## 4 Influence of Magnetic Fluid Dynamic Viscosity on External Magnetic Field  
As the external magnetic field is applied to the magnetic fluid, the magnetic fluid particles will be magnetized. The magnetized particles will rotate under the effect of the torque induced by the magnetic field. Then, the speed difference between the particles and magnetic fluid increases to increase the viscosity. Hence, when the magnetic field is applied to the magnetic fluid, according to Shliomis' formula, the magnetic fluid viscosity can be expressed as [12]  
\[\eta_{H}=\eta+\frac{3}{2}\phi\frac{0.5\alpha L(\alpha)}{1+0.5\alpha L(\alpha)} \left(\sin^{2}\beta_{0}\right)\eta_{c}=\eta+\Delta\eta_{H}, \tag{11}\]  
where \(\eta\) is the viscosity of the magnetic fluid without any magnetic field, \(\varphi\) is the volume fraction of magnetic fluid, \(\alpha\) is the ratio between the magnetic potential and the thermal motion, \(L(\alpha)\) is the Langevin function, \(\beta_{0}\) is the angle between the external magnetic field and vortex vector of the magnetic particles, \(\eta_{c}\) is the viscosity of the carrier fluid, and \(\Delta\eta_{H}\) is the incremental viscosity of the magnetic fluid as the external magnetic field is applied to the magnetic fluid.  
According to the Langevin equation, the viscosity increment \(\Delta\eta_{H}\) produced by the magnetic fluid can be rewritten as [12]  
\[\Delta\eta_{H}=\frac{1.5\pi d_{p}{}^{3}\mu_{0}(\mu_{r}-1)H^{2}\phi}{12k_{0}T \phi+\pi d_{p}{}^{3}\mu_{0}(\mu_{r}-1)H^{2}}\eta_{c}, \tag{12}\]  
where \(k_{0}=1.380648\times 10^{-23}\) is the Boltzmann's constant, \(T\) is the absolute temperature of the fluid, \(d_{p}\) is the average diameter of the aggregates of the magnetic fluid particles, \(\mu_{0}=4\pi\times 10^{-7}\) H/m is the vacuum permeability, \(\mu_{r}\) is the relative permeability coefficient, and \(H\) is the magnetic field strength in the unit of A/m.  
Considering the thickness of the magnetic fluid-coated surfactant, the magnetic fluid viscosity without an external magnetic field can be written as [12]  
\[\eta=\frac{\eta_{c}}{1-2.5\Big{(}1+\frac{\delta}{r_{p}}\Big{)}^{3}\phi+1.55 \Big{(}1+\frac{\delta}{r_{p}}\Big{)}^{6}\phi^{2}}, \tag{13}\]  
where \(\delta\) is the thickness of the surfactant and \(r_{p}\) is the average radius of the aggregates.  
Substituting (12) and (13) into (11), the viscosity coefficient, \(\eta_{H}\), of the magnetic fluid under the external magnetic field can be rewritten as (14)  
\[\eta_{H}=\eta_{c}\left[\frac{1}{1-2.5\Big{(}1+\frac{\delta}{r_{p}}\Big{)}^{3 }\phi+1.55\Big{(}1+\frac{\delta}{r_{p}}\Big{)}^{6}\phi^{2}}+\frac{1.5\pi d_{p }{}^{3}\mu_{0}(\mu_{r}-1)H^{2}\phi}{12k_{0}T\phi+\pi d_{p}{}^{3}\mu_{0}(\mu_{ r}-1)H^{2}}\right]. \tag{14}\]  
It can be seen from (14) that, as the external magnetic field \(H\) increases, the magnetic fluid viscosity increases. When the external magnetic field reaches a certain intensity, the viscosity increasing trend becomes smaller, and becomes stable because magnetic particles in the magnetic fluid tend to stabilize. In addition, the oil film pressure of the magnetic fluid will gradually increase with the increase in the external magnetic field. The magnetic fluid viscosity and oil film pressure have a proportional relationship. Thus, the capacity of the ferrofluid lubricated hydrodynamic journal bearing under the effect of an external magnetic field can be improved.  
However, the increase in bearing capacity of a dynamic pressure bearing means that the rigidity of the dynamic pressure bearing is improved. According to the threshold value of the unstable critical speed (2), when \(\lambda\) and \(M\) are kept constant, increasing \(K_{1}\) can effectively improve the unstable critical speed of the rotor. Applying a magnetic field to the ferrofluid can increase its viscosity; then, the property of the hydrodynamic bearing will be improved. The frequency of the oil whirl and oil whip will increase, and the rotor will become more stable.