A river is dened by the domain D ={ (x, y)|| y| < 1, < x < }. A factory spills a

Question

A river is dened by the domain D ={ (x, y)|| y| < 1, < x < }. A factory spills a contaminant into the river. The contaminant is further spread and convected by the ow in the river. The velocity eld of the uid in the river is only in the x direction. The concentration of the contaminant at a point (x, y) in the river and at time is denoted by u(x, y,). Conservation of matter and momentum implies that u satises the rst-order PDE

u (y^2 1)ux =0.

The initial condition is u(x, y,0)=(e^y)e^x2.

(a) Find the concentration u for all (x, y,).

(b) A sh lives near the point (x, y)=(2,0) at the river. The sh can tolerate contaminant concentration levels up to 0.5. If the concentration exceeds this level, the sh will die at once. Will the sh survive? If yes, explain why. If no, nd the time in which the sh will die. Hint Notice that y appears in the PDE just as a parameter.

Explanation / Answer

we solve it for some y=y_0

solution for u_t + cu_x = 0 is

f(x-ct)

So here so is

e^y_0 e^{-(x-(1-y_0^2) t )^2}

b) y_0 = 0 here

x = 2

f(t) = e^{-(2-t)^2}

f(t) = 0.5 => -(2-t)^2 = ln 0.5

(2-t)^2 = ln 2

2- t = 1/2 ln 2

t = 2- 1/2 ln 2

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