Cylindrical Polar Coordinates for Conservation of Mass

In fluid dynamics, we often encounter situations where using Cartesian coordinates $(x,y,z)$ is not the most convenient or natural choice. In cases involving rotational symmetry (like flow through pipes or around circular objects), it is advantageous to use cylindrical polar coordinates $(r,\theta ,z)$.

Each component is defined as follows:

• $r$: Radial distance from axis (distance from the origin in the plane)
• $\theta$: Angular coordinate (measured counterclockwise from the $x$-axis)
• $z$: Height or axial coordinate (same as in Cartesian)

The velocity components in cylindrical coordinates are:

$$\mathbf{V}=V_r{\mathbf{e}}_r+V_{\theta }{\mathbf{e}}_{\theta }+V_z{\mathbf{e}}_z$$

The divergence operator in cylindrical coordinates is given by:

$$\begin{array}{rl}\nabla \cdot \mathbf{V}& =\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\\ & =\frac{1}{r}\frac{\partial (r{V}_r)}{\partial r}+\frac{1}{r}\frac{\partial V_{\theta }}{\partial \theta }+\frac{\partial V_z}{\partial z}\end{array}$$

The factor of $\frac{1}{r}$ in the first two terms adjusts for the radial and azimuthal coordinate system.

Recall that the Differential Form of Conservation of Mass can be simplified to:

$$\frac{\partial \rho }{\partial t}+\nabla \cdot \rho \mathbf{V}=0$$

In cylindrical polar coordinates, this is now

$$\frac{\partial \rho }{\partial t}+\frac{1}{r}\frac{\partial }{\partial r}(r\rho V_r)+\frac{1}{r}\frac{\partial }{\partial \theta }(\rho V_{\theta })+\frac{\partial }{\partial z}(\rho V_z)=0$$