1. Really try to figure this out yourself: don't just look it up in a book.
(a) Attempt to write the vector (cross) product of two vectors a and b
in indicial notation. First write down the cross product explicitly in terms of
a1, a2, a3,
b1, b2, b3.
Then try to write the i th component compactly using the summation convention.
See if it might be convenient to define a special object (tensor) &epsilonijk
which is +1 if ijk is a cyclic permutation of 123, -1 if ijk is a non-cyclic permutation of 123, and 0 if any two of i, j, k, are the same.
(b) Write the curl of vector field u in indicial notation.
2. Analysis of pipe flow experimental results:
instructions here.
Data here OpenOffice, MS Excel, HTML. (Note: these updated since last week.)
To confirm pipe radii, make your own measurements on these photos:
"medium" tube,
the straw ("fat tube") and the "really tiny" one (2nd from right, I think - but remotely possible it was the 3rd from right).
Cross-section of measuring cylinder: If you look at the last 2 columns of the data sheet,
you will see that hf-h0=7cm. Therefore 600cm3 = A * 7cm.
Documentary photos here.
3. Force exerted on wall in Rayleigh impulsive flow. (Segel, Section 3.2.3)
(a) Write down u(y,t) for Rayleigh impulsive flow.
(b) Compute the stress on the plate at y=0 (as a function of time).
(c) Compute the total momentum transfer between the plate and the fluid
(per unit area) during the time-interval [0,T],
confiming that (although the stress goes to infinity as t goes to 0 from
above) this total is a finite quantity (for any T).
(d) What happens to your answer in (c) as T goes to infinity? Does this
make sense? Explain.
