DOC PREVIEW
MIT 6 001 - Streaming Interactions

This preview shows page 1-2-3-4-5-34-35-36-37-68-69-70-71-72 out of 72 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 72 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

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.11Streaming Interactions------L~5~711.1 IntroductionSome of the most significant interactions between continua having large relative velocities in-volve charged particle beams accelerated under near-vacuum conditions. Thus, the first part of thischapter gives some background in electron beam dynamics. The charged particles of Chap. 5 become a con-tinuum in their own right because their inertia is dominant. Section 11.2, on the laws and theorems fora charged particle gas, draws on the fluid mechanics of Chap. 7, and leads to the steady electron flowsconsidered in Secs. 11.3 and 11.4. Flows illustrated in these latter sections are typical of thosefound in magnetrons and in electric and magnetic electron beam lenses. Pictured as they are inLagrangian coordinates, the motions appear to be time varying. But, if viewed in Eulerian coordinates,the electron flows of these sections are steady and might be considered in Chap. 9. The remaining sec-tions relate not only to electron beams, but to electromechanical continua introduced in previouschapters.Sections 11.6-11.10 have as a common theme the use of the method of characteristics to understanddynamics in "real" space and time. The approach is restricted to two dimensions, here one space andthe other time, but makes it possible to investigate such nonlinear phenomena as shock formation andnonlinear space-charge oscillations. Thus it is that these sections are concerned with quasi-one-dimensional models. As pointed out in Sec. 4.12, the small-amplitude limits of these models are identi-cal with the long-wave limits of two- and three-dimensional models. Thus, the quasi-one-dimensionalmodel represents what physical content there is to the dominant modes from the infinite number ofspatial modes of a linear system. However, nonlinear phenomena can be incorporated into the quasi-one-dimensional model.In addition to giving the opportunity to develop nonlinear phenomena, the method of characteristicsgives the opportunity to explore the implications of causality for longitudinal boundary conditions andthe general domain of dependence of a response pictured in the z-t plane. This gives an alternative tothe complex-wave point of view, taken up in the remainder of the chapter, in appreciating the differencebetween absolute and convective instabilities and between evanescent and amplifying waves. The proto-type configurations examined in Sec. 11.10 are analogous to traveling-wave electron beam or beam plasmasystems taken up in later sections.Sections 11.11-11.17 return to a theme of complex waves. Spatial transients in the sinusoidalsteady state are considered in Sec. 5.17 with the tacit assumption that the response decays away fromthe excitation source. As illustrated in these sections, the response could just as well amplify fromthe region of excitation. How is an evanescent wave, which simply decays from the region of excitation,to be distinguished from one that amplifies? Temporal transients are first introduced in Sec. 5.15,and instability, defined as an unbounded response in time, illustrated in Sec. 8.9. In a system thatis infinitely long in the longitudinal direction, a dispersion relation that gives "unstable" w's forreal k's can either imply that the response is unbounded in time at a given fixed location, or thatthere is unlimited growth for an observer moving with the response. In a given situation, how is anabsolute instability to be distinguished from one that is convective? For special hyperbolic systems,these questions are answered in terms of the method of characteristics in Sec. 11.10. Sections 11.11and 11.12 are devoted to the alternative of answering these questions in terms of complex waves. Theremaining sections illustrate with classic examples.BALLISTIC CONTINUA11.2 Charged Particles in Vacuum; Electron BeamsEquations of Motion: In terms of the Eulerian coordinates of Sec. 2.4, Newton's law for a par-ticle having mass m and charge q, subject to the Lorentz force (Eq. 3.2.1), ism(- + v-Vv) = q(E + vx -po ) (1)Multiplied by the particle number density, n, this expression is almost what would be written todescribe a fluid. The pressure and viscous stress terms are absent from Eq. 1. To each point inspace is ascribed the velocity, v, of the particle that happens to be at that point at the giveninstant in time.Because the pressure and viscous stresses are absent, much of the literature of electron beamspictures the motions in Lagrangian terms, as discussed in Sec. 2.4. Then, the initial coordinates ofeach particle are the independent variables as the partial derivative with respect to time is taken:m = q(E + v x ý H) (2)at0A.t11.1Secs. 11.1 & 11.2Thus, for example, in cylindrical coordinates the equations of motion for a particle having the instan-taneous position (r,e,z) ared2r de 2 + Hrd dE (3)dt2 dt m r m oz dt2de d"2r+ 2 dr d8 1 d ( 2 2 -- de = (d Hr dz oH dr-(4)dt2 dt dt r dt ordt oz dt2d z qE -1 H Hr d (5)2 dtm z m or dtwhere the second terms on the left in Eqs. 3 and 4 are respectively the centripetal and Coriolisaccelerations of rigid-body mechanics.The dynamics of interest can be pictured as EQS with an imposed magnetic field. In Sec. 3.4 itis argued that in EQS systems, the magnetic force is negligible compared to the electric force. Now,the particles of interest include electrons or ions in vacuum. Their velocities can easily be largeenough to make magnetic forces due to the imposed field important. The arguments of Sec. 3.4 show thatthe part of the force attributable to a magnetic field induced by the displacement current (or the cur-rent density associated with the accumulation of net charge) is still negligible provided that times ofinterest are long compared to the transit time of an electromagnetic wave.The laws required to complete the description are usually written in Eulerian coordinates, muchas in the description of charged migrating and diffusing


View Full Document

MIT 6 001 - Streaming Interactions

Documents in this Course
Quiz 1

Quiz 1

6 pages

Databases

Databases

12 pages

rec20

rec20

2 pages

Quiz II

Quiz II

15 pages

Streams

Streams

5 pages

Load more
Download Streaming Interactions
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Streaming Interactions and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Streaming Interactions 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?