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While studying a text on fluid mechanics, I came across the following:

The continuity equation for a plane incompressible flow in polar form is

$$\frac{1}{r}\frac{\partial}{\partial r}(rv_r)+\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}=0$$

How did it come about? There was no clear explanation in the text.

I know the continuity equation for an incompressible flow in rectangular coordinates is $u_x+v_y=0$, where $u$ and $v$ are the horizontal and vertical velocity components, respectively.

Bernhard Listing
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The continuity equation for an incompressible flow in Cartesian Coordinates is: $$\nabla \cdot \vec{v} = 0$$

The derivation of the continuity equation for an incompressible flow in a different set of coordinates would require us to derive it from "scratch" by applying the conservation of mass to a control volume in the non-Cartesian coordinates.

Alternatively, you can apply a change of basis to the Cartesian result. Just be mindful that the notation $\nabla \cdot \vec{v}$ is not really a true dot product, it is convenient shorthand to denote the divergence of a vector (see for example: divergence in polar coordinates).

STOI
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