## 3 Results and Discussion  
### Magnetic Fluid Lubrication Viscosity Characteristics  
The Roelands [10,14] viscosity equation is often used when studying fluid viscosity, where temperature and pressure are the environmental factors that must be considered:  
\[\eta_{t}(T,P)=\eta_{0}exp\Bigg{\{}(ln\eta_{0}+9.67)\Bigg{[}\Big{(}1+5.1\times 1 0^{9}P\Big{)}^{z}\times\Big{(}\frac{T-138}{T_{0}-138}\Big{)}^{-S_{0}}-1\Bigg{]} \Bigg{\}} \tag{2}\]  
\(T_{0}\) is the initial temperature in K, \(\eta_{0}\) is the viscosity at \(T_{0}\) and zero pressure, \(z\) and \(S_{0}\) are the dimensionless viscous pressure coefficient and the dimensionless viscous temperature coefficient, respectively, generally taken as 0.1\(\sim\)1.5. The application of Equation (2) requires the following:  
1. The volume force and inertial force of the magnetic fluid are negligible;
2. The radius of the journal and bearing is much larger than the film thickness;
3. No-slip phenomenon at the magnetic fluid lubrication interface;
4. The lubrication film does not change along the radial pressure. For magnetic fluids with a certain volume fraction of magnetic particles, when the magnetic particle volume fraction \(\phi>2\%\), the coefficient of magnetic fluid dynamic viscosity can be expressed using the [PERSON] formula modified by [PERSON] [24,25]:  
\[\frac{\eta_{f/0}}{\eta_{c}}=\frac{1}{1+2.5\phi-1.55\phi^{2}} \tag{3}\]  
When considering the thickness of the surfactant coating on the magnetic particles, it is equivalent to increasing the radius of the magnetic particles, and the radius of the magnetic particles coated with the surfactant is a (1 + _c/r\({}_{p}\)_) times the radius of the pure \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Outer Diameter** & **Inner Diameter** & **Width (B/mm)** & **Orthopaedic Force** & **Operating** \\
**(D/mm)** & **(d/mm)** & **(A/m)** & **Temperature (\({}^{\circ}\)C)** \\ \hline
57 & 41 & 16 & 87,000 & \(\leq\)100 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic parameters of permanent magnet rings. Figure 3: Magnetic fluid lubricated bearing structure. magnetic particles, then the volume fraction is \((1+c/\tau_{p})^{3}\) times. The Einstein viscosity of the surfactant coated magnetic particles is modified to:  
\[\frac{\eta_{f0}}{\eta_{c}}=\frac{1}{1+2.5\Big{(}1+\frac{c}{\tau_{p}}\Big{)}^{3} \phi-1.55\Big{(}1+\frac{c}{\tau_{p}}\Big{)}^{6}\phi^{2}} \tag{4}\]  
where \(\eta_{f0}\) is the magnetic fluid viscosity coefficient in the absence of a magnetic field, \(\eta_{c}\) is the base carrier fluid kinetic viscosity coefficient, \(\phi\) is the magnetic particle volume fraction, c is the surfactant thickness, \(r_{p}\) is the average radius of the magnetic particle surfactant together, the main technical parameters of the magnetic fluid are shown in Table 3, then the magnetic fluid viscosity expression in the absence of a magnetic field is:  
\[ \begin{split}&\eta_{t0}(T,P)=\frac{1}{1+2.5\Big{(}1+\frac{c}{ \tau_{p}}\Big{)}^{3}\phi-1.55\Big{(}1+\frac{c}{\tau_{p}}\Big{)}^{6}\phi^{2}} \eta_{0}\times\\ & exp\bigg{\{}\Big{(}1n\eta_{0}+9.67\Big{)}\Big{[}\big{(}1+5.1 \times 10^{9}P\big{)}^{2}\times\Big{(}\frac{T-138}{T_{0}-138}\Big{)}^{-S_{0}}-1 \Big{]}\bigg{\}}\end{split}  \tag{5}\]  
Equation (5) was calculated using MATLAB to obtain the temperature and pressure dependence of the viscosity of the magnetic fluid lubrication film without the action of a magnetic field as shown in Figure 4. As can be seen from the Figure 4, the viscosity of the magnetic fluid decreases exponentially with increasing temperature and increases linearly with increasing pressure. The main reason for this is that the viscosity of the fluid depends mainly on its intermolecular internal friction, and the resistance to the flow of the magnetic fluid is mainly due to the relative friction between its internal molecules and between the nanoparticles.