Explanation Write the expression of Laplace equation in cylindrical coordinate. ∇ ϕ = 1 r ∂ ∂ r ( r u r ) + 1 r 2 ∂ ∂ θ ( r u θ ) + ∂ u z ∂ z ....... (I) Here, the velocity potential function is ϕ , the radius of the cylinder is r , velocity component in radial direction is u r , velocity component in angular direction is u θ , and velocity component in axial direction is u z . Write the expression of component of velocity vector in cylindrical coordinates ( r , θ , z ) and ( u r , u θ , u z ) in radial direction. u r = ∂ ϕ ∂ r ....... (II) Here, the radial component is u r , the radius of cylinder is r , and the velocity potential function is ϕ . Write the expression of component of velocity vector in cylindrical coordinates ( r , θ , z ) and ( u r , u θ , u z ) in angular direction. u θ = 1 r ∂ ϕ ∂ θ ....... (III) Here, the angular component is u θ , the radius of cylinder is r , and the velocity potential function is ϕ . Write the expression of component of velocity vector in cylindrical coordinates ( r , θ , z ) and ( u r , u θ , u z ) in axial direction. u z = ∂ ϕ ∂ z ....... (IV) Here, the radius of cylinder is r , the axial component is u z , and the velocity potential function is ϕ . Write the expression of potential operator ∇ for cylindrical coordinate. ∇ = ( ∂ ∂ r + 1 r ∂ ∂ θ + ∂ ∂ z )

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Chapter 10, Problem 67P

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The Laplace equation is valid in an irrotational region of flow.

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Chapter 10, Problem 66P

Chapter 10, Problem 68P

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The Laplace equation is valid in an irrotational region of flow.

Solution Summary: The author explains the Laplace equation is valid in an irrotational region of flow. Write the expression of velocity vector in cylindrical coordinates.

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The Laplace equation is valid in an irrotational region of flow.

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