PS_4_2.006S08 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 2.006...

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v2 2.006F08 1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 2.006 Thermal Fluids II Problem Set 4 Issued: Thursday, February 28, 2008 Due: Thursday, March 6, 2008 Problem 0 Please read the White Chapter 7, review course notes Chapter 9. Problem 1 (SAHG 8.3) Please consider a laminar steady flow past an infinite plate. Now, however, fluid is withdrawn by a steady constant suction through the plate, which is slightly porous. In this case, the boundary layer does not grow with distance along the plate but remains constant, so that ± u / ± x = 0 . a) Starting with the boundary layer equations, please show that u ( y ) = u 0 1 ± exp ² 0 y v ³ ´ µ · ¸ ³ ´ µ · ¸ where ± 0 is the constant suction velocity at the plate and is a negative value. b) Please find the displacement and momentum thickness. c) Please find the drag on one side of the plate (the plate length is L).
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v2 2.006F08 2 Problem 2 Air at 20 C and 1 atmosphere pressure flows through a 0.15 m diameter smooth tube with a center velocity of 8 m/s. a) Assume the logarithmic overlap relation is valid at the center of the pipe. Please use the center velocity of the pipe to iteratively solve for the shear velocity u* in the pipe. b) What is the shear stress at the wall? c) What is the pressure drop per unit length in the pipe? d) Apply the momentum equation to a disk-shaped control volume centered on the axis of the pipe using the time-average flows in the pipe. Using this model, relate the effective shear stress on the constant radius surface of the disk to the local time-averaged pressure gradient in the pipe as a function of the radius of the control volume. Please determine the effective shear stress on a constant radius surface as a function of radius for the conditions above. Please graph your result.
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