This research paper provides a new result on induced fluid flow in a circular cylinder under the influence of a rotating magnetic field. Maxwell’s induction equation and asymptotic technique are used to obtain the new result and it is observed that the convective term which Mofatt neglected in his work is significant in the boundary layer

Ayeni, B. O.

Fenuga, O. J.

English

Faculty of Science, Federal University of Lafia

FULafia Journal of Science & Technology Vol. 4 No.2   December 2018131NEW RESULT ON A FLUID FLOW INDUCED  BY A ROTATING MAGNETIC FIELD*1Fenuga O.J  and  2Ayeni B. O1Department of Mathematics, University of Lagos, Nigeria2Department of  Mathematics, Tai Solarin University of  Education, Ogun State, Nigeria.*Corresponding Email: ofenuga@unilag.edu.ngManuscript Received: 12/08/2018     Accepted: 15/08/2018      Published: December 2018ABSTRACTThis research paper provides a new result on induced fluid flow in a circular cylinder under the influence of a rotating magnetic field.  Maxwell’s induction equation and asymptotic technique are used to obtain the new result and it is observed that the convective term which Mofatt neglected in his work is significant in the boundary layer.Key words: Induced fluid flow ,  Maxwell’s Induction equation, Rotating Magnetic field, Convective term, Boundary layer ISSN (Print): 24490954ISSN (Online): 26364972ENGINEERINGFULafia Journal of Science & Technology Vol. 4 No.2   December 2018132INTRODUCTIONIn a work by Moffatt(1967), he considered the effect of a rotating magnetic field on an infinitely long circular cylinder of conducting fluid. He assumed a low magnetic Reynolds number Rm and obtained an exact expression for the magnetic field in terms of Bessel’s function. He found out that the rate of vorticity generation by the Lorentz force was independent of time and that this steady vorticity source produced a rigid body rotation in the interior of the cylinder inside a viscous-magnetic boundary layer.  Nigam (1969) examined a spherical container of a conducting fluid in a rotating magnetic field but his analysis was identical to that of Moffatt. Dahlberg (1972) showed that the rate of vorticity generation in a circular cylinder was steady for all rotation rates of the magnetic field and he then derived a general solution for the vorticity field. The stability of this solution for the case of a slowly rotating field was examined by Richardson (1974) who concluded that instability would occur for every small magnetic field strength. Sneyd (1978) studied the effects of a rotating magnetic field on the container of conducting fluid. He assumed a small magnetic Reaynold number and the frequency of alternation or rotation was rapid and so the magnetic field was confined in a thin layer on the surface of the container. He used a boundary layer analysis to find the rate of vorticity generation due to Lorentz force and used a perturbation analysis to study the flow induced in a slightly distorted circular cylinder by a rotating field. His results showed the development of complex flow in the viscous-magnetic boundary layer which may be unstable. Paul and Singh (1998) have shown the analytical solution of laminar and have fully developed free convective flow between two coaxial vertical cylinders partially filled with a porous medium and clear fluid. Yan (2000) has considered viscous flow about a submerged circular cylinder, which oscillates with steady current. Several solutions in the case of hydromagnetic free-convective flows have been obtained by Chandran et al., (1993; 1996; 1998; 2001), for different physical situations of flow formations. Stiller and Franar (2006), studied early stage of turbulent flow driven by a rotating magnetic field via direct simulation and electric potential measurement for the case of a cylindrical geometry. The numerical results show that the undisturbed flow remain stable up to the linear stability limit whereas small perturbation may initiate nonlinear transition at subcritical Taylor numbersThis work extends the work of Moffatt (1974), by providing a new result and also shows  that the convection term which [4] neglected in his work is not negligible  in the boundary layer.MATERIALS AND METHODSThe Maxwell induction equation used in this work is:     .............................(2.1)where  B  is the rotating magnetic field u  is the velocity of the moving fluid η  is the magnetic diffusivity of the fluid t  is the time ∇  is the gradient operator.  .(2.2)Using ( )zr BBBB    ,   , θ=  , then (2.1) becomes  (2.3)Then considering the r-component in (2.3) we have     ....................(2.4)Then using the dimensionless variables for  parameters in (2.4), we have               ......(2.5)Substituting (2.5) in (2.4) we obtain  ....(2.6)Let   and  Then (2.6) becomes       .......(2.7)NEW RESULT ON A FLUID FLOW INDUCED  BY A ROTATING MAGNETIC FIELDFULafia Journal of Science & Technology Vol. 4 No.2   December 2018133Method of SolutionUsing asymptotic expansion for θuB r     and     we have..(2.8)Substituting (2.8) in (2.7) and equating the coefficients of  mR1  and the constant term we obtain respectively. .....(2.9)and  ......(2.10)Equation (2.9) was solved by [4] and he obtained a unique solution for 0rB   but he neglected the convection term ( )Bu ××∇   which resulted in equation (2.10). This paper now extends the work of [4] by solving for 1rB   in equation (2.10).RESULTS AND DISCUSSIONTheorem 4.1Consider the equation.....(4.1.1)  ,1a=λ together with ( ) ( ) ( ) ,0   , ,0   ,    ,00   ,111=∞== tBtaBrB rrr  Then 1rB   has a unique solutionProofLet  Then (4.1.1) becomes whose solution is:.(4.1.2)where  and      Hence, the solution is unique and the proof is complete.Theorem 4.2 1rB  is positiveProofRecall  (from [4])Then substitute for ( )τξ  ,f   in (4.1.2) we have:(4.2.1)This completes the proof. Hence  ( )Bu ××∇  is not negligible in the boundary layer near  ar = .CONCLUSION The results extends the work of [4]  by obtaining  which was not in [4] and showing that there exists a unique solution   for the stated problem and that the convection term ( )Bu   ××∇   which [4] neglected in his research work is not negligible in the boundary layer near ar =ACKNOWLEDGEMENTThe authors appreciate the comments of the reviewers in improving the quality of the paper.Conflict of InterestThe authors declare that there is no conflict of interest REFERENCESBatchelor, G.K (1967). An introduction to fluid mechanics. Cambridge university press, New York.Chandran, P, Sacheti, N.C . and Singh, A.K(1993). Effect of rotation on unsteady hydrodynamic Couette flow.  Astrophysics and Space Science, Vol. 202, pages 1-10.Chandran, P, Sacheti, N.C . and Singh, A.K(1996). Hydromagnetic flow and heat transfer past a continuously  moving  porous boundary. International communications in Heat and Mass  transfer, Vol. 23,pages 889-898. Chandran, P, Sacheti, N.C and Singh, A.K (1998). Unsteady hydromagnetic free convection flow with heat  flux and accelerated boundary motion. Jounal of physical science of Japan, Vol.67,pages 124-129.NEW RESULT ON A FLUID FLOW INDUCED  BY A ROTATING MAGNETIC FIELDFULafia Journal of Science & Technology Vol. 4 No.2   December 2018134Chandran, P, Sacheti, N.C . and Singh, A.K(2001). A unified approach to analytical solution of a hydromagnetic  free convection flow. Scientific Mathematicae,Vol. 53,pages467-476.Dahlberg, E (1972). Vorticity generation in a circular cylinder by a rotating magnetic field.Moffatt H. K. (1965). On fluid flow induced by a rotating magnetic field. Journal of fluid mechanics, vol.22,  part 3, pages 521-528.Nigam G. D (1969). Flow of a fluid in a spherical container in the presence of rotating magnetic field. Journal  of mathematical physics science, volume 3, page 205.Paul, T. and Singh,A.K(1998): Natural convection between coaxial vertical cylinders partially filled with a  porous material. Forschung im Ingenieurwesen, Vol.64, Pages 157-162.Richardson A. T (1974). On the stability of a magnetically driven rotating fluid flow. Journal of fluid mechanics,  vol. 63, page 593.Sneyd A. D (1978). Fluid flow induced by a rapidly alternating magnetic field. Journal of Mechanics, volume  92, part 1, pages 35-51.Stiller, J and Frana .K.(2006): Transitional and weakly turbulent flow in a rotating magnetic field. AIP Physics  of fluids, Vol.18(074105)Yan, B (2000). Unsteady viscous flow about a submerged circular cylinder with a steady current. Fluid Dynamic  Reseach,Vol.26,pages 69-94.Young, E.C (1972). An introduction to partial differential equations. Allyn and Bacoon inc. BostonNEW RESULT ON A FLUID FLOW INDUCED  BY A ROTATING MAGNETIC FIELD

NEW RESULT ON A FLUID FLOW INDUCED BY A ROTATING MAGNETIC FIELD

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NEW RESULT ON A FLUID FLOW INDUCED BY A ROTATING MAGNETIC FIELD

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Faculty of Science, Federal University of Lafia