5 10/1

1. Divide the fluid region outside a square cylinder into little finite elements of size $\Delta{x}\times\Delta{y}$. For a typical such element, write a finite element discretization for the continuity equation. 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.

2. Write a finite element discretization for the $x$-momentum equation for a little finite element 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.

3. 5.2 Make sure to write the full equations before assuming a radial flow. Make that a neat graph, and include the streamlines.

4. 5.3. This is two-dimensional Poiseuille flow (in a duct instead of a pipe). $T_{ij}$ is the book’s notation for the complete surface stress including the pressure,
   
   $$T_{ij} = -p \delta_{ij} + \tau_{ij}$$
   
   
   where $\delta_{ij}$ is called the Kronecker delta or unit matrix. So the book is really saying the pressure is $-5$ and there is an additional viscous stress $\tau_{xy}=-2\mu v_0y/h^2$. Watch it, the expression $n_j\tau_{ji}$ gives the stress components in the $x,y,z$-axis system.

5. 5.6. $Z$ is the height $h$. The final sentence is to be shown by you based on the obtained result. Hints: take the curl of the equation and simplify. Formulae for nabla are in the vector analysis section of math handbooks. If there is a density gradient, then the density is not constant. And neither is the pressure. $T_{ij}$ is the book’s notation for the complete surface stress, so the book is saying there is no viscous stress. (That is self-evident anyway, since a still fluid cannot have a strain rate to create viscous forces.)