Explanation Given information: The flow is two dimensional, incompressible and steady. Write the expression for stream function of r component of velocity. u r = V ˙ / L 2 π r ...... (I) Here, volumetric flow rate is V ˙ ,length of flow is L and radius of flow channel is r . Substitute 1 r ∂ ψ ∂ θ for u r in equation (I). 1 r ∂ ψ ∂ θ = V ˙ / L 2 π r ∂ ψ ∂ θ = V ˙ / L 2 π ∂ ψ = V ˙ / L 2 π ∂ θ ...... (II) Integrate the Equation (II). ∫ ∂ ψ = ∫ V ˙ / L 2 π ∂ θ ψ = V ˙ / L 2 π ( θ ) + f ( r ) ...... (III) Here, arbitrary constant function of integration is f ( r ) . Differentiate equation (III) with respect to r . ∂ ψ ∂ r = ∂ ∂ r [ V ˙ / L 2 π ( θ ) + f ( r ) ] ∂ ψ ∂ r = f ′ ( r ) ...... (IV) Substitute − u θ for ∂ ψ ∂ r in equation (IV). − u θ = f ′ ( r ) f ′ ( r ) = − u θ ...... (V) Substitute 0 for − u θ in Equation (V). f ′ ( r ) = 0 ....... (VI) Integrate Equation (VI) with respect to r ∫ f ′ ( r ) = ∫ 0 f ( r ) = C ...... (VII) Here, constant of integration is C . Substitute 0 for f ( r ) in Equation (III). ψ = V ˙ / L 2 π ( θ ) + 0 = V ˙ / L 2 π ( θ ) ...... (VIII) Here, f ( r ) is 0 for the stream line at origin. Write the expression for velocity potential function of radial component of velocity. u r = V ˙ / L 2 π r

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ISBN: 2810022150991

Author: CENGEL

Publisher: MCG

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Chapter 10, Problem 60P

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The velocity potential function.

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The stream function. The velocity potential function.

Solution Summary: The author explains the stream function and velocity potential function. The flow is two dimensional, incompressible and steady.