# Advanced 2D Computational Fluid Dynamics Model of an External Gear Pump Considering Relief Grooves
## 3 CFD Model  
### Theoretical Details of the CFD Model Development  
The following assumptions were made when creating the model:  
* The flow process is 2D;
* The working fluid is _Newtonian_;
* The working fluid is incompressible;
* The Body forces are negligible;
* The viscous heating is not considered.  
The CFD model realization is based on the main terms from computational fluid mechanics [53; 54] integrated in ANSYS Fluent(r).  
The _Continuity equation_ for compressible working fluid can be present in the following form:  
\[\frac{\partial\rho}{\partial t}+\overset{\rightarrow}{\
abla}\left(\rho \overset{\rightarrow}{V}\right)=0, \tag{1}\]  
where \(\rho\) is a density of working fluid, \(\
abla\) is divergence and \(\overset{\rightarrow}{V}\) is a velocity vector. For the steady state of compressible working fluid \(\frac{\partial}{\partial t}=0\) for any variables, from which it follows that  
\[\overset{\rightarrow}{\
abla}\left(\rho\overset{\rightarrow}{V}\right)=0. \tag{2}\]  
However, the working fluid is incompressible, \(\rho=\text{const}\), therefore \(\frac{\partial\rho}{\partial t}\cong 0\). In this case, the _continuity equation_ can be expressed as  
\[\overset{\rightarrow}{\
abla}\times\overset{\rightarrow}{V}=0. \tag{3}\]  
Figure 6: Advanced 2D model with relief grooves composed of radial (0.5 \(\times\) 0.25 mm) and tangential (119.7\({}^{\circ}\) and 79.5\({}^{\circ}\)) channels. Lengths are given in millimeters.  
For the incompressible fluid, the formal form is  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{ \partial z}=0, \tag{4}\]  
where \(u\), \(v\), and \(w\) are the components of the velocity.  
In a _Cartesian_ coordinate system, the _Navier-Stokes_ equations for incompressible fluid have a main vector form:  
\[\rho\left(\frac{\partial^{\rightarrow}V}{\partial t}+\left(\overset{ \rightarrow}{V}\overset{\rightarrow}{\
abla}\right)\overset{ \rightarrow}{V}\right)=-\overset{\rightarrow}{\
abla}P+\rho\overset{ \rightarrow}{g}+\mu\,\
abla^{2}\overset{\rightarrow}{V}, \tag{5}\]  
where \(P\) is the pressure, \(g\) is the gravity, and \(\mu\) is the dynamic viscosity. It is valid for the second assumption (_Newtonian_ fluid), which can be expressed for each of the coordinate axes:  
The \(X\) component of the equation is  
\[\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{ \partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=-\frac{\partial P }{\partial x}+\rho g_{x}+\mu\left(\frac{\partial^{2}u}{\partial x^{2}}+v\frac {\partial^{2}u}{\partial y^{2}}+w\frac{\partial^{2}u}{\partial z^{2}}\right). \tag{6}\]  
The \(Y\) component of the equation is  
\[\rho\left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac {\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)=-\frac{ \partial P}{\partial y}+\rho g_{y}+\mu\left(\frac{\partial^{2}v}{\partial x^{2} }+v\frac{\partial^{2}v}{\partial y^{2}}+w\frac{\partial^{2}v}{\partial z^{2}} \right), \tag{7}\]  
where \(g_{x}\) and \(g_{y}\) are the \(x\) and \(y\) directions of gravity. The \(Z\) component of the _Navier-Stokes_ equation is not applicable to the first assumption (2D model).  
The turbulence \(k\)-\(\varepsilon\) (kinetic energy) equations take the following general form:  
\[\frac{\partial}{\partial t}(\rho k)+\
abla\left(\partial k\overset{ \rightarrow}{V}\right)=\
abla\left[\left(\mu+\frac{\mu_{i}}{\sigma_{k}} \right)\
abla k\right]+G_{k}+G_{b}-\rho\varepsilon-Y_{M}+S_{k}, \tag{8}\]  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\
abla\left(\partial k\overset{ \rightarrow}{V}\right)=\
abla\left[\left(\mu+\frac{\mu_{i}}{\sigma_{k}} \right)\
abla\varepsilon\right]+C_{1\varepsilon}\