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.7Laws, Relations Laws, Approximations andRelations of Fluid MechanicsApproximations andof Fluid MechanicsI77.1 IntroductionThe following. chapters carry the subject of continuum electromechanics to its third level. Notonly do the field sources assume distributions consistent with deformations of the support medium, themedium is itself free to respond to the associated electromagnetic forces. For gases and liquids, aswell as fluid-like continua such as certain plasma models and electron beams, this response must be con-sistent with the mechanical laws and relations now derived. The role of this chapter is the mechanicalanalogue of the electromagnetic one played by Chap. 2.The chapter is organized so that Secs. 7.2-7.9 are sufficient background in incompressible inviscidfluid mechanics to proceed directly with related electromechanical studies. An even wider range of elec-tromechanical coupling mechanisms than might be imagined at this point are tied to fluid interfaces.This makes fluid interfaces (Sec. 7.5), surface tension (Sec. 7.6) and jump conditions (Sec. 7.7) ap-propriate for early discussion.Compressibility and related acoustic phenomena are taken up in Secs. 7.10-7.12. Then, contribu-tions of fluid friction, the consequence of fluid viscosity, are taken up in Secs. 7.13-7.17. Theresulting Navier-Stokes's equations are summarized in Sec. 7.16.Overlaying the derivation of the laws of fluid mechanics is the development of relations that playa role in the following chapters for describing the continuum mechanics that is analogous to that forthe electric and magnetic transfer relations in the preceding chapters. Transfer relations describingan incompressible and inviscid inertial continuum (Sec. 7.9) will be used many times. Also for futurereference are the relations of Sec. 7.11, which embody the acoustic phenomena associated with compres-sibility, those of Sec. 7.19, which establish the interplay between viscous and inertial effects, andof Sec. 7.20, which describe "creep flow," in which fluid friction overwhelms inertia.Viscous diffusion, the diffusion of vorticity, has considerable analogy to magnetic diffusion.Thus, the studies of Chap. 6 are a useful background for understanding the interplay of inertia and fluidfriction.This chapter is largely concerned with general laws and relations. The chapters which follow makeextensive use of these results in specific case studies.Chapter 2 begins with a discussion of the two quasistatic limits of the general laws of electro-dynamics, identifying rate processes brought in by electrical dissipation in each of these approxima-tions. This chapter ends with a similar discussion.7.2 Conservation of MassWith the mass per unit volume of a continuous medium defined as p, a statement of mass conservationfor a volume V of fixed identity isd f pdV = 0 (1)Here, the volume V is defined such that it always encloses the same material. The surface S enclosingthe materials therefore moves with the material, and the velocity v is the velocity of surface and mate-rial alike.With the integral theorem of Eq. 2.6.5, it is possible to express Eq. 1 as the integral form ofmass conservation:t (2)dV + pv.nda = 0 V SWritten in this form, the law applies for V and S either fixed or enclosing material of fixed identity.Using Gauss' theorem, the surface integral can again be expressed as a volume integral, so that the equa-tion involves one integral over the volume, V. Because V is arbitrary, it follows that the integrandmust vanish:pV.~= Vp+ 0 (3)DtThis is the required differential law of mass conservation.Incompressibility: If fluid motions are typified by times that are long compared to the transittime of an acoustic wave through a length typifying the system, for important classes of flows the massdensity in the vicinity of a given fluid particle remains constant. In view of the definition of theJSecs. 7.1 & 7.2convection derivative, Sec. 2.4, this means that0DtFor incompressible motions, the mass density evolves much as the free charge density in an insulatingfluid (Sec. 5.10). If fluid particles of interest originate where the mass density is uniform, itdensity in the region occupied by this same fluid at a later time is also uniform.follows that the mass case.or "uniform" density 4, p constant, is a special "homogeneous" = the solution to Eq. Thus, of From Eqs. 3 and 4 it follows from conservation mass that for an incompressible fluidV*. = 0 (5)whether the fluid is homogeneous or not.The quasistatic nature of the incompressible model is investigated in Secs. 7.12 and 7.22.7.3 Conservation of MomentumBecause momentum is a vector field, rather than a scalar one, it is convenient to deal with itsindividual components in Cartesian coordinates. Of course, this in no way restricts the validity ofthe resulting equation of motion.Again, with the understanding that the volume V always encloses the same material, and hence thatits surface deforms with the local velocity of the material, conservation of momentum for the ith com-ponent isd vidV = FidV (1)The integral on the right represents contributions to the total force acting on the volume thatcome from the surrounding material (viscous and pressure forces) and from "external" sources, such asgravity and electromagnetic fields.Use of the integral theorem, Eq. 2.6.5, gives the integral law for conservation of momentum:dV + pv.at 1vnda = FidV (2)SGauss' theorem, Eq. 2.6.2, makes possible a conversion of the surface integral to a volume integral:Jv(- + V.pvi4)dV = FidV V at fV I (3)iExpansion of terms on the left givesv[ + + p .vi dV = vFidV (4)Again, the integrand of the volume integrations collected together must vanish, but note that conservationof mass, Eq. 7.2.3, requires that the first term in brackets vanish. Thus, the differential law repre-senting conservation of momentum isav 4.p [ + v V =Vv] F (5)On the left is
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