Figure 1

Integration over $y$ is carried out from 0 up to $A\left(x_z\right)$ after transformation to the coordinates $(z,y)$.Reuse & Permissions

Figure 2

Hyperboloidal cone with cylindrical symmetry.Reuse & Permissions

Figure 3

The curves with constant $z=z(x,r)$ (thin full lines) in the hyperboloid. The functions $z(x,r)$ have the form (2.21) and obey the boundary condition (2.17) at $\lambda =1$. The thick lines depict the walls of the hyperboloid for ${\eta }_0=0.4$ and 0.7. The dashed curves are ellipses, defined as $\xi =\mathrm{const}$ according to the defining relation (4.8). While the ellipses are orthogonal to the hyperboloidal boundaries of any ${\eta }_0$, the curves $z=z(x,r)$ are orthogonal only to the boundary of specific ${\eta }_0$, for which they have been calculated.Reuse & Permissions

Figure 4

Total stationary flux through the bottleneck of the hyperboloidal cone $J_0$ (normalized by $4aD{\rho }_0$) versus the parameter ${\eta }_0$. The variational method using the natural coordinates (4.8) gives the exact result (4.15). Extended Fick-Jacobs equation based on the $\lambda$-expansion of the transformation relation $z=z(x,r)$ leads to values that are close to the exact ones for ${\eta }_0>0.5$ (Ext FJ). For comparison, there are also plots of $J_0$ coming from the Fick-Jacobs equation (FJ), the Fick-Jacobs equation with the first order correction (Mod1), steady-state approximation up to the second order (2nd) and Zwanzig’s intuitive modification of the corrected Fick-Jacobs equation (Mod2) [2] in the graph.Reuse & Permissions