Approximations of the generalized Fick-Jacobs equation

We analyze the generalized Fick-Jacobs equation, obtained by a rigorous mapping of the diffusion equation in a quasi-one-dimensional (quasi-1D) (narrow 2D or 3D) channel with varying cross section $A\left(x\right)$ onto the longitudinal coordinate $x$. We show that for constructing approximations and understanding their applicability in practice, it is crucial to study the 2D (3D) density inside the channel in the regime of stationary flow. We present algorithms enabling us to derive approximate formulas for the effective diffusion coefficient involving derivatives of $A\left(x\right)$ higher than $A^{\prime }\left(x\right)$ and give examples for 2D channels. Effects of the boundary conditions at the ends of a finite channel and the case of nonsmooth $A\left(x\right)$ are also discussed.

Figure 1

Linear approximation takes the linearized boundary (thick dashed line) instead of $A\left(x\right)$ in the vicinity of a chosen $x=\overline{x}$. So the real isodensities (solid lines) are replaced by the circles (dashed lines), corresponding to the 2D density in the linear cone.

Figure 2

Approximation of the boundary $y=A\left(x\right)$ (thick solid line) at $x=\overline{x}$ by a circle (thick dashed line) of radius $R_0$ and centered at $(0,y_0)$. The thin solid circles represent the isodensities in the circular cavity; the thin dashed lines are circles orthogonal to them.

Figure 3

Hyperbolic cones (3.18) for $a=1$ and various $\gamma$ (the dotted and solid thick lines). The thin solid lines are the exact isodensities for any $\gamma$ (ellipses); the thin dashed lines are circles, replacing the ellipses in the LA and CA; small dots depict hyperbolic isodensities corresponding to the Mod2 approximation (3.15) (drawn for $\gamma =1$).

Figure 4

$D\left(x\right)$ for the hyperbolic cone (3.18) for $a=1$ and $\gamma =2$. The solid line represents the exact solution , the thick dashed line is the LA (1.12), and the thick dotted line is the CA (3.11). The thin dashed line depicts Mod 2, Eq. (3.15).

Figure 5

Testing channels defined by (3.20) (thick lines). The solid thin lines describe the exact isodensities; the dashed lines are the isodensities approximated by circles for $g\to 1$.

Figure 6

The effective diffusion coefficient $D\left(x\right)$ for the channel bounded by the function (3.20) for $g=0.87$ and 1, calculated by using all derived formulas. The thick solid line is the exact solution (3.21). The thick dashed line is the LA (1.12); the thick dotted line is the CA (3.11). The thin lines depict the approximations (3.15, 3.16, 3.17). The approximation by hyperbola (Mod 2 or $n=2$) corresponds to the dashed line; the dotted line plots $n=3$ and the solid line $n=4$.

Figure 7

Finite 2D linear cone bounded by the thick solid lines; $x_L<x<x_R$. The mapping procedure finds the circular (thin dashed lines) isodensities, corresponding to the uncut cone (bounded by the thick dashed lines). The true isodensities (thin solid lines) in the finite cone have to respect the BC at its ends (dotted lines) $\rho (x_{L,R},y)={\rho }_{L,R}$, not depending on $y$.

Figure 8

Effective diffusion coefficient $D\left(x\right)$ for the finite linear cone $A\left(x\right)=x$; $0<x<1$ (solid line) compared with the exact result for the uncut cone (dashed line), the constant $D\left(x\right)=\pi ∕4$.