Explanation Given information: The steady, two dimensional incompressible velocity fields is V → = ( u , v ) = ( 1 2 a y 2 + b ) i → + ( − a x y + c ) j → Here, 1 2 a y 2 + b = u and − a x y + c = v . For the irrotational flow the vorticity must be equal to zero. Since, the flow is in the x-y plane the only non zero component in the z -direction. Write the expression for the vorticity of the flow. ζ z = ∂ v ∂ x − ∂ u ∂ y ...... (I) Write the expression for the velocity component in x direction. v = − a x y + c ...... (II) Differentiate the equation (II) with respect to x . ∂ v ∂ x = − ∂ ( − a x y ) ∂ x + ∂ c ∂ x = ∂ ( a x y ) ∂ x + ∂ c ∂ x = a y Write the expression for the velocity component in y direction. u = 1 2 a y 2 + b ...... (III) Differentiate the equation (III) with respect to y. ∂ u ∂ y = ∂ ( 1 2 a y 2 + b ) ∂ y = ∂ ( 1 2 a y 2 ) ∂ y + ∂ ( b ) ∂ y = 1 2 2 a y = a y Substitute a y for ∂ u ∂ y and a y for ∂ v ∂ x in Equation (I). ζ z = a y − a y = 0 Since, the vorticity is zero hence the flow is irrotational. Write the expression for the velocity component in terms of the potential function in y direction. u = ∂ ϕ ∂ x ...... (IV) Write the expression for the velocity component in terms of the potential function in x direction. v = ∂ ϕ ∂ y ...... (V) Substitute 1 2 a y 2 + b for u in Equation (IV). 1 2 a y 2 + b = ∂ ϕ ∂ x

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ISBN: 9781259877766

Author: CENGEL

Publisher: MCG

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Fluid Kinematics

The branch of science that deals with objects which are either in motion or stationary are studied under the branch of classical mechanics. Fluid mechanics is a subcategory of classical mechanics that deals with the behavior of fluids at rest (fluid stat…

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Chapter 10, Problem 58P

To determine

Whether the flow is irrotational or not.

The expression for the velocity potential function for irrotational flow.

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Chapter 10, Problem 57P

Chapter 10, Problem 59P

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Whether the flow is irrotational or not. The expression for the velocity potential function for irrotational flow.

Solution Summary: The author explains the expression for the velocity potential function for irrotational flow. Since the flow is in the x-y plane, the vorticity must be equal to zero.