The interested equation is advection-diffusion equation. One of the canonical example is Navier-Stokes equations. However, I would like to let the coefficient of diffusion constant goes to zero, $\epsilon \rightarrow 0$, while mesh sizes, $ \Delta x, \Delta dy \rightarrow 0 $, and time steps,$\Delta t \rightarrow 0$, decreasing towards zero. Then, I can obtain inviscid compressible equation which can be an example for the Euler equation of gas dynamics. The following equation is scalar advection-diffusion equation in 2D cartesian coordinates.
$$ \frac{df}{dt} + U\frac{df}{dx} + V\frac{df}{dy} = \epsilon \left( \frac{d^2f}{dx^2} + \frac{d^2f}{dy^2} \right)$$
The main interest is to find exact formulation for the Euler equations of gas dynamics in cylindrical coordinates $(r, \theta)$ with artificial diffusion part.
Edit : The main equation that I am interested in the Euler equations of gas dynamics in 2D. The cartesian vector form can be stated as $$\begin{pmatrix} \rho \\ \rho u \\ \rho v \\ E \end{pmatrix}_t + \begin{pmatrix} \rho u \\ \rho u^2 + p \\ \rho u v \\ u(E + p) \end{pmatrix}_x + \begin{pmatrix} \rho v\\ \rho u v \\ \rho v^2 + p \\ v(E + p) \end{pmatrix}_y = \textbf{0}$$
Here, conservation of mass, momentums in -x and -y directions and energy equations can be seen in a matrix form. However, I would like to add artificial viscosity with small constant factor as in Navier-Stokes equations to smoothen the possible oscillations that will occur during numerical simulation. For this reason, I need to know the vector Laplacian form in cylindrical coordinates for Navier-Stokes equations. Then I can apply the above theory that was stated.