Chain rule applies to Navier-Stokes Equations in polar coordinates

Question

Based on the fact that $p=p(r)$ only, consider the Navier-Stokes equations and show that the $e_\theta$ component reduces to

I can get to this if the following is true

$\frac {1} {r} \frac {\partial} {\partial r} ( r \frac {\partial u_\theta} {\partial r}) = \frac {1} {r} \frac {\partial u_\theta} {\partial r} + \frac {\partial ^2 u_\theta} {\partial r^2}$

But I have no clue as to how the left which I can get to from N-S equations in polar coordinates, can become the right

Answer 1

It's just a simple chain rule $$ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u_\theta}{\partial r}\right)=\frac{1}{r}\left(\frac{\partial r}{\partial r}\frac{\partial u_\theta}{\partial r}+r\frac{\partial}{\partial r}\frac{\partial u_\theta}{\partial r}\right)=\frac{1}{r}\left(\frac{\partial u_\theta}{\partial r}+r\frac{\partial^2 u_\theta}{\partial r^2}\right)=\frac{1}{r}\frac{\partial u_\theta}{\partial r} +\frac{\partial^2 u_\theta}{\partial r^2}$$