11 11/20 M

1. 7.6. Do not ignore gravity, but assume the pipe is horizontal. Careful, the gravity vector is not constant in polar coordinates. Find the components using geometry or from $\vec{g}=-g\nabla{h}$. Assume $x$ is horizontal like $z$ and $y$ vertically upwards. 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. What is the force required to pull the rod through the axis, per unit length?