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.8Statics and Dynamics of SystemsHaving a Static Equilibriumli-~· .8.1 IntroductionIn general, it is not possible for a fluid to be at rest while subject to an electric or magneticforce density. Yet, when a field is used to levitate, shape or confine a fluid, it is a static equi-librium that is often desired. The next section begins by identifying the electromechanical conditionsrequired if a state of static equilibrium is to be achieved. Then, the following three sectionsexemplify typical ways in which these conditions are met. From the mathematical viewpoint, the subjectbecomes more demanding if the material deformations have a significant effect on the field. Thesesections begin with certain cases where the fields are not influenced by the fluid, and end with modelsthat require numerical solution.The magnetization and polarization static equilibria of Sec. 8.3 also offer the opportunity toexplore the attributes of the various force densities from Chap. 3, to exemplify how entirely differentdistributions of force density can result in the same incompressible fluid response and to emphasizethe necessity for using a consistent force density and stress tensor.Given a static equilibrium, is it stable? This is one of the questions addressed by the remainingsections, which concern themselves with the dynamics that result if an equilibrium is disturbed. Sometypes of electromechanical coupling take place in regions having uniform properties. These are exem-plified in Secs. 8.6-8.8. However, most involve inhomogeneities. The piecewise homogeneous modelsdeveloped in Secs. 8.9-8.16 are chosen to exemplify the range of electromechanical models that can bepictured in this way.The last sections, on smoothly inhomogeneous systems, serve as an introduction to a viewpointthat could equally well be exemplified by a range of electromechanical models. Once it is realizedthat the smoothly inhomogeneous systems can be regarded as a limit of the piecewise inhomogeneous sys-tems, it becomes clear that all of the models developed in this chapter have counterparts in this domain.The five electromechanical models that are a recurring theme throughout this chapter are sum-marized in Table 8.1.1.Table 8.1.1. Electromechanical models.Model ApproximationMagnetization (MQS) or polarization (EQS) No free current or chargeInstantaneous magnetization or polarizationFlux conserving (MQS) T << TCharge conserving (EQS) T << T or TmigInstantaneous magnetic diffusion (MQS) T >> TmInstantaneous charge relaxation (EQS) T >> Te or TmigMagnetization and polarization models for incompressible motions require an inhomogeneity in mag-netic or electric properties. The remaining interactions involve free currents or charges which gener-ally bring in some form of magnetic diffusion or charge relaxation (or migration). How such rateprocesses come into the electromechanics is explicitly illustrated in the sections on homogeneous sys-tems, Secs. 8.6 and 8.7. However, in the more complex inhomogeneous systems, the last four models ofTable 8.1.1 not only result in analytical simplifications, but give insights that would be difficultto glean from a more general but complicated description. "Constant potential" continua fall in thecategory of instantaneous charge relaxation models.STATIC EQUILIBRIA8.2 Conditions for Static EquilibriaOften overlooked as an essential part of fluid mechanics is the subject of fluid statics. A re-minder of the significance of the subject is the equilibrium between the gravitational force densityand the hydrostatic fluid pressure involved in the design of a large dam. On the scale of the earth'ssurface, where g is essentially constant, the gravitational force acting on a homogeneous fluidobviously is of a type that can result in a static equilibrium.Except for scale, electric and magnetic forces might well have been the basis for Moses' partingof the Red Sea. Fields offer alternatives to gravity in the orientation, levitation, shaping orSecs. 8.1 & 8.2Fig.8.2.1.(a)Electricfieldusedtoshapea"lens"ofconductingliquidrestingon apoolofliquidmetal.Moltenplasticsandglassaresufficientlyconductingthattheycanbere-~l(0)(b)«[9I(c)(d)gardedas"perfect"conductors.(b)Polarizationforcesusedtoorientahighlyinsulatingliquidinthetopofatankregardlessofgravity.The schememightbeusedforprovidinganartificialbottomincryogenicfuelstoragetanksunderthezero-gravityconditionsofspace.(c)Liquidmetallevitatorthatmakesusedofforcesinducedbyatime-varyingmag-neticfield.Athighfrequencies,thefluxisexcludedfromthemetal,andhencethefieldstendtowardaconditionofzeroshearingsurfaceforcedensity.(d)Cross-sectionalviewofaxisymmetricmagneticcircuitandmagnetizableshaftwithmagnetizablefluidusedtosealpenetrationofrotatingshaftthroughvacuumcontainment.1-3otherwisecontrollingofstaticfluidconfigurations.ExamplesareshowninFig.8.2.1.Forwhatforcedistributionscaneachelementofafluidbeinstaticequilibrium?Iftheex-ternalelectricormagneticforcedensityisFe,thentheforceequationreducesto++-V'(p -pg·r)(1)ThisexpressionisalimitingformofEq.7.4.4withthevelocityzero.Evenifeffectsofviscosity1.J.R.Melcher,D.S. Guttman andM.Hurwitz,"Die1ectrophoreticOrientation,"J.SpacecraftandRocketsi,25(1969).2.E.C.Okressetal.,"ElectromagneticLevitationofSolidandMoltenMetals,"J.Appl.Phys.Q,545(1952).3.R.E.Rosensweig,G.Misko1czy andF.D.Ezekiel,"Magnetic-FluidSeals,"MachineDesignMarch281968. ' ,Sec.8.28.2are included in the model, because v = 0, Eq. 1 still represents the static equilibrium. Thus, it isalso the static limit oZ Eq. 7.4.4. The curl of a gradient is zero. So, the curl of Eq. 1 gives anecessary condition on Fe for static equilibrium:V x Fe = 0 (2)To achieve a static equilibrium, the force density must be the gradient of a scalar, -VS. Then Eq. 1becomesV(p -pg'r + S) = 0 (3)which will be recognized as Eq. 7.8.4 in the limit v = 0.More often than not, in an electromagnetic field a fluid does not reach a static
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