8 10/20 W

1. 7.9. You can assume that the film thickness is so small that the curvature of the pipe wall can be ignored. In that case, it becomes 2D steady flow in the $x$-direction along a flat wall of spanwise length $2\pi r_0$ in the $z$-direction. Take the $x$-axis downwards. Assume $v=0$ (vertical streamlines), $u=u(x,y)$ and $w=0$ (two-dimensional flow), and that $p=p(x,y,z)$. Everything else must be derived; derive both pressure and velocity field. Do not ignore gravity. For the boundary conditions at the free surface, assume that the liquid meets air of zero density and constant pressure $p_a$ there. Air of zero density cannot exert nonzero tangential forces. Also write appropriate boundary conditions where the fluid meets the cylinder surface.

2. 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.

3. 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?

4. 7.1a Assume only that the cylindrical velocity components only depend on $r$, $v_r,v_\theta,v_z=v_r,v_\theta,v_z(r)$, 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.)

5. 7.1b Continuing the previous question, derive the velocity and pressure fields.