Explanation Given information: Write the expression for Navier-Stokes equation for cylindrical coordinates along θ direction. μ [ ∂ ∂ r ( 1 r ∂ ∂ r ( r u θ ) ) ] = 0 ....... (I) Here, the dynamic viscosity is μ , velocity along θ direction is u θ ,the radius is r . Write the expression for gap between the outer and inner radius. h = R o − R i Here, the gap between the outer and inner radius h , the inner radius is R i and the outer radius is R o . Write the expression for distance from one end of the wall. y = R o − h Write the expression for speed of the upper plate. V = R i ω i Here, the angular velocity at inner radius is ω i . Calculation: Integrate Equation (I). ∫ μ [ ∂ ∂ r ( 1 r ∂ ∂ r ( r u θ ) ) ] = ∫ 0 μ r ∂ ∂ r ( r u θ ) = C 1 ∂ ( r u θ ) = C 1 μ r ∂ r ....... (II) Here, the integration constant is C 1 . Integrate Equation (II). ∫ ∂ ( 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 ......(III) Here, the integration constant is C 2 . Apply boundary condition in Equation (III). Substitute 0 for u θ and R o for r in Equation (III). u θ = C 1 2 μ ( r ) + C 2 r 0 = C 1 2 μ ( R o ) + C 2 R o − C 2 R o = C 1 2 μ ( R o ) C 2 = − ( C 1 2 μ ) ( R o ) 2 ...... (IV) Substitute ω i R i for u θ , − ( C 1 2 μ ) ( R o ) 2 for C 2 and R i for r in Equation (III). u θ = C 1 2 μ ( r ) + C 2 r ω i R i = C 1 2 μ ( R i ) + − ( C 1 2 μ ) ( R o ) 2 R i ω i R i = C 1 2 μ ( R i 2 − R o 2 R i ) C 1 = 2 μ ω i R i 2 R i 2 − R 0 2 Substitute 2 μ ω i R i 2 R i 2 − R 0 2 for C 1 in Equation (IV) To determine (b) The type of flow when the outer wall approaches to infinity and the inner cylinder radius is very small.

Fluid Mechanics: Fundamentals and Applications

4th Edition

ISBN: 9781259696534

Author: Yunus A. Cengel Dr., John M. Cimbala

Publisher: McGraw-Hill Education

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Chapter 9, Problem 95P

To determine

(a)

The velocity profile approach from the outer cylinder wall to the inner cylinder wall.

To determine

(b)

The type of flow when the outer wall approaches to infinity and the inner cylinder radius is very small.

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Chapter 9, Problem 94P

Chapter 9, Problem 96P

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(a) The velocity profile approach from the outer cylinder wall to the inner cylinder wall.

Solution Summary: The author illustrates the Navier-Stokes equation for cylindrical coordinates along theta direction. Write the expression for distance from one end of the wall.