Mass transport equation Cartesian to polar coordinates

Question

Can someone please advise on how to transform the following equation to polar coordinates? $$\frac{\partial \rho(x,t)}{\partial t}=v\frac{\partial \left(\rho(x,t) L(x)\right)}{\partial x}+D\frac{\partial}{\partial x}\left(L(x)\frac{\partial \rho(x,t)}{\partial x}\right)$$

Answer 1

This is a convection-diffusion equation $$ \frac{\partial \rho}{\partial t} = \nabla\cdot(\underbrace{D\, {\boldsymbol \nabla}\rho - \rho{\boldsymbol v}}_{-\boldsymbol f}) , $$ where $\rho$ is a species concentration (in mass transfer), $D$ is the diffusivity, ${\boldsymbol v} = v_r {\boldsymbol e}_r + v_\theta {\boldsymbol e}_\theta$ is the velocity field, and ${\boldsymbol f} = f_r {\boldsymbol e}_r + f_\theta {\boldsymbol e}_\theta$ is the flux. The differential operators write $$ {\boldsymbol \nabla}\rho = \partial_r\rho\, {\boldsymbol e}_r + \frac{\partial_\theta\rho}{r}\, {\boldsymbol e}_\theta, \qquad\text{and}\qquad \nabla\cdot {\boldsymbol f} = \frac{\partial_r (r f_r)}{r} + \frac{\partial_\theta f_\theta}{r} $$ with the flux components $f_r = \rho v_r - D\partial_r\rho$ and $f_\theta = \rho v_\theta - D \partial_\theta\rho / r$. If $D$ and $v$ do not depend on space (as seems to be the case here), we have $$ \frac{\partial \rho}{\partial t} = \frac{D \partial_r (r \partial_r\rho) - v_r \partial_r(r\rho)}{r} + \frac{D/r\, \partial_{\theta\theta} \rho - v_\theta \partial_\theta\rho }{r} . $$