Deriving the Lamm Equation
Start with the continuity equation (conservation of mass):
\[
\left(\frac{\partial c}{\partial t}\right)_{t} = -\text{Div}[J_{total}],
\]
with \(J_{total = J_{sed} + J_{dif}}\), where:
\[
J_{sed} = c\frac{\partial r}{\partial t} = c\omega^{2}sr,
\]
and
\[
J_{dif} = -D\left(\frac{dc}{dr}\right).
\]
After subsituting, we arrive at:
\[
\left(\frac{\partial c}{\partial t}\right)_{r}= -\nabla \left[c\omega^{2}sr^{2}-Df\left(\frac{dc}{dr}\right)_{t}\right].
\]
When we convert this equation to use cylindrical equations, we arrive at the Lamm equation, described by Ole Lamm in 1929, which explains the process of sediementation and diffusion in a sector-shaped cell:
\[
\frac{\partial{c}}{\partial{t}} = D \left(\frac{\partial^2{c}}{\partial{t}^2} + \frac{1}{r}\frac{\partial{c}}{\partial{t}}\right) - s \omega^2 \left(r\frac{\partial{c}}{\partial{t}} + 2c\right).
\]
\[
\frac{\partial{c}}{\partial{t}} = -\frac{\partial}{r\partial r} \
eft{c\omega^{2}sr^{2} - Dr \left(\frac{dr}{dr}\right)_{t} \right}
\]
For multiple non-interacting species, we can write simultaneous Lamm equations:
\[
\sum \frac{\partial{c}}{\partial{t}} = -\sum \frac{\partial}{r\partial r} \
eft{c\omega^{2}sr^{2} - Dr \left(\frac{dr}{dr}\right)_{t} \right}
\]