MIT OpenCourseWare http://ocw.mit.edu Continuum Electromechanics For any use or distribution of this textbook, please cite as follows: Melcher, James R. Continuum Electromechanics. Cambridge, MA: MIT Press, 1981. Copyright Massachusetts Institute of Technology. ISBN: 9780262131650. Also available online from MIT OpenCourseWare at http://ocw.mit.edu (accessed MM DD, YYYY) under Creative Commons license Attribution-NonCommercial-Share Alike. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.6Magnetic Diffusion and InductionInteractions6.1 IntroductionExcept that magnetoquasistatic rather than electroquasistatic systems are considered, in this chap-ter electromechanical phenomena are studied from the same viewpoint as in Chap. 5. Material deformationsare again prescribed (kinematic) while the magnetic field sources, the distributions of current or mag-netization density, evolve in a dynamical manner that is self-consistently described throughout thevolume of interest. Most of the discussion in this chapter relates to magnetic diffusion with materialconvection.In practical terms, this chapter takes leave of the windings and associated slip rings or commu-tators used in Chap. 4 to constrain current distributions in moving elements and takes up conductors inwhich the currents seek a distribution consistent with the magnetoquasistatic field laws and the imposedmotion. The magnetic induction machine is an important example. Most often encountered as a rotatingmachine, it might also have as a zioving member a "linear" sheet of metal or even a liquid. The study oftemporal and spatial transients and of boundary layer models in Secs. 6.9-6.11 is pertinent to the linearinduction machines, whether they be applied to train propulsion or manufacture of sheet metal. The"deep conductor" interactions considered in Secs. 6.6 and 6.7 give insights concerning liquid-metal in-duction pumping, a topic continued in Chap. 9.The boundary conditions and transfer relations summarized in Secs. 6.3 and 6.5 are a basic resourcefor developing analytical models representing systems suggested by the case studies of Secs. 6.4 and6.6. Similarly, the dissipation and skin-effect relations developed in Secs. 6.7 and 6.8 are designedto be of general applicability.Much of the magnetic diffusion phenomena developed in this chapter, the mathematical relations aswell as the physical insights, pertain as well to the diffusion of molecules or of heat. Hence, divi-dends from an investment in this chapter are in part collectable in Chap. 9. In addition, what inSec. 6.2 is a theorem concerning the conservation of flux for material surfaces of fixed identity, inChaps. 7 and 9 relates to fluid mechanics and becomes Kelvin's vorticity theorem. Diffusion of vorticity,a momentum transfer process in fluids taken up in Chap. 7,has much in common with magnetic diffusion.The conduction model in this chapter is exclusively ohmic. The model is especially appropriate inthe relatively highly conducting materials of interest if magnetic diffusion effects are an issue.Typically, conductors are solid or liquid metals, or perhaps highly ionized gases. The development ispurposely one that parallels the sections on ohmic conductors in electroquasistatic systems, Secs. 5.10-5.16. A comparative study of electroquasistatic and magnetoquasistatic rate processes, models andexamples results in the recognition of both analogies and contrasts.Although resistive types of induction interactions are by far the most common, time-average forcescan be developed through phase shifts created by other types of loss mechanisms. The important exampleof magnetization hysteresis interactions is used in Sec. 6.12 to exemplify not only how time-averagemagnetization forces can be developed, but by analogy, how polarization interactions can be created inan electroquasistatic context.6.2 Magnetic Diffusion in Moving MediaFor a material at rest in the primed frame of reference, Ohm's law isJI= E' (1)where the conductivity 0 is in general a function of position and time. This law, introduced in Sec. 3.3,implies at least two charge-carrier species and a Hall parameter (Eq. 3.3.4) that is small compared tounity. Use of the field transformations If = -(Eq. 2.5.11b) and E' = + v x poH (Eq. 2.5.12b) ex-presses Eq. 1 in the laboratory frame of reference,Jf = a(E + v x p) (2)where v is the velocity of the material having the conductivity a. This generalization of Ohm's law torepresent conduction in a moving material is clearly valid provided that the material is moving with aconstant velocity. But the law will be used throughout this chapter for materials that are accelerating.The assumption ismade that accelerations have a negligible effect on the processes responsible for theconduction, for example, in a metallic conductor, that the acceleration of the ponderable material hasa negligible effect on electronic motions.Solution of Eq. 2 for E gives an expression that can be substituted into Faraday's law, Eq. 2.3.25b,to obtainSecs. 6.1 & 6.2Vx f B + (x =-- + V x (v •B) (3)where the definition B E Vo(H + M) has been used.The embodiment of Ohm's and Faraday's laws, represented by Eq. 3, has a simple physical signifi-cance best seen by considering the integral form of these same laws. With E' replaced using Eq. 1,Faraday's integral law, Eq. 2.7.3b, becomes÷Jf + dd-" dt B nda (4)C SFig. 6.2.1Surface of fixedidentity.C SIn writing this equation, the surface S enclosed by the contour C, Fig. 6.2.1, is one of fixed identity(one attached to the deforming material), so v = v .(The same expression would be obtained by inte-grating Eq. 3 over a surface of fixed identity and applying the generalized Leibnitz rule, Eq. 2.6.4.)According to Eq. 4, the dissipation of total flux linked by a surface of fixed identity is propor-tional to the "iR" drop around the contour of fixed identity enclosing the surface. The statement is ageneralization of one representing an ideal deforming inductor having the terminal variables (X,i)shorted by a resistance R:dX (5)iR =.dtFig. 6.2.2Circuit equivalentto C in Fig. 6.2.1.In the limit of "infinite" conductivity, the flux intercepted by a surface of fixed identity is invariant.Equations 3 and 4 represent the same laws, so if the left side of Eq. 3 is negligible, it too impliesthat the flux linking a contour of fixed identity is conserved. The circuit helps to emphasize that inmost of
View Full Document
Unlocking...