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1. 7.5. You may assume that the velocity and pressure are independent of the angular and axial positions in cylindrical coordinates. Also assume that the flow is steady with no velocity component in the axial direction. Do not assume the radial velocity is zero.

2. In 7.5, what is the power needed to keep the rod rotating, per unit axial length? What is the pressure difference between the surfaces of the pipe and the rod?

3. 7.6. Do not ignore gravity, but assume the pipe is horizontal. Careful, the gravity vector is not constant in polar coordinates. Do not ignore the pressure gradients: assume the pressure can be any function $p=p(r,\theta,z,t)$. 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.

4. 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 along a flat wall of spanwise length $2\pi r_0$ in the $z$-direction. Take the $x$-axis downwards. Assume $\vec v=\vec v(y)$ only (developed flow), $w=0$ (two-dimensional flow), and $p=p(x,y,z)$. Everything else must be derived; give 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. Also write appropriate boundary conditions where the fluid meets the cylinder surface.