MAE 5130: VISCOUS FLOWS Homework #3 Solutions Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

PROBLEM 3.2 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 Sum of separate solutions for moving inner or outer cylinder Superposition is possible because N/S equations are linear for this flow

PROBLEM 3.2: NON-DIMENSIONAL REPRESENTATION Radius non-dimensionalized by r0 u non-dimensionalized by U0

PROBLEMS 3-14 and 3-22: OSEEN AND TAYLOR VORTICES G0=C=1.0, n=1x10-5 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

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

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

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