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 7For Section 7.2:Prob. 7.2.1 In Sec. 3.7, ci is defined such that in the conservative subsystem, Eq. 3.7.3 holds.Show that ai satisfies Eq. 7.2.3 with p+ai. Further, show that if a "specific" property Si is definedsuch that Bi pai, then by virture of conservation of mass, the convective derivative of Si is zero.For Section 7.6:Prob. 7.6.1 Show that Eq (hb of Tah1P 7_6_2 is correctProb. 7.6.2 Show that Eqs. (j) and (2) fromTable 7.6.2 are correct.Prob. 7.6.3 A pair of bubbles are formed with thetube-valve system shown in the figure. Bubble 1 isblown by closing valve V2 and opening Vl. Then, Vlis closed and V2 opened so that the second bubble isfilled. Each bubble can be regarded as having a con-stant surface tension y. With the bubbles having thesame initial radius Eo, when t = 0, both valves areopened (with the upper inlet closed off). The objectof the following steps is to describe the resultingdynamics.(a) Flow through the tube that connects the bubblesis modeled as being fully developed and viscousdominated. Hence, for a length of tube k havinginner radius R and with a viscosity of the gas , Fig. P7.6.3the volume rate of flow is related to the pressure difference byR4 (Pa-Pb) 3Qv = 8 .m /secThe inertia of the gas and bubble is ignored, as is that of the surrounding air. Find an equationof motion for the bubble radius E1 .(b) With the bubbles initially of equal radius Eo, there is a slight departure of the radius of oneof the bubbles from eauilibrium. What hapDens?(c) In physical terms, explain the result of (b).For Section 7.8:Prob. 7.8.1 A conduit forming a closed loop consists of a pair oftubes having cross-sections with areas Ar and Ay .These are arrangedas shown with a fluid having density pb filling the lower half and asecond fluid having density Pa filling the upper half. The object ofthe following steps is to determine the dynamics of the fluid, speci-fically the time dependence of the interfacial positions Er and E .(a) Use mass conservation to relate the displacements (Er', E ) toeach other and to the fluid velocities (vr, vk) on the rightand left respectively. Assume that the fluid is inviscidand has a uniform profile over the cross-section of a tube.(b) Use Bernoulli's equation, Eq. 7.8.5 to relate quantitiesevaluated at the interfaces in the lower fluid, and in theupper fluid.(c) Write the boundary conditions that relate quantities acrossthe interfaces.(d) Show that these laws combine to give an equation of motionfor the right interface having the formd2r i dr 2 d 2 Fig.P7.m 2+2(Pb Pa) - (-) + Kr = 0 Fig. P7.Problems for Chap. 7uc7.43Prob. 7.8.1 (continued)What are the effective mass per unit length, m, and "spring-constant" K?(e) Now, assume that the departures from equilibrium are small (linearize) and determine thenatural frequencies of the system. Under what conditions will the system be unstable?(f) A U tube is filled with water and open to the air. With a length of water in the tube (of uniformcross-section), Z, what are the natural frequencies?Prob. 7.8.2 A hemispherical object rests on a flat plate.Fluid passes over and around the sphere with a velocity thatis to be determined. The flow is uniform but a function of g1time far from the hemisphere. 1rV= r(a) Note Eq. 7.8.11 and subsequent discussion. Find theinviscid velocity and velocity potential on the hemi-spherical surface. Z(b)Find the pressure distribution on the hemisphere.(c) Whatthatthatwereis the lift force on the hemisphere? (Assume Fig. P7.8.2the pressure inside the sphere is the same asat the stagnation point r = R, 9 = i just outside the sphere, as would be the case if therea small hole through the shell at this point.)Prob. 7.8.3 An electromagnetic rocket constrained by a test stand is shown in the figure. In theinterior region there is a space occupied by an apparatus that produces a normal surface force densityTn on the surface Si. A tube connects this space to the outside, and hence equalizes the pressuresinside Si and outside the rocket. The fluid inside Si and outside the rocket has negligible massdensity. There are no external forces in the fluid bulk. Thus the pressure in the surroundinghomogeneous fluid is p=Tn.The volume is largeenough that the fluid inside the rocket hasnegligible velocity and an essentially steady Aflow condition prevails. It is expelled through .the throat and reaches a point where its veloc-:. n .J "ity U is essentially uniform and x directed; .the pressure is equal to that of the surround-." P=ings (say p=0O) and the cross-sectional area isA. Gravitational effects are negligible. UseEqs. 7.8.5, 7.3.2 and 7.4.3 to find the total Fig. 7.8.3force on the rocket in terms of Tn and A.For Section 7.9:Prob. 7.9.1 In Sec. 2.17, conservation of electric energy is used to derive reciprocity conditions forthe flux-potential transfer relations. The object here is a similar derivation for the transfer rela-tions of Table 7.9.1 based on conservation of kinetic energy. Start with the assumption that for aninviscid incompressible fluid having uniform mass density, the change in kinetic energy is the resultof displacements at the a and 8 planes.S(Wkin) -p6. ndaDerive reciprocity conditions similar to Eq. 2.17.10.Prob. 7.9.2 An annular region of incompressibleinviscid fluid is bounded by outer and inner coaxialboundaries of radius a and 8 respectively, asshown in the figure. Hence, the configurationis similar to the circular cylindrical case ofTable 7.9.1. However, rather than being in astate of uniform axial motion when in equilibrium,the fluid here is rotating. This equilibriumrotation is rigid body and could be establishedby spinning a cylinder of fluid for a long enoughtime that viscous shear stresses could transmitthe motion to the fluid volume. Ignore gravita-tional effects.Fig. P7.9.2Problems 'for Chap. 7zK~dIn`FýOýr7.44'~Prob. 7.9.2 (continued)(a)What is the vorticity of the
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