7 10/25 W

1. Write down the worked-out mathematical expressions for the integrals requested in question 5.1. This is a good exercise in identifying various surface and volume integrals in integral conservation laws. Explain their physical meaning, if any. Don't worry about actually doing the integrations. However, show integrands and limits completely worked out.

   Take the surfaces $S_{I}$, $S_{II}$, $S_{III}$, and $S_{IV}$ to be one unit length in the $z$-direction. (To figure out the correct direction of the normal vector $\vec n$ at a given surface point, note that the control volume in this case is the right half of the region in between two cylinders of radii $r_0$ and $R_0$ and of unit length in the $z$-direction. The vector $\vec n$ is a unit normal vector sticking out of this control volume.)

2. Write a finite volume discretization for the $x$-momentum equation for the little finite volume in polar coordinates. Just like the continuity equation done in class, your final equation should only involve pressures, densities, and velocities at the center points of the finite volumes. Ignore the viscous stresses for now.

   The unknown velocities used in the computation should be taken to be the polar components $v_r$ and $v_\theta$. But momentum conservation for $x$-momentum is asked. (Conservation of $r$-momentum or $\theta$-momentum would be complete nonsense.) So you will need to write the $x$-component of velocity in terms of the polar unknowns. Note that in Cartesian coordinates, the polar unit vectors are given by
   

   $${\widehat \imath}_r=\cos(\theta){\widehat \imath}+ \sin(\th... ... \sin(\theta){\widehat \imath}+ \cos(\theta){\widehat \jmath}$$
   
   
   These should be able to allow you to evaluate the $x$-components of velocity and pressure forces that you need.

3. Assuming that there are known viscous stresses at the centers of the sides of the finite element, what additional terms do you get in the obtained equation due to viscous forces? Assume the stress tensor is given in polar form. (So $\tau_{rr}$, $\tau_{r\theta}$, etcetera.) Once again you will need the polar unit vectors to get x-components of the forces.