Industrial Mathematics - Alexiades
            Project 2: Ostwald Ripening Model
              Fate of 2 crystal sizes
Consider an Ostwald ripening experiment with crystals of two initial sizes:  
x1*=0.05  and x2*=0.09 microns, and parameter values
     k=5.e7, μ=1.e-3, c*=7.52e-7, Γ=4.e-3, c0=1.05c*
The goal is to simulate this experiment computationally to verify
the theoretical predictions.  Here are the tasks that need to be carried out.

1. Write down the mathematical problem that models the experiment
   (physical assumptions, ODEs, initial conditions, parameters).
   Describe what you expect to happen qualitatively and why, 
   and what you plan to do to simulate the experiment. 
   [the steps are described below]

2. Find the dissolution time, t1, of the crystal that dissolves 
   first, and the size of the surviving crystal at that time.
   [ Explain what you'll do, then do it ].
   Determine t1 and x2(t1) as accurately as you can. 
   Plot x1(t) and x2(t) up to time t1.

3. For times beyond t1, there is only one crystal size.
   Find the two equillibria for this situation, and predict
   the eventual fate of the survivor on the basis of theory.
   Does the prediction depend on the accuracy of the values found in 2 ?

4. Calculate the evolution of x2(t) beyond time t1 to verify the
   prediction. Show the evolution of the two sizes in a time plot.  

5. Draw conclusions about the experiment and the usefulness of
   the analysis and simulation.   

The above steps and results should be presented as a formal report to your boss,
as if your job depended on it. It should be self-contained and independent from Project 1.
You may reuse wording from your (revised) Project 1, as appropriate,
but do NOT assume that your boss has seen Project 1.
Do NOT list the above tasks as exercises to be solved, work them into your narrative
to guide the reader as to what you are doing.
Please submit the Report on paper.