## 2 Modeling and Simulation Method
### Model Solution Method  
#### 2.2.1 Simulation Program and Solution Methods  
To solve this model, commercial software 3D-Magnetostatic and FLUENT coupled with MHD module packaged in ANSYS are adopted. 3D-Magnetostatic is used to calculate the magnetic field distribution of the fluid channel including the one \(B_{y}\) in the MHD pump and the one \(B_{z}\) through the bottom channel. Then the magnetic simulation results are exported to the computation fluid domain in FLUENT as the external magnetic field data. The computation mesh created with ANSYS ICEM is fully structured. The problem involves the moving parts and it is the flow through the moving parts that is of interest. Therefore, we chose the rotating case as the moving reference frame and the relative velocity formulations are applied. The user-defined function (UDF) is used to simulate the time-varying vibration of the case. At each time step, the spatial discretization based on the Green--Gauss approach is performed. Fluxes in momentum and electrodynamics are evaluated by means of second-order upwind discretization. Time integration is performed utilizing first-order implicit scheme. 2.2 The Solving Method of Interaction between Flow Field and Electromagnetic Field and Boundary Conditions  
In studying the interaction between the flow field and electromagnetic field, it is critical to know the current density \(\mathbf{J}\) due to induction. Since the physical processes of the MHD pump and sensing part are quite different, the different interaction methods and boundary types must be chosen [31; 32]. Generally, two approaches may be used to evaluate the current density in FLUENT. One is through the solution of a magnetic induction equation; the other is through solving an electric potential. Boundary conditions in the two methods are set quite different in the MHD module coupled with FLUENT. In the first approach, the magnetic induction equation is derived from [PERSON]'s law and [PERSON]'s equation. The Equation (5) can be written as:  
\[\frac{\partial\mathbf{B}}{\partial t}+(\mathbf{u}\cdot\
abla)\mathbf{B}=\frac {1}{\mu\sigma}\
abla^{2}\mathbf{B}+(\mathbf{B}\cdot\
abla)\mathbf{u}, \tag{7}\]  
with \(\mathbf{B}=\mathbf{B}_{0}+\mathbf{b}\), the induction Equation (7) can be defined as (8) in the conductive medium:  
\[\frac{\partial\mathbf{b}}{\partial t}+(\mathbf{u}\cdot\
abla)\mathbf{b}=\frac {1}{\mu\sigma}\
abla^{2}\mathbf{b}+((\mathbf{B}_{0}+\mathbf{b})\cdot\
abla)- (\mathbf{u}\cdot\
abla)\mathbf{B}_{0}, \tag{8}\]and the current density is given by:  
\[\mathbf{j}=\frac{1}{\mu}\
abla\times(\mathbf{B}_{0}+\mathbf{b}) \tag{9}\]  
The boundary conditions of magnetic field at conducting and insulating walls in the first approach are respectively denoted as (10) and (11):  
\[\frac{\partial\mathbf{b}}{\partial\mathbf{n}}=0 \tag{10}\]  
\[\frac{\partial\mathbf{b}}{\partial\mathbf{\tau}}=0 \tag{11}\]  
The first magnetic approach is suitable for the electric induction by the conductive fluid motion under the magnetic field following [PERSON]'s law. The generation of induced electromotive voltage in the annual bottom sensing channel should be simulated using the first approach. Thus, the boundary of conductive internal and external walls in the bottom sensing part should be given as (10) and the upper and bottom insulating ones can be regarded as (11). On the other hand, the second approach for the current density is to solve the electric potential equation and calculate the current density using [PERSON]'s law. For a static field and assuming \(\mathbf{b}\ll\mathbf{B_{0}}\), [PERSON]'s law can be written as (12), when the magnetic Reynolds number \(\mathrm{Re}_{m}=\mu\sigma\mathrm{\textit{U}}_{0}l\) is far less than 1.