# Influence of the Radial Gap on the External Gear Pump Performance
## 2 Design Features and CFD Model
### CFD Model  
The following assumptions were made in the process of creating the 2D CFD model of the investigated external gear pump: the working fluid is _Newtonian_ (hydraulic oil), the fluid is incompressible, the flow process is 2D, the viscous heating is not considered, and the body forces are negligible. The use of the 2D CFD model is based on the main computational fluid mechanics terms, which are embedded in Ansys Fluent(r) software [21, 48, 49]. For the compressible fluid, the _continuity equation_ can be presented with the following expression:  
\[\frac{\partial\rho}{\partial t}+\overrightarrow{\
abla}\cdot(\rho\overrightarrow {V})=0, \tag{1}\]  
where \(\overrightarrow{V}\) is a velocity vector, \(\
abla\) is the divergence, and \(\rho\) is the density of the working fluid. In the case of a steady state in the compressible working fluid of \(\frac{\partial}{\partial t}=0\) for any variables, then,  
\[\overrightarrow{\
abla}\cdot(\rho\overrightarrow{V})=0. \tag{2}\]  
In connection with the made assumption, the working fluid is incompressible (\(\rho=\text{const}\)); therefore, \(\frac{\partial\rho}{\partial t}\cong 0\). Thus, the _continuity equation_ can be present as follows,  
\[\overrightarrow{\
abla}\times\overrightarrow{V}=0. \tag{3}\]  
The formal form in the case of incompressible fluid is the following,  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w }{\partial z}=0, \tag{4}\]  
where \(u\), \(v\), and \(w\) are the velocity along the corresponding axes. The _Navier-Stokes_ equations for incompressible fluid has a main vector form in a _Cartesian_ coordinate system:  
\[\rho\left(\frac{\partial\overrightarrow{V}}{\partial t}+\left(\overrightarrow {V}\overrightarrow{\
abla}\right)\overrightarrow{V}\right)=-\overrightarrow {\
abla}P+\rho\overrightarrow{\delta}+\mu\
abla^{2}\overrightarrow{V}, \tag{5}\]  
Figure 2: **2D geometrical models:** (**a**) classical model. Flow direction, rotation direction, inlet, and outlet pressure (\(p_{in}\) and \(p_{out}\)), and some dimensions (in mm) are shown; (**b**) advanced model. Relief grooves are shown (in green). where \(\mu\) is the dynamic viscosity, \(g\) is the gravity, and \(P\) is the pressure. This expression follows from the assumption for the _Newtonian_ fluid. For each of the coordinate axes, it can be expressed as follows:  
\(X\) component of the equation:  
\[\rho\bigg{(}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\bigg{)}=-\frac{ \partial P}{\partial x}+\rho g_{x}+\mu\bigg{(}\frac{\partial^{2}u}{\partial x ^{2}}+v\frac{\partial^{2}u}{\partial y^{2}}+w\frac{\partial^{2}u}{\partial z^{ 2}}\bigg{)}. \tag{6}\]  
\(Y\) component of the equation:  
\[\rho\bigg{(}\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\bigg{)}=-\frac{ \partial P}{\partial y}+\rho g_{y}+\mu\bigg{(}\frac{\partial^{2}v}{\partial x ^{2}}+v\frac{\partial^{2}v}{\partial y^{2}}+w\frac{\partial^{2}v}{\partial z ^{2}}\bigg{)}, \tag{7}\]  
where \(g_{x}\) and \(g_{y}\) are \(x\) and \(y\) gravity directions. The \(Z\) component of the _Navier-Stokes_ equation is not applicable to this assumption for the 2D model.