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4. (12 points) Suppose we have an infinitely large 2-dimensional pool of water and at the origin (0,0), a small drop of ink at time t = 0. The ink will slowly mix with water, and the concentration p(x, y, t) of ink (mass of ink per volume of water) at any point in space (r, y) and time t > 0 can be modelled by the equation 1 p(x,y,t) (r2+y?) -e 4nt (a) (4 points) Show that the total mass of ink in the pool, at any time t > 0, is equal to 1. (b) (2 points) Suppose this pool has a current (the water is flowing) with constant velocity ū= (U1, U2). Rewrite the formula for the concentration p(x, y,t) (Hint: The ink drop would travel to point (ut, uzt) at time t and so your new formula of concentration should just be shift of the original formula.) (c) (4 points) Consider you are filming with a circular camera of radius R to measure the total mass of ink. To do so, you must follow the path of the ink droplet. Give a formula for the total mass inside this moving domain for any time t. (Hint: At a fixed time t, consider where the domain should be centred to follow the ink in the current.) (d) (2 points) Calculate the rate of change of the average concentration.
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