2 9/12

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, here is another way to do it. The equation for $x$ is
   
   $$\frac{Dx}{Dt} = -y.$$
   
   
   If you differentiate this once with respect to time, the right hand side becomes $Dy/Dt$ and you can then get rid of $y$ using the other equation. That produces a second order equation for $x$ that you can solve. Then go back to the original ODE above 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? (Hint: remember that the isobars are normal to the gradient of the pressure. The gradient of the pressure is the force per unit area, which is density times acceleration. So you can find the direction of the gradient of the pressure.)

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 $r,\theta$?