Characteristic time scales for diffusion processes through layers and across interfaces

This paper presents a simple tool for characterizing the time scale for continuum diffusion processes through layered heterogeneous media. This mathematical problem is motivated by several practical applications such as heat transport in composite materials, flow in layered aquifers, and drug diffusion through the layers of the skin. In such processes, the physical properties of the medium vary across layers and internal boundary conditions apply at the interfaces between adjacent layers. To characterize the time scale, we use the concept of mean action time, which provides the mean time scale at each position in the medium by utilizing the fact that the transition of the transient solution of the underlying partial differential equation model, from initial state to steady state, can be represented as a cumulative distribution function of time. Using this concept, we define the characteristic time scale for a multilayer diffusion process as the maximum value of the mean action time across the layered medium. For given initial conditions and internal and external boundary conditions, this approach leads to simple algebraic expressions for characterizing the time scale that depend on the physical and geometrical properties of the medium, such as the diffusivities and lengths of the layers. Numerical examples demonstrate that these expressions provide useful insight into explaining how the parameters in the model affect the time it takes for a multilayer diffusion process to reach steady state.

Figure 1

Geometrical interpretations of MAT. Example curves for the diffusion model described by Eqs. (1) and (13) with $u_b=1$ and $\ell =1$ for (a, c, e) $D=0.1$ and (b, d, f) $D=0.4$: (a)–(b) Plot of the transient solution $u(t;x)$ at $t=[0.01,0.2,0.4,0.8,1.5,3.0]$ (blue solid lines) and at the characteristic time scale $t=M\left(\ell \right)={\ell }^2/\left(2D\right)$ as in Eq. (14) (black dashed line). Each plot also includes the steady state solution $u_{\infty }\left(x\right)=1$ (red dash-dot line) and a black arrow indicating the direction of increasing time $t$. (c)–(d) Plot of the cumulative distribution function $F(t;x)$ (2) at $x=\ell$ and $0\le t\le t_d=20$. The shaded area depicts the geometrical interpretation of MAT described in Eq. (7) while the vertical black line represents MAT at $x=\ell$. (e)–(f) Representation of MAT as the mean value of the inverse function $v\left[u\right(t;x\left)\right]$ that maps the transient solution $u(t;x)$ to $t$ as described in Eq. (8). The inverse function $v\left[u\right(t;x\left)\right]$ is plotted at $x=\ell$ and MAT at $x=\ell$ is represented by a horizontal dashed black line.

Figure 2

Characterizing the time required for a multilayer diffusion process to reach steady state. Plot of the transient solution $u_i(t;x)$ at $t=[0.01,0.2,0.4,0.8,1.5]$ (blue solid lines) and $t=\tau$ (black dashed line) for $i=1,\cdots ,m$ of the multilayer diffusion model (15, 16, 17, 18, 19, 20) for the nine different diffusive processes (A)–(I) discussed in Sec. 4. Each plot includes the characteristic time scale ($\tau$), the steady state solution $u_{i,\infty }\left(x\right)$ (23) for $i=1,\cdots ,m$ (red dash-dot line) and a black arrow indicating the direction of increasing time $t$.