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.9Electromechanical Flows9.1 IntroductionThe dynamics of fluids perturbed from static equilibria, considered in Chap. 8, illustrate mechan-ical and electromechanical rate processes. Identified with these processes are characteristic times.Approximations are then motivated by recognizing the hierarchy of these times and the temporal range ofinterest. For example, if the response to a sinusoidal steady-state drive having the frequency w is ofinterest, the range is in the neighborhood of T = l/w. Even for a temporal transient, where the naturalfrequencies are at the outset unknown, approximations are eventually justified by seeing where the re-ciprocal of a given natural frequency fits into the hierarchy of characteristic times.In this chapter, it is steady flows and the establishment of such flows that is of interest.Typically, the characteristic times from Chap. 8 are now to be compared to a transport time, Z/u.The recognition of simplifying approximations becomes even more important, because nonlinear equa-tions are likely to be an essential part of a model.The requirement for a static equilibrium that force densities be irrotational is emphasized inSec. 8.2. Taken up in Sec. 9.2 is the question, what types of flow can result from application of suchforce densities? This is the first of 11 sections devoted to homogeneous flows, where such propertiesas the mass density and electrical conductivity are uniform throughout the flow region.Some of the most practical interactions between fields and fluids can be represented by a forcedensity or surface force density that is determined without regard for the fluid motion or geometry.Such imposed surface and volume force density flows are the subject of Secs. 9.3-9.8. In Sec. 9.3,fully developed flows are described in such a way that their application to a wide range of problemsshould be evident. By way of illustration, surface coupled and volume coupled electric and magneticflows are then discussed in Secs. 9.4 and 9.5. Liquid metal magnetohydrodynamic induction pumpsusually fit the model of Sec. 9.5.To appreciate a fully developed flow, it is necessary to consider the flow development. InSec. 9.6, this is done by examining the temporal transient that results as a closed system is suddenlyturned on and the steady flow allowed to establish itself. Then, in terms of boundary layers, thespatial transient is discussed. In addition to its application to surface coupled flows, illustratedin Sec. 9.7, the boundary layer model is applied to a self-consistent bulk coupled flow in Sec. 9.12.In Secs. 9.6 and 9.7, viscous diffusion is of interest, both fluid inertia and viscosity are import-ant and times of interest are, by definition, on the order of the viscous diffusion time.Illustrated in Sec. 9.8 are an important class of electromechanical models in which the bulk flowis described by linear equations. Here, transport times are long compared to the viscous diffusiontime and "creep flow" prevails.The self-consistent imposed field flows of Secs. 9.9-9.12 give the opportunity to broaden therange of dynamical processes. In the first two of these sections, magnetohydrodynamic processes aretaken up. The magnetic diffusion time is short compared to the other times of interest, the viscousdiffusion time and the magneto-inertial time. These sections first illustrate how the field altersfully developed flows and then considers how the electromechanics contributes to temporal flow develop-ment. The electrohydrodynamic approximation discussed and illustrated in the last two sections of thispart is based on having a self-precipitation time for unipolar charges that is long compared to othertimes of interest, for example, an electroviscous time.With the introduction of inhomogeneity come more characteristic dynamical times. These areillustrated for systems having a static equilibrium and abrupt discontinuities in properties inSecs. 8.9-8.16. Typically, the associated characteristic times represent propagation of surfacewaves. Smoothly distributed inhomogeneities, Secs. 8.17-8.18, give rise to related internal waveswith their characteristic times. The flow models developed in Sec. 9.13 and illustrated in Sec. 9.14incorporate wave phenomena similar to those from Chap. 8. The wave phenomena show up in steady flowsituations through critical conditions, often expressed in terms of the ratio of a convective velocityto a wave velocity, i.e., as a Mach number. In essence these numbers are the ratio of transport timesto wave transit times. Times of interest in these sections, which reflect the existence of waves, area capillary time 'T = y/'Pi2 (Sec. 8.9), a gravity time Tg = /g (Pb -9 Pa)/(Pb + Pa)(Sec. 8.9) andvarious magneto- and electro-inertial times (Secs. 8.10-8.15).In view of Sec. 8.8 on magneto-acoustic and electro-acoustic waves, it should be expected thatadditional times introduced in the remaining sections on compressible flow are the transit times foracoustic and acoustic related waves. Sections 9.15 and 9.17 bring into the discussion the additionalphysical laws required to represent interactions with the internal energy subsystem of a gas. Here,the energy equation is derived and thermodynamic variables needed in subsequent sections defined. Theselaws are not only necessary for the description of thermal-to-electrical energy conversion (to be takenSec. 9.1up in Secs. 9.21-9.23), but also can be used to describe convective heat transfer.The quasi-one-dimensional model introduced in Sec. 9.19 is the basis for the various energy con-version systems discussed in the remaining sections. Once again, even in steady flows, the role of wavepropagation is unavoidable. As in Sec. 9.14, flow through energy conversion devices is dependent on thefluid velocity relative to a wave velocity. This time, the waves are acoustic related.In Secs. 9.21 and 9.23, the energy conversion process is again highlighted. These
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