## 2 Modeling and Simulation Method
### Model Solution Method
#### 2.2.1 Simulation Program and Solution Methods
\[\mathbf{j}=\sigma(-\
abla\boldsymbol{\varphi}+(\mathbf{u}\times\mathbf{B})) \tag{12}\]  
For sufficiently conducting media, the principle of conservation of the electric charge gives:  
\[\
abla\cdot\mathbf{j}=0 \tag{13}\]  
The electric potential equation is therefore given by:  
\[\
abla^{2}\boldsymbol{\varphi}=\
abla\cdot(\mathbf{u}\times\mathbf{B_{0}}) \tag{14}\]  
The boundary conditions of magnetic field at conducting and insulating walls in the second approach are respectively denoted as (15) and (16):  
\[\boldsymbol{\varphi}=\boldsymbol{\varphi_{0}}\;\;\sigma\;\;\mathbf{J}= \mathbf{J_{0}} \tag{15}\]  
\[\frac{\partial\boldsymbol{\varphi}}{\partial\mathbf{n}}=(\mathbf{u}\times \mathbf{B})_{boundary}\cdot\mathbf{n} \tag{16}\]  
Compared with the first magnetic method, the second electric method is applicable for analyzing the conductor's force and motion driven by of its energized current and magnetic field. The fluid flow driven by the MHD pump in the middle channel Figure 2 should be simulated using the second approach. Thus, the boundary of energized electrodes in the middle channel for the MHD pump should be denoted as (15) and the other insulating ones can be set as (16). In summary, the simulation of MHD pump for radial velocity generation in FLUENT with the help of MHD module is made using the electric approach according to the Equations (3), (4), (6), (13) and (14) with the boundary conditions (15) and (16). On the other hand, the MHD sensing part is simulated using the magnetic approach according to the Equation (3), (4), (6), (8), and (9) with the boundary conditions (10) and (11). Specifically, the boundary type of energized electrodes in the MHD pump is a fixed electric current condition as \(\mathbf{J}=\mathbf{J_{0}}\) as Equation (15), while the internal and external electrode in the sensing channel should be given as Equation (10) for the unknown induction electromotive force and current. However, only one kind of interaction method can be chosen in the same simulation using the MHD module of FLUENT. To solve the boundary type alternating process, a combined computation method is put forward and illustrated in \[\frac{\partial\mathbf{b}}{\partial t}+(\mathbf{u}\cdot\
abla)\mathbf{b}=\frac {1}{\mu\sigma}\
abla^{2}\mathbf{b}+((\mathbf{B}_{0}+\mathbf{b})\cdot\
abla)- (\mathbf{u}\cdot\
abla)\mathbf{B}_{0}, \tag{8}\]0 In the software, the radial velocity driven by the MHD pump is simulated in the whole fluid domain using the second electric approach as shown in Figure 3a. Obviously, the radial velocity distribution in the bottom channel is difficult to export and reuse for the next sensing simulation directly. The stable calculation results of static pressure \(P\) and velocity \(u_{z}\) on the cross sections are adopted as the velocity inlet boundary to repeat the induced radial velocity in the bottom channel. The cross-plane should be selected in the pumped stability fluid region. Then the first magnetic approach is chosen in the sensing simulation of the fluid domain shown in Figure 3b, in which the radial velocity distribution is reproduced by the velocity inlet in the middle and outside channels. The other walls except internal and external electrodes are set as insulating walls with the boundary condition expressed as (11). At last, the generated voltage of the C-MHD ARS with the designated current in MHD pump can be given.