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# hw_05 - Due In class Friday October 5 2007 Name ME 521 Fall...

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Due: In class, Friday October 5, 2007 Name For instructor or TA use only: Problem Score Points 1 15 2 15 3 50 4 20 ME 521 Fall Semester, 2007 Homework Set # 5 Professor J. M. Cimbala Total: 100 1 . (15 pts) Text problem 3.1 . For consistency, do three parts: ( a ) Show that x 2 y 2 = constant along streamlines two ways: (i) Integrate Eq. (3.37). (ii) Integrate Eq. (3.36) to find an equation for ψ as a function of x and y , then show that x 2 y 2 = constant along lines of constant ψ . ( b ) Prove that this velocity field is indeed irrotational. ( c ) Sketch a few streamlines. 2 . (15 pts) In class, we used a surface integral and Gauss’s law for a material volume to show that the total rate of work per unit volume done by surface stresses on a fluid is equal to ( 1 1,1 1 u u dx + 3 3,1 1 u u dx + ) ji i j u x τ . Derive the same expression by considering the work done on each face of an infinitesimal rectangular fluid element of dimensions dx 1 by dx 2 by dx 3 . For consistency, use the sketch provided here as a guide, where the velocities and stresses on the two faces normal to the x 1

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hw_05 - Due In class Friday October 5 2007 Name ME 521 Fall...

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