I'm currently working on an assignment where I end up with the following differential equation in cylindrical coordinates:
$$ 0 = \frac{\partial}{\partial r}\left(r \frac{\partial w_\varphi}{\partial r}\right), $$
where $r$ is the radius and $w_\varphi$ describes the angular velocity of a stirred liquid.
If I assume $r$ to be a "regular" variable and I apply the chain rule I get the following equation:
$$ 0 = \frac{\partial^2 w}{\partial r^2}r + \frac{\partial w}{\partial r} \tag{1} $$
The last task of the assignment is to compare the results of this equation with the results obtained from simplifying the Navier-Stokes equation (basically the holy grail of fluid mechanics) and to explain possible differences. The equation I obtain from the Navier-Stokes equation looks like this:
$$ 0 = \frac{\partial^2w}{\partial r^2} r + \frac{\partial w}{\partial r} - \frac{w}{r} \tag{2} $$
so I get an extra $-\frac{w}{r}$ out of it.
I'm fairly certain that both equations are correct, or at least the intended outcome of their task, because my professor said that everything was looking good up until now.
The question is of course where does this extra $-\frac{w}{r}$ come from?
My professor gave me the hint of looking at the $r$ variable more closely and to try and think of how it would change in carthesian coordinates. Or to look at it in a vector form, so using $\vec{r} = \begin{pmatrix} r \> cos(\varphi)\\r \> sin(\varphi)\end{pmatrix}$ instead of just $r$ to account for the moving unit vectors in cylindrical coordinates.
To be honest my courses in calculus happened quite a while ago and I forgot a good portion of them so I'm a bit confused by this problem and don't really know where to begin. If anyone has an idea on how to solve or approach this problem I would be very grateful.
EDIT:
Upon further research I found a chapter out of an ODE Textbook in which it seems the same or at least a similar problem is posed as an exercise. Unfortunately I couldn't find any solutions for the exercises of the book, but maybe it can shed some light into what my question is.
https://faculty.missouri.edu/~chiconec/pdf/water1.pdf
The exercise in question is 6.7