## 3 Simulation Results and Analysis
### Simulation Results and Analysis on the MHD Pump  
In the previous simulation on the radial velocity generation, it has been confirmed that the fluid flow can be induced above a starting current. In theory, the energized current \(I_{y}\) along the forward or reverse direction can drive the flow to form the circumfluence. However, the homogeneity and stability of the radial velocity \(u_{r}\) in the sensing channel of C-MHD ARS need to be given more attention. To validate, the simulation with the energized current \(I=3\)A along different directions is made using the electric approach in the fluid domain, as seen in Figure 3a, without any angular velocity. In the simulation, the mesh coordinate of the sensing bottom channel is along the axis \(z\), from \(-0.002\) m to \(0\) m. We chose the middle plane, \(z=-0.001\)m, to provide an analysis using the graph of the velocity shown in Figure 6. When the current \(I_{y}+\) is energized under the orthogonal imposed magnetic field \(B_{x}+\), as shown in Figure 5, the flow is driven upward in the middle channel and the radial flow in the sensing bottom channel is centripetal. In Figure 6a, the radial velocity is increased along the direction of the centripetal and the velocity of the vector is evenly distributed toward the center. Notably, the relative inward circumferential velocity needs to be more accelerated according to the sensing effect based on the Coriolis effect. On the other hand, when the energized current \(I_{y}-\) is set, the direction of the radial velocity contains inconsistencies. In detail, the vector along the centripetal direction appears near the coordinate \(y=0\), which is quite different from the other positions along the centrifugal direction, as shown in Figure 6b. That is to say, the Coriolis force is confused in this case, which is not suitable to compensate the MHD effect for MHD ARS at low frequency. This may be attributed to the property of the heavy fluid, which makes it difficult for the fluid in the outside channel to overcome the gravity and flow upward evenly. Moreover, the distribution of the velocity vector may be related to the cuboid shape of the MHD pump, which needs to be improved further.  
Likewise, the quantitative analysis of the velocity \(u_{z}\) in the center of the MHD pump is made under a different imposed current \(I\). The results are plotted in Figure 7. By the least squares fit,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mechanical Structure Parameters** & **(mm)** & **Physical Parameters of Conducting Fluid** & **Galinstan** \\ \hline Inner radius of annular channel & 4 & density \(\rho\) (kg/m\({}^{3}\)) & 6.4 \(\times\) 10\({}^{3}\) \\ Outer radius of annular channel & 14.5 & dynamic viscosity \(v\cdot\rho\) (kg/cm\({}^{3}\)) & 0.0024 \\ Height of upper and bottom channel & 2 & electric conductivity \(\sigma\) (s/m) & 3.46 \(\times\) 10\({}^{6}\) \\ Thickness of outside channel & 2 & Magnetic permeability \(\mu\) (H/m) & 1.257 \(\times\) 10\({}^{-6}\) \\ Thickness of central channel & 2 & melting point (\({}^{\circ}\)C) & \(-\)19 \\ Width of central channel & 10 & boiling point (\({}^{\circ}\)C) & \(>\)1300 \\ Total height & 37 & Gravity & \(\surd\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main mechanical structure parameters and physical parameters of the fluid channel.  
Figure 6: Contours of the radial velocity and the vector graph of velocity on the middle plane \(z=-0.001\) m with the imposed current \(I_{y}=3\) A and \(I_{y}=-3\) A in a different direction.  
this relationship can be expressed as (17). The tangent is given by the rising speed at the starting current at about \(I=1\) A. With the increase of current, the incremental of the velocity \(u_{z}\) decreases. Naturally, the viscous force is proportional to the velocity and has an obstruction effect on the flow.  
\[u_{z}=-0.79644e^{(-I/9.1134)}+0.83667 \tag{17}\]