Note: $J_i$ is relative to a molar average velocity. So, our conversion to $j_i$ and the use of a mass average velocity is not fully consistent.
Note: In combustion, we often have lots of $N_2$ and using Fick's law instead of a full multicomponent treatment is not that bad.
$$D_{i,e} = \left[\frac{(1-x_i)}{\sum_{j=1,j\ne i}^n \frac{x_j}{\mathcal{D}_{i,j}}}\right]$$The following illustrates the importance of using the "full" Fick's law form given above. This is from Pitsch and Peters (1998).
Also, $$ y_i \rightarrow -D\nabla y_i$$ $$ T \rightarrow -\alpha\nabla T$$ $$ v \rightarrow -\nu\nabla v$$
$\alpha = \lambda/(\rho c_p)$
$\nu = \mu/\rho$
$D$, $\alpha$, $\nu$ all have units of $m/s^2$
For constant properties, the unsteady diffusion equation for some scalar $\eta$, with diffusivity $\Gamma$ is