Governing equations and boundary conditions for a steady-state compressible viscous flow in an axisymmetric annular orifice I'm trying to simulate a 2D axisymmetric model of steady-state compressible viscous flow using Mathematica, but I get some errors. There is a chance that I'm making some mistakes with the governing equations and/or the boundary conditions, as suggested in some posts (e.g. here). 
Geometry:
The geometry is an axisymmetric tubular step about the x axis forming an annular orifice in 3D.

Assumptions:


*

*Axisymmetric about the x axis

*conduction and radiation are negligable

*Newtonian fluid with constant viscosity $\eta$

*Ideal gas (air)

*Steady-state


Writing the conservation equations in cylindrical form for the axisymmetric problem:
Conservation of mass:
$$ \frac{\partial}{\partial x}\left(  \rho v_x \right)+\frac{1}{r}\frac{\partial }{\partial r}\left(r \rho v_r\right)=0 \tag{1}$$
Conservation of linear momentum in axial direction:
$$\frac{\partial}{\partial x}\left( \rho v_x^2+\mathring{R} \rho T \right)+\frac{1}{r}\frac{\partial}{\partial r}\left( r \left( \rho v_r v_x + \eta \frac{\partial v_x}{\partial r} \right)\right) \tag{2}$$
Conservation of linear momentum in radial direction:
$$
\frac{\partial}{\partial x}\left( \rho v_x v_r+\eta \frac{\partial v_r}{\partial x} \right)+
\frac{1}{r}\frac{\partial}{\partial r}\left( r \left( \rho v_r ^2 +\mathring{R} \rho T \right) \right)=0 \tag{3}$$
And energy balance (conservation of heat) more info here:
$$\rho c_v\left(
 v_x  \frac{\partial T}{\partial x}
+ v_r \frac{\partial T}{\partial r}
\right)+
\mathring{R} \rho T
 \left(
\frac{1}{r}\frac{\partial}{\partial r} \left( r v_r \right)+
\frac{\partial v_x}{\partial x}
\right)+
 \eta \left(
2 \left( \frac{\partial v_x}{\partial x} \right)^2+
2 \left( \frac{\partial v_r}{\partial r} \right)^2+
\left( \frac{\partial v_r}{\partial x}+ \frac{\partial v_x}{\partial r} \right)^2 \\
-\frac{2}{3}\left(
 \frac{1}{r} \frac{\partial}{\partial r}\left( r v_r \right) +
\frac{\partial v_x}{\partial x}
 \right)^2
\right)=0 \tag{4}$$
The boundary conditions as I presume are:


*

*(1) Inlet: uniform pressure $P=P_1$

*(2) Outlet: uniform pressure $P=P_0$

*(3) Axis of symmetry $v_r=0$ and $\frac{\partial *}{\partial r}=0$ for all variables $v_r$, $v_x$, $T$ and $\rho$

*(4) no slip walls $v_*=0$


I would appreciate if you could help me know if I'm writing the equations correctly or I'm making any apparent mistakes? Are the boundary conditions enough/correct?
 A: After doing quite some research I think I have the correct equations. Unfortunately all of the equations above (except for the continuity) are wrong. 
Linear momentum:
The linear momentum conservation equation in the most compact and general form can be written as:
$$\rho \frac{D \check{v}}{D t}=\check{\nabla} \check{\sigma} \tag{5}$$
Where the $\check{\sigma}$ is the Cauchy stress tensor $\boldsymbol{\sigma}$ in matrix form. The Cauchy stress tensor can be divided into a volumetric/dilatational part and a deviatoric part:
$$\check{\sigma}= \check{\tau} -P\check{I} \tag{6}$$
In a perfect fluid P is the hydrostatic pressure and $\boldsymbol{\tau}$ is the viscous stress tensor.
Constitutive equation:
In an ideal Newtonian fluid viscous stress tensor can be calculated from:
$$ \check{\tau}=\eta\left(
\check{\nabla}^T \check{v}+
\left(
\check{\nabla}^T \check{v}
\right)^T
\right)
+\lambda 
\left(\check{\nabla}\check{v}^T\right)
\check{I} \tag{7}$$
Where $\eta$ is the absolute/dynamic viscosity and $\lambda$ is the bulk viscosity defined as:
$$\lambda=\kappa-\frac{2}{3}\eta \tag{8}$$
Where $\kappa$ is the dilation/expansion viscosity, which from Stokes’ hypothesis for monatomic gases at low density it is negligible impyling: 
$$ \check{\tau}=\eta\left(
\check{\nabla}^T \check{v}+
\left(
\check{\nabla}^T \check{v}
\right)^T
-\frac{2}{3}
\left(\check{\nabla}.\check{v}^T\right)
\check{I}
\right)  \tag{9}$$
$\eta$ is not necessarily constant. There are some empirical models for example Sutherland:
$$\eta  \approx C_S \frac{T^{\left(3/2\right)}}{T+T_S} \tag{10}$$
Where $C_S$ and $T_S$ are constants. 
Combining the equations 5, 6 and 9 and also considering the axisymmetry, ideal gas and steady-state assumptions equations of linear momentum in axial and radial directions can be written as: 
$$\begin{gather*}
\rho \left( 
v_r \frac{\partial v_r}{\partial r}
+v_x \frac{\partial v_r}{\partial x}
\right)
=
\frac{\partial}{\partial r} \left(
\eta \left(
-\frac{2}{3}\left(
\frac{1}{r}\frac{\partial}{\partial r} \left( r v_r \right) +\frac{\partial v_x}{\partial x}
\right) 
+2 \frac{\partial v_r}{\partial r}
\right)
\right) \\
+\frac{\partial}{\partial x}\left(
\eta\left(
\frac{\partial v_r}{\partial x}+\frac{\partial v_x}{\partial r}
\right)
\right)+
\frac{2 \eta}{r}\left(
\frac{\partial v_r}{\partial r}-
\frac{v_r}{r}
\right)
-\frac{\partial }{\partial r}\left(\mathring{R}\rho T \right)
\end{gather*} \tag{11}$$
In radial direction and 
$$\begin{gather*}
\rho \left( 
v_r \frac{\partial v_x}{\partial r}
+v_x \frac{\partial v_x}{\partial x}
\right)
=
\frac{\partial}{\partial x} \left(
\eta \left(
-\frac{2}{3}\left(
\frac{1}{r}\frac{\partial}{\partial r} \left( r v_r \right) +\frac{\partial v_x}{\partial x}
\right) 
+2 \frac{\partial v_x}{\partial x}
\right)
\right) \\
+\frac{\partial}{\partial r}\left(
\eta\left(
\frac{\partial v_r}{\partial x}+\frac{\partial v_x}{\partial r}
\right)
\right)+
\frac{ \eta}{r}\left(
\frac{\partial v_r}{\partial x}+
\frac{\partial v_x}{\partial r}
\right)
-\frac{\partial }{\partial x}\left(\mathring{R}\rho T \right)
\end{gather*} \tag{12}$$
In axial direction. (ref1, ref2)
Energy equation:
I'm still not completely sure about the correct form of the energy equation as I have explained here. But assuming my post here is valid then the energy equation in compact form can be written as: 
$$ \rho \frac{D e}{D t}=\check{\sigma} : \check{\nabla}^T \check{v} \tag{13}$$
Considering that for an ideal gas $e=c_v T$ (more info here and here) and axisymmetry and steady-state assumptions it expands to:
$$ \begin{gather*}
\rho c_v\left(
 v_x  \frac{\partial T}{\partial x}
+ v_r \frac{\partial T}{\partial r}
\right)+
\mathring{R} \rho T
 \left(
\frac{1}{r}\frac{\partial}{\partial r} \left( r v_r \right)+
\frac{\partial v_x}{\partial x}
\right)= \\
 \eta \left(
2 \left( \frac{\partial v_x}{\partial x} \right)^2+
2 \left( \frac{\partial v_r}{\partial r} \right)^2+
\left( \frac{\partial v_r}{\partial x}+ \frac{\partial v_x}{\partial r} \right)^2
-\frac{2}{3}\left(
 \frac{1}{r} \frac{\partial}{\partial r}\left( r v_r \right) +
\frac{\partial v_x}{\partial x}
 \right)^2
\right)
\end{gather*} \tag{14}$$
