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1. 7.6. Do not ignore gravity, but assume the pipe is horizontal. Do not use the effective pressure. Careful, the gravity vector is not constant in polar coordinates. Do not ignore the pressure gradients: assume the pressure can be any function $p=p(r,\theta,z,t)$ and derive anything else. Merely assume that the pressure distribution at the end of the pipe and rod combination is the same as the one at the start. For the velocity assume $v_r=v_\theta=0$ and $v_z=v_z(r,z)$. Anything else must be derived. Give both velocity and pressure field.

2. For the case of question 7.6, what is the force required to pull the rod through the axis, per unit length? In 7.9, (previous homework), what is the net downward shear force on the pipe? Does the simple answer surprise you? Why not?

3. 7.1a Assume only that the velocity only depends on $r$, $\vec v=\vec v(r)$, that $p=p(r,\theta,z,t)$ is arbitrary, and that the pipe is horizontal. Use the effective pressure. Show that two velocity components must be zero. (You should be able to show that the effective pressure is independent of $\theta$ from the appropriate momentum equation by noting that $p$ at $\theta=2\pi$ must be the same as at $\theta=0$; otherwise just assume it is. Also note that the velocity can obviously not be infinitely large on the pipe centerline.)

4. 7.1b Continuing the previous question, derive the velocity and pressure fields.

5. 7.4. Argue your answer. In what terms would you ballpark the answer? What is the importance of the pressure level? Of the flow velocity? What are the relevant values involved? What are the most important uncertainties? You might want to think of what the right answer for the head loss would be if there is no flow.