## 2 Mathematical Formulation
### Continuum Mechanics  
The mathematical formulation that describes physical problems is the basis of computational numerical models. In this study, the continuum mechanics regarding a solid volume and a fluid describes the problem addressed. The used description comes predominantly from the literature of [35]. It is assumed a continuous medium consisting of a linear elastic solid material, subject to mass forces \(\mathbf{b}(\mathbf{x},t)\) inside its volume \(V\) and surface forces \(\mathbf{t}(\mathbf{x},t)\) on its boundary \(\partial V\). Thus, the governing equations to evaluate displacements \(\mathbf{u}(\mathbf{x},t)\), strains \(\varepsilon(\mathbf{x},t)\) and stresses \(\boldsymbol{\sigma}(\mathbf{x},t)\) through time \(t\) are:  
\[ \begin{array}{c}\boldsymbol{\
abla}.\boldsymbol{\sigma}(\mathbf{x},t)+\rho_ {0}\mathbf{b}(\mathbf{x},t)=\rho_{0}\frac{\partial^{2}\mathbf{u}(\mathbf{x},t) }{\partial t^{2}}\\ \boldsymbol{\sigma}(\mathbf{x},t)=\lambda T(\boldsymbol{\varepsilon})\mathbf{ 1}+2\mu\varepsilon\end{array} \quad\left\} \begin{array}{c}\text{Cauchy's Equation}\\ \text{Constitutive Equation}\\ \text{Geometric Equation}\end{array} \right.\\ \varepsilon(\mathbf{x},t)=\boldsymbol{\
abla}^{S}\mathbf{u}(\mathbf{x},t)= \frac{1}{2}(\mathbf{u}\otimes\boldsymbol{\
abla}+\boldsymbol{\
abla}\otimes \mathbf{u})\end{array} \tag{1}\]  
The solid boundary \(\Gamma\equiv\partial V\) considers the displacement boundary \(\Gamma_{u}\) and the stress boundary \(\Gamma_{\sigma}\) as the following:  
\[ \begin{array}{c}\Gamma_{u}:\mathbf{u}=\mathbf{u}^{*}\\ \Gamma_{\sigma}:\mathbf{t}^{*}=\boldsymbol{\sigma}\cdot\mathbf{n}\end{array}  \right\}\text{Boundary Conditions in Space} \tag{2}\]  
The initial and boundary conditions are:  
\[ \begin{array}{c}\mathbf{u}(\mathbf{x},0)=\mathbf{0}\\ \dot{\mathbf{u}}(\mathbf{x},0)=\mathbf{v}_{0}\end{array} \quad\text{Initial Conditions} \tag{3}\]  
\(T(\boldsymbol{\epsilon})\) corresponds to the trace of the strain tensor \(\varepsilon\), \(\lambda\) and \(\mu\) are known as Lame constants, which characterize the elastic behavior of the material and are obtained experimentally. In addition, \(\mathbf{1}\) designates a unit tensor of a second-order tensor. Finally, \(\boldsymbol{\
abla}\) is the gradient or Nabla operator, and \(\rho_{0}\) is the density or specific mass of the linear elastic solid. The operator \(\otimes\) denotes a tensor product since \(\mathbf{u}\) is a vector, also referred to as a first-order tensor. The fixed boundary condition is also referred to as a Dirichlet condition, which \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Authors** & **Lubricating Regime** & **Lubricant Type** & **Viscosity (N.s/mm\({}^{\text{2}}\))** & **Solid Material** & **Roughness** & **Sliding Speed (mm/s)** & **Friction Coefficient** \\ \hline
[20] & elastohydrodynamic contact (transition of EHD to mixed lubricated) & oil & 0.2337 & & & & \\  & elastohydrodynamic & oil & (at 24 \({}^{\circ}\)C) and 0.0243 & steel & - & 20 to 2000 & 0.01 \\ \hline
[21] & elastohydrodynamic lubrication (EHL), & engine oil & 0.1204 & Cu–Al, Quaker States SAE & Pb–Cu–Al, Sn–Al–Si & - & 147 and 10,000 & - \\  & boundary & 15W–40 & (at 100 \({}^{\circ}\)C) & & & & \\ \hline
[22] & mixed lubrication and dry friction & PAO4 oil & 0.168 & & & & \\  & and dry friction & PAO4 oil & (at 24 \({}^{\circ}\)C) and 0.0039 & carbon steel & \(R_{a}=0.02\)\(\mu\)m & 100 & 0.11 to 0.6 \\  & elastohydrodynamic lubrication (EHL) & Seven types of grease & - & steel & \(R_{q}<10\) nm & 6.3 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. would be the prescribed displacement condition, while the state described by the derivative of the displacement is also known as a Neumann condition. We assume a continuous medium consisting of an incompressible Newtonian viscous fluid in the barotropic and transient regime, subject to viscous and inertial forces within its material volume \(V_{t}\) and surface traction or pressure forces at its boundary \(\partial V_{t}\).