Steady Flow of Couple Stress Fluid through a Rectangular Channel Under Transverse Magnetic Field

M. Pavankumar Reddy, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 174-181 ISSN: 2395-3519

International Journal of Advanced Trends in Computer Applications www.ijatca.com

Steady Flow of Couple Stress Fluid through a Rectangular Channel Under Transverse Magnetic Field 1

M. Pavankumar Reddy, 2J.V. Ramana Murthy 1,2 Department of Mathematics, National Institute of Technology Warangal, Warangal-506004, Telangana, India 1 mprnitw@gmail.com, 2jvr@nitw.ac.in

Abstract: In this paper, we have considered the steady and an incompressible conducting couple stress fluid flow in the presence of transverse magnetic field through a rectangular channel with uniform cross–section. The induced magnetic field is neglected. We consider the case that there is no externally applied electric field. Under these conditions, we get 4th order PDE for velocity w along the axis of the rectangular tube. The usual no slip and hyper stick boundary conditions are used to obtain the solution for w. We obtained the velocity w in terms of Fourier series. Skin friction on the walls and volumetric flow rate are obtained in terms of physical parameters like couple stress parameter and Hartmann number. The effects of these parameters on skin friction and volumetric flow rate are studied through graphs.

Keywords: Couple stress fluid, Skin friction, Rectangular channel, Magnetic field.

I.

INTRODUCTION

The steady flow of a conducting fluid through a straight avenue under a uniform transverse magnetic field presents one of the elementary problems in magneto hydrodynamics. Magnetic flow in a rectangular channel is a classical problem that has significant applications in magneto hydrodynamic power generators and pumps etc. Nowadays, magnetic field has earned great value due to widespread applications in industry and bioengineering, such as electrostatic precipitation, power generators, petroleum industry, aerodynamic heating, the purification of molten metals from nonmetallic materials, polymer technology and fluid droplet sprays. Hartmann (1937) was the first person to obtain a solution for this type of flows to compare with his experimental results on mercury. Hartmann and Lazarus (1937) studied the impact of a transverse uniform magnetic field on the flow of a viscous incompressible electrically conducting fluid between two infinite parallel stagnant and insulating plates. An approximate method of solution has given by Tani (1962) for the steady laminar incompressible flow of an electrically conducting fluid through a straight avenue of arbitrary cross section with conducting or nonconducting walls in the presence of a uniform

transverse magnetic field based on a minimum principle. Ahmed and Attia (1998) further studied the viscous and joule dissipation effects under an external uniform magnetic field in an eccentric annulus of an electrically conducting incompressible fluid. Abel et al.

(2004) studied the momentum, mass and heat transfer past a stretching sheet using the Walters-B viscoelastic model in the presence of a transverse magnetic field.

(Ahmed and Attia, 2000; Attia, 2005) studied the MHD flow and heat transfer of a viscous incompressible fluid through a rectangular duct. Hassan and Attia (2002) considered the transient Hartmann flow of a dusty incompressible fluid in a rectangular channel under the influence of an applied uniform magnetic field. The steady flow of Micropolar Fluid with Suction under transverse magnetic field in a rectangular channel was studied by Ramana Murthy et al. (2011).

Srnivasacharya and Shiferaw (2008) studied the steady flow of an electrically conducting and incompressible micropolar fluid flow through a rectangular channel taking into consideration the Hall and ionic effects.

In the above studies of non–Newtonian fluids with MHD effect, couple stress fluids have not been considered. Stokes (1966) introduced the theory of couple stresses and gave the simplest generalization of

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M. Pavankumar Reddy, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 174-181 ISSN: 2395-3519

the viscous fluid theory that maintains the couple stresses and body couples. One of the applications of couple stress fluid is its use to study the mechanism of lubrication of synovial joints (1999), which has grown to be the object of scientific research. In Recent past, the significance of couple stress fluid flows in chemical engineering applications involving liquid crystals, polymeric suspensions (2007), polymer-thickened oils and physiological fluid mechanics (1994) were attracted by researchers. These fluids are also used vigorously in the tribology of thrust bearings (2009) and the lubrication of engine rod bearings (2008).

Couple stress fluids are not much complicated as compared to micropolar fluids(2007). As the microstructure was not available at the kinematic level, hence kinematics of such fluids were explained using the velocity field. Stokes' problems were studied by Devakar and Iyengar (2008) under the isothermal conditions for an incompressible couple stress fluid.

The magnetic field effects in 3D flow subject to convective boundary condition were investigated by Hayat et al. (2015) for couple stress nanofluid over a nonlinear stretched surface. Srinivasacharya and Kaladhar (2012) studied the mixed convection flow of couple stress fluid with soret and dufour effects in a non-Darcy porous medium. The inclined magnetic field characteristics of couple stress material in a porous medium was recently inspected by Ramesh (2016) in peristaltic flow. The peristaltic flows were investigated

by Reddy et al. (2005) in a rectangular duct. As far as the authors know, the magneto hydrodynamic flow of couple stress fluid through a rectangular channel has not been treated analytically.

Hence, in this paper, our objective is to study the flow of the magneto hydrodynamic couple stress fluid through a rectangular channel. We have used Cartesian co-ordinate system for formulating the mathematical equations and obtained the exact solution for velocity.

Skin friction on the walls and volumetric flow rate are obtained in terms of physical parameters like couple stress parameter and Hartmann number. We have studied the effects of these parameters on volumetric flow rate, skin friction and illuminated the results through graphs.

II. MATHEMATICAL FORMULATION

(2) where Q is the velocity, P is the pressure, đ?œŒ is the density, đ?œ‡ is the viscosity coefficient, Ρ is the couple stress viscosity parameter.

An incompressible and couple stress fluid flow through a channel is considered with uniform rectangular cross section with side lengths a and b. Using a Cartesian coordinate system (X,Y,Z) with center of rectangular cross section as origin and the axis of the tube as Z axis along which the flow is assumed. H0, a constant magnetic field in the perpendicular direction to the flow is applied. Along the rectangular tube a constant pressure gradient causes generation of the flow in it.

We made an assumption that the induced magnetic and electrical fields are negligible.

Figure 1: Flow configuration in a rectangular channel We take by the geometry of the problem given in fig. 1 and nature of the flow

Q  Wk ,

H  H0 j ,

J 

ď ł c

Q  H0 j  

ď ł c

U 0 H 0Wi

where W=W(X, Y)

J H   Hence

ď ł c

U 0 H 02Wk and

Q.ďƒ‘0Q  0

Now equation (2) reduces to

ďƒ‘ 0 P  ď ­ďƒ‘ 02 Q  ď ¨ďƒ‘ 04Q 

ď ł

c

H 02Q (3)

The following non-dimensional scheme is introduced:

X  ax, Y  ay, W  U0 w, Z  az and P  ď ˛U 02 p

The coupled equations for steady, incompressible and couple stress fluid flow with transverse magnetic field are given by

ďƒ‘0 .Q  0

ď ˛Q.ďƒ‘0Q  ď€­ďƒ‘0 P ď ­ďƒ‘02Q ď€­ď ¨ďƒ‘04Q  J ď‚´ H

whereU0 an average entrance velocity.Substituting these in equation (3) we obtain

ďƒ‘4 w  Sďƒ‘2 w  SM 2 w  L0

(1)

(4)

The equation (4) is solved with no slip boundary conditions: www.ijatca.com

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M. Pavankumar Reddy, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 174-181 ISSN: 2395-3519

w  0 on x  1 and y   y0 where y0 =

b a

M

(5)

and hyper stick boundary conditions:

Hartmann

1 1 w 1 w Q  i  j 0 y   y0 2 2 y 2 x on and x  1 respectively (6) 2 a S ,

Equation (4) can be written as:



where couple stress parameter

U 0 a , 

Re  number

number

 H 02 a 2 c

and

dp L0  SL  Re.S dz =constant

(2  12 )(2  22 )w  L0

Reynolds

where

12  22  S ,

1222  SM 2

III. SOLUTION OF THE PROBLEM Let us choose   L  f ( y ) cos r x  g n ( x) cos tn y  n  n M 2 n1 n 1 r (2n  1) rn  , tn  n 2 y0 . Substituting (7) in (4) we get, where

w



r





f  2rn2 f nii  f n(iv ) cos rn x   tn4 g n  2tn2 g nii  g n(iv ) cos tn y

4 n n

n 1





(7)

n 1

     S   rn2 f n  f nii cos rn x   tn2 g n  g nii cos tn y   SM 2   f n cos rn x  g n cos tn y   0 n 1 n 1  n 1  ( iv ) 2 ii 4 2 2  f n  (2rn  S ) f n  (rn  Srn  SM ) f n  0 (8)









g (iv)  (2t 2  S ) g ii  (t 4  St 2  SM 2 ) gn  0

n n n n and n Equations (8) and (9) can be written as

D

2

 un2

 D

2



 vn2 f n  0

D and

2 1

d d D D1  dy and dx where 2 2 2 2 un  vn  S  2rn  1  22  2rn2 ,

  n2

 D

2 1

(9)



 n2 gn  0

(10)

un2 vn2  rn4  Srn2  SM 2  rn4  (12  22 )rn2  1222

n2   n2  S  2tn2  12  22  2tn2 , un2  12  rn2 ,

n2  n2  tn4  Stn2  SM 2  tn4  (12  22 )tn2  1222 vn2  22  rn2 and n2  12  tn2 ,  n2  22  tn2 .

Solving (10) we get

cosh un y cosh vn y cosh  n x cosh n x  Bn gn ( x)  Cn  Dn cosh un y0 cosh vn y0 and cosh  n cosh n   L cosh un y cosh vn y   w   2    An  Bn  cos rn x M cosh un y0 cosh vn y0  n 1  f n ( y)  An

  cosh  n x cosh  n x     Cn  Dn  cos tn y cosh  n cosh  n  n 1 

(11)



L    Cn  Dn  cos tn y 2 n 1 By no slip condition on x  1, w  0 gives M

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(12)

176

M. Pavankumar Reddy, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 174-181 ISSN: 2395-3519  L   An  Bn  cos rn x  2 y   y0 , w  0 n 1 Again by no slip condition on gives M (13) w y   y0 , 0 y By hyper-stick condition, on which gives 

 A u

n n

n 1

tanh un y0  Bn vn tanh vn y0  cos rn x   cosh  n x cosh  n x  n    Cn  Dn  tn (1)  0 cosh  cosh  n 1  n n 

w 0 x which gives

x  1, Similarly by hyper-stick condition on 



 A n 1



n

(14)

cosh un y cosh vn y  n  Bn  rn (1) cosh un y0 cosh vn y0  

   Cn n tanh  n  Dn  n tanh  n  cos tn y  0 n 1

(15)

Using the orthogonality property, we have

An  Bn  From (13) we obtain,

1 y0

y0

 cos t y n

cos tm y dy   mn

 yo

2L (1)n1  M 2 rn

2L (1) n1  An M 2 rn

(16)

2L Dn  2 (1)n1  Cn M rn

(17)

 2r (1) n 1 2rn (1) n 1  t m  Cm n 2  D  m  m  rn2  m2  rn2  

(18)

 2t (1) n 1 2t ( 1) n 1  rm  Am n 2 2  Bm n 2 2  um  t n vm  tn  

(19)

Bn 

2L Cn  Dn  2 (1)n1  M rn From (12) we obtain, From (14) we obtain, 

Anun tanh un y0  Bn vn tanh vn y0    1

m 1

m 1

From (15) we obtain,

Cn n tanh  n  Dn  n tanh  n 

1 y0



  1 m 1

m 1

Substituting (16) and (17) in (18) and (19) we get   2r r ( 2   2 )(1) m  n  An  un tanh un y0  vn tanh vn y0    Cm  m n 2 m 2 m 2 2  m 1  y0 ( m  rn )(  m  rn )    4 Lrn (1) n 1  2 Lvn (1) n 1 tanh vn y0    2 2 2  M 2 rn m 1  y0 M (  m  rn ) 

 2r r (u  v )(1)  Cn  n tanh  n   n tanh  n    Am  m2 n 2 2 2 2  m 1  y0 (um  tn )(vm  tn )    4 Lr (1) n 1  2 L n (1) n 1 tanh  n    2 2n 2 2   M 2 rn m 1  y0 M (vm  t n )  

and

2 m

2 m

(20)

mn

(21)

Equations (20) and (21) are in the form

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M. Pavankumar Reddy, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 174-181 ISSN: 2395-3519 

a1n An   Cmenm  b1n m 1

(22)



c1n Cn   Am f nm  d1n m 1

where

a1n  un tanh un y0  vn tanh vn y0 c1n   n tanh  n   n tanh  n ,

(23) ,

2rm rn ( m2  m2 )(1)m n 2rm rn (um2  vm2 )(1)m n enm  f nm  2 2 2 2 2 y0 ( m2  rn2 )( m2  rn2 ) , y0 (um  tn )(vm  tn ) ,   4 Lrn (1) n 1  2 Lvn (1) n 1 tanh vn y0 b1n     2 2 2  M 2 rn m 1  y0 M (  m  rn )  ,

 4 Lrn (1) n 1  2 L n (1) n 1 tanh  n d1n    2 2 2 2   M 2 rn m 1  y0 M (vm  t n )  and . 

Eliminating Cn from equations (22) and (23) we get 

a m 1

nm

A  bn m

anm where

(24)  f kn  a 1   n  e nk c1 if n  m k 1  k    e f km if n  m nk   c1k k 1



bn  b1n   enm and

m 1

d1m c1m .

The equation (24) is an infinite system of equations in An. We truncate the system to n =10 and solve for An. Then from (22) Cn can be found. From (16), Bn can be found and from (17) Dn can be found. Hence all the coefficients in (11) for the velocity w are now known.

IV. RESULTS AND DISCUSSION For particular value of physical parameters S and M, the values of 1 and 2 are calculated using the quadratic equation

 2  S   SM 2  0 .

Then un, vn, n and n are found. Now velocity w is computed using (11). The effects of physical parameters S and M on velocity, volumetric flow rate and skin friction are found. We can observe that for a

Velocity w: In fig. 2, velocity contours at different values of M for a fixed value of S=50 are shown. We notice that as M increases, fluid is having high velocity near the walls and more and more fluid is drifted towards walls of the channel and the center of the channel being maintained flat. In the figure, black region shows low values and bright region indicates high values of w. To show this clearly w is plotted in fig. 3 at fixed values of cross-sections for y=0.25, y=0.5, y=0.75 and y=0.9.

2 fixed S value, to get real values of , S  4 M .

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M. Pavankumar Reddy, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 174-181 ISSN: 2395-3519

Figure 2: For S = 50, Velocity w(x, y) at M = 1.732, M = 2.236, M = 2.645.

Figure 3: At S=50 & M=1.732, M=2.236, M=2.645, w(x, y) at cross-sections y=0.25, y=0.5, y=0.75 & y=0.9.

Volumetric Flow rate: Volumetric flow rate V (non-dimensional) is given by 1 y0

V

 w dydx

1  y0



 4 Ly0 Tanhvn y0 Tanh n Tanh n  (1) n 1  Tanhun y0  4  Bn  y0Cn  y0 Dn  An   2 M rn  un vn n n  n 1

In fig. 4, volumetric flow rate V is shown at different values of magnetic parameter M. It is observed that as M increases, volumetric flow rate decreases drastically. But when M is fixed, as S increases, volumetric flow rate is almost constant.

Figure 4: Volumetric flow rate vs Couple stress parameter at different values of magnetic parameter M.

Skin friction: Skin friction is the force acting on the surface per unit area. It is obtained from constitutive equation of couple stress fluid.

T *   Ip  2  E 

1 I  div M 2

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M. Pavankumar Reddy, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 174-181 ISSN: 2395-3519

1 W  Q T  M  mI  4   W  4  (  W ) 2 where with .   m  4(    ) W 4  W  4  W  1,1 1,2 2,1  M   4W2,1  4 W1,2 m  4W2,2  0 0  For our problem,

  2W  2W div M  2   i X  Y Hence

 j  and

0  0 m 

  0   I  div M  2   0  2   W  X 

 2W   X   2W  0  Y    2W  0  Y  0



These equations give non-dimensional stress T=T*a/U0 and

T13 

  e 2    e  T23   w  2 w   w   w x  S y  S  and 

C  T13 y   y0 C f  T23 Hence the skin friction on faces x  1 is f and on faces is . x   1 On ,    2 cosh un y 2 cosh vn y  n 1 e  rn  An 1 cosh u  Bn 2 cosh v  (1)  Cf    n n   S n 1  2 2  Cn 1  nTanh n  Dn 2  nTanh n  cos tn y  

This skin friction is function of y locally. Hence we find average skin friction

1  2 y0

y0

C

f

dy

 y0

.

At S = 50, e = 0.5, at different values of M2 the average skin friction is tabulated in table 1.

Table 1: Average skin friction values for different values of M2 at S = 50, e = 0.5

M2 3 5 7 171.483 104.032 75.0971 Avrage Cf From this we observe that as Hartmann number M increases, skin friction decreases.

V. CONCLUSION By applying Magnetic field, for couple stress fluids the volumetric flow rate and skin friction on the walls are controlled i.e., decrease.

For a fixed value of M, the effect of Couple stress parameter on volumetric flow rate is almost nil. Skin friction is inversely proportional to couple stress parameter.

Conflict of Interests The author does not have any conflict of interest regarding the publication of paper.

Acknowledgement The Author expresses his earnest thanks to the reviewers for their valuable remarks and suggestions for improvement of the paper.

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