Due: In class, Friday November 9, 2007 Name For instructor or TA use only: Problem Score Points 1 15 2 15 3 20 4 20 5 15 6 15 ME 521Fall Semester, 2007 Homework Set # 9 Professor J. M. CimbalaTotal: 100 1. (15 pts) Text problem 3.8. Note: Draw a sketch of a 3-D fluid element of dimensions (dr, rdθ, dz), then analyze the three components individually, one face at a time. Check your answers with those given in Appendix B1. 2. (15 pts) Consider the incompressible Navier-Stokes equation in planar (2-D) polar coordinates, as given in Appendix B2. Notice that the viscous term, , contains some “extra” terms, involving 2uν∇G2rur−and 22rur∂∂. Have you ever wondered where these extra terms come from? Now you will find out. (a) Show that unit vectors and ereJJGare functions of , and that in fact /ree∂ ∂=JJGJGJand /r∂∂=−GJ. (b) Expand the viscous term carefully, using rruue ue=+GJG, and show that the equations in Appendix B2 are indeed correct, including the “extra” terms.
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potential flow, stream function, Professor J. M. Cimbala, 2-D potential flow