Jtdylktuy

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 ?

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.