## 2 Mathematical Formulation
### Continuum Mechanics
The fluid is a particular case of a continuous medium, described by its constitutive equations. Thus, the equations that analyze density \(\rho_{f}(\mathbf{x},t)\), velocity \(\mathbf{v}_{f}(\mathbf{x},t)\), stresses \(\sigma_{f}(\mathbf{x},t)\), and pressures \(p_{f}(\mathbf{x},t)\) through initial conditions \(t=0\) and time \(t\) outline the fluid mechanics problem. The boundary conditions of the fluid mechanics problem are related to the spatial or Eulerian description when analyzing a fixed control volume in space. There is the boundary condition with the velocity value determined in some boundary parts \(\Gamma_{\mathrm{v}}\) and the adhesion condition due to the viscous nature of the fluid. Furthermore, there is an impenetrability condition in some parts of the boundary (walls) \(\Gamma_{v_{n}}\). In addition to these conditions concerning the fluid velocity, there are boundary conditions \(\Gamma_{\sigma}\) associated with stresses or pressures. According to the problem, the thermodynamic pressure or the pressure in the fluid can describe the stress vector. Thus, it will represent a portion of the normal component of the stress vector \(\mathbf{t}_{f}\) over a section of the boundary \(\Gamma_{p}\). Therefore, the equations that best govern the fluid mechanics problem, taking into account an incompressible Newtonian viscous fluid in the barotropic and transient regimes, are:  
\[ \begin{array}{rll}\mathbf{\
abla}\cdot\mathbf{v}_{f}=0\\ -\mathbf{\
abla}p_{f}+\mu_{f}\mathbf{\Delta v}_{f}+\rho_{f}\mathbf{b}_{f}&=& \rho_{f}\frac{\mathrm{d}\mathbf{v}_{f}}{\mathrm{d}t}\\ \sigma_{f}=-p_{f}\mathbf{1}+2\mu_{f}\mathbf{d}\\ \rho_{f}=\rho_{f}\left(p_{f}\right)\end{array}\begin{array}{rll}\text{Continuity Equation}\\ \text{Navier-Stokes Equation}\\ \text{Constitutive Equation}\end{array}  \tag{4}\]  
\[ \begin{array}{rll}\mathbf{v}_{f}(\mathbf{x},t)=\bar{\mathbf{v}}(\mathbf{x}, t)&\forall\,\mathbf{x}\,e\,\Gamma_{\mathrm{v}}\\ \mathbf{v}_{n}(\mathbf{x},t)=\mathbf{v}_{r}\cdot\mathbf{n}=\left(\mathbf{v}_{f }-\mathbf{v}^{*}\right)\cdot\mathbf{n}=0&\forall\,\mathbf{x}\,e\,\Gamma_{ \mathrm{v}_{n}}\\ \mathbf{v}_{r}(\mathbf{x},t)=\mathbf{v}_{f}-\mathbf{v}^{*}=0\Rightarrow\mathbf{ v}_{f}=\mathbf{v}^{*}&\forall\,\mathbf{x}\,e\,\Gamma_{\mathrm{v}}\\ p(\mathbf{x},t)=p^{*}(\mathbf{x},t)&\forall\,\mathbf{x}\,e\,\Gamma_{p}\end{array}  \tag{5}\]  
where \(\mathbf{b}_{f}\) represents the vector of mass forces acting on the fluid, and \(\mu_{f}\) is the constant of proportionality. It can be called the first viscosity coefficient, absolute viscosity, or dynamic viscosity. \(\bar{\mathbf{v}}(\mathbf{x},t)\) corresponds to the prescribed value of the velocity at a given point on the boundary, representing the Dirichlet boundary condition. \(\mathbf{v}_{n}\) represents the component of the fluid rate in the direction normal to the border. \(\mathbf{v}_{r}\) is the relative fluid-wall velocity, and \(\mathbf{v}^{*}\) is the impermeable-wall velocity. The general solution of the equations consists primarily in determining the hydrodynamic pressure field generated along the lubricated region. For a complete description of this problem, it is necessary to define the boundary conditions from which the pressure field develops. In this case, a pressure boundary condition (\(p^{*}(\mathbf{x},t)\) in the \(\Gamma_{p}\)) is applied as a Dirichlet condition.