This assignment is optional, and can be used to replace a previous assignment that you didn't do so well on.

As you probably know, the oil tanker "Prestige" just sank in deep water (10,000 feet deep) off the coast of Spain, carrying 20 million gallons of fuel oil.  The oil tanks are currently intact, but are expected to leak (or perhaps even burst suddenly) sooner or later, thanks to the corrosion of the salt water and the high pressures involved, leading to what some are calling an ecological disaster much bigger than the Exxon Valdez spill in Alaska a few years back.  What do you think?

1) Draw up a mathematical model for the leaking of the oil over time.  Make reasonable assumptions about how the tanks will eventually leak or burst. Based on these assumptions, write down a differential equation that describes the leakage as a function of time.  Note that this is an ORDINARY differential equation, not a partial differential equation.  We are looking at what happens in a single spot -- the wreckage of the tanker -- and all quantities are functions of time.

Solve this differential equations, by hand if possible and on the computer if not.  Describe qualitatively what happens.

Feel free to assign numbers arbitrarily to the parameters in your differential equation.  If this were a serious modeling project you'd be expected to do some research on what those numbers really are.  For purposes of this homework, however, you can pull them out of the blue.

2) Write up a mathematical model for the dispersion of the oil that is leaked.  Depending on how ambitious you are, you may want to treat the ocean as being 1 dimensional (up-current vs. down-current), as 2-dimensional (north, south, east and west), or as 3-dimensional (depth, too, remembering that oil is lighter than water).  Use the results of Problem 1 as the source of oil, which you can concentrate in a small region around the wreck. (This source is part of the "k(t,x)" term.  The other part is the rate at which oil is being removed from the ocean, either by washing ashore, forming tar-like compounds that sink to the bottom, breaking down into harmless compounds or evaporating. That part is NOT localized, and should be a function of the oil density)

The goal here isn't to SOLVE the resulting advection/diffusion equation, but to write it down in all its glory, with a real understanding of what all the different terms mean.  

3) If you are feeling especially ambitious, try to solve your equations from (2) on the computer, and report on what you found.

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