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MAE 5130: VISCOUS FLOWS
Homework #3 Solutions
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
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PROBLEM 3.2
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Consider axial Couette flow of Fig. 3-3 with both cylinders moving
– Find velocity distribution u(r) and plot for (a) U1=U0, (b) U1=-U0, and (c ) U1=-2U0
– 0: inner cylinder coordinate, 1: outer cylinder coordinate
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Sum of separate solutions for moving inner or outer cylinder
Superposition is possible because N/S equations are linear for this flow
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PROBLEM 3.2: NON-DIMENSIONAL REPRESENTATION
Radius non-dimensionalized by r0
u non-dimensionalized by U0
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PROBLEMS 3-14 and 3-22: OSEEN AND TAYLOR VORTICES
G0=C=1.0, n=1x10-5
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Taylor profiles are flatter than Oseen
Taylor profiles are less peaked
Decay by dropping quickly near the axis with a reversal in vorticity away from the axis
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PROBLEM 3-32: JEFFERY-HAMEL WEDGE FLOW, Re=0, a≠0
• Case where a = 0º is the Poiseuille parabola for channel flow
• Case where a = 90º is the separation point
• For a > 90º, separation or backflow must occur in a diverging flow even at zero
Reynolds number
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PROBLEM 3-55
a
b
• Both circumscribed forces are greater than the actual forces
• Both inscribed forces are less than the actual forces
• The ratios are significantly different from unity and vary by a factor of 2 to 3
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EXAMPLE: DRAG CALCULATION FROM A WAKE
• A uniform stream flows past an immersed object as shown above, producing a
broad, low-velocity wake, which is idealized as a V-shape.
• Pressures at p1 and p2 are essentially equal
• The flow is 2D and incompressible, with width b into page
• Derive a formula for drag force on the object and obtain expression for drag
coefficient, CD
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