Before the derivation from a physical view, we'd like introduce the full compressible Naiver-Stokes(-Fourier) system in $\Omega \subset \R^d$ directly: \begin{equation} \label{NSF} \begin{cases} \partial_t \rho +\nabla \cdot (\rho u )=0, \\ \partial_t (\rho u)+\nabla \cdot (\rho u\otimes u ) = \rho g+ f+ \nabla \cdot \sigma,\\ \partial_t ( E)+\nabla \cdot ( Eu)+\nabla \cdot q=\rho g\cdot u+ f\cdot u+\rho h+\nabla\cdot (\sigma\cdot u ) \end{cases} \end{equation} usually with rheological assumption: \begin{equation} \label{rheological} \sigma =-pI+\tau, \end{equation} and total energy \begin{equation} \label{total energy} E=\frac 12\rho |u|^2+\rho e. \end{equation} The physical meanings of quantities in the system are listed below:

Here we introduce the cylindrical coordinate system for vector fields to describe axisymmetric flows, and derive the expressions of classical differential operators in cylindrical coordinates. Coordinate transform Denote the standard basis as $e_1,e_2,e_3$ and $x_1,x_2,x_3$ the coordinate in $\R^3$. Let $$(r,\theta,h)=\br{\sqrt{x_1^2+x_2^2}, \arctan \br{\frac {x_2}{x_1} },x_3}\Longleftrightarrow (x_1,x_2,x_3)=(r\cos \theta, r\sin \theta,h).$$This induce the Jacobi matrix between $(x_1,x_2,x_3)\sim (r,\theta,z)$ and its inverse \begin{equation*} J\eqqcolon \begin{pmatrix} \dfrac{\partial r }{\partial x_1 } & \dfrac{\partial r }{\partial x_2 } & \dfrac{\partial r }{\partial x_3 }\\ \dfrac{\partial \theta }{\partial x_1 } & \dfrac{\partial \theta }{\partial x_2 } & \dfrac{\partial \theta }{\partial x_3 } \\ \dfrac{\partial h }{\partial x_1 } & \dfrac{\partial h }{\partial x_2 } & \dfrac{\partial h }{\partial x_3 } \\ \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin\theta & 0 \\ -\dfrac{\sin \theta }{r} & \dfrac{\cos \theta }{r} & 0\\ 0 & 0 & 1 \end{pmatrix} \Longleftrightarrow J^{-1}= \begin{pmatrix} \dfrac{\partial x_1 }{\partial r } & \dfrac{\partial x_1 }{\partial \theta } & \dfrac{\partial x_1 }{\partial h } \\ \dfrac{\partial x_2 }{\partial r } & \dfrac{\partial x_2 }{\partial \theta } & \dfrac{\partial x_2 }{\partial h } \\ \dfrac{\partial x_3 }{\partial r } & \dfrac{\partial x_3 }{\partial \theta } & \dfrac{\partial x_3 }{\partial h } \\ \end{pmatrix}= \begin{pmatrix} \cos \theta & -r\sin \theta & 0\\ \sin \theta & r \cos \theta & 0\\ 0 & 0 & 1 \end{pmatrix}. \end{equation*}And we choose the another orthonormal basis as \begin{equation} \label{cylinderical-jacobi} \begin{pmatrix} e_r \\ e_\theta \\ e_h \end{pmatrix} =\begin{pmatrix} \cos \theta & \sin \theta & 0\\ -\sin \theta & \cos \theta & 0\\ 0 & 0 &1\\ \end{pmatrix} \begin{pmatrix} e_1\\ e_2\\…