# Experimental Verification of the CFD Model of the Squeeze Film Lifting Effect
## 2 Materials and Methods
### Description of the CFD Model  
Calculations were conducted using commercially available software [25]. The model describes the air film between the membrane and the lifted object. Due to software limitations, a circular sector of the film was utilized instead of a 2D grid. Due to the symmetry of the phenomenon, a cyclical model was adapted. Consequently, the movement of the lifted object was reduced to vertical motion above the membrane, preventing any horizontal movement of the object. The utilized sector had a central angle of \(\alpha=1^{\circ}\), an initial height of \(H=7\)\(\upmu\)m, and a radius equivalent to that of the modeled scenario. The authors used a structural mesh; mesh independence was conducted, and an exemplary result of maximum pressure is presented in Figure 5. The mesh has been divided into 450 finite volumes in the radial direction and 18 in the film thickness direction. The time step was set based on the mesh independence study as 1/120 part of one cycle of inducer motion. The scheme of the model division is presented in Figure 6. Sym1 and Sym2 regions in the model describe cyclic symmetry faces. The bottom wall describes the deflecting surface of the membrane. The deflection geometry was realized using two modes previously described.  
Figure 4: Prototype of the membrane with thrust ring: (**a**) disassembled; (**b**) assembled with a steel membrane; (**c**) assembled with PZT element.  
Figure 5: Peak pressure observed in the film over a single cycle: (**a**) mesh independence study; (**b**) timestep independence study.  
The surface of the lifted object in the model is represented as a Rigid Body and is represented by the Top surface. The object's kinematics is obtained by solving the motion equation. The previous assumption of symmetry allows for 1 DOF motion described by the equation of motion (1).  
\[m\ddot{z}=F_{zero}-mg \tag{1}\]  
where \(m\) is the mass of the lifted object; \(\ddot{z}\) is the acceleration of the lifted object, \(F_{zero}\) are the sum of the forces due to the air gap, \(g\) is a gravitational acceleration \(g=9.81\) kg \(\frac{m}{s^{2}}\). The motion of both the membrane and the lifted objects generates a deformation of the model. With the use of the moving mesh technique, used structural mesh has been preserved. The model uses the Second Order Backward Euler transient scheme. The convergence criterion for the calculations is set by obtaining RMS of below \(10^{-6}\). During the calculation, the finite volume method was adopted, solving continuity and ideal gas state equations: mass conservation Equation (2); momentum Equation (3); and ideal gas Equation (4). The tensor used in the momentum equation is given by Equation (5).  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\overrightarrow{v}\right)=0 \tag{2}\]  
\[\frac{\partial\left(\rho\overrightarrow{v}\right)}{\partial t}+\
abla\cdot \left(\rho\overrightarrow{v}\times\overrightarrow{v}\right)=-\
abla p+ \
abla\cdot\tau \tag{3}\]  
\[\rho=\frac{Mp}{RT} \tag{4}\]  
\[\tau=\mu\left(\
abla\overrightarrow{v}+\left(\
abla\overrightarrow{v} \right)^{\top}-\frac{2}{3}\delta\
abla\cdot\overrightarrow{v}\right) \tag{5}\]  
where the Identity matrix is designated \(\delta\). In Equations (2)-(5), the mathematical operators are as follows: \(\
abla\)--divergence; \(\overrightarrow{v}\times\overrightarrow{v}\) is the dyadic operator of velocity vectors \(\overrightarrow{v}\); \(\
abla\) is gradient; \(\rho\)--density of the fluid; \(p\)--pressure; \(\tau\)--stress tensor; \(M\)--gas molar mas; \(R\)--gas constant \(R=8.31446\frac{1}{\text{K}\text{mol}}\); \(T\)--temperature; \(\overrightarrow{v}\)--velocity vector; \(\mu\)--dynamic viscosity.  
Figure 6: Grid of the FVM of the film: (**a**) idea of the division; (**b**) model drawn in real proportions between angle and radius.