Question

# a. (50 points) Please read and understand Example 9-13 to Example 9-19. By doing so, you will automatically get 20 points. I hope you all will spend enough time on

this topic, as Navier-Stokes Equation is among the most important equations in fluid mechanics. N-Sequation still remains minimally understood, and is currently one of the 6 unsolved Millennium Prize Problems (one of them was solved in 2003 by Grigori Perelman on Poincaré conjecture). That being said, anyone who can solve the N-S equation fully will get \$1 million dollars awarded by the Clay Mathematics Institute. Want to become a millionaire? For more information, please refer to: b. (50-points) Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe of diameter D or radius R = D/2 inclined at angle a (Figure P9-103, E-book) There is no applied pressure gradient ( = 0). Instead, the fluid flowsƏxdown the pipe due to gravity alone. We adopt the coordinate system shown, with x down the axis of the pipe. Derive an expression for the x-component of velocity u as a function of radius r and the other parameters of the problems (30-point). Calculate the volume flow rate (10 point) and average axial velocity through the pipe (10-point).

Fig: 1

Fig: 2