## 3 Results and Discussion
### Magnetic Fluid Lubrication Viscosity Characteristics
As the temperature increases, the molecular and magnetic nanoparticle motion within the magnetic fluid  
Figure 5 shows the viscosity versus pressure of a magnetic fluid at different temperatures without the effect of a magnetic field. From Figure 5, it can be seen that the viscosity of the magnetic fluid decreases significantly at increasing temperatures. As the pressure increases, the viscosity of the magnetic fluid at 20 \({}^{\circ}\)C and 40 \({}^{\circ}\)C has a clear tendency to rise, while at 60 \({}^{\circ}\)C and 80 \({}^{\circ}\)C the viscosity of the magnetic fluid does not tend to rise, and at 100 \({}^{\circ}\)C the viscosity of the magnetic fluid barely rises, remaining at 0.2-0.3 Pa-s. Under the action of the applied magnetic field, the magnetic particles in the magnetic fluid are arranged into a chain structure with the same direction as the magnetic field strength by the magnetic moment, and its viscosity is greater than that of the absence of magnetic field strength. The modified equation for the viscosity of the magnetic fluid under the action of an applied magnetic field is:  
\[\eta_{h}(T,P,H)=(\eta+\Delta\eta_{h}(H))\cdot k_{1} \tag{6}\]  
where \(\eta\) is the magnetic fluid viscosity without magnetic field, \(\Delta\eta_{h}\) is the magnetic fluid viscosity increment with magnetic field, both in Pa-s, \(k_{1}\) is the ratio of experimental viscosity increment to theoretical viscosity increment, generally taken as 0.5 to 3. For ease of calculation, the \(k_{1}\) is 1. According to the rotational viscosity theory of [PERSON] [26; 27] and [PERSON]'s theory, the relationship between the incremental viscosity \(\Delta\eta_{h}\) of the magnetic fluid and the viscosity of the base carrier fluid is given by:  
\[\frac{\Delta\eta_{h}}{\eta_{t}}=\frac{1.5}{\frac{3k_{0}T}{\phi}+\frac{3k_{0} T}{2\pi\eta_{h}^{3}\mu_{0}(\mu_{r}-1)H^{2}}} \tag{7}\]  
where \(k_{0}\) is the Boltzmann constant with a value of 1.380649 \(\times\) 10-23, \(\mu_{0}\) is the vacuum permeability, \(\mu_{r}\) is the relative permeability, and \(H\) is the magnetic field strength in A/m. The magnetic induction strength \(B\) and the magnetic field strength \(H\) satisfy:  
\[B=\mu_{r}H \tag{8}\]  
Figure 5: Magnetic fluid lubrication film viscosity–temperature and viscosity–pressure characteristics. The expression for the viscosity of a magnetic fluid under the action of a magnetic field is obtained by combining Equations (5)-(8):  
\[ \begin{split}\eta_{b}(T,P,B)=\eta_{0}exp\bigg{\{}&\big{(} \ln\eta_{0}+9.67\big{)}\bigg{[}\big{(}1+5.1\times 10^{9}P\big{)}^{2}\times\big{(} \frac{T-138}{T_{0}-138}\big{)}^{-5_{0}}-1\bigg{]}\bigg{\}}\times\\ &\bigg{[}\frac{1}{1+2\,5\big{(}1+\frac{c}{T_{p}}\big{)}^{3}\phi- 1.55\big{(}1+\frac{c}{T_{p}}\big{)}^{6}\phi^{2}}+\frac{1.5k_{1}}{\frac{1}{ \phi}+\frac{3k_{0}T_{p}^{2}}{2\pi\eta_{0}^{2}(p-1)B^{2}}}\bigg{]}\end{split}  \tag{9}\]  
Given a magnetic fluid pressure of 5 MPa, Equation (9) was calculated using MATLAB to obtain the relationship between the viscosity of the magnetic fluid lubrication film under the action of a magnetic field and the variation in temperature and magnetic field as shown in Figure 6. As can be seen in Figure 6, the viscosity of the magnetic fluid gradually increases with increasing magnetic induction, and the growth trend is similar to the magnetization curve, and the viscosity of the magnetic fluid stabilizes when the magnetic induction is greater than 0.4 T. This also indicates that the saturation magnetization strength of this magnetic fluid is around 0.4 T. Under the action of the magnetic field, the magnetic particles within the magnetic fluid are arranged in a chain-like structure along the magnetic field direction by the magnetic moment, which increases the resistance to flow, the viscosity of the magnetic fluid increases and the load-bearing capacity increases [28; 29]. Figure 7 shows the variation in magnetic fluid viscosity with magnetic field at different temperatures.