# How to transform this Diffusion equation ?

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.