Instantaneous Point-source Solution - IfH

Section 3: Similarity solution to the one-dimensional diffusion equation 9
3. Similarity solution to the one-dimensional diffusion
equation
Consider the one-dimensional inviscid problem of a narrow, infinite
pipe (radius a) as depicted in Figure 3. A mass of tracer, M, is
injected uniformly across the cross-section of area A = πa 2 at the
point x = 0 at time t = 0. We seek a solution for the spread of tracer
in time due to pure diffusion.
The governing equation is
∂C
∂t = D ∂2 C
∂x 2 (11)
which requires two boundary conditions and an initial condition:
• As boundary conditions, we impose that the concentration at
±∞ remain zero:
C(±∞, t) = 0. (12)
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