Magneto hydrodynamic Fluid Flow past Contracting Surface Taking Account of Hall Current

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1 ISSN: ISO 900:008 Certified Volue 7, Issue 3, May 08 Magneto hydrodynaic Fluid Flow past Contracting Surface Taing Account of Hall Current Sauel K. Muondwe, Mathew Kinyanu David Theur Kang ethe Giterere Abstract: An unsteady, lainar hydrodynaic flow of an incopressible, viscous and electrically conducting Newtonian fluid over a porous contracting sheet in a rotating syste considering hall current effects and heat source has been studied. The governing syste of partial differential equations is transfored to diensionless equations. The diensionless equations are then solved nuerically using the finite difference ethod. Using graphs the effects of various paraeters on velocity, teperature and concentration are discussed The results obtained here are useful in applications on wire and glass fiber drawing in cooling of nuclear reactors, cooling of etallic plates and etallurgy. Index Ters MHD, Rotation, Hall current, contracting surfaces. I. INTRODUCTION Fluid is defined as a substance that defors continuously when acted on by a shearing stress of any agnitude. Fluids are classified as liquids and gases. Fluids can also be categorized as Newtonian and non-newtonian fluids. For Newtonian fluids the shearing stress is linearly related to the rate of shearing strain but for non-newtonian fluids shearing stress is not linearly related to the rate of shearing strain. Water, air, ercury, erosene and thin lubricating oils are Newtonian fluids whereas paints, coal tar, blood and grease are non-newtonian fluids. Investigation of hydro agnetic natural convection flow with heat and ass transfer in porous edia has been studied by several researchers due to its applications in geophysics, astrophysics, aeronautics, eteorology, electronics, cheical, and etallurgy and petroleu industries. [] studied free convective ass transfer flow past infinite vertical porous plate with variable suction and Soret effect and concluded that increase in agnetic intensity contribute to the decrease in the velocity. [] Analyzed radiation effects on MHD free convection flow of Kuvshinshii fluid with ass transfer past a vertical porous plate through a porous edia. [] Concluded that teperature decreases with increase in radiation paraeter. Most researchers ignore Hall current while applying Oh s law because it has no significant effect for sall and average values of the agnetic field. The effects of Hall current are very iportant in the presence of a strong agnetic field, because for strong agnetic field electroagnetic force is proinent. [3] studied Hall effect on transient MHD flow past an ipulsively started vertical plate in a porous ediu with raped teperature, rotation and heat absorption and concluded that for both raped teperature and isotheral plates the priary and secondary otions are accelerated due Hall current, further they noted that sin friction (drag force due to priary velocity) rises with increasing values of agnetic paraeter for raped teperature. [4] Studied effect of Hall current, radiation and rotation on natural convection heat and ass transfer flow past a oving vertical plate. [4] Concluded that an increase in rotation paraeter causes a decrease in priary velocity, secondary velocity in the region away fro the plate and reverse effect is observed on secondary velocity in a region near the plate. [5] Studied unsteady hydro-agnetic couette flow of a viscous, incopressible and electrically conducting fluid between two infinitely long parallel porous plates, taing Hall current into account, in the presence of a transverse agnetic field. Fluid flow within the channel is induced due to ipulsive oveent of the lower plate of the channel Unifor agnetic fields is in the direction orthogonal to the pereable plates, unifor suction and inection through the plates are applied. [5] Concluded that when the agnetic field is considered to be fixed relative to the plate, the flow is accelerated by the agnetic field. However, the agnetic field retards the fluid flow when the agnetic field is fixed relative to the flow. [6] Studied Hall effect on MHD ixed convective flow of a viscous incopressible fluid past a vertical porous plate iersed in a porous ediu with heat source/sin. [6] Concluded that sin friction decreases for Hartan nuber less than critical value, afterwards it starts to increase. 4

2 ISSN: ISO 900:008 Certified Volue 7, Issue 3, May 08 [7] Studied effects of Hall current, rotation and Soret effects on MHD free convection heat and ass transfer flow past an accelerated vertical plate through a porous ediu. [7] concluded that; Hall current tends to accelerate secondary fluid velocity throughout the boundary layer region whereas it has a reverse effect on the priary fluid velocity throughout the boundary layer region; Rotation tends to accelerate secondary fluid velocity throughout the boundary layer region whereas it has a reverse effect on the priary fluid velocity throughout the boundary layer region; [8] analyzed effects of heat source/sin on MHD flow and heat transfer over a shrining sheet with ass suction, and concluded that increase in ass suction paraeter it causes oentu boundary layer thicness thinner. [9] Studied fluid flow and heat transfer of Nano fluid containing otile gyrotactic icro-organis passing a nonlinear stressing vertical sheet in the presence of non- unifor agnetic field. [0] studied Cobined effects of variable agnetic field and porous ediu on the flow of MHD fluid due to exponentially shrining sheet and concluded that the suction paraeter and porous ediu pereability increase the velocity profiles, whereas, agnetic field has reverse effect, the Prandtl nuber and suction paraeter have a reducing effect on the teperature profiles. The ai of this wor is therefore to study MHD stoes fluid flow proble and cobined effects of various paraeters on velocity, teperature and concentration past a porous contracting surface in a rotating syste taing account of Hall current. II. MATHEMATICAL FORMULATIONS Consider an unsteady MHD lainar boundary layer flow of an incopressible, electrically conducting, and viscous Newtonian fluid past a contracting electrically non-conducting sheet ebedded in porous edia with heat and ass transfer in a rotating syste. The contracting sheet is pereable to allow for a possible suction and is v cx continuously contracting in the x-axis direction in the plane with a velocity, where c is negative for contacting plate and n is nonlinear contracting paraeter. At distance L units away, a second ipereable and electrically non conducting sheet is placed parallel to the contracting sheet for closed flow. The syste is rotated with constant angular velocity Ω; the y axis is taen to be infinite. The pressure gradient is in the positive x-axis direction, and is directed upward parallel to the direction of gravity. Additionally, a variable agnetic field is applied noral to the two plates. g X axis n Ipereable plate H Contracting plate Main flow direction Fig : Flow configuration If Q is the teperature dependent voluetric heat generation paraeter, then 0 and 0 e e J J H E eq H () H Q represents a heat source, Q represents a heat sin. Taing Hall current into account, the generalized Oh s law can be written as 0 In equation () the electron pressure gradient (for wealy ionized fluid), ion slip and thero electric effects are neglected. Under these assuptions, and in the absence of electric field, equation () becoes J J H v () x y e 0 5

3 ISSN: ISO 900:008 Certified Volue 7, Issue 3, May 08 J J H u (3) y x e 0 Where e e is the Hall paraeter? Solving equations () and (3) for and yields eh0 J x v u (4) eh0 J y v u (5) The Lorentz force J B yields J B i J B (6) y 0 x 0 If Q is the teperature dependent voluetric heat generation paraeter, then 0 and 0 Q represents a heat source, Q represents a heat sin. Incorporating equations (4) to (6) into the equations of oentu e energy and concentration, the following respective equations of oentu, energy and concentration are obtained u u u u u v H u 0 v v u v u t x z x z (7) g T T g C C v v v v v v H u 0 u v v v u t x z x z Equation of energy: 6 T 3c 3 p z T T T f T T D T C C u 0 t x z cp x z cscp x z Q0 u v H0 u v T T cp cp cp z z T Equation of concentration: C C C C C D T T T C C u 0 D u 0 t x z x z T x z x z The initial and boundary conditions: t 0: u 0, v 0, 0, T 0, H 0, C 0 at 0 z L t 0 : u u, v 0, T T, C C, H H0, at, x 0 Channel entrance n t 0 : u cx, v 0, T Tw, C Cw, H H 0x, at, z 0 (porous wall) t 0 : u 0, v 0, T T, C C, H H0, at, z L (ipereable wall) () The non-diensional Variables are defined as follows u u, U v v, U w 0 U, t w Ut L, Governing equations in non-diensional for z z L, n x x L, T T T T T w, H (8) (9) (0) H0H () 6

4 ISSN: ISO 900:008 Certified Volue 7, Issue 3, May 08 u u u u u u R 0rv Xiu Mu v u t x z Re x z Gr T Gr C c (3) v v v v v v Eru Xiu Mu v u t x z R x z (4) Equation of energy T T T T T C C Ec u v u 0 D fr t x z R pr x z x z R z z QT 4 T RR u v 3NPr R z (5) Equation of concentration C C C C C Sr T T u 0 t x z Re Sc x z Re x z (6) QH 0 e H Where Q is the non-diensional heat source/sin paraeter. Where M is the agnetic field uc U paraeter, paraeter, Gr p UL Re is the Reynolds nuber, gl T U w T p L Xi is the pereability U U is the local teperature Grashof nuber, Ro is rotational paraeter. c p Where Pr is prandlt nuber, N f f 3 4 T U Dufour nuber, Ec c is the Ecert nuber, R p T represents the teperature differences. T w. T is the radiation paraeter, D f D c c s f p C T w w C T e H is the Joule heating paraeter and T c T The initial and boundary conditions in non-diensional for: t 0 : u 0, v 0, T 0, C 0, H 0, at, z 0 at 0 z L t t t 0 : u, v 0, T, C, H n clx 0 : u, v 0, T, C, H x U, at, z 0 Channel entrance 0 : u 0, v 0, T, C, H, at, z L (Ipereable wall) n III. METHOD OF SOLUTIONS p, at, z 0 (Porous wall) (7) The equations are solved using finite differences ethod and ipleented in MATLAB. The finite difference ethod replaces each PDE with a discrete approxiation for space and tie doain. is 7

5 ISSN: ISO 900:008 Certified Volue 7, Issue 3, May 08 The Finite Difference for of the equation of oentu along the x axis is U U U U i, U U i, U Ui, U U i, R0 U w0 Vi, Vi, t x z Ui, U Ui, Ui, U Ui, U i, U U U ( x) Re ( z) Re U U i, i, i, Xi M Gr Grc U U U U Ti, Ti, C C Vi, Vi, U U (8) Maing U the subect of the forula t U U U U U U U x Re i, i, i, i, t t z Re x Ui, Ui, Ui, U Ui, U Ui, U Ui, tw0 txi tm tgr tgrc Ui, U Ui, U U Ti, Ti, C C z tr0 t w0t t t Vi, Vi, Vi, Vi, U / ( U x z Re( z) Re( x) txi tm t ) Moentu equation along y-axis Vi, V Vi, V i, Vi, V i, Vi, Vi, Vi, V i, U w0 t x z R0 U U V i, V V i, V i, V V i, ( x) Re ( z) Re V i, V Vi, Vi, V Vi, Vi, Vi, Vi, Vi, Maing V Vi, Vi, U U the subect of the forula Xi M t t V V V V V V V V V V V V x Re z Re i, i, i, i, i, i, i, i, t tw 0 txi tm U Vi, Vi, Vi, Vi, Vi, Vi, Vi, Vi, x z tr 0 0,,,,, / ( w t t t t Ui Ui Vi Ui Ui Vi, x z Re( z) t txi tm t Re( x) (9) (0) ) () 8

6 ISSN: ISO 900:008 Certified Volue 7, Issue 3, May 08 Energy equation in finite difference for Ti, T Ti, T i, Ti, T i, Ti, Ti, Ti, T i, U w0 t x z Ti, Ti, Ti, Ti, Ti, Ti, ( x) Pr Re T T T ( z) Pr Re Ec U Ui, U U i, Vi, Vi, Vi, V i, T T T Re x z i, i, i, i, U U Vi, V T Re i 3 Pr Re, Ti, Ti, Ti, Ti, Ti, R Ni z Ci, Ci, Ci, Ci, Ci, C i, Ci, Ci, Ci, Ci, Ci, C i, Q Df Re D Re f T T x z () Maing T the subect of the forula t T T T T T T T x Pr Re i, i, i, i, t t z Pr Re x Ti, Ti, Ti, Ti, Ti, U Ti, Ti, Ti, tw 4t z z Ni Pr Re 0 Ti, Ti, Ti, T i, Ti, Ti, Ti, Ti, tec U Ui, U U i, Vi, Vi, Vi, V i, U U Vi, V tr Re Re x z td f Ci, C Ci, Ci, Ci, C i, Ci, C Ci, Ci, Ci, C i, Re x x Qt t w0t t t 4t Qt Ti, / U x z Pr Re( z ) Pr Re( x ) 3z Ni Pr Re Concentration equation in finite difference for C C C C i, C C i, C Ci, C C i, U w0 t x z Ci, Ci, Ci, Ci, Ci, Ci, ( x) Sc Re C C C ( z) Sc Re C C C i, i, i, i, Sr Ti, Ti, Ti, Ti, Ti, T i, Ti, Ti, Ti, Ti, Ti, T i, Re x z (3) (4) 9

7 Maing C ISSN: ISO 900:008 Certified the subect of the forula Volue 7, Issue 3, May 08 t C C C C C C C x Sc Re i, i, i, i, t t z Sc Re x Ci, Ci, Ci, Ci, Ci, U Ci, C Ci, tw tsr z x Re 0 Ci, C Ci, T i, Ti, Ti, Ti, Ti, Ti, tsr T T T T T T z Re i, i, i, i, t w0t t t / U Re( ) Re( ) x z Sc z Sc x (5) IV. RESULTS AND DISUSSION In order to get a physical insight into the proble at hand, the velocity, teperature and concentration have been analyzed by assigning nuerical values to various non diensional paraeters. These paraeters are input into a coputer progra where each paraeter is varied at a tie. The various paraeters that have been varied include, Magnetic field M, Rotational paraeter Ro, Pereability paraeter X, tie t,, Radiation paraeter, Prandtl nuber and suction paraeter. curve M Ro t i iii iv v ii i Viii ix vi Vii Fig : Priary velocity profiles for different values of Magnetic Paraeter M, Rotation paraeter Ro, tie t and Hall paraeter for heat source and positive Grashof nuber. Fro figure and 3 it is noted that in the presence of a heat source Q 0, An increase in Magnetic Paraeter M leads to a decrease in the agnitude of priary and secondary velocity profiles. An increase in the Rotation paraeter Ro leads to a decrease in the priary and an increase in secondary velocity profiles. This is because when the frictional layer at the oving plate is suddenly set into the rotation then the Coriolis force tends to oppose priary velocity by generating cross flow i.e. secondary velocity. An increase in tie causes an increase in priary 0

8 ISSN: ISO 900:008 Certified Volue 7, Issue 3, May 08 and secondary velocity profiles. An increase in Hall paraeter leads to a slight increase in the agnitude of priary velocity profiles and a decrease in secondary velocity profiles. curve M Ro t i Vii Vi v iv iii ii i viii ix Fig 3: Secondary velocity profiles for different values of Magnetic Paraeter M, Rotation paraeter Ro, tie t and Hall paraeter for heat source and positive Grashof nuber. curve M Ro t i iii ii i iv vi Viii ix v vii Fig 4: Teperature profiles for different values of Magnetic Paraeter M, Rotation paraeter Ro, tie t and Hall paraeter for heat source and positive Grashof nuber. Fro figure 4 it is noted that an increase in the agnetic paraeter causes decrease in the teperature profiles. An increase in rotation paraeter causes an increase in teperature profiles. An increase in tie causes an increase in the teperature profiles. An increase in the Hall paraeter i causes an increase in the teperature profiles. Increasing i decreases the conductivity of the fluid, resulting in an increase in Joule heating, leading to a thicer theral boundary layer

9 ISSN: ISO 900:008 Certified Volue 7, Issue 3, May 08 curve M Ro t i v vi viii vii ix iii i iv ii Fig 5: Concentration profiles for different values of Magnetic Paraeter M, Rotation paraeter Ro, tie t and Hall paraeter for heat source and positive Grashof nuber. Fro figures 5it is noted that an increase in the agnetic paraeter causes an increase in the concentration profiles. An increase in rotation paraeter causes an increase in concentration profiles. An increase in tie causes a decrease in the concentration profiles. An increase in the Hall paraeter i causes decrease in the Concentration profiles. curves Xi Ni Wo Pr vi iv ii vii ix v i iii viii Fig 6: Priary velocity profiles for different values pereability paraeter X i, radiation paraeter N, suction Paraeter Wo, Prandtl nuber Pr for heat source and positive Grashof nuber. Fro figure 6 and 7it is noted that an increase in the Pereability Paraeter Xi leads to a decrease in the velocity. In the presence of a heat source, an increase in the Radiation Paraeter N leads to an increase in the agnitude of both priary and secondary velocity. An increase in suction paraeter causes a decrease in both priary and secondary velocity profiles. Increasing prandtl nuber leads to an increase in both priary and secondary velocity profiles.

10 ISSN: ISO 900:008 Certified Volue 7, Issue 3, May 08 curve Xi Ni Wo Pr vi iv ii vii ix v i iii viii Fig 7: Secondary velocity profiles different values pereability paraeter X i, radiation paraeter N, suction Paraeter Wo, Prandtl nuber Pr for heat source and positive Grashof nuber. curves Xi Ni Wo Pr viii v ii iii i vi iv vii ix Fig 8: Teperature profiles for different values pereability paraeter X i, radiation paraeter N, suction Paraeter Wo, Prandtl nuber Pr for for heat source and positive Grashof nuber. Fro figure 8 it is noted that an increase in pereability Xi causes a decrease in the teperature profiles. An increase in Radiation paraeter causes a decrease in teperature profiles. An increase in suction paraeter causes a decrease in teperature profiles. An increase in Prandtl nuber causes an increase teperature profiles. Figure 9 it is noted that an increase in Xi reduces the rate of species transportation fro the surface of the contracting sheet, leading to an increase in the concentration profiles. An increase in radiation paraeter leads to an increase in the concentration profiles. An increase in suction paraeter causes an increase in concentration profiles. An increase in Prandtl nuber causes a decrease in concentration profiles. Prandtl nuber Pr, is a easure of relative iportance of viscosity and theral conductivity of the fluid. As Pr increases, the viscous effects increase thereby reducing species olecular activity. This leads to decrease in ass average velocity that in turn leads to a decrease in concentration profiles. 3

11 ISSN: ISO 900:008 Certified Volue 7, Issue 3, May 08 curves Xi Ni Wo Pr ii vii viii v ix iii vi iv i Fig 9: Concentration profiles for different values pereability paraeter X i, radiation paraeter N, suction Paraeter Wo, Prandtl nuber Pr for heat source and positive Grashof nuber. V. CONCLUSION The study of velocity field, variation of teperature and concentration past contracting rotating porous ediu has been carried out. The PDE s governing the flow is highly non linear and coupled, and the equations have been solved by using the finite difference ethod. Codes are run for sall values of tie and there is no significant difference in the results obtained. Soe of the findings are: i. Priary velocity rises with an increase in tie, radiation paraeter, Hall paraeter and Prandtl nuber. ii. Priary velocity reduces with an increase Hartan nuber, pereability paraeter, Rotation paraeter, suction paraeter. iii. Secondary velocity rises with an increase in tie, radiation paraeter, Rotation paraeter, Hall paraeter and Prandtl nuber. iv. Secondary velocity reduces with an increase in Hartan nuber, pereability paraeter, suction paraeter. v. Teperature rises with an increase in tie, Rotation paraeter, Hall paraeter and Prandtl nuber vi. Teperature reduces with increase Hartan nuber, pereability paraeter, suction paraeter and radiation paraeter. The present wor can provide basis for further research by including flow that involves non Newtonian fluids. ACKNOWLEDGEMENT The Authors sincerely thans Departent ebers of Pure and Applied Matheatics of Joo Kenyatta University and Dedan Kiathi University of Technology for their useful discussions and guidance. Roan Sybol NOMENCLATURE Quantity Concentration susceptibility, g Acceleration due to gravity vector, [ s - ] H Magnetic flux density along the x- axis [wb - ] h Specific enthalpy, [ Jg - - ] J Current density, [A - ] L distance between vertical sheets, [] Heat generation constant, T Theral conductivity of porous ediu, [w - - ] T Absolute free teperature of the fluid, [] Absolute teperature of the upper sheet [] Absolute teperature of the contracting sheet, [] c Contracting rate, [s - ] C p Specific heat at a constant pressure, [Jg - - ] 4

12 ISSN: ISO 900:008 Certified Volue 7, Issue 3, May 08 Darcy pereability, [ ] Mean absorption coefficient of the fluid, Molecular diffusion coefficient, Theral diffusion coefficient, Diensionless heat source GREEK SYMBOL QUANTITY β Voluetric co efficient of theral expansion, [ - ] Coefficient of theral expansion due to concentration gradient, [ - ] Stefan-Boltzann constant, Fluid density, [g -3 ] µ Coefficient of viscosity, [g - s Angular velocity, [ ] σ Electrical conductivity, [ ] Magnetic pereability, Gradient operator REFERENCES [] Seethaahalashi. B.N., Prasad. G.Raan (0). MHD Free Convective Mass Transfer Flow Past an Infinite Vertical Porous Plate with Variable Suction and Soret Effect, Asian Journal of Current Engineering and Matheatics, 3: [] Vidyasagar & B.Raana., (03). Radiation Effects on MHD Free Convective Flow of Krushinshii Fluid with Mass Transfer Past a Vertical Porous Plate through Porous Media, Asian Journal of Current Engineering and Matheatics, : [3] Ahed.N and Das.K., (03). Hall Effect on Transient MHD Flow past an Ipulsively Started Vertical Plate in a Porous, Applied Matheatical Sciences, 7(5): [4] G. S. Seith, S.Sarar and S.M.Hussain.(04). Effect of hall current, radiation and rotation on natural convection heat and ass transfer flow past a oving vertical plate, Ain shas engineering ournal, 5: [5] Muhur P. (963). Unsteady hydro-agnetic couette flow of a viscous, incopressible and electrically conducting fluid between two infinitely long parallel porous plates. J. Phys. Soc. Jpn., 8, [6] Bhupendra Kuar Shara, Abhay Kuar Jha, R.C.Chaudhary (007), Hall effect on MHD ixed convective flow of a viscous incopressible fluid past a vertical porous plate iersed in a porous ediu with heat source/sin., General Physics-Magneto hydrodynaic, 5(7): [7] D. Sara, K.K. Pandit, (06) Effects of Hall current, rotation and Soret effects on MHD free convection heat and ass transfer flow past an accelerated vertical plate through a porous ediu, Ain Shas Engineering Journal, :-6. [8] Bhattacharyya.K. (0). Effects of Heat Source/Sin on MHD Flow and Heat Transfer Over a Shrining Sheet with Mass Suction, Cheical Engineering Research Bulletin, 5:-7. [9] S. A. M. Mehryan, Farshad Moradi Kashool M. Soltani, Kaaran Raaheifar (06). Fluid flow and heat transfer of Nano fluid containing otile gyrotactic icro-organis passing a nonlinear stressing vertical sheet in the presence of non- unifor agnetic field ; nuerical approach, PLOS/ one ournal,(6):-3. [0] Raesh K. (05), Cobined effects of variable agnetic field and porous ediu on the flow of MHD fluid due to exponentially shrining sheet, International Journal of Matheatic Archives,6(6):8-6. AUTHOR BIOGRAPHY Sauel anyi Muondwe obtained his Msc. In Applied Matheatics fro Joo Kenyatta University of Agriculture and Technology (JKUAT), Kenya in 04 and PhD in Applied Matheatics is in progress at 5

13 ISSN: ISO 900:008 Certified Volue 7, Issue 3, May 08 JKUAT. Presently he is woring as an assistant lecture at Dedan Kiathi University of Technology. He has published six papers in international Journals. His Research area is in MHD and Fluid Dynaics. Professor Mathew Ngugi Kinyanui Obtained his MSc. In Applied Matheatics fro Kenyatta University, Kenya in 989 and a PhD in Applied Matheatics fro Joo Kenyatta University of Agriculture and Technology (JKUAT), Kenya in 998. Presently he is woring as a professor of Matheatics at JKUAT where he is also director of Post Graduate Studies. He has published over fifty papers in international Journals. He has also guided any students in Masters and PhD courses. His Research area is in MHD and Fluid Dynaics. MHD and Fluid Dynaics. Professor David Theuri. Obtained his Msc. In Applied Matheatics fro Joo Kenyatta University of Agriculture and Technology (JKUAT), Kenya in 99 and PhD in Applied Matheatics fro JKUAT 003. Presently he is woring as a professor of Matheatics at JKUAT. He has published over ten papers in international Journals. He has also guided any students in Masters and PhD courses. His Research area is in Dr. Kang ethe Giterere obtained his MSc. In Applied Matheatics fro Joo Kenyatta University of Agriculture and Technology (JKUAT), Kenya in 007 and a PhD in Applied Matheatics fro the sae university in 0. Presently he is woring as a Lecturer at JKUAT. He has published six papers in international ournals and guided any students in Masters courses. His area of research is MHD and Fluid Dynaics. 6