# Derivation of the diffusion-transport equation

Think of a flowing stream, and imagine the motion of a certain amount of substance in a portion of the stream. You’d notice two things about it: it would diffuse—that is, the concentration of it would lessen in the area in which you originally spotted it—and it would move down the stream at the stream’s velocity, call it $$v$$. The equation that describes the amount of substance in any given place $$x$$ in the stream at $$t$$ is called the concentration equation (or the diffusion-transport equation). Let’s derive it!

From our description above, we have the conservation equation

Writing this in mathematical notation, we have

for $$c$$ the channel capacity, $$\rho$$ the density of the substance, $$A$$ the cross-sectional area of the segment under consideration, and $$k$$ the conductivity of the medium in the channel. This is now an integral equation for the amount of substance in a certain area, but we’d prefer a differential equation for ease of solution. We can get there using the mean value theorem. Assuming continuity of a function $$f$$, we have

for some $$\xi \in (a,b)$$. Applying this to our integral equation and writing

by assumption of the “niceness” of $$u$$ gives

Dividing through by $$\Delta x$$, letting $$\Delta x \rightarrow 0$$, and rearranging terms gives the diffusion-transport equation:

Pretty neat! One could easily incorporate generation of material within the stream by adding to the integral equation the accumulation term

for $$f$$ some generation kernel. The differential equation would then have the term $$\frac{1}{c\rho}f(x,t)$$ added at the end.

Written on June 19, 2016