Effective diffusion in the region between two surfaces

The purpose of this paper is to provide a formula for the effective diffusion operator obtained by projecting the three-dimensional diffusion equation onto a two-dimensional plane, assuming reflective boundary conditions at two surfaces in three-dimensional space and a finite transversal stabilization rate.

Figure 1

A two-dimensional channel with a bump in (a) and its effective diffusion coefficient $\mathcal{D}$ in (b). The vertical width of the channel is equal to 1, and its central line is shown in black. Where the inclination of the channel is 0 (or close 0), the value of $\mathcal{D}$ is 1 (or close to 1). Where the absolute value of the channel's inclination is maximal, the effective diffusion reaches its minimum values.

Figure 2

Vertical stabilization function $\mathcal{E}=\mathcal{E}\left(t\right)$ on the channel's region above the interval $[-12,12]$. At time $t=0$ this function has value 0 since by construction the initial concentration density is constant with respect to $y$.

Figure 3

Comparison at $t=1/2$ of the effective concentration density $p$ (solid curve) and the solution $\rho$ (dashed curve) of Eq. (4) having initial condition $\rho (x,0)=p(x,0)$ given by Eq. (10). The net effect is that the presence of the bump in the channel has delayed the evolution of the diffusion process in the $x$ direction, in comparison with the case of a channel with straight parallel walls to the $x$ axis.

Figure 4

Recovering the effective diffusion coefficient ${\mathcal{D}}_P$ from a solution of the diffusion Eq. (20). In (a) we show the functions in the numerator (solid line) and the denominator (dashed line) of Eq. (11) for ${\mathcal{D}}_P$. In (b) we show the graphs of ${\mathcal{D}}_P$ (solid line) and $\mathcal{D}$ given by Eq. (6) (dashed line).

Figure 5

Channel of constant width $w=1$ along the normal lines of a curve with a bump.

Figure 6

Comparison of the effective diffusion density (at $t=1/2)$ obtained by analytically solving Eq. (12) (dashed line) and the numerical computation of the integral 13 (solid line). The initial conditions used are those given by Eq. (8). The close fit of the curves is explained by the fact that when using adequate coordinates the finite and infinite transversal diffusion rate cases give to the same formula.

Figure 7

Approximation of a channel by an angular sector, which is formed by the tangent lines of the channel's walls at two points in a line of projection.

Figure 8

Intersecting planes and the corresponding orthonormal frames $\mathit{x},\mathit{y},\mathit{z}$ and $\mathit{X},\mathit{Y},\mathit{Z}$. The unit vector $\mathit{n}$ spans the intersection the planes, $\mathit{z}$ is the direction of projection, $\mathit{x}=(\mathit{n}\times \mathit{z})/|\mathit{n}\times \mathit{z}|$ and $\mathit{y}=\mathit{z}\times \mathit{x}$. The frame $\mathit{X},\mathit{Y},\mathit{Z}$ is obtained from the frame $\mathit{x},\mathit{y},\mathit{z}$ by a rotation by an angle $\psi$ that fixes the $\mathit{x}$ axis. The angle $\psi$ (which we will refer to as the tilt) is such that after the rotation is applied to $\mathit{y}$ it aligns with $\mathit{n}$.

Figure 9

The equations of any pair of planes in three-dimensional space have the form $Z=m_{1}X$ and $Z=m_{2}X$ in the $X,Y,Z$ coordinates of the frame $\mathit{X},\mathit{Y},\mathit{Z}$ depicted in Fig. 8. For any fixed value $Y_0$ the two planes intersect the plane $Y=Y_0$ in two nonparallel lines passing through the origin. The field ${\mathit{J}}_0$ shown in the picture is the gradient field of the harmonic function $P_0=log(X^2+Y^2)$.

Figure 10

The equations of any pair of planes in three-dimensional space have the form $Z=m_{1}X$ and $Z=m_{2}X$ in the $X,Y,Z$ coordinates of the frame $\mathit{X},\mathit{Y},\mathit{Z}$ depicted in Fig. 8. For any fixed value $X_0$ the two planes intersect the plane $X=X_0$ in two parallel lines to $\mathit{Y}$. The field ${\mathit{J}}_{\pi /2}$ shown in the picture is the gradient field of the harmonic function $P_{\pi /2}=Y$.

Figure 11

The principal response directions are spanned by unit vectors $e_{\mathcal{D},1}$ and $e_{\mathcal{D},2}$ that map under $\mathcal{D}$ to the major and minor axes of the effective diffusion ellipsoid. When the gradient of $p(x,y,t)/w(x,y)$ is aligned with these directions the relative flux magnitude $\left|\mathcal{D}\mathbf{\nabla }(p/w)\right|/\left|\mathbf{\nabla }(p/w)\right|$ takes its maximum and minimum value, respectively.

Figure 12

Effective diffusion ellipsoids for $\psi =0$ and $D_0=1$. The levels curves of $\omega$ in (a) correspond to ellipsoids in (b). As we move along the ray $m_1=m_2=\mu \ge 0$ we obtain ellipsoids with semi-axis $1/(1+{\mu }^2)$ and 1.

Figure 13

The effective diffusion ellipsoids obtained by letting $m_1=m_2=1$ and varying the tilt parameter $\psi$ in the range between $-2$ and 2. For other values of $m_1$ and $m_2$ the picture is qualitatively the same as the one shown.

Figure 14

Radial effective diffusion function of the surfaces given by the formulas $z_1=sin\left(\sqrt{x^2+y^2}\right)-3/2\text{and}{z}_2=cos\left(2\sqrt{x^2+y^2}\right)+3/2$.

Figure 15

Two orthogonal planar waves in (a) given by the Eqs. (36), and the corresponding tilt function $\psi$ in (b). These functions extend periodically in both $x$ and $y$ axes.

Figure 16

The eigenvalue functions of the symmetric factor of the effective diffusion matrix associated to the planar orthogonal waves given by the Eqs. (36). In (a) we show the largest eigenvalue and in (b) the smallest.

Figure 17

The vector field $\mathbf{\nabla }w$ in (a) and ${\mathbf{\nabla }}^{\perp }w$ in (b) of the width function $w=z_2-z_1$ for the surfaces Eq. (36). These fields form the bases where we have computed the effective diffusion matrix.

Figure 18

Principal response fields of the surfaces in Eq. (36). When the gradient of $p(x,y,t)/w(x,y)$ is aligned with the line field in (a) the relative flux magnitude takes its maximum value. Similarly, when the gradient of $p(x,y,t)/w(x,y)$ is aligned with the line field in (b) the relative flux magnitude takes it minimum value.

Figure 19

Effective concentration density for a numerical solution of the diffusion Eq. (20) at time $t=1/5$ and with reflective boundary conditions on the surfaces defined by Eq. (36). The initial condition used was a three-dimensional scaled spherically symmetric Gaussian distribution centered at $(0,0,3/2)$ and with variance equal to $1/2$.

Figure 20

Gradient field of $p(x,y,t)/w(x,y)$ in (a) and the effective flux $\mathit{j}(x,y,t)$ in (b), for a numerical solution of the diffusion Eq. (20) at time $t=9/10$. Reflective boundary conditions are assumed on the surfaces defined by Eq. (36).