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1. Assume that a flow enters a two dimensional duct of constant area. If no boundary layers developed along the wall, the centerline velocity of the flow would stay constant. Assuming that a Blasius boundary layer develops along each wall, what is the correct expression for the centerline velocity?

2. Continuing the previous question. Approximate the Blasius velocity profile to be parabolic up to $\eta=3$, and constant from there on. At what point along the duct would you estimate that developed flow starts based on that approximation? Sketch the velocity profile at this point, as well as at the start of the duct and at the point of the duct where the range $0\le\eta\le3$ corresponds to $\frac18$ of the duct height accurately in a single graph. Remember the previous question while doing this!

3. Write down the vorticity for Stokes flow around a sphere. The velocity field was given in homework question 6.1, and a more extensive discussion is in section 21.8. Sketch some typical lines of constant vorticity, in particular $\omega=0.25$, 0.5, 0.75 and 0.99 $\omega_{\rm {max}}$, where $\omega_{\rm {max}}$ is the maximum vorticity. Now compare this very low-Reynolds number vorticity field with that of high Reynolds number boundary flow. As the boundary layer solution, you can use the error function profile one of the previous homework, and assume that $\sqrt{4{\nu}t}$ is say one tenth of the radius. (The fact that it is a cylinder instead of a sphere makes no important difference here.) You can use the boundary layer approximation for the vorticity here.

4. Streamlines for very low Reynolds number, very viscous Stokes flow look superficially the same as those for high Reynolds number ideal inviscid flows: both are symmetric front/rear. But do they really look the same? Plot their streamlines reasonably accurately. They are given in sections 19.8 and 21.8. You could use some plotting package to plot them. Alternatively, you could figure out where the streamlines through the points $r=1.25$, 1.5, 1.75, and 2 sphere radii from the center in the symmetry plane end up far upstream and downstream, and then sketch the streamlines as well as possible based on that info.