G.C.Hazarika /International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 1, Issue 5, 2014, pp.46-51
Effect of magnetic field on flow near a axisymmetric Stagnation point
on a moving cylinder
G.C. Hazarika*
Aparajita Sarmah**
Department of Mathematics, Dibrugarh University, Assam 786004
*E-mail: gchazarika@gmail.com
**aparajitasarmah@gmail.com
Abstract
The flow of an electrically conducting fluid in the vicinity of an axisymmetric stagnation point on a moving
cylinder under the influence of magnetic field is studied. The governing partial differential equations are
transformed to ordinary differential equations and the resulting initial value problem is solved numerically by
using shooting method. Effects of different parameters such as Reynolds number, Hartmann number and
Magnetic Reynolds number on the flow have been studied numerically. The effects of all the parameters are
quite significant on velocity and magnetic fields as evidence from the numerical results.
Keywords: Stagnation point ,Hartmann number, Magnetic Reynolds number
1. Introduction:
The classical two dimensional stagnation point flow impinging on a flat plate, first considered
by Hiemanz(1911),was extended to axisymmetric case by Homann(1936).The steady two
dimensional flow over a semi infinite flat surface with mass and heat transfer characteristics was
considered by Chamkha et al(2000) .The problem of MHD steady laminar two dimensional
stagnation flow of a viscous incompressible electrically conducting fluid of a variable thermal
conductivity over a stretching sheet was solved by Sharma et al (2008) using shooting method.
Wang (1974) was first to find exact solution for the problem of axisymmetric stagnation flow on an
infinite stationary circular cylinder. Gorla(1976,1977,1978,1979) in a series of papers, studied the
steady and unsteady flows over a circular cylinder in the vicinity of the stagnation point for the
cases of constant axial movement, and the special case of axial harmonic motion of a non-rotating
cylinder. Recently, Cunning et al (1998)have considered the stagnation flow problem on a rotating
circular cylinder with constant angular velocity including of suction and blowing with constant rate.
Takhar et al (1999) have also investigated the unsteady viscous flow in the vicinity of an
axisymmetric stagnation point of an infinite circular cylinder when both the cylinder and the free
stream velocities vary as a same function of time. Attia (2007) presented the axisymmetric
stagnation point flow towards a stretching surface in the presence of uniform magnetic field with
heat generation. Unsteady stagnation point flow over a plate moving along the direction of flow
impingment has been studied by Fang et al(2011).
The present work has been undertaken in order to study the flow and the effect of magnetic field
near an axisymmetric stagnation point on a moving cylinder. Mathematical formulation of the
problem under consideration is presented and a system of similarity transformation is applied to
reduce the system of partial differential equation and their boundary condition describing the
problem into an ordinary differential equation. The system of ordinary differential equation is solved
numerically by shooting method. The effect of different parameters such as Reynolds Number,
Hartmann number, Magnetic Reynolds Number on the flow has been studied numerically. The
variation of the velocity distribution has been illustrated.
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G.C.Hazarika /International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 1, Issue 5, 2014, pp.46-51
2. Formulation of the Problem:
Consider the steady,laminar ,incompressible flow of a Newtonian fluid at an axisymmetric
stagnation point on a moving cylinder. The flow is axisymmetric about the z-axis. The stagnation line
is at z = a and r = a (the radius of the cylinder).The cylindrical co-ordinates are ( r , q , z ) and the
corresponding velocity components given by (u , v , w) .The magnetic components are given by
( Br , 0, Bz ) . Figure 1.1 shows the coordinate system and the flow model on a moving cylinder. The
equation governing the flow of an incompressible axisymmetric, steady, laminar fluid are follows
Figure 1.1:Coordinate system and flow model
Mass Equation
u ¶u ¶w
+
+
=0
r ¶r ¶z
Momentum Equations in radial and axial direction
¶u
¶u
¶ 2u 1 ¶u u ¶ 2u s
s
1 ¶p
+w = +g ( 2 +
- 2 + 2 ) + wBr Bz - uBz 2
u
¶r
¶z
¶r
¶z
r ¶r r
r ¶r
r
r
w
1 ¶p
¶w
¶w
¶ 2 w 1 ¶w ¶ 2 w s
s
+u
=+g( 2 +
+ 2 ) - wBr 2 + uBr Bz
¶z
¶r
¶r
r ¶z
r ¶r ¶z
r
r
Maxwell’s equation
ur
Ñ 2 .B = 0
Magnetic induction equations in radial and axial directions
¶
( wBr - uBz ) - h mÑ 2 Br = 0
¶z
1 ¶
{r ( wBr - uBz )} + h mÑ 2 Bz = 0
r ¶r
The boundary conditions for the problem under the consideration are given by
A t r = a, u = 0, w = W0 , Br = Br 0 , Bz = Bz 0
At r = a , w = 2 Az , Br = Br ¥, Bz = Bz¥
Where m the dynamic viscosity , r is the density, p is fluid pressure , u is the velocity
component in r-direction and w in z -direction.
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G.C.Hazarika /International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 1, Issue 5, 2014, pp.46-51
Solution of the Problem
Let us consider the following similarity transformation for velocity and magnetic field
r
h = ( )2
a
u = - Aah -1/ 2 f (h )
w = 2 Azf '(h )
Br = - B0 ah -1/ 2 g (h )
Bz = 2 B0 zg '(h )
Where f are velocity distribution radial and g are magnetic field distribution in radial direction,
h is the dimentionless co-ordinate.
Under the above transformation the continuity equation is identically satisfied.
To eliminate p we note that
¶ ¶p
¶ ¶p
( )= ( )
¶z ¶r
¶r ¶z
Using above momentum equations become
f IV = -2 f "' h-1 + .5Re( f ' f "- ff "')h-1 + .25M 2a2 ( f " g 2 - f ' gg '- f ' g 2h-1 + fg '2 - fgg "+ fgg ' h-1 )h-2
The magnetic induction equations become
g '' = .5Rm ( f ' g - fg ')h-1 + .25 g ' h-1 - .5 g h-2
Where the prime denotes the differentiation with respect to h
And Re =
U 0 ar
, the Re ynolds number
m
M = B0 a
Pm =
s
, the Hartmann number
rg
g
, the magnetic Pr andlt number
hm
Where the boundary conditions are reduced to
At h = 1 , f = 0, f ' = 0, g = Bo
At h = a , f ' = 1, g = B1
3. Results and Discussion:
The problem under investigation is solved numerically using fourth order Runge –Kutta
shooting method. The dimensionless velocity in radial and axial direction and magnetic field
distribution for different values of the parameters viz M (Hartmann number), Rm (Magnetic
Reynolds number), Re (Reynolds number) are obtained and presented in the graphs.
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G.C.Hazarika /International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 1, Issue 5, 2014, pp.46-51
Variation of radial and axial velocity , magnetic field distribution for different values of
Hartmann number are shown in the figures (1-3). It is observed that magnitudes velocity and
magnetic field distribution decreases significantly as Hartmann number increases. Thus magnetic
field can be utilized to deaccelerate the stagnation point flow. Figures (4) shows the variation of
radial and axial velocity for different Values of Reynolds number. Variation of axial and magnetic
field distribution for different values of Reynolds number are shown in the figure 5 and figure 6 and
7. Here it is seen that axial and radial velocity and magnetic field distribution goes on increasing as
Reynolds number increases. In figure (8-10) ,the variation of radial and axial velocity and magnetic
field distribution for different values of Reynolds number are shown. Here radial and axial velocity
and magnetic field distribution increases for increasing values of Magnetic Reynolds number.
1
1.2
-------------f'
-.-.-.-.-.-. f
M=1, 2, 3, 4
0.8
M=1, 5, 9, 13
1
0.8
0.4
0.6
f'
f f
0.6
0.2
0.4
0
0.2
-0.2
0
-0.4
1
1.5
2
2.5
3
3.5
4
4.5
-0.2
5
1
1.5
2
h
3
3.5
4
h
Fig 1: Variation of radial and axial velocity
Fig 2: Variation of axial velocity for different
for different values of Hartmann number
values of Hartmann number
1
0.7
-------------- f'
-.-.-.-..-.-. f
0.6
0.8
0.5
0.6
0.4
Re=0.1, 0.3, 0.5, 0.5, 0.7, 0.9
0.4
f f'
g
2.5
0.3
0.2
0.2
0
M=1, 5, 9, 13
Bo=0.5, B1=0
0.1
-0.2
0
-0.1
1
1.5
2
2.5
3
3.5
4
h
Fig 3: Variation of magnetic field distribution
for different values of Hartmann number
-0.4
1
1.5
2
2.5
3
3.5
4
4.5
5
h
Fig 4: Variation of radial and axial velocity for
different values of Reynolds number
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G.C.Hazarika /International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 1, Issue 5, 2014, pp.46-51
0.7
1.2
Re=1, 5, 9, 13, 17
0.6
0.8
0.5
0.6
0.4
Re=.1, .3, .5, .7
Bo=0, B1=0.5
f'
g
1
0.4
0.3
0.2
0.2
0
0.1
-0.2
1
1.5
2
2.5
3
3.5
0
4
1
1.5
2
2.5
3
3.5
4
4.5
5
h
h
Fig 5: Variation of axial velocity for
Fig 6: Variation of for magnetic field distribution
different values of Reynolds number
different values of Reynolds number
0.7
1
0.6
0.8
0.5
Rm=1, 2, 3, 4
0.6
0.4
f , f'
g
0.4
0.3
0.2
0.2
Re=1, 6
Bo=0.5, B1=0
0
0.1
-0.2
0
-0.1
1
1.5
2
2.5
3
3.5
-0.4
4
1
1.5
2
2.5
3
3.5
4
4.5
5
h
h
Fig 7: Variation of magnetic field distribution for
Fig 8: Variation of radial and axial velocity for
for different values of Reynolds number
different values of magnetic Reynolds number
0.7
1.2
Rm=2, 6, 10
0.6
0.8
0.5
0.6
0.4
Rm=0.5, 1.0, 1.5, 2.0
Bo=0, B1=0.5
f'
g
1
0.4
0.3
0.2
0.2
0
0.1
-0.2
1
1.5
2
2.5
3
3.5
h
Fig 9: Variation of axial velocity for different
values of magnetic Reynolds number
4
0
1
1.5
2
2.5
3
3.5
4
4.5
5
h
Fig 10: Variation of magnetic field distribution
different values of magnetic Reynolds number
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G.C.Hazarika /International Journal of Modern Sciences and Engineering Technology (IJMSET)
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Volume 1, Issue 5, 2014, pp.46-51
4. Conclusion:
The influence of magnetic field near an axisymmetric stagnation point on a moving cylinder is
examined. From the analysis we can conclude that a significant variation in the flow takes place due
to magnetic field. When the magnetic parameter, the Hartmann number increases, the fluid velocity
decreases in axisymmetric stagnation point flow. Thus the velocity profile and magnetic field
distribution are affected by the dimensionless parameters.
5. Acknowledgements:
The author is thankful to the reviewers for their valuable suggestions to enhance the quality of the
article.
6. References:
1.
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3.
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AUTHOR’S BRIEF BIOGRAPHY
Prof. G.C.Hazarika: He is Professor and Head in Department of Mathematics ,
Dibrugarh University, Dibrugarh, Assam , India. He is a person who contribute to
popularize Mathematics in Assam. His yeoman services as a Computer
Programmer, Lecturer, Reader, Professor, Director, College Development
Council and Directori/c Centre for Computer Studies, Dibrugarh University and is
the key person to introduce the subject computer science in Dibrugarh
University. He has nearly 19 PhDs to his credit more than 70 papers were
published in various esteemed reputed International Journals. He is Member of
Various Professional Bodies. He is a Gold Medalist in M.Sc Examination.
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