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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.

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  • $\begingroup$ 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. $\endgroup$ – nluigi Dec 5 '15 at 22:31
  • $\begingroup$ @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. $\endgroup$ – John H. K. Dec 6 '15 at 8:35
  • $\begingroup$ 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. $\endgroup$ – nluigi Dec 6 '15 at 11:02
  • $\begingroup$ Yes, I assume they're completely miscible. $\endgroup$ – John H. K. Dec 6 '15 at 15:18
  • $\begingroup$ 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. $\endgroup$ – nluigi Dec 7 '15 at 17:17
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This is a difficult question to answer in general.

Some point to consider are :

  • Differences in density will cause one fluid to tend to sink under the other, this will me faster or slower depending on their relative densities. This immediately introduces some energy into the system. BUT their viscosities will also affect how that affects mixing. For example if you had water and ethanol, both fairy dense, low viscosity and miscible you can imagine that the you will have transient turbulence as one flows below the other which will speed up mixing a lot.

  • Convection : any small differences in temperature/pressure between the fluids will create currents and pressure waves which may speed up mixing.

In some ways this is beyond the scope of an engineering question because the approximations required for an 'ideal' answer are completely unrealistic. To me this is a purely mathematica exercise rather than engineering per se.

In terms of ideal gasses you are probably bes off looking at the average time it take for a gas molecule to cross from one side to the other so the average velocity component in the x axis as a function of temperature would give you some approximation. Although the conditions for two liquids (frequent collisions) compared to two low pressure gasses (few collisions may give different results)

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