## 2 Mathematical Formulation
### Coupled System  
There are two ways to introduce lubrication into the solid-fluid problem: one is the coupled treatment of the problem. The other is related to the fact that lubrication generally reduces the coefficient of friction \(f\). Thus, some parameters can be incorporated into the constitutive friction relations, best discussed in the bibliography by [PERSON] [36].  
When treating lubrication as a coupled problem, the relevant equations for the fluid film are given. These are derived from the classical Navier-Stokes equations, introducing several simplifications inherent to the problem, such as considering that the flow at the interface is laminar and incompressible. Additionally, if the nonlinear convective term in the Navier-Stokes equations can be neglected, as well as the inertia terms, the general equations reduce to the Reynolds equation:  
\[\frac{\partial}{\partial x}\left(\frac{h_{s}^{3}}{\mu_{f}}\frac{\partial p_{N }}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{h_{s}^{3}}{\mu_ {f}}\frac{\partial p_{N}}{\partial y}\right)=6\mathrm{v}_{x}\left(\frac{ \partial p_{N}}{\partial x}\right) \tag{7}\]  
This equation is valid for stationary processes when a constant relative slip velocity \(\mathrm{v}_{x}\) is employed and for \(\delta>0\), or \(g>0\) (see Figure 1). The variable \(h_{s}\) characterizes the height of the gap, which may depend on the deformations of the solids. Hence, a coupled nonlinear problem with the coupling terms being the contact pressure \(p_{N}\) and the deformation-dependent height \(h_{s}\) is constituted.  
Figure 1: Lubrication interface definition.  
According to [PERSON] and [PERSON] [42], coupled systems and formulations are those applicable to multiple domains and dependent variables that often describe different physical phenomena. No domain can be solved separately from the others, just as it can explicitly eliminate no set of dependent variables at the differential equation level.  
Therefore, the dynamic fluid-structure interaction problem cannot solve the fluid and solid formulations separately due to the unknown forces at the interface. This coupling can be weak or strong, depending on the degree of interaction. Furthermore, this coupling occurs at the interfaces between domains through the boundary conditions imposed there. So, the displacement of the solid \((\mathbf{u}(\mathbf{x},t))\) influences and interacts with the generation of pressures in the fluid \(\left(p_{f}(\mathbf{x},t)\right)\), according to the following equations.  
\[\frac{\partial p_{f}}{\partial n}=-\rho_{f}\dot{\mathbf{v}}_{fn}=-\rho_{f} \dot{\mathbf{v}}_{f}\mathbf{n} \tag{8}\]  
\[\dot{\mathbf{v}}_{fn}=\dot{u}_{n}=\dot{\mathbf{u}}\mathbf{n} \tag{9}\]  
The weak form of the fluid in the coupled system will be given by  
\[\int\limits_{B_{f}}\!\!\delta p_{f}\
abla\Big{(}\
abla p_{f}\Big{)}dB+\int \limits_{\Gamma_{c}}\!\!\delta p\dot{\mathbf{u}}d\Gamma=0 \tag{10}\]  
where \(B_{f}\) represents the fluid domain, the coupled problem is discretized into the standard form for the approximate displacement vector and the pressure in the fluid, using the nodal parameters of each field and the appropriate shape functions. Then, they are entered into the discrete equations for the solid and fluid, connected through the coupling term that appears due to the pressures.