2. Consider the following velocity components for an incompressible, plane flow:
V, = A r 'I + B r cos0 (15)
Ve = B r sine
Where A and B are constants. Determine the corresponding stream function
3. Consider a swirling motion superimposed upon a circular-duct flow by letting the velocity components take the form:
tr, =0; ue = ue (r, t); uz = uz (r)
For these velocities, are the governing eq. linear? Why? Show by substituting in the various components of the N-S equations that the axial flow uz is unaffected by the swirl, so that an arbitrary ue can be added without changing the Poiseuille distribution. What makes this possible? No need to solve the eauatinnt4. A vertical shaft weighing w per unit length (see figure at the side) is sliding down concentrically inside a pipe as shown in the figure below. Oil separates the two members.
a) Write the governing cq. and indicate which terms are important.
b) Clearly write down the h.c.
c) Solve for the velocity distribution.
d) Determine the tenninal velocity VT of the shaft by solving the Wavier-Stokes equations.
(Hint Determine the relevant forces acting (and where they act) when terminal velocity is reached. That will be one of your b.c. for the problem)
5. Consider the laminar flow of an incompressible fluid past a flat plate at y = 0. The boundary layer velocity profile is approximated as:
U 8 = I for 05 y5 Sand u =U for y >8.
Determine the wall shear stress using the momentum integral equation. Compare the various boundary layer parameter values with those of Blasius's. (15)
6. Air enters a circular duct of cross-sectional area A. subsonically from standard atmosphere and flows isentropically through a choked converging-diverging duct. The area A varies as: A = 0.1 + x2. where x is the distance along the duct (-0.5m < x < 03m). What are the inlet and exit Mach numbers? List the Mach number variation along the dud for dx = 0.1 m. (15)
