Cmjv01i02p0177 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol. 1(2), 177-188 (2010).

Unsteady pulsatile hydro magentic poiseuille flow of a couple stress fluid through a porous medium with periodic body acceleration Dr. S.V. SUNEETHA1, Dr. M. VEERA KRISHNA 1 and Prof. R. SIVA PRASAD2 1

Department of Mathematics, Rayalaseema University, KURNOOL (A.P) - 518002 (INDIA) (veerakrishna_maths@yahoo.com; svsmaths@gmail.com) 2 Department of Mathematics, Sri Krishnadevaraya University, Anantapur (A.P) - 515002 (INDIA) (resprasad_racharla@yahoo.co.in) ABSTRACT In this paper, we discuss an analytical study of unsteady magneto hydro dynamic poiseuille flow of an incompressible electrically conducting couple stress fluid through a porous medium between the parallel plates taking into account pulsation of the pressure gradient effect and under the influence of periodic body acceleration with phase difference ď Ś . The solution of the problem is obtained with the help of perturbation technique. Analytical expression is given for the velocity field, and the effects of the various governing parameters entering into the problem are discussed with the help of graphs. The shear stresses on the boundaries and the discharge between the plates are also obtained analytically and their behaviour computationally discussed with different variations in the governing parameters in detail. Key words: Poiseuille flow, unsteady flows, couple stress fluids, periodic body acceleration, parallel plate channels, porous medium.

1. INTRODUCTION A fluid flow driven by a pulsatile pressure gradient through porous media is of great interest in physiology

and biomedical engineering. Such a study has application in the dialysis of blood through artificial kidneys or blood flow in the lung alveolar sheet. Ahmadi and Manvi2 derived a general

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

178

S.V. Suneetha et al., J.Comp.&Math.Sci. Vol.1(2), 177-188 (2010).

equation of motion for flow through porous medium and applied it to some fundamental flow problems. Rapits 8 has studied the flow of a polar fluid through a porous medium, taking angular velocity into account. The problem of peristaltic transport in a cylindrical tube through a porous medium has been investigated by El-Shehawey and El-Sebaei7, their results show that the fluid phase means axial velocity increases with increasing the permeability parameter. Afifi and Gad1 have studied the flow of a Newtonian, incompressible fluid under the effect of transverse magnetic field through a porous medium between infinite parallel walls on which a sinusoidal traveling wave is imposed. The flow characteristics of a Casson fluid in a tube filled with a homogenous porous medium was investigated by Dash et.al6. Bhuyan Hazarika 4 has studied the pulsatile flow of blood in a porous channel in the presence of transverse magnetic field. The flows in bends and branches are of interest in a physiological context for several reasons. The additional energy losses due to the local disturbances of the flow are of interest in calculating the air flow in the lungs and in wave-propagation models of the arterial system. The details of the pressure and shear stress distribution on the walls of a bend or bifurcation are of interest in the study of parthenogenesis because it appears that the localization of plaques is related to the local flow patterns. In vascular surgery questions arise, such as what is the best

angle for vascular graft to enter an existing artery in a coronary bypass (Skalak, R. and Nihat Ozkaya)12. The theory of laminar, steady one-dimensional gravity flow of a non-Newtonian fluid along a solid plane surface for a fluid exhibiting slope at the wall has been studied by Astarita et.al3. Suzuki and Tanaka 13 have carried out some experiments on non-Newtonian fluid along an inclined plane. The flow of Rivlin-Ericksen incompressible fluid through an inclined channel with two parallel flat walls under the influence of magnetic field has been studied by Rathod. P and Shrikanth11. Rathod and Shrikanth9 have studied the MHD flow of Rivlin-Ericksen fluid between two infinite parallel inclined plates. The gravity flow of a fluid with couple stress along an inclined plane at an angle with horizontal has been studied by Chaturani and Upadhya 5. Rathod and Thippeswamy10 have studied the pulsatile flow of blood through a closed rectangular channel in the presence of micro-organisms for gravity flow along an inclined channel. Hence, it appears that inclined plane is a useful device to study some properties of non-Newtonian fluids. In this paper, we discuss an analytical study of unsteady magneto hydrodynamic poiseuille flow of an incompressible electrically conducting couple stress fluid through a porous medium between the parallel plates taking into account pulsation of the pressure gradient effect and under the influence of periodic body acceleration with phase difference ď Ś.

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

S.V. Suneetha et al., J.Comp.&Math.Sci. Vol.1(2), 177-188 (2010). 2. Formulation and solution of the p ro b l em :



179

η  4u in the above equation gives ρ z 4

We consider the unsteady hydro magnetic flow of a couple stress fluid through a porous medium induced by the pulsation of the pressure gradient. The plates are assumed to be electrically insulated. The flow takes place with uniform magnetic field and under the influence of periodic body acceleration with phase difference .

the effect of couple stresses,  is the density of the fluid, µe is the magnetic permeability,  is the coefficient of kinematic viscosity, k is the permeability of the medium, Ho is the applied magnetic field. All the physical quantities in the above equation have their usual meaning.

We choose Cartesian co-ordinate system O(x, y, z) such that the boundary walls y=0 and y=h. In such a way that the xz-plane is taken on the lower plate and this y-axis is normal to the plates. A uniform magnetic field is acting along the y-axis. The induced magnetic field is assumed to be negligible and also the flow in the porous medium is assumed to be fully developed. The periodic body acceleration is assumed to be G=g 0 cos  where, g 0 is the amplitude of the body acceleration and  is its phase difference. Under these assumptions the unsteady hydro magnetic equations governing the couple stress fluid flow in the absence of body force f and body moment I are

The boundary conditions are

2 2 2 4 u 1 p  u η  u σ µe H 0   2   u t ρ x y ρ y 4 ρ

 u  g 0 cos  (2.1) k Where, u (y, t) is the velocity, the term

u0

y0

(2.2)

u0

yh

(2.3)

Since the couple stresses vanish at both the plates which in turn, implies that

 2u 0 y 2

y0

( 2. 4)

 2u 0 y 2

yh

(2.5)

Introducing non-dimensional variables are

x x*  , h

y y*  , h

u* 

u h * t , t   2 ,  h

h2 , 

p* 

p h2  2

,* 

Using the non-dimensional variables (dropping asterisks), we obtain

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

180

S.V. Suneetha et al., J.Comp.&Math.Sci. Vol.1(2), 177-188 (2010).

u  4 u  2u  4  a 2 2  ( M 2  D 1 )a 2 u t y y   a2

p  a 2 G cos  x

h ρ is the couple stress 

parameter 2

σ µe H 0 h M2  is the Hartman number ρ

D 1 

h2 is the inverse Darcy parameter k

g0 h3 G 2 

 a 2 G cos 

(2.6)

Where a 2 

 p   p     a 2      ei  t    x  s  x  o 

is the body acceleration

The equation (2.12) can be solved by using the following perturbation technique

u  u s  u o e i t

u 0 u 0

y 0 y1

at at

2 u 0 y 2

y 0

( 2.7) ( 2.8)

( 2 .9 )

 u 0 y 2

y 1

(2 .10)

For the pulsation pressure gradient



p  p   p  i t     e x  x  s  x o

(2.11)

Equation (2.6) reduces to the form 4

u  u  u  4  a 2 2  ( M 2  D 1 )a 2 u t  y y

(2.13)

Substituting the equation (2.13) in (2.12) and equating like terms on both sides

d 4u s d 2 us 2  a  ( M 2  D 1 )a 2 u s 4 2 dy dy

p   a 2    a 2 G cos  (2.14)  x  s

parameter Corresponding the non-dimensional boundary conditions are given by

(2 .12)

d 4u o d 2 uo 2  a  ( M 2  D 1  i )a 2 u o 4 2 dy dy p   a 2   (2.15)  x  o Subjected to the boundary conditions And

us  0

y0

(2.16)

us  0

y 1

(2.17)

2 us 0 y 2

y 0

(2 .18)

2 us 0 y 2

y 1

( 2.19)

And

uo  0

y0

(2 .20)

uo  0

y 1

( 2.21)

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

S.V. Suneetha et al., J.Comp.&Math.Sci. Vol.1(2), 177-188 (2010).

 2u o 0 y 2

y0

( 2.22)

 2u o 0 y 2

y 1

( 2.23)

 p   p    ps and    po  x  s  x  o

 du   du   and U     dy  y 0  dy  y 1

form as L  

The solutions of the equations (2.14) and (2.15) subjected to the boundary conditions (2.16) to (2.23) give the velocity distribution of the fluid under consideration.

u  C1e m y  C2 e m y  C3 e m y  C 4e  m y 1

p s  G cos   M 2  D 1 5

po  i t e 1 M  D  i 

The non-dimensional discharge between the plates per unit depth is given by Q 1

Q   u ( y,t ) dy 0

3. Results and Discussion

  C 5 em y  C6 e m y  C7 e m y  C8 e m  

a 2  a 4  4 a 2 ( M 2  D 1  i) m6  2 The shear stresses on the lower and upper plates are given in dimension less

Let 



181

(2.24)

Where, the constants C1 , C2 ,..........C 8 are given in appendix.

m1 

a 2  a 4  4 a 2 ( M 2  D 1 ) , 2

m2 

a 2  a 4  4a 2 ( M 2  D1 ) 2

a 2  a 4  4 a 2 ( M 2  D 1  i) m5  , 2

From the linear momentum equations, we may note that if the magnitude of the body acceleration dominates over the axial pressure gradient then the velocity u is positive and the flow takes place from left to right. In case of the magnitude of pressure gradient is more then the body acceleration, then u is negative and the flow takes place from right to left. In general the magnitude of velocity u increases from zero the state of rest on the lower boundary (y=0) to a maximum in the upper half region and later gradually reduces to rest on the upper boundary (y=1). The flow governing the non-dimensional parameters namely viz. a couple stress parameter, M the Hartmann number, D-1 the inverse Darcy parameter, G body acceleration para-

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

182

S.V. Suneetha et al., J.Comp.&Math.Sci. Vol.1(2), 177-188 (2010).

meter, Po the amplitude of pulsation pressure gradient. Fig. (1-3) represent the velocity profiles for the pulsation pressure gradient dominates the body acceleration parameter and which corresponds to (t =1,  

 ,   60 0 ) 2

with variations in the governing parameters while fixing the other parameters and the figures (4-6) represents the reverse case with flow taking place from right to left. Fig. (1 and 4) illustrates the magnitude of the velocity u enhances with increasing the couple stress 0

0 0

0.2

0.4

0.6

0.8

0.5

-0.05 -0.1

-0.1 D‾¹=2000

a=0.5

-0.2

a=1

a=4 -0.3

-0.15

D‾¹=4000

D‾¹=6000

-0.2

a=8

D‾¹=8000 -0.25

-0.4

-0.3 -0.5

-0.35 y

Fig. 1. The velocity profile for u with a M=2, G=1, D-1 =2000, Po=Ps =10

Fig. 3. The velocity profile for u with D-1 M=2, a=0.5, G=1, Po=Ps =10. 0.45

0 0

0. 2

0.4

0.6

0.8

0.4

-0. 05 0.35

-0.1

0.3

M=2 -0.15

a=1

M=8

0.2

a=4

M=10

0.15

M=5

u -0.2

a=0.5

0.25

-0. 25

a=8

0.1

-0.3

0.05

-0. 35

0 0

0.2

0.4

0.6

0.8

y y

Fig. 2. The velocity profile for u with M a=0.5, G=1, D-1 =2000, Po=Ps=10

Fig. 4. The velocity profile for u with a M=2, G=1, D-1 =2000, Po=Ps =10

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

S.V. Suneetha et al., J.Comp.&Math.Sci. Vol.1(2), 177-188 (2010).

0.4 0.35 0.3 0.25

M=2 M=5

0.2

M=8 M=10

0.15 0.1 0.05 0 0

0.2

0.4

0.6

0.8

Fig. 5. The velocity profile for u with M a =0.5, G=1, D-1= 2000, Po=Ps =10 0.4 0.35 0.3 0.25 u

D‾¹=2000 D‾¹=4000

0.2

D‾¹=6000

0.15

D‾¹=8000

0.1 0.05 0 0

0.5

Fig. 6. The velocity profile for u with D-1 M=2, a=0.5, G=1, Po=Ps =10. Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

183

184

S.V. Suneetha et al., J.Comp.&Math.Sci. Vol.1(2), 177-188 (2010).

0.45

0.35

0.25 G=0 0.15

G=1

G=2 0.05

G=3 G=4

- 0.05 0

0.2

0.4

0.6

0.8

-0.15

- 0.25 y

Fig. 7. The velocity profile for u with G. M = 2, a = 0.5, D-1= 2000, Po = Ps = 10, 0 0

0.5

-0.1 -0.2 Po=10 Po=25

u -0.3

Po=50 Po=100

-0.4 -0.5 -0.6 y

Fig. 8. The velocity profile for u with Po. M= 2, a=0.5, G=1, D-1 =2000, Ps =10, Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

S.V. Suneetha et al., J.Comp.&Math.Sci. Vol.1(2), 177-188 (2010).

185

Table I: The shear stresses on the upper plate a

0.5

III

0.004568 0.005682

0.007856

0.003566

0.002678 0.008315

VII

0.024568 0.003675 0.002453

VIII

0.002466 0.003675

0.005667

0.002214

0.002056 0.006218

0.009245 0.002275 0.001814

0.001483 0.002783

0.003568

0.001215

0.001056 0.004568

0.006275 0.001218 0.000935

III

VII

VIII

D-1

2000

4000

6000

2000

III

VII

VIII

0.5

-0.00324

-0.00569

-0.00678

-0.00325

-0.00268

-0.00568

-0.00922

-0.00305

-0.00145

-0.00578

-0.00667

-0.00724

-0.00483

-0.00448

-0.00832

-0.01335

-0.00468

-0.00281

-0.00845

-0.00984

-0.01458

-0.00622

-0.00586

-0.00998

-0.02688

-0.00625

-0.00485

III

VII

VIII

Table II: The shear stresses on the lower plate.

M D

-1

2000

4000

6000

2000

III

VII

VIII

0.5

0.151466 0.131455

0.114589

0.112145

0.093145 0.115408

0.084565 0.200555 0.351456

0.245565 0.224566

0.205675

0.206658

0.124885 0.208859

0.166508 0.324652 0.467865

0.305678 0.284599

0.256679

0.267568

0.200586 0.246678

0.206486 0.512085 0.685865

III

VII

VIII

Table III: The Discharge Q

M D

-1

2000

4000

6000

2000

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

186

S.V. Suneetha et al., J.Comp.&Math.Sci. Vol.1(2), 177-188 (2010).

parameters “a” and fixing the other parameters. The magnitude of the velocity u reduces with increasing the intensity of the magnetic field for the irrespective of the flow takes place from left to right and vice-versa. This has been displayed in the figures 2 and 5. From figures (3 and 6), it is evident that the magnitude of the velocity u decreases with increasing the inverse Darcy parameter D -1 . Hence lesser the permeability of the porous medium greater the retardation experienced by the flow in the entire flow field. The velocity profile (fig. 7) exhibits how the velocity u influenced with the body acceleration parameter G. We may observe that the negative pressure gradient in the momentum equation balances the body acceleration term and hence in the absence of any other extraneous forces the fluid is at rest, since the channel walls are at rest. However, when the body acceleration dominates the pulsation pressure gradient, the magnitude of the velocity component u enhances with increase in G in the entire flow field. Likewise it is interesting to note that when the pulsation pressure gradient dominates the body acceleration, an increase in G the magnitude of the velocity u reduces in the entire flow field. The Fig. (8) illustrates the magnitude of the velocity u enhances with increase in the amplitude of pulsation of pressure gradient. The shear stresses have been evaluated on the boundaries and tabulated in the tables 1 and 2. The magnitude of the stresses on either plate enhances with

increase in body acceleration parameter G and the Hartmann number M, and it reduces with increase in the amplitude of pulsation pressure gradient and the inverse Darcy parameter D-1 fixing the other parameters. Thus lesser the permeability lower the stresses on the boundaries, also the magnitude of the stresses on the lower boundary is far lesser than the corresponding magnitudes on the upper boundary. We observe that the stresses reduces on the upper boundary while enhances on the lower boundary with increase in the couple stress parameter ‘a’. The discharge Q between the plates enhances with increase in ‘a’ and Po, and reduces with increase M, D-1 and G. 4. CONCLUSIONS 1. The magnitude of the velocity enhances with increase in the couple stress parameter ‘a’ and the amplitude of pulsation pressure gradient. 2. The magnitude of the velocity reduces with increase in the Hartmann number M and the inverse Darcy parameter D-1. 3. When the body acceleration dominates the pulsation pressure gradient, the magnitude of the velocity u enhances with increase in the body acceleration parameter G, while pulsation pressure gradient dominates body acceleration the magnitude of the velocity reduces with increase in G. 4. The magnitude of the stresses on either plate enhances with increase in G and M, and it reduces with increase in the amplitude pulsation pressure gradient and D-1. The stress

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

S.V. Suneetha et al., J.Comp.&Math.Sci. Vol.1(2), 177-188 (2010). reduces on the upper boundary and enhances on the lower boundary with increase in the couple stress parameter ‘a’. 5. The discharge Q between the plates enhances with increase in a and Po, and reduces with increase in M, D-1 and G. REFERENCES 1. Afifi, N.A.S. and Gad, N.S., “Interaction of peristaltic of peristaltic flow with pulsatile magneto-fluid through a porous medium”, Acta Mechanica, 149, 229-237 (2001). 2. Ahmadi, G and Manvi, R, “Equation of motion for viscous flow through a rigid porous medium”, Ind. J. Tech., 9, 441-444 (1971). 3. Astarita, G., Mariucxi, G. and Palumbo, G., “Non-Bewtonian gravity flow on inclined surfaces”. Ind. Eng. Chem. Fundam, 3, 333 (1964). 4. Bhuyan, B. C. and Hazarika, G. C. “Effects of magnetic field on pulsatile flow of blood in a porous channel”, Bioscience Research Bulletin, 17 (2), 105-112 (2001). 5. Chaturani, P. and Upadhya, V.S., “Gravity flow of a fluid with couple stress along an inclined plane with application to blood flow”, Biorheology, 14, 237-246 (1977). 6. Dash, R.K., Mehta, K.N. and Jayaraman. G., “Casson fluid flow in a pipe filled with a homogeneous porous medium”, Int. J. Eng. Sci., 34, 1145-1156 (1996). 7. El-Shehawey, E. F. and El-Sebaei,

187

Wahed., “Peristaltic transport in a cylindrical tube through a porous medium”, Int. J. Math. & Math. Sci., 24(4), 217-230 (2000). 8. Rapits. A., “Effects of couple stress on the flow through a porous medium”, Rheol. Acta, 21, 736-737 (1982). 9. Rathod, V.P. and Hossurker Shrikanth, G. “MHD flow of Revlin Ericksen fluid between two infinite parallel inclined plates”, The Mathematics Education, XXXII (4), 227-232 (1998). 10. Rathod, V.P. and Thippeswamy, G., “Gravity flow of pulsatile blood through closed rectangular inclined channel with micro-organisms”. The Mathematics Education. XXXIII (1), 40-49 (1999). 11. Rathod, V.P. and Hossurker Shrikanth, G., “MHD flow of Revilin Ericksen fluid through an inclined channel”. Bull. Of Pure & Appl. Sci., 17 E (1), 125-134 (1998). 12. Skalak, R. and Nihat Ozkaya, “Bio fluid Mechanics”, Dept. of Civil Engineering and Engineering Mechanics, Coulumbia University, New York. 167 (2000). 13. Suzuki, A. and Tanka, T., “Measurement of flow properties of powers along an inclined plane”, I & EC Fundamental, 10, 84 (1971). APPENDIX

p  G cos    C1   C2  C3  C 4  s 2  M  D 1  

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

188

S.V. Suneetha et al., J.Comp.&Math.Sci. Vol.1(2), 177-188 (2010).

 1   m1  p  G cos   m  e  e m1 C 3  e  m2  e m1 C 4   s 2  1 e 1  m1  1 e e   M D   d d d d  d1 d 6 po   C3   2 C 4  3 , C 4  3 4 , C 5   C 6  C 7  C 8  2  1 d1 d1 d 1d 5  d 2 d 4 M  D  i   C2 













po  1   m5   m  e  e m5 C 7  e  m6  e m5 C8   2  1 e 5  m5  1 e e   M  D   i   ' ' ' ' ' ' d d d  d1 d 6 d C 7   2' C8  3' , C8  3' 4' d1 d1 d1 d 5  d 2' d 4' C6 









( m22  m12 )(e m1  e  m1 ) ( e m2  e  m2 ) 2 2 d1  , d  ( m  m )  2 2 1 (e m2  e m1 ) (e m2  e m1 )







p s  G cos  ( m12  m22 )(1  e m1 ) e m2 (m12  m22 )(e  m1  e m1 ) 2 d3    m1 , d 4  M 2  D 1 (e m2  e m1 ) (e m2  e m1 )

d5 

(m22 e  m2  m12 e m1 )(e m2  e m1 )  (e  m2  e m1 )(m22 e m2  m12 e m1 ) (e m2  e m1 )

d6 

p s  G cos  (e m2  e m1 )( m12 e m1 )  (1  e m1 )(m22 e m2  m12 e m1 )  M 2  D 1 (e m2  e m1 )

( m62  m52 )(e m5  e  m5 ) ' (e m6  e  m6 ) 2 2 d  , d 2  (m6  m5 ) (e m6  e m5 ) (e m6  e m5 ) ' 1



pO (m52  m62 )(1  e m5 ) e m6 ( m52  m62 )(e m5  e m5 ) 2 ' d  2   m5 , d 4  M  D 1  i (e m6  e m5 ) (e m6  e m5 ) ' 3

d 5' 

(m62 e  m6  m52 e m5 )(e m6  e m5 )  (e  m6  e m5 )(m62 e m6  m52 e m5 ) (e m6  e m5 )

d 6' 

po (e m6  e m5 )( m52 e m5 )  (1  e m5 )(m62 e m6  m52 e m5 )  M 2  D 1  i (e m 6  e m 5 ) Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

