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1. Find the circulation around a circle around the origin for the difused vortex flow of question 4.2c by directly integrating the line integral. Also find it by integrating the vorticity in accordance with Stokes' theorem. Do you get the same result? If not, explain why not.

2. Find the circulation around a square around the origin for the line vortex flow of question 4.2b by directly integrating the line integral. Also find it by integrating the vorticity in accordance with Stokes' theorem. Do you get the same result? If not, explain why not. (Hint: Note that the flow of the previous question is the same as the one here if seen from a large distance. So, if you do the integrals of the previous question over a large circle and you look at them from far away, things would look the same.)

3. 5.1 (b). Also do and explain $\int_{\rm FR} \rho\;dV$. Note that this is incompressible inviscid flow. The viscous flow is much more complex.

4. If the jet leaves a rocket through an area of 0.5 m$^2$ at a velocity of 500 m/s relative to the rocket, and the exit density is 0.5 kg/m$^3$, what can you say about the total mass of the rocket?

5. Find the average exit velocity in the pipe flow of question 5.13. Do the same for the flow of 5.14.

6. If in question 5.12, the liquid comes out of the tube of radius R with a Poiseuille axial velocity
   
   $$v_z = V_{\max} \left(1 - \frac{r^2}{R^2}\right),$$
   
   
   and unknown radial and swirl velocity components, and with density $\rho$, then derive the mass flowing out of the tube per unit time. If at the short distance below the pipe, the stream has contracted to a radius $R_c$ and the velocity has become uniform and equal to $\vec v = U{\hat \imath}_z$, then what is the value of the constant $U$ in terms of the other parameters?

7. Suppose you want to compute the flow in a square region. Show how you can reduce the number of unknowns to a finite set by restricting the computation to a finite set of points. Formulate an equation for each (non boundary) point based on the law of mass conservation (continuity).