= Conformal map equation for planar Darcy growth
{title2=$\operatorname{Re}(\zeta F_\zeta\overline{F_t})=Q\quad(|\zeta|=1)$}
Let $F(\zeta,t)$ map the <unit disk> conformally to a planar <Darcy flow> region, with $F(0,t)=0$ at a point source of <flux> $2\pi Q$. With zero <velocity potential> on the <boundary>, the logarithmic source potential becomes $\phi=Q\log|\zeta|$. Differentiating the inverse map at fixed physical position gives $\phi_t=-Q\operatorname{Re}(F_t/(\zeta F_\zeta))$, while $|\nabla\phi|^2=Q^2/|\zeta F_\zeta|^2$. The <boundary> law $\phi_t+|\nabla\phi|^2=0$ therefore gives the displayed equation, for nonzero $Q$.
The vector $\zeta F_\zeta\,d\theta$ is outward normal times arclength on $\zeta=e^{i\theta}$. Consequently the equation gives the <area> growth law $d|D|/dt=2\pi Q$. For a circular map $F=R(t)\zeta$, it reduces to $R\dot R=Q$, and hence $R(t)^2=R(0)^2+2Qt$.
Back to article page
