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I have a diffusion equation:

$$ \nabla^2C = \frac{\partial C}{\partial t}$$.

Now I would like to transform this equation into a co-ordiante frame which movies with the unperturbed palaner interface($z=0$ ) with steady state veloity $v_o$, i.e. $z = z -v_0t$.

Can anyone explain how to go about it ?

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The current variables are the position $\mathbf{x}$ and the time $t$, that will be transformed to $\mathbf{x}'=\mathbf{x}-\mathbf{v}_0 t$ and $t'=t$. According to the chain rule, $$ \frac{\partial}{\partial t} = \frac{\partial t'}{\partial t} \frac{\partial}{\partial t'} + \frac{\partial x'}{\partial t} \frac{\partial}{\partial x'} + \frac{\partial y'}{\partial t} \frac{\partial}{\partial y'}+ \frac{\partial z'}{\partial t} \frac{\partial}{\partial z'} = \frac{\partial}{\partial t'} + \mathbf{v}_0 \cdot \nabla $$

Therefore, the equation is now $$ \nabla^2 C = \frac{\partial C}{\partial t} + \mathbf{v}_0 \cdot \nabla C, $$ i.e., in the moving frame the diffusion equation takes the form of an advection-diffusion equation. Naturally, using $\mathbf{v}_0=0$ in the equation leads back to the original diffusion equation.

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