MIT OpenCourseWare http://ocw.mit.edu Electromechanical Dynamics For any use or distribution of this textbook, please cite as follows: Woodson, Herbert H., and James R. Melcher. Electromechanical Dynamics. 3 vols. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-NonCommercial-Share Alike For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/termsElectromechanical Coupling with Viscous FluidsPROBLEMS14.1. Rework the example of Sec. 14.2.1 with the applied flux density in the X2 direction.Assume that no current can flow in the x3 direction. In particular obtain expressions forvelocity profile (like Eq. 14.2.4), voltage (like Eq. 14.2.7), traction (like Eq. 14.2.10), andtotal power per unit area (like Eq. 14.2.12). Make plots of voltage and loss per unit areafor the constants of Table 14.2.1 and compare the results with those plotted in Fig. 14.2.3.14.2. A viscous liquid flows through a circular pipe, as shown in Fig. 14P.2. At the inletthe pressure is uniform and equal top1,and at the outlet it is still uniform, butp2.The volumerate of flow is Qm/lsec. Under the assumption that the flow is axisymmetric and steady andthat the velocity is low enough that the fluid can be considered incompressible, find thevelocity profile vz(r). Hint. Look for solutions where v = v2(r)iz and p = p(z).Fig. 14P.214.3. The channel shown in Fig. 14P.3 contains a viscous fluid of conductivity a moving inthe xz-direction. You are to analyze this problem using the Hartmann flow solutions(Section 14.2.2). The highly conducting electrodes are connected by a load resistance R.(a) Given the pressure drop from inlet to outlet, the dimensions of the system, fieldBo, and conductivity a, what is the power dissipated in the resistance?(b) What value of R should be used to dissipate the largest possible power in theload?(c) If the fluid is mercury, Bo = 20,000 gauss, d = 1cm, I = 1 m, and w = 10 cm,what is the Hartmann number? What is the value of the optimum resistancefound in part (b)?Perfectly conductingFig.
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