# How long does it take for two initially seperate fluids to reach a certain homogenity across their container?

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.

• Typically the Schmidt number for fluids is on the order of 1 for gases and 1000 for liquids. Given the viscosities you can estimate the diffusion coefficients I suppose. – nluigi Dec 5 '15 at 22:31
• @nluigi Well, I think I'll try that. If the resulting time is within the order of magnitude of the real time, it should be enough as an estimation. – John H. K. Dec 6 '15 at 8:35
• You will likely only be able to do an analysis to engineering accuracy (within order of magnitude of the exact answer) anyway. Can you assume well-mixedness of the compartments? That would also greatly simplify the problem. – nluigi Dec 6 '15 at 11:02
• Yes, I assume they're completely miscible. – John H. K. Dec 6 '15 at 15:18
• Ok but does the mixing in the compartments go fast? Then you could write a balance like: $${dc_1 \over dt}=-k_1a(c_1-c_2)$$ and similar for $c_2$ and solve for the concentrations. – nluigi Dec 7 '15 at 17:17