Instantaneous Point-source Solution - IfH
Instantaneous Point-source Solution - IfH
Instantaneous Point-source Solution - IfH
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Section 3: Similarity solution to the one-dimensional diffusion equation 16<br />
To find C 1 we must use the conservation of mass:<br />
∫<br />
M = C(x, t)dV<br />
=<br />
= M<br />
which gives the constraint<br />
∫ ∞<br />
−∞<br />
V<br />
∫ ∞ ∫ a<br />
−∞ 0<br />
∫ ∞<br />
−∞<br />
C(η)2πrdr √ Dtdη<br />
f(η)dη<br />
(<br />
C 1 exp − η )<br />
dη = 1. (31)<br />
4<br />
To solve this integral we need to make one more change of variables<br />
to remove the 1/4 from the exponential. Thus, we introduce ζ such<br />
that<br />
ζ 2 = 1 4 η2 (32)<br />
2dζ = dη. (33)<br />
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