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323 Chapter 4 x Differential Relations for a Fluid Particle 4.35 From the Navier-Stokes equations for incompressible flow in polar coordinates (App. E for cylindrical coordinates), find the most general case of purely circulating motion XT(r), Xr Xz 0, for flow with no slip between two fixed concentric cylinders, as in Fig. P4.35. υθ (r) r r=a No slip r=b Fig. P4.35 Solution: The preliminary work for this problem is identical to Prob. 4.32 on an earlier page. That is, there are two possible solutions for purely circulating motion XT(r), hence C1r vT C2 , subject to vT (a) 0 r This requires C1 C2 0, or vT Cy(2h – y) v w C1b C2 /b 0 (no steady motion possible between fixed walls) Ans. 4.36 A constant-thickness film of viscous liquid flows in laminar motion down a plate inclined at angle T, as in Fig. P4.36. The velocity profile is u C1a C2 /a and vT (b) 0 y g h 0 (y) Find the constant C in terms of the specific weight and viscosity and the angle T. Find the volume flux Q per unit width in terms of these parameters. θ x Fig. P4.36 Solution: There is atmospheric pressure all along the surface at y h, hence wp/wx The x-momentum equation can easily be evaluated from the known velocity profile: § wu wu· v ¸ © wx wy¹ U¨u wp Ug x P 2 u, or: 0 0 Ug sinT + P (2C) wx U g sinT Solve for C Ans. (a) 2P The flow rate per unit width is found by integrating the velocity profile and using C: h Q ³ 0 h u dy ³ 0 Cy(2h y) dy 2 3 Ch 3 U gh 3 sinT per unit width Ans. (b) 3P 0.