Explanation Given information: The irrotational line vortex of strength Γ in the x y or r θ , the velocity components are u r = ∂ ϕ ∂ r = 1 r ∂ ϕ ∂ θ = 0 and u θ = 1 r ∂ ϕ ∂ θ = − ∂ ϕ ∂ r = Γ 2 π r . Write the expression of radial component of velocity. u r = 0 ....... (I) Now, radial component of velocity potential function is 0 , so the velocity potential function is the function of θ . Write the expression of velocity potential function. ϕ = f ( θ ) ....... (II) Here, the velocity potential function is ϕ and angular function is f ( θ ) . Write the expression of angular component of the velocity. u θ = 1 r ∂ ϕ ∂ θ ....... (III) Here, the velocity potential function is ϕ , and the radius of cylinder is r , the Substitute Γ 2 π r for u θ in Equation u θ in Equation (III) Γ 2 π r = 1 r ∂ ϕ ∂ θ Γ 2 π = ∂ ϕ ∂ θ ∂ ϕ = ( Γ 2 π ) ∂ θ Integrate both side of the above equation. ∫ ∂ ϕ = ∫ ( Γ 2 π ) ∂ θ ϕ = ( Γ 2 π ) θ + C Neglect the integration constant C

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 10, Problem 68P

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The expression for the velocity potential function and stream function for the line vortex.

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Chapter 10, Problem 67P

Chapter 10, Problem 69P

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Fluid Mechanics: Fundamentals and Applications

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The expression for the velocity potential function and stream function for the line vortex.

Solution Summary: The author describes the velocity potential function and stream function for the irrotational line vortex of strength Gamma .

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The expression for the velocity potential function and stream function for the line vortex.

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Chapter 10 Solutions