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Ae311_fall_2011_hw7

# Ae311_fall_2011_hw7 - p d x where x is along the axis of...

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Prof. Daniel J. Bodony AE311, Fall 2011 HW #7 Due Wednesday 7 December 2011 in the AE311 Drop Box by 5 pm Note: all of these problems assume the steady flow of an incompressible fluid with constant density ρ and constant viscosity μ . Problem 1 Consider the laminar flow of a fluid layer falling down a plane inclined at an angle θ with the horizontal under the influence of gravity. If h is the thickness of the layer in the fully developed stage, show that the velocity distribution is u = ρ g sin θ 2 μ ( h 2 - y 2 ) where the x -axis points along the free surface and the y -axis towards the plane. Show that the volume flow rate per unit width is Q = ρ gh 3 sin θ 3 μ and the frictional stress on the wall is τ w = ρ gh sin θ. free surface (where τ = 0) g y x h θ (Over)

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Problem 2 Consider the steady laminar flow through the annular space formed by two coaxial tubes. The flow is along the axis of the tubes and is maintained by a constant pressure gradient d
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Unformatted text preview: p / d x , where x is along the axis of the tubes. Show that the velocity profile at any radius r is u ( r ) = 1 4 μ ∞ d p d x b r 2-a 2-b 2-a 2 ln( b / a ) ln r a B where a is the radius of the inner tube and b is the radius of the outer tube. Find the radius at which the maximum velocity is reached, the volume rate of flow, and the stress distribution. b a u ( r ) Problem 3 A long vertical cylinder of radius b rotates with angular velocity Ω concentrically outside a smaller stationary cylinder of radius a . The annular space is filled with fluid of viscosity μ ∞ . Show that the steady state velocity distribution is u θ = r 2-a 2 b 2-a 2 b 2 Ω r . Show that the torque exerted on either cylinder, per unit length, equals 4 πμ ∞ Ω a 2 b 2 / ( b 2-a 2 ). b a u ( r ) Ω θ...
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Ae311_fall_2011_hw7 - p d x where x is along the axis of...

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