2 9/13

1. 4.1

2. A two-dimensional flow field is given in Eulerian (and Cartesian) coordinates by:
   
   $$u = -y \qquad v = x$$
   
   
   Integrate the Cartesian particle path of a typical particle in this flow, assuming that the particle is initially at the point $x=\xi$ and $y=\eta$. Sketch the particle path. Write down the Lagrangian description of this flow. (Hint: If you do not remember how to solve systems of ordinary differential equation, differentiate the ODE for $x$ once and then get rid of $y$ in the equation using the other equation. Solve that 2nd order ODE for $x$. Then go back to the original ODE to figure out what $y$ is. Then apply the initial conditions.)

3. What is the acceleration vector of the fluid particles for the flow above? So what do you think about the pressure field? How do isobars look?

4. Find the streamlines for the flow above from solving
   
   $$d\vec r // \vec v, \quad dt=0$$
   
   
   Do you think this flow would be easier to solve in polar coordinates $r,\theta$?