520 ADV FLUID MECH I  
                                  Fall Term 2003

                  
Hwk 2 (Due Th Oct 10)

1. Show that rotation of a 2D Cartesian system has the Kronecker delta tensor for the fundamental tensor.

2. Find the fundamental matrix g_pq for a two dimensional homogeneous coordinate system q_i where the new coordinate system has the horizontal axis rotated by angle a but the vertical axis is left alone.

3. Find the fundamental metric of a polar coordinate surface.

4. Find how the unit vectors in the r and theta directions depend on r and q.

5. Show d_ij is an isotropic tensor, i.e. show d_kl (d_ki+c_ki) (d_lj+c_lj) = d_ij
(hint: remember |c| is small)

6. Find the path-, streak-, and stream- lines for a 2D velocity field that is vertical for 0<t<t₁ and horizontal for t₁<t.

7. Prove the corollary to the Reynolds Transport theorem.

8. Kitchen Experiment: Find how the kinematics of water leaving a faucet compare to that of a solid particle. Is the acceleration equal to gravity? For this experiment you do not want an aerator and may want to make all measurements before a droplet forms.