Imagine the following initial condition:
Two completely miscible fluids A and B with different dynamic viscosities $\eta_A$ and $\eta_B$ are seperated in a container with fixed Volume $V_0$ and length $L$. The temperature $T_e$ is constant across the container, the time is $t_0=0$. The mass fraction $w_B(x)$ at this point in time is a step function (or a convenient approximation of it), as shown in the sketch below.
With those definitions in mind, what is the time $t_m$ at which the mass fraction $w_B$ is within $\pm \delta \%$ of its equilibrium value across the whole container?
The first thing which came to mind was the diffusion equation: $$\frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla \cdot \big[ D(\phi,\mathbf{r}) \ \nabla\phi(\mathbf{r},t) \big]$$ From there, I probably 'just' need an approximation for $D(\phi(\mathbf{r},t))$ and solve the differential equation. Is this the right direction? What approximations would be appropriate?
The Stoke-Einstein-Equation enables me to calculate $D$: $$ D = \frac{k_\mathrm{B} \cdot T}{6 \pi \cdot \eta \cdot r}$$ but I don't know the radius $r$. I just dont know how to move on.