Explanation Given information: Write the expression for the Navier-Stokes equation. μ [ ∂ ∂ r ( 1 r ∂ ∂ r ( r u θ ) ) ] = 0 ....... (I) Here, the dynamic viscosity is μ , the radius is r and the velocity along θ is u θ . Calculation: Integrate Equation (I). ∫ μ [ ∂ ∂ r ( 1 r ∂ ∂ r ( r u θ ) ) ] = ∫ 0 μ r ∂ ∂ r ( r u θ ) = C 1 ....... (II) μ r ∂ ∂ r ( r u θ ) = C 1 ∂ ( r u θ ) = C 1 μ r ∂ r ....... (III) Here, the integration constant is C 1 . Integrate Equation (3). ∫ ∂ ( r u θ ) = ∫ C 1 μ r ∂ r r u θ = C 1 μ ( r 2 2 ) + C 2 u θ = C 1 2 μ ( r ) + C 2 r ....... (IV) Here, the integration constant is C 2 . Apply boundary condition. Substitute ω i R i for u θ and R i for r in Equation (IV). ω i R i = C 1 2 μ ( R i ) + C 2 R i ....... (V) Apply boundary condition. Substitute ω o R o for u θ and R o for r in Equation (IV). ω o R o = C 1 2 μ ( R o ) + C 2 R o ω o R o 2 = C 1 R o 2 + C 2 2 μ C 2 = 2 μ ω o R o 2 − C 1 R o 2 ....... (VI) Substitute 2 μ ω o R o 2 − C 1 R o 2 for C 2 in Equation (V). ω i R i = C 1 2 μ ( R i ) + 2 μ ω o R o 2 − C 1 R o 2 R i ω i R i 2 = C 1 2 μ ( R i ) 2 + 2 μ ω o R o 2 − C 1 R o 2 C 1 = 2 μ ( ω o R o 2 − ω i R i 2 ) R o 2 − R i 2 Substitute 2 μ ( ω o R o 2 − ω i R i 2 ) R o 2 − R i 2 for C 1 in Equation (VI)

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ISBN: 9781266016042

Author: CENGEL

Publisher: MCG CUSTOM

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Chapter 9, Problem 96P

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The expression for velocity component as a function of radius.

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