Explanation Given information: Gap between plate is h , and forced pressure gradient is ∂ P ∂ x . Write the expression forvelocity along x-direction. u = 1 2 μ d P d x ( y 2 − h y ) ........... (I) Here, gap between plates along y-direction is h , fluid viscosity is μ . Write the expression of stream function along y-direction. ∂ ψ ∂ y = u ........... (II) Here, stream function is ψ . Write the expression for stream function along x-direction. ∂ ψ ∂ x = − v ........... (III) Here, velocity along y-direction is v . Substitute 1 2 μ d P d x ( y 2 − h y ) for u in Equation (II). ∂ ψ ∂ y = 1 2 μ d P d x ( y 2 − h y ) ...........(IV) Integrate Equation (IV) partially with respect to y . ψ = ∫ 1 2 μ d P d x ( y 2 − h y ) ∂ y = 1 2 μ d P d x ∫ ( y 2 − h y ) ∂ y = 1 2 μ d P d x ( y 3 3 − h y 2 2 ) + g ( x ) ........... (V) Here, constant of integration is g ( x ) DifferentiateEquation (V) partially with respect to x . ∂ ψ ∂ x = g ' ( x ) ........... (VI) Substitute − v for ∂ ψ ∂ x in equation (VI). g ' ( x ) = − v ........... (VII) Substitute 0 for v in Equation (VII). g ' ( x ) = 0 ........... (VIII) Integrate Equation (VIII) g ( x ) = 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 9, Problem 48P

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The expression for ψ in vertical direction.

The value of stream function along top wall.

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Chapter 9, Problem 47EP

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The expression for ψ in vertical direction. The value of stream function along top wall.

Solution Summary: The author describes the expression for psi in vertical direction, and the value of stream function along top wall.