In this post the fact that the mass continuity equation in a mixture of gases has no diffusion term, i.e., $$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\vec{v})=0$$ has been discussed. Nevertheless, I'm struggling to come up with a proof starting from the convection-diffusion-reaction equation for each species. It seems to me that the starting point should be $$\frac{\partial n_{j}}{\partial t}+\nabla\cdot(n_{j}\vec{v}_{j}-D_{j}\nabla n_{j})=S_{j}$$ where $\vec{v}_{j}$ as opposed to $\vec v$ denotes the convective velocity vector of species $j$. I can define $n_{j}=\omega_{j}n$ where $\omega_{j}$ denotes the partial fraction of the fluid/gas made by species $j$ and $n$ denotes total denisty such that $$\sum_{j}\frac{\partial n_{j}}{\partial t}=\sum_{j}\frac{\partial}{\partial t} (\omega_{j}n)=\sum_{j}\omega_{j}\frac{\partial}{\partial t} n+\sum_{j}n\frac{\partial}{\partial t} \omega_{j}=\frac{\partial n}{\partial t}\sum_{j}\omega_{j}+n\frac{\partial}{\partial t}\sum_{j} \omega_{j}=\frac{\partial n}{\partial t}$$ I can go ahead with the same analysis for the convective term by defining $$\vec v=\frac{\sum_{j}n_{j}\vec{v}_{j}}{n}$$ and since mass is not produced $\sum_{j}S_{j}=0$ for the source term but the cancellation of the diffusion term still eludes me. I would appreciate any help with this final part. I'm curious if Maxwell-Stefan relation is necessary here.
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