Cmjv01i03p0274

J. Comp. & Math. Sci. Vol. 1(3), 274-277 (2010).

Flow of viscous conducting fluid through triangular tube MADAN LAL and SARVESH KUMAR Department of Applied Maths, MJP Rohilkhand University, Bareilly 243006 (India) ABSTRACT MHD flow of a conducting viscous incompressible fluid through porous media in an equilateral triangular tube has been discussed by using trilinear co-ordinate system.

Key Words : MHD Flow, Viscous Fluid, Porous Media, Triangular Tube.

1. INTRODUCTION Flow problems through porous media were originally based upon Darey’s experimental law which was later generalized by Brinkman 2 .

Yamammoto and Iwamuro6 expressed the equations of flow through a highly porous medium under the influence of temperature difference. Raptis and Perdikis5 studied the free convective flow through a highly porous medium bounded by an infinite vertical porous plate with constant suction. The propagation of thermoconvective waves (TCW) in a binary mixture was investigated by Bhattacharyya and Gupta 1. Jana and Das 4, Ghosh and Sengupta 3 have discussed different problems of interaction between electromagnetic field and the flow of conducting fluids.

In this paper, hydromagnetic flow of a viscous conducting incompressible fluid has been discussed through porous media in an equilateral triangular tube. The problem has been made to solve by using the trilinear co-ordinate system neglecting the effect due to induced magnetic field.

In the last section, it has been shown that the flow becomes steady after a long time if the pressure gradient is transient in nature and flow also dies out in above conditions.

2. Mathematical Analysis The flow is considered laminar along the axis of the tube which is taken along x-axis and assumed following nomenclature.

 B

Magnetic field vector

Journal of Computer and Mathematical Sciences Vol. 1 Issue 3, 30 April, 2010 Pages (274-402)

Madan Lal et al., J.Comp.&Math.Sci. Vol.1(3), 274-277 (2010)

 E  q

Velocity of the fluid

p= = µ= k= =

Pressure Density of the fluid Coefficient of viscosity Permeability of the medium Electrical conductivity

 J

Electric field vector

(    )u 

    B     ,   , B  B y z k  

 B and  

In view of above notations and neglecting the effects due to perturbation of the field as well as induced magnetic field, the basic equations of motion for the flow of conducting viscous incompressible fluid in a porous medium, are

   q 1      J B   p  .q  q (2.1) t  k k   (2.2) div q  0

If the field vectors are considered as:

(2.3)

and



1 p  f ( t)  C  x

(2.4)

In view of above relations, equation (2.1) generates the following two equations

C (   )u1   

 

The initial and boundary conditions are considered as below:

q

u1  0 on T: boundary of the tube (2.8) u2  0 u2 = 0

for t  0

(2.9)

Let O be the centre of equilateral triangular tube PQR which is assumed to be at origin. The y and z axis are perpendicular and parallel to the side QR. If  i (i = 1, 2, 3) denotes the perpendicular from any point within the triangle on the sides QR, RP and PQ respectively. Let ‘a’ be the length of each side of the triangle whose inradius is r. i.e.

  r  y

where C is constant, also u(x, y, z) = u1 + u2 (2.5)

2

(2.7)

where

Current density

 q  { u( x, y, z),0,0}  E  (0,0,0)  B  (0, B2 , B3 )

u  f (t) t

275

(2.6)

y  z   y    r   z     r 

having  i  a

   

(2.10)

(2.11)

Using above trilinear co-ordinate

Journal of Computer and Mathematical Sciences Vol. 1 Issue 3, 30 April, 2010 Pages (274-402)

276

Madan Lal et al., J.Comp.&Math.Sci. Vol.1(3), 274-277 (2010)

system, we have

      i  i 

 

  i 



 j 

i j

  i  j (2. 12)

With the help of above transformation, equations (2.6) and (2.7) takes the form 

 i 





 u    i  i 



 j 



 u  u i  j

i j

C    where u   ,

(2. 13)

  i  ,

i  ,, (2.14)

and 

 i 

  u      i  i 



 j 

  u  u  i  j

n   wherein      In the consequences of equation (2.11), we have

FG H

C C    

where u2  0 ,

 i  0,

i  12 , ,3

(2.16) Solution of equation (2.13) satisfying equation (2.14) can be taken as 



u  



n 

i 

A n sin

n  i 

(2.17)

Therefore, equation (2.13) becomes 



  n 

i 

A n sin

n  i C   

(2.18)





i

(2.19)

i 

sin e series, above equation becomes

C C       n 



 i 

sin

 n  i 

(2. 20)

Equating equations (2.18) and (2.19),

An 

C   

(2.21)

Therefore, the solution of equation (2.13) becomes

u  (2. 15)



Expres sin g (  0  2 i ); i  1,2,3 as fourier

i j

u    f (t) t

IJ K b   g

 C C        i    i 

C n    n 



 i 

 n i sin n 

(2.22)

Similarly, the solution of equation (2.15) satisfying condition given in equation (2.16) can be assumed as 



u  



n 

i 

Bn ( t )sin

n i 

(2. 23)

Using equation (2.23) in equation (2.15), following differential equation has obtained

B'n ( t )   Bn ( t ) 

 f ( t) n

(2.24)

Taking Laplace Transform regarding ‘s’ as parameter

Journal of Computer and Mathematical Sciences Vol. 1 Issue 3, 30 April, 2010 Pages (274-402)

Madan Lal et al., J.Comp.&Math.Sci. Vol.1(3), 274-277 (2010)

~ B n (s) 

2. If the flow is unsteady :

2f (s) n(   s)

(2.25) In this case the pressure gradient is assumed to be transient in nature,

Using Inverse Laplace Transform

 Bn ( t )  n

z

t

 ( t  u)

e





 i 

 n

z

t



f ( u)du

(2. 26)

e

 n

z

t



e

   q   n



i 

 n i  sin  n  n

ni f ( u)du sin 





0

e( t  u) eudu

et  et (1  )



 i 

FG H

IJ K

 et  et n i sin n (  )   (3.2)

OP (2.28) Q

1. If the flow is steady: In the case of steady flow f(t) becomes independent of time and is assumed to be zero, therefore equation (2.28) reduces to 

 i 

 n i sin n 

CONCLUSIONS If the transient pressure gradient is small then the flow becomes steady after a long time. It is also noticed that the flow ultimately dies out for a very large value of t.

i 

3. DISCUSSION

 C  q   n



z

t

(2.27)



 ( t  u )

0

e( u  ) f ( u)du 

f ( u)du sin

Putting the values of u1 from equation (2.22) and u2 from equation (2.27) in equation (2.5),the solution of equation (2.1) becomes

LM N

z

t

C 

Therefore, equation (6.5) becomes ( t  u)

n i 

  C  q    n

f ( t )  et ,

(i.e.)

Using above value of Bn(t) in equation (2.23), the solution of equation (2.15) has obtained,

  u    n

277

(3.1)

REFERENCES 1. S.N. Bhattacharyya and A.S. Gupta, Physics Fluids, 20, 3215 (1985).

2. H.C. Brinkman, Appld. Sc. Res. A., 1, 17 (1947).

3. S. S. Ghosh and P. R. Sengupta, Jour. Sci. Res. 10, 117 (1988).

4. S. K. Jana and B. Das, Ind. Jour.

Theo. Phys. 2, 33 (1985).

5. A.A. Raptis and C.P. Perdikis, Int. J. Engng. Sci. 23, 51 (1985).

6. K. Yamammoto and N. Iwamuro, J. Engng. Maths, 10, 41 (1976).

Journal of Computer and Mathematical Sciences Vol. 1 Issue 3, 30 April, 2010 Pages (274-402)