Derivation of Navier-Stokes Equation in cylindrical coordinates

The Navier-Stokes equations are derived in cylindrical coordinates . Since the strain rate tensor \({\bf D}=({\bf L}+{\bf L}^T)/2\), if \({\bf V}=v_r{\bf e_r}+v_\theta{\bf v_\theta}+v_z{\bf e_z}\) (where \(L=\nabla{V}\) is the gradient tensor of velocity vector ), the strain rate and stress tensors are \[\begin{split}&{\bf D}=\begin{bmatrix}\dfrac{\partial v_r}{\partial r}&\dfrac{1}{2}\left(\dfrac{\partial v_r}{r\partial\theta}-\dfrac{v_\theta}{r}+\dfrac{\partial v_\theta}{\partial r}\right)&\dfrac{1}{2}\left(\dfrac{\partial v_r}{\partial z}+\dfrac{\partial v_z}{\partial r}\right)\\\dfrac{1}{2}\left(\dfrac{\partial v_r}{r\partial\theta}-\dfrac{v_\theta}{r}+\dfrac{\partial v_\theta}{\partial r}\right)&\dfrac{\partial v_\theta}{r\partial\theta}+\dfrac{v_r}{r}&\dfrac{1}{2}\left(\dfrac{\partial v_\theta}{\partial z}+\dfrac{\partial v_z}{r\partial\theta}\right)\\\dfrac{1}{2}\left(\dfrac{\partial v_r}{\partial z}+\dfrac{\partial v_z}{\partial r}\right)&\dfrac{1}{...