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 (1). ∫ 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: 2810022150991

Author: CENGEL

Publisher: MCG

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Chapter 9, Problem 97P

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The expression for velocity component as a function of radius.

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Chapter 9, Problem 96P

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Question 2 (40 Points) Consider the following double pendulum-like system with links ₁ and 12. The angles 0 and & could have angular velocities ėêk and êk, respectively, where ②k is a unit vector that points out of the page and is perpendicular to x and y. They could also have angular accelerations Ök and êk. The angle is defined relative to the angle 0. The link 12 is a spring and can extend or compress at a rate of 12. It can also have a rate of extension or compression Ï2. li y êr1 êe 12 χ 3 еф er2 ده لج 1) Express the velocity of the mass in terms of the unit vectors ê0, êr1, êø, and êr2, and any extension/contraction of the links (e.g.,. i; and Ï¿) (12 Points) 2) Express the acceleration of the mass in terms of the unit vectors ê¤, ê×1, êp, and êÃ2, and any extension/contraction of the links (e.g.,. İ; and Ï¿) (12 Points) 3) Express the velocity of the mass in terms of unit vectors î and ĵ that point in the x and y directions, respectively. Also include the appropriate,…

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The expression for velocity component as a function of radius.

Solution Summary: The author explains the expression for velocity component as a function of radius and the Navier-Stokes equation.

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