Universality in edge-source diffusion dynamics

We show that in edge-source diffusion dynamics the integrated concentration $N\left(t\right)$ has a universal dependence with a characteristic time scale $\tau =(\mathcal{A}∕\mathcal{P}{)}^2\pi ∕(4D)$, where $D$ is the diffusion constant while $\mathcal{A}$ and $\mathcal{P}$ are the cross-sectional area and perimeter of the domain, respectively. For the short-time dynamics we find a universal square-root asymptotic dependence $N\left(t\right)=N_0\sqrt{t∕\tau }$ while in the long-time dynamics $N\left(t\right)$ saturates exponentially at $N_0$. The exponential saturation is a general feature while the associated coefficients are weakly geometry dependent.

Figure 1

A sketch of the cross section $\Omega$ (white) with boundary $\partial \Omega$ (black curve) in the $xy$ plane of a rod with translational invariance in the $z$ direction. At time $t=0$, where the concentration $c(\mathbf{r},t)$ inside $\Omega$ is zero, diffusion is suddenly turned on from the outside, where the concentration is kept at the constant value $c_0$ (gray).

Figure 2

A linear-log plot of $N\left(t\right)∕N_0$ as a function of $t∕\tau$ for a circular cross section (white circles) and for an arbitrarily shaped cross section (gray stars); both shapes are shown in the insets. The solid line shows the exact result for the circle, i.e., the first 1000 terms in the infinite series Eq. (14), the dashed line shows the short-time asymptotic expression Eq. (5), the dot-dashed line shows the long-time asymptotic expression Eq. (12) for the circle with ${\alpha }_1={\gamma }_{0,1}^2\pi ∕16$ and ${\beta }_1=4∕{\gamma }_{0,1}^2$, while the data points are the results of time-dependent finite-element simulations.

Figure 3

Plot of the deviation $\mathrm{\Delta }N\left(t\right)∕N_0$ from the circular case, Eq. (15), as a function of $t∕\tau$ for different cross sections (see insets). The data points are results of finite-element simulations. The ellipses have eccentricities 1, 1.5, 2, and 2.5, the circle sectors have angles $\pi ,2\pi ∕3,\pi ∕2$, and $\pi ∕3$, and the rectangles have aspect ratios 1, 1.5, 2, and $3$. Note how the maximal deviation is less than $\pm 0.03$ or $\pm 3\%$.

Figure 4

Plots as in Fig. 3 but here for two extreme geometries: a sector of a circular disk with angle $2\arctan (1∕30)$ and a rectangle with aspect ratio $1∕30$ (see insets). While these two shapes have nearly the same area and perimeter, the former fills more slowly than the circular cross section and the latter faster. The maximal deviations are $-6\%$ and $+10\%$, respectively.