## 4 Design and Analysis of Magnetic Fields
### Flow Analysis of Magnetorheological Fluid  
According to Equations (3) and (20), the apparent viscosity \(\eta\) of the MRF after yielding is  
\[\eta=\frac{\tau_{y}(H)}{\dot{\gamma}}+k\big{|}\dot{\gamma}\big{|}^{m-1} \tag{35}\]  
From Equation (35), it can be observed that the apparent viscosity of MRF is inversely proportional to the shear strain rate. Along with the increased magnetic field intensity, the effect of zero-field viscosity \(\eta_{0}\) on the apparent viscosity becomes smaller. When the magnetic field intensity is 10 kA/m, 30 kA/m, 50 kA/m and 70 kA/m, respectively, the relationships between apparent viscosity and the shear strain rate of the MRF were as shown in Figure 8.  
Figure 8: Relationship between apparent viscosity and shear strain rate. As can be seen from Figure 8, along with the increase of magnetic field intensity, the effect of the shear strain rate on the apparent viscosity increases. The value of the apparent viscosity tends to stabilize when the shear rate is greater than 800 s\({}^{-1}\). To facilitate the analysis of the plug flow region (the unyielding region of the MRF), the ratio of the minimum shear yield stress \(\tau_{\text{min}}\) required by the load to the MRF magnetic shear yield stress \(\tau_{y}(H)\) is defined as the plug flow coefficient \(\lambda\). Then, \(\lambda\) can be expressed as  
\[\lambda=\frac{\tau_{y}(H)}{\tau_{\text{min}}} \tag{36}\]  
The magnetic field intensity data at 3A in Figure 7a was brought into Equations (26) and (36) to obtain the plug flow coefficient in the working gap, as shown in Figure 9, where \(h_{z}\) is the distance across the gap, which indicates the distance between the plane where the MRF is located and the input disc. As can be seen from Figure 9, when \(\lambda\geq 1\), the magnetorheological fluid is in the unyielding state, and it flows rigidly in the working gap. Since \(\lambda<<1\) near the wall of the working gap of the transmission, the magnetorheological fluid near the wall of the working gap is in the yielding flow state. According to the plug flow coefficient in Figure 9a, the flow velocity distribution of the MRF in the working gap was plotted as Figure 9b. From Figure 9b, it can be seen that the rotational speed of the disc-type magnetorheological transmission with the coil placed at one side is more related to the thickness of the gap, and the flow velocity distribution of the MRF in the unyielding region (regions A and B) is closer to the rotational speed of the input disc. In regions E, C and D, the magnetic field intensity is low, which leads to the yielding of the MRF, and the MRF is in a viscoplastic flow state. The flow velocity distribution in the cross section where the radius of disc \(r\) is 104 and 81 mm is intercepted in regions E and C, respectively, as shown in Figure 10.  
The flow velocity of MRF varies widely along the thickness direction in the E, B, and C regions, the flow velocity does not decrease according to the linear law because the  
Figure 10: Flow of MRF in the working gap: (**a**) region E, (**b**) region C.  
Figure 9: Flow of MRF in the working gap. (**a**) Plug flow coefficient of MRF, (**b**) flow velocity of MRF. viscosity of the MRF undergoes shear thinning with the increase to the shear strain rate. The MRF in this region is not fully cured, which is one of the reasons for the unstable transmission performance. According to the analysis of Figure 5, when the load torque is further increased, the magnetorheological fluid in the working gap changes from a rigid to a viscoplastic flow state. The load torque of the transmission was set to 137.7 N\(\cdot\)m, the speed difference \(\Delta\omega\) = 22.74 rad/s was known from Equations (18) and (19), and the yield thickness was 0.52 mm. The flow of MRF in the working gap is shown in Figure 11.  
As can be seen from Figure 11, the magnetic field intensity near the input disc was slightly greater than near the output disc when the shear yield stress required by the load exceeded the magnetic shear yield stress of the MRF.