it. In this frame of reference, the flow is a pure shear flow (or stagnation flow), and the velocity field is of the form $(u_x=-\dot{\epsilon}x,u_y=\dot{\epsilon}y)$, where $\dot{\epsilon}$ is the stretching rate. The thickness of the sheet $s(t)$ is given by $s(t)=s_0\mathrm{exp}\left(-\int_0^t\dot{\epsilon}(t')\mathrm{d}t'\right)$, and is assumed to be much smaller than the radius of curvature of the sheet. Stretching will tend to align the direction of the gradient of $\chi$ with the compression direction, and we can therefore consider the sheet and the scalar field to be locally invariant in the stretching direction. With this assumption and the velocity field given above, the scalar transport equation \[eq:AD\] becomes $$\frac{\partial \chi_i(x,t)}{\partial t}+\dot{\epsilon}x\frac{\partial \chi_i(x,t)}{\partial x}=K_i\frac{\partial^2\chi_i(x,t)}{\partial x^2}.
\label{eq:AD1D}$$ We transform these equations into diffusion equations with the following change of variable [@ranz_1979]: $$\begin{aligned}
\label{eq:adim_1}
\xi&=&\frac{x}{s(t)},\\
\label{eq:adim_2}
\tau&=&K_1\int_0^t\frac{\mathrm{d}t'}{s(t')^2}.\end{aligned}$$ This amounts to normalize lengths by $s(t)$, which sets the thickness of the sheet to $\xi=1$, and time by a diffusion time scale based on the diffusivity inside the sheet (phase 1). With this change of variable, equation \[eq:AD1D\] reduces to $$\frac{\partial \chi_i(\xi,\tau)}{\partial \tau}=\frac{K_i}{K_1}\frac{\partial^2\chi_i(\xi,\tau)}{\partial \xi^2}.
\label{eq:D1D}$$ The scalar field is initialized with a difference $\Delta\chi$ between the phases such as $$\begin{aligned}
\chi_1(\vert\xi\vert \leq 1,\tau=0)&=&\Delta\chi,\\
\chi_2(\vert\xi\vert > 1,\tau=0)&=&0.\end{aligned}$$ The 