et al. [9] studied the effect of radiation on temperature and velocity in electrodynamics froth flow process. They concluded that the velocity and temperature increases as the radiation parameter increases. They carried out a similar calculation for the simpler case of a homogeneous flow without porous medium. The convective magnetohydrodynamic two phase flow and heat transfer of a fluid in an inclined channel was investigated by Malashetty et al. [10]. Attia [11] has investigated an unsteady MHD Couette flow and heat transfer of dusty fluid with variable physical properties. Keeping above facts and applications, in the present paper we discuss the laminar convective flow of a dusty viscous fluid through a porous medium of non-conducting walls in the presence of inclined magnetic field with volume fraction, heat source and considering porous parameter. Here we analyze the effect of velocity, velocity of the particle phase , temperature of the dusty fluid at the inclined magnetic field with various parameters as t (time), M (Magnetic parameter), Pr(Prandtl number) ,Gr(Grashof number), S(Heat source parameter), θ(Inclined magnetic field angle), ε3(Porous parameter), Φ(Volume fraction of dusty particals), p(Pressure gradient).

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1.2 MATHEMATICAL FORMULATION In Cartesian co-ordinate system, we consider unsteady laminar flow of a dusty, incompressible, Newtonian, electrically conducting, viscous fluid through a porous medium of uniform cross section h, when one wall of the channel is fixed and the other is oscillating with time about a constant non-zero mean. Initially at t ≤0, the channel wall as well as the fluid is assumed to be at the same temperature T0 . When t>0, the temperature of the channel wall is instantaneously raised to

Tw

which oscillate with time and is thereafter maintained constant. Let x-axis be along

the fluid flow at the fixed wall and y-axis perpendicular to it. An inclined magnetic field is applied to the flow along y direction with the heat source.

1.2.1 Assumptions: The governing equations are written based on the following assumptions: The dust particles are solid, spherical, non-conducting, and equal in size and uniformly distributed in the flow region. (i) The density of dust particles is constant and the temperature between the particles is uniform throughout the motion. (ii) The interaction between the particles, chemical reaction between the particles and liquid has not been considered to avoid multiple equations. (iii) The volume occupied by the particles per unit volume of the mixture, (i.e., volume fraction of dust particles) and mass concentration have been taken into consideration. . (iv) The dust concentration is so small so that it is not disturbing the continuity and hydro magnetic effects. This means that the continuity equation is satisfied.

1.2.2 Governing equations of the flow The fluid flow is governed by the momentum and energy equation under the above assumptions is

 u  1 p  2u 1    v 2  g   T  T0    t   x y  1 

 KN 0  KN 0c2 H 02  v  u  u  uSin 2     K1    

(1)

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 K v   p  2u       g   T  T0     v  u   2 t N 0 m  x y  m

(2)

T   2T Q0   T  T0  t  C p y 2  C p

(3)

The boundary conditions to the problem are:

t  0; u  y, t   v  y, t   0,

T  y, t   0 for 0  y  1

t  0; u  y, t   v  y, t   0,

T  y, t   0 at y  0

u  y, t   v  y, t   1   eint

T  y, t   1   eint at y  1

Where m

u  y, t  is the velocity of the fluid and v  y, t  is velocity of the dust particles,

is the mass of each dust particle,

temperature ,

(4)

N0

is the number density of the dust particle ,

T0 is the initial temperature, Tw

T is the

is the raised temperature , φ is the volume fraction

of the dust particle f is mass concentration of dust particle,  volumetric coefficient of the 

thermal expansion, K1 is the porous parameter , K is the stoke’s resistance coefficient,  is the electrical conductivity of the fluid, c is the magnetic permeability, magnetic field induction,

Q0 is

heat source, H 0 is the

C p is the specific heat at constant pressure,

 is the thermal

conductivity. The problem is simplified by writing the equations in the following non dimensional. Here the characteristic length is taken to be h and the characteristic velocity is v.

x * y * h2 p * vt * uh * T  T0 x  , y  p ,  2 , t  2 , u  ,T  h h v h v Tw  T0 *

And v* 

vh



(5)

Substituting the above non-dimensional parameters equation (6.5) in the governing equations (13) (after removing asterisks), we get

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u  p  2u    Gr T  1 v   u t  x  y2 Where

f

(6)

  1   2 M   3  constant

 p  2u  v      2  GrT     u  v  t  x y 

(7)

T 1  2T   ST t Pr y 2

(8)

Where

Gr 

1 

g   Tw  T0  h3

2

(Grashof number)

f m 1  ,1  2 ,  2  , M '  c2 h2 H 02 1 1    Kh 1 

M  M ' Sin2 (Magnetic parameter), f 

3 

Pr 

 h2

K1 1   

(Porous Parameter),  

mN 0



(Mass concentration of dust particles),

f (Concentration resistance ratio), 1

c p Q h2 (Prandtl number), S  0 (Heat source parameter)  vC p

The corresponding non-dimensional boundary conditions are:

t  0; u  y, t   v  y, t   0,

T  y, t   0 for

t  0; u  y, t   v  y, t   0,

T  y, t   0 at

u  y, t   v  y, t   1   eint ,

0  y 1 y  0.

(9)

T  y, t   1   eint , at y  1

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1.3 SOLUTION OF THE PROBLEM To solve the equations (6-8) we use the below equations introduced by Soundalgekar and Bhat equations when

  1 .

u ( y, t )  u0 ( y )   u1  y  eint v  y, t   v0  y    v1  y  eint

(10)

T  y, t   T0  y    T1  y  eint After substituting equation (10) in equations (6-8), we can write

u0"  y   1   2 M   3  u0  y   1 0  y   P  GrT0  y 

(11)

 v0  y    u0  y    u0"  y   P  GrT0  y 

(12)

T0"  y   Pr ST0 ( y)  0

(13)

u1"  y   1   2 M   3  in  u1  y   1v1  y   Gr  y  T1  y 

   nif  v1  y    u1  y    u1''  y   GrT1  y  T1"  y   (in  S ) prT1  y   0

(14) (15) (16)

The corresponding boundary conditions becomes

u0  y   u1  y   v0  y   v1  y   0, T0  y   T1  y   0 at y = o u0  y   u1  y   v0  y   v1  y   1, T0  y   T1  y   1 at y = 1

(17)

On solving equation (13) with the help of boundary conditions (17), we get

T0  y  

SinhL1 y SinhL1

(18)

Substituting equation (18) in equations (11) and (12) 21


u0"  y   1   2 M   3  u0  y   1v0  y   P  Gr 

 v0  y    u0  y    u0"  y   P  Gr 

SinhL1 y SinhL1

SinhL1 y   SinhL1 

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(19)

(20)

Substituting equation (20) in equation (19), we obtain

u0"  y   A2u0  y   P  Gr

Where A2 

SinhL1 y SinhL1

(21)

  2 M   3      1 

By solving equation (21) with the boundary conditions (17), we get

u0  y  

sinh L1 y p Ay p   sinh Ay e  1  1  A3  2  e A  1   A3 2  A A sinh L1   sinh A

The first and second order partial derivatives of

u0'  y  

(22)

( ) are

L cosh L1 y p Ay  p  A cosh Ay e   1  A3  2  e A  1   A3 1  A A sinh L1   sinh A

2 L 2 sinh L1 y p   A sinh Ay u0''  y   P  e Ay   1  A3  2  e A  1   A3 1 A sinh L1   sinh A

(23)

(24)

Substituting equations (22) and (24) in equation (12) we obtain

v0  y  

Where

A1 p Ay sinh L1 y p   sinh Ay e  1  A1 1  A3  2  e A  1   A1 A3 2  A A sinh L1   sinh A

(25)

Gr    A1  1  A2  A3  2 L1  A2   ,

By solving equation (16) with the boundary conditions (17), we get

22


T1  y  

Vol.1, No. 1, 16-33 May, 2013 ISSN: 2336-0054

sinh L0 y sinh L0

(26)

Substituting equation (26) in equations (14) and (15) we obtain

u1"  y   1   2 M   3  in  u1  y   1 1  y   Gr 

   nif  v1  y    u1  y    u1''  y   Gr 

sinh L0 y sinh L0

(27)

sinh L0 y   sinh L0 

(28)

Substituting equation (28) in equation (27) we obtain

u1''  y   B 2u1  y   Gr

sinh L0 y sinh L0

(29)

On solving equation (29), with the boundary conditions (17), we get

u1  y   1  B2 

sinh L0 y sinh By  B2 sinh B sinh L0

The first and second order partial derivatives

u1'  y   1  B2 

(30)

( ) are given by

L cosh L0 y B cosh By  B2 0 sinh B sinh L0

(31)

L20 sinh L0 y B 2 sinh By u  y   1  B2   B2 sinh B sinh L0 '' 1

(32)

Substituting equations (31) and (32) in equation (28), we obtain

 sinh L0 y  sinh By v1  y   A2 B0 1  B2   B2  sinh B sinh L0  

Gr    A2  1  B 2  , B2  2 L0  B 2   

and

(33)

B0 

    nif   2  n2 f 2

Substituting the equations (22) and (30) in the equation (10), we obtain 23


u  y, t  

Vol.1, No. 1, 16-33 May, 2013 ISSN: 2336-0054

sinh L1 y p Ay p   sinh Ay e  1  1  A3  2  e A  1   A3 2  A A sinh L1   sinh A

 sinh L0 y  int sinh By  1  B2   B2 e sinh B sinh L0  

(34)

Substituting the equations (25) and (33) in the equation (10)

v  y, t  

A1 p Ay sinh L1 y p   sinh Ay e  1  A1 1  A3  2  e A  1   A1 A3 2  A A sinh L1   sinh A  sinh L0 y  int sinh By  A2 B0 1  B2   B2 e sinh B sinh L0  

(35)

By substituting equations (18) and (26) in equation (10) we obtain

T  y, t  

sinh L0 y int sinh L1 y  e sinh L1 sinh L0

(36)

Hence the equations (34)-(36) represent velocity of the fluid, velocity of the dust particle and temperature respectively.

1.4 RESULTS AND DISCUSSION The numerical values in the following figures were illustrated at ϕ = 0.005, n=1,   0.5 ,

1  0.1 .The numerical computations for the velocity of the fluid, velocity of the particle phase and temperature of the dusty fluid at an inclined magnetic field has been carried out for the various parameters t, M, f , Pr , Gr , S,  ,  3 ,  ,p. Fig. 1 shows the velocity profiles of the fluid and of the particle phase for different values of f. The velocity of the fluid and of the particle phase decreases as f increases. It is evident that the velocity of particle phase bit slower than the fluid phase.Fig. 2 depicts the velocity profiles of the fluid phase and of the particle phase for different values of M. It is clear that velocity decreases with increase of magnetic parameter M when Gr > 0.Fig. 3 shows the velocity profiles of the fluid phase and the particle phase for different values of  3 . The velocity of the fluid and of 24


the particle phase decreases as

3

Vol.1, No. 1, 16-33 May, 2013 ISSN: 2336-0054

increases. Fig. 4 represents the velocity profiles of the fluid

and of the particle phase for different values of  . The velocity of the fluid and of the particle phase decrease as increase in  , velocity of the both the fluid and of the particles decreases. It is interesting to note that after reaching the peak point 1the particle phase velocity increases with the increase of  . From fig. 5 we observe that there is a gradual decrease of fluid and particle velocity with the increase of time.Fig. 6 shows the velocity profiles of the fluid and the velocity profiles of the particle phase for different values of S . The velocity for both the fluid and the particle phase decreases with the increase of S . But after certain period it is reversed in both the cases of fluid and particle. Fig. 7 we observe that the increase in Gr causes gradual increase in fluid velocity and particle velocity.Fig. 8 shows the velocity profiles of the fluid and of the particle phase for different values of Pr. The velocity of both the fluid and the particle phase decreases as Pr increases. It is evident that the velocity of particle phase bit faster than fluid phase.Figs. 9 and 10 it is observed that an increase in φ causes an increase in fluid velocity and particle velocity. Figs. 11 and 12 shows the temperature distribution of the dusty fluid for different values of Pr and S. It is observed that by increase in Pr and S, the temperature of the dusty fluid decreases. But from fig 11 it is evident that temperature distribution is linear when Pr=0.71 and at S=1.Fig.13 shows the temperature distribution of the dusty fluid for different values of t. It is evident that an increase of time causes the decrease in temperature of the dusty fluid.

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1.6 fluid phase velocity_________ dust phase velocity................

1.4 1.2 1

u,v

0.8 f = 0.2,0.4,0.6,0.8

0.6 0.4 0.2 0 -0.2

0

0.1

0.2

0.3

0.4

0.5 y

0.6

0.7

0.8

0.9

1

Fig 1.Variation of velocity for different values of f. When M=2, t=0.001, Pr=0.71, S=4,

3  1 .

1.5

fluid phase velocity______ dust phase velocity..........

u,v

1

M=0.5,1.0,1.5,2.0 0.5

0

0

0.1

0.2

0.3

0.4

0.5 y

0.6

0.7

0.8

0.9

1

Fig 2. Variation of velocity for different values of M When f=0.8, t=0.001, Pr=0.71, S=4,

3  1 .

26


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1.6 1.4

fluid phase dust phase velocity.................

1.2 1 0.8

 3 =1, 3, 5,

u,v 0.6

7

0.4 0.2 0 -0.2 0

0.1

0.2

0.3

0.4

0.5 y

0.6

0.7

0.8

0.9

1

Fig 3. Variation of velocity for different values of

3

When f=0.8, t=0.001, Pr=0.71, S=4, M=2.

1.6 1.4

fluid phase velocity _________ dust phase velocity ................

1.2 1 0.8 u,v



0.6

   

, , , 6 4 3 2

0.4 0.2 0 -0.2 0

0.1

0.2

0.3

0.4

0.5 y

0.6

0.7

0.8

0.9

1

Fig 4. Variation of velocity for different values of  When Gr=5, f=0.2, t=0.001, Pr=0.71, S=4, M=4.

3  1

27


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1.6 fluid phase velocity________ dust phase velocity.............

1.4 1.2 1 0.8

u,v

t=0.001,1.0,1.5,2.0

0.6 0.4 0.2 0 -0.2

0

0.1

0.2

0.3

0.4

0.5 y

0.6

0.7

0.8

0.9

1

Fig 5. Variation of velocity for different values of When Gr=5, f=0.2, Pr=0.71 ,S=4,M=4,

t

3  1

3 2.5

fluid phase velocity________ dust phase velocity..............

2 1.5 S=1,2,4,6 u,v

1

S=1,2,4,6

0.5 0 -0.5 -1

0

0.2

0.4

0.6

0.8

1 y

1.2

1.4

1.6

1.8

2

Fig 6. Variation of velocity for different values of S When f=0.2, Pr=0.71, Gr=5, t=.001, M=1,

3  1

28


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1.8 1.6

fluid phase velocity______ dust phase velocity..........

1.4 1.2

u,v

1 0.8 0.6 0.4 0.2

Gr = 3,5,10,15

0 -0.2

0

0.1

0.2

0.3

0.4

0.5 y

0.6

0.7

0.8

0.9

1

Fig 7. Variation of velocity for different values of Gr When f=0.2, Pr=0.71, S=3, t=0.001, M=1,

3  1

1.2 fluid phase velocity_________ dust phase velocity................

1

0.8

u,v

0.6 Pr = 0.71,1.0,7.0,11.4

0.4

0.2

0

-0.2

0

0.1

0.2

0.3

0.4

0.5 y

0.6

0.7

0.8

0.9

1

Fig 8. Variation of velocity for different values of Pr When f=0.2, Gr=5, S=3, t=0.001, M=1,

3  1

29


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6 dust phase velocity......... 5

4

v

3

 =0.8, 0.6, 0.4, 0.01

2

1

0

0

0.1

0.2

0.3

0.4

0.5 y

0.6

0.7

0.8

0.9

1

Fig 9. Variation of partical velocity for different values of  When f=0.2, Pr=0.71, Gr=5, S=3, t=0.001, M=1,

3  4

1. fluid phase velocity_____ 1.

1

0. u 0.

0.

 = 0.01, 0.4, 0.6,

0.

0.8 0

0

0.

0.

0.

0.

0. y

0.

0.

0.

0.

1

Fig 10. Variation of fluid velocity for different values of  When f=0.2, Pr=0.71, Gr=5, S=3, t=0.001, M=1,

3  4

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Vol.1, No. 1, 16-33 May, 2013 ISSN: 2336-0054

1.4 Pr=0.71 Pr=1.0 Pr=7.0 Pr=11.4

1.2

1

T

0.8

0.6

0.4

0.2

0

0

0.1

0.2

0.3

0.4

0.5 y

0.6

0.7

0.8

0.9

1

Fig 11. Variation of temparature field for different values of Pr When f=0.2, Gr=5, S=3, t=0.001, M=2,

3  4

1.5 S=1 S=3 S=5 S=7

T

1

0.5

0

0

0.1

0.2

0.3

0.4

0.5 y

0.6

0.7

0.8

0.9

1

Fig 12. Variation of temparature field for different values of S When f=0.2, Gr=5, Pr=0.71, t=0.001, M=2,

3  4

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Vol.1, No. 1, 16-33 May, 2013 ISSN: 2336-0054

1.5 t=0.1 t=0.5 t=1.0 t=1.5

T

1

0.5

0

0

0.1

0.2

0.3

0.4

0.5 y

0.6

0.7

0.8

0.9

1

Fig 13. Variation of temparature field for different values of When Gr=0.5, f=0.2, Pr=0.71, S=3, M=2,

t

3  1

1.5 REFERENCES [1]. V. Ramamurthy, Free convection effects on the Stokes problem for an infinite vertical plate in a dusty fluid, J. Math. Phys. 24 (1990) 297. [2]. A.E. Gill, The boundary layer region for convection in a rectangular cavity, J. Fluid Mech. 26 (1966) 515. [3]. H.A.Attia, N.A. Kotb, MHD flow between two parallel plates with heat transfer, Acta Mech. 117 (1996) 215–220. [4]. S.C. Mishra, P. Talukdar, D. Trimis, and F. Durst, Two-dimensional transient conduction and radiation heat transfer with temperature dependent thermal conductivity, International Communications in Heat and Mass Transfer, 32 (2005) 305–314, [5]. S.Chakraborty, MHD flow and heat transfer of a dusty viscoelastic stratified fluid down an inclined channel in porous medium under variable viscosity,” Theoretical and Applied Mechanics, 26 (2001) 1–14. [6]. S.K.Nag and R.N. Jana Couette flow of a dusty Gas, Acta Mechanica, Springer verlag. 33 (1979)179-180.

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[7]. B.C. Ghosh, hydromagnetic flow of a dusty viscoelastic Maxwell fluid through a rectangular channel, Int. J. Appl.Mech. Eng. 7 (2000) 591–618. [8]. A. J. Chamkha,Unsteady hydromagnetic flow and heat transfer from a nonisothermal stretching sheet immersed in a porous medium,” International Communications in Heat and Mass Transfer, 25 (1998) 899–906. [9]. E.R.G.Eckert and M.Faghri, Viscous heating of high Prandtl number fluids with temperature-dependent viscosity, Int’l Journal of Heat and Mass Transfer 29 (1986) 1177–1183. [10]. M.S.Malashetty, J. C. Umavathi, and J. Prathap Kumar, Convective magnetohydrodynamic fluid flow and heat transfer in an inclined channel, Heat and Mass Transfer, 37 (2001) 259–264. [11]. H. A. Attia,Unsteady MHD couette flow and heat transfer of dusty fluid with variable physical properties,Applied Mathematics and Computation, 177 (2006) 308– 318.

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