p0504 - 1 R 2 ∂ ∂ R p R 2 u R Q 1 R sin φ ∂ u θ...

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5.4. CHAPTER 5, PROBLEM 4 453 5.4 Chapter 5, Problem 4 5.4(a): For incompressible flow, the continuity equation in cylindrical coordinates is 1 r r ( ru r )+ 1 r u θ ∂θ + w z =0 Hence, when u θ = w =0 , all that remains is 1 r r ( ru r )=0 Integrating over r , the most general form of the radial velocity is u r ( r, θ ,z )= f ( θ ,z ) r where f ( θ ,z ) is a function of integration. 5.4(b): For incompressible flow, the continuity equation in spherical coordinates is
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Unformatted text preview: 1 R 2 ∂ ∂ R p R 2 u R Q + 1 R sin φ ∂ u θ ∂θ + 1 R sin φ ∂ ∂φ ( u φ sin φ ) = 0 Hence, when u θ = u φ = 0 , all that remains is 1 R 2 ∂ ∂ R p R 2 u R Q = 0 Integrating over R , the most general form of the radial velocity is u R ( R, θ , φ ) = g ( θ , φ ) R 2 where g ( θ , φ ) is a function of integration....
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