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. © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “IJMSET promotes research nature, Research nature enriches the world’s future” 46 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. © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “IJMSET promotes research nature, Research nature enriches the world’s future” 47 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. © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “IJMSET promotes research nature, Research nature enriches the world’s future” 48 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 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “IJMSET promotes research nature, Research nature enriches the world’s future” 49 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 © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “IJMSET promotes research nature, Research nature enriches the world’s future” 50 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 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. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. Hiemenz,K.and Grenzchicht, D.,[1911] “An Einem in den Gleichformingen Flussigkeitsstrom Eingetauchten Geraden Kreiszylinder”, DinglersPolytech. J., 326, 321-410. Homann, F. Z.,[1936] “Der Einfluss Grosser Zahighkeitbei der Strmung um den Zylinder und um die Kugel”, Zeitsch.Angew. Math. Mech,. 16, 153-164. Chamkha, A.J. and C. 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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. © IJMSET-Advanced Scientific Research Forum (ASRF), All Rights Reserved “IJMSET promotes research nature, Research nature enriches the world’s future” 51