Problem 5

        In this problem we examine the behavior of a dilute 
suspension of particles which is being sheared.  As the 
suspension is sheared (the motion which is produced 
when a fluid is confined between two concentric cylinders 
and one of them is rotated) the particles will tumble over 
one another.  This will lead to a random walk of the 
particles which can be characterized by a diffusion 
coefficient, much like a molecular diffusivity.  We can 
measure this quantity - the shear-induced self-diffusivity - 
by examining the random walk of a single tracer particle in 
a suspension of otherwise identical particles.  Because the 
random motion is due to the interaction of the tracer with 
other particles, the diffusivity will be identically zero if the 
concentration of the other particles is zero.
        A current area of research is the determination of 
the dependence of the self-diffusivity on concentration in 
the dilute limit.  If the particles are perfect spheres, theory 
suggests that the leading order term in the diffusivity 
should go as c^2 where c is the concentration.  Other 
models suggest that if the spheres are not perfect, the 
diffusivity should be proportional to c to leading order.  A 
couple of years ago, we measured the diffusivity in the 
dilute limit for a suspension of slightly prolate spheroids 
(sort of slightly stretched out spheres).  The results 
were as follows:

c       diffusivity     error
0.01    2.37e-4         2.2e-5
0.025   5.63e-4         6.5e-5
0.05    1.42e-3         1.6e-4
0.075   2.27e-3         4.0e-4
0.10    4.06e-3         7.6e-4
0.15    9.96e-3         1.8e-3

The errors given above are the one standard deviation errors
in the measured diffusivities calculated from the statistics
governing the measurement process.

        Using this data, fit a constitutive equation for the 
diffusivity of the form:

        Diffusivity = c * x(1) + c^2 * x(2) + c^3 * x(3)

using weighted linear regression, and determine the error 
in each of the fitting coefficients.  In particular, determine 
if the difference between the O(c) coefficient x(1) and zero is 
statistically significant.

For graphics I would like you to plot up the data with errorbars
(use the errorbar command), together with the fitted model
(solid line) and 95% model confidence interval (dashed lines).
Be sure to label everything using the legend command.  Finally,
the behavior is a bit clearer (graphically, anyway) if you plot up
D/c rather than the diffusivity itself.

Hint:  You -must- use weighted linear regression for this 
problem.

As an interesting side-note, the magnitude of the diffusivity
measured in these experiments was -much- larger than could
be accounted for by theory (at the time, anyway).  We published
the results, as well as similar values measured for perfect spheres,
and they stood as an anomaly for about 10 years.  It wasn't 
until this past year (11/07) that the results were finally explained.
It turned out that it was the theory that was wrong - that two
perfect spheres interacting in the presence of a wall -will- give
rise to the O(c) term, and of precisely the magnitude measured
in the experiments.  We (and others) are still exploring the
ramifications of this new diffusion mechanism.