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.Problems for Chapter 5For Section 5.3:Prob. 5.3.1 For flow and field that are two-dimensional and representedthe definitions suggested by Table 2.18.1, show that lines along which tTare represented by Eq. 5.3.13a.Prob. 5.3.2 For flow and field that are axisymmetric in cylindrical cocthat case in Table 2.18.1, show that lines along which the charge densit3Eq. 5.3.13b.For Section 5.4:Prob. 5.4.1 Gas passes through the planarchannel shown in Fig. P5.4.1 with the velocity4U(x/d)[1 -x/d]iL. An electric field isimposed by placing the lower plane at poten-tial V relative to the upper one. Betweentx= 0 and 4= a on this lower plane, posi-tively charted particles having mobility bare injected through a metallic grid. Agoal is to determine the current i collectedby an electrode imbedded opposite the injec-tion grid. It is presumed that the potentialof this electrode remains essentially zero.(a) Use the result of Prob. 5.3.1 to showthat the injected particles follow thecharacteristic linesU 2 2x bV-2 - x (1 - -) + y = constantd 3d d(b) Show that the current-voltage relation isbV 2 Ud[a 3 (bV/d)0,itýFig. P!, V > 2-Ud2/baV (< Ud2/ba3Prob. 5.4.2 The potential of a spherical particle having radiusR is constrained to be0(r=R) = Vcos8(This could be accomplished by making the surface from electrodesegments, properly constrained in potential.) The sphere is sur-rounded by fluid generally moving in the z direction. The flowis solenoidal and irrotational, consistent with its being inviscidand entering at z + -m without.rotation. (See Fig. P5.4.1. Suchflows are taken up in Chap. 7.) The fluid flow velocity is givenv -rv ; R=]cosev = = -UR [ 2There are no other sources of field than those on the sphereitself. The following steps establish the electrical current onthe sphere created by ions entering uniformly with the fluid atZ + -m(a) Assume that the contribution of the ion space charge to the field isand v in terms of AE and AV.5.71Prob. 5.4.2 (continued)(b) Find the expression for the particle trajectories in the formr Vbf(R' 8, UR) = constant(c) Assume that V > 0 and that the ions are positive. Find the critical points in the region outsidethe sphere.3 3(d) Plot the characteristic lines in two cases: for bV/RU < -and for bV/RU > 1. Identify thecritical points in the case where they exist in the region outside the sphere.(e) Find the current i to the particle as a function of bV/RU. (Be sure to identify any "break points"in this V-i relation.Prob. 5.4.3 A circular cylindrical conductor havingradius a has the potential V relative to a surroundingcoaxial cage having radius Ro (Fig. P5.4.3). Hence it -imposes an electric field E = (V/r)/ln(Ro/a) on the air ,in the region a < r < Ro.The wind passing perpendicular -"to this conductor has the velocity /2 2v = -U(1 -cos ri + U(1 + ) sin 0 i2 r 2 er rconsistent with an inviscid model. (Thus, there is a finitetangential wind velocity at the surface of the conductor.)Charged particles enter uniformly at the appropriate \%"infinity." This might be a model for the contamination %, ROof a high-voltage d-c conductor by naturally charged dust.(a) Consider two cases: (i) conductor and particles of thesame polarity and, (ii) conductor and particles of oppo-site polarity. This is equivalent to taking the particles Fig. P5.4.3as positive and V as positive or negative. Find the criticalpoints (lines).(b) Find the characteristic lines and sketch them for the two cases.(c) Determine the electrical current to the conductor as a function of V.Prob. 5.4.4 Fluid enters the region between the electrodes shown in Fig. P5.4.4 through a slit at thetop (where x = c). The system extends a length k into the paper and the volume rate of flow throughthe slit is Qv m3/sec. The electrodes to left and rightrespectively are located at xy = -a2and xy = a2 andhave the constant potentials -Vo and Vo.The elec-trodes in the plane x = 0 are essentially grounded,with the one between x = -a and x = a used to collectthe current i. Entrained in the gas as it enters atx = c is a charge density that is uniform over thecross section at that location. The charge densityis po. The fluid velocity is4-_t tv = 2C(xix -yiy)(a) What is the constant C?(b) Find the critical lines, if any.(c) Given a certain volume rate of flow Qv, find thecurrentL I L Lhe centLe eIecLrode as a LunICL onof bV,,where b is the mobility of the charged Fig. P5.4.4particles. Present i(bVo) as a dimensionedsketch. (Assume that Qv and Vo, as well as the charge density po, are positive.)For Section 5.5:Prob. 5.5.1 For a "drop" in an ambient electric field and flow as discussed in this section, bothpositive and negative "ions" are present simultaneously. The objective here is to make a chargingdiagram patterned after those of Figs. 5.5.3 and 5.5.4. Because there are now two differentProblems lor Chap. 5IIIIrI5.72Prob. 5.5.1 (continued)mobilities, b+ and b_, it is best to make the abscissa the imposed electric field E. Construct thecharging diagram, including charging trajectories, showing final values of charge. (With bipolarcharging, the final charge can be less than qc in magnitude. Expressions should be derived for theselimiting values of charge.)Prob. 5.5.2 The objective is to determine the charging diagrams, Figs. 5.5.3 and 5.5.4, with the lowReynolds number flow represented by Eq. 5.5.5 replaced by an inviscid flow. (See Sec. 7.8 for discussionof this class of flows.) Important here is the fact that such a flow can have a finite tangential veloc-ity on a rigid boundary. The fluid velocity is given here asR3v =-U[ -R3 os + U[+ ] sinr r 2r(a) Find A and the general characteristic equation that replaces Eq. 5.5.6.(b) Because both tangential and normal velocity are zero on the surface of the "drop" for the lowReynolds number flow, the points on the surface described by Eq. 5.5.10 are critical points.With an inviscid flow, matters are not so simple. Show that, as before,
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