# Tribological Analysis of Steels in Fuel Environments: Impact of Alloy Content and Hardness
## 2 Materials and Experimental Procedures
### Lubrication Analysis  
To understand the lubrication status and degree of asperity interaction of the tribo-pair during the reciprocating motion in the tests, transient elastohydrodynamic lubrication (EHL) analyses were conducted with a mixed EHL model including a mass-conservation algorithm. This mixed EHL model and its numerical schemes were developed based on previous studies [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29] with an efficient procedure for contact elasticity [29; 30; 31; 32; 33]. With the \(x\)-axis pointing along the moving direction and the \(y\)-axis the lateral direction, pressure distribution \(p\) between the tribo-pair is governed by the Reynolds equation:  
\[\frac{\partial}{\partial x}\left(\frac{\rho h^{3}}{12\eta}\frac{\partial p}{ \partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\rho h^{3}}{12\eta} \frac{\partial p}{\partial y}\right)=\frac{u_{1}}{2}\frac{\partial(\rho h)}{ \partial x}+\frac{\partial(\rho h)}{\partial t} \tag{2}\]  
where \(h\), \(\rho\), and \(\eta\) are the geometric gap between the two surfaces, lubricant density, and viscosity, respectively. In the tests, the disk was stationary, while the velocity of the ball, \(u_{1}\), was a sinusoidal function with a maximum speed of 0.628 m/s in the middle of the stroke. \(\theta\) is the fractional film content [19], i.e., the ratio between the accumulated thickness occupied by the lubricant and the gap at that location. Gap \(h(x,\ y,\ t)\) includes the combined surface elastic deformation, \(v(x,\ y,\ t)\), of the two surfaces, the spherical geometry at the vicinity of the contact, \(\frac{x^{2}+y^{2}}{2R}\), and the initial separation at the center of the EHL zone, \(h_{0}\). \[h(x,\ y,\ t)=h_{0}(t)+\frac{x^{2}+y^{2}}{2R}+v(x,\ y,\ t) \tag{3}\]  
Surface was not considered for simplicity. The density, \(\rho\), and viscosity, \(\eta\), of ethanol were modeled based on the work reported in [34] at different pressures and temperatures. However, experimental data for high pressure (beyond 200 MPa) were not available; therefore, values were extrapolated outside the low-pressure range using models as follows. Tabulated data at 40 \({}^{\circ}\)C degrees were provided by Prof. [PERSON] based on formulae in [34] in a private communication. The variation of lubricant density \(\rho\) with pressure \(p\) was curve-fit with the Dowson-Higginson relationship:  
\[\rho=772.1\bigg{(}1+\frac{1.01\times 10^{-9}p}{1+2.16\times 10^{-9}p}\bigg{)} \tag{4}\]  
in the unit of kg/m\({}^{3}\). The viscosity data were curve-fitted with the same relationship in the unit of Pa.s:  
\[\eta=8.195\times 10^{-4}\bigg{(}1+\frac{5.61\times 10^{-9}p}{1+0.402\times 10^{-9 }p}\bigg{)} \tag{5}\]  
On the other hand, the pressure-viscosity coefficient of 5.08 GPa\({}^{-1}\) was obtained for ethanol with the exponential viscosity model [35] from data up to 0.1 GPa, and it was used in the Hamrock-Dowson equation to find initial values of the central film thickness in the numerical simulations. The properties of decane at different temperatures for pressure up to 89 MPa can be obtained from the Webbook of NIST ([[https://webbook.nist.gov/chemistry/fluid/](https://webbook.nist.gov/chemistry/fluid/)]([https://webbook.nist.gov/chemistry/fluid/](https://webbook.nist.gov/chemistry/fluid/)), accessed on 22 February 2024). Under four different temperatures, [PERSON] et al. [36] reported data including higher pressure situations and presented the following curve-fitted equations. The density of decane is:  
\[\frac{\rho}{\rho_{0}}=\bigg{(}1-0.207\mathrm{log}_{10}\frac{10^{-6}p+B}{0.1+B} \bigg{)}^{-1} \tag{6}\]  
in the unit of kg/m\({}^{3}\) with pressure in Pa.