## 2 Numerical Method  
### Traction Inverter Cooling System  
In the present study, the MHD pump-based cooling system is developed for the thermal management of traction inverters. The cooling performance of a 22 kW traction inverter is numerically investigated considering a MHD pump-based cooling system. The computational domain comprises a 22 kW traction inverter, a cooling channel, and an MHD pump as shown in Figure 1. The specification of the traction inverter is extracted from reference [32]. The cooling channel has a non-uniform cross-section with a length of 420 mm and a height of 10 mm. The detailed view of cooling channel with internal fins is depicted in Figure 2. The rectangular straight fins are provided to enable the uniform distribution of coolant inside the channel and improve the heat transfer. The larger cross-section of the cooling channel with fins is provided just below the traction inverter to achieve the desired cooling performance. The water and water-based Fe\({}_{3}\)O\({}_{4}\) ferrofluid are considered as coolants to dissipate the heat from the traction inverter. The thermophysical properties of Fe\({}_{3}\)O\({}_{4}\) nanoparticles, water, and solid domains are shown in Table 1[33; 34; 35]. The MHD pump is placed at the inlet of the cooling channel to provide the traction inverter with cooling. The MHD pump consists of two 20 mm \(\times\) 10 mm N-35 magnets and copper electrodes with a 4 mm\({}^{2}\) cross-section, connected at the side outer walls of cooling channel. In the MHD pump, ferrofluid, as an electrically conducting fluid, is used to generate the unidirectional current. A strong magnetic field that is parallel to the current is applied to the fluid. The Lorentz force, whose direction is determined by the cross product of the vectors of the current and magnetic field, is produced by the interaction of an orthogonal magnetic field, an electric field, and ion relative motion. The fluid is efficiently pushed when the mechanism that holds the electromagnet and the enclosure is significant [18]. Based on this concept in the proposed MHD pump, the magnetic field is enabled by both magnets arranged across the cooling channel on opposite sides and the electric field is enabled by passing the current through the electrode. The Lorentz force in the perpendicular direction is generated, which forces the ferrofluid from the inlet to the outlet of the cooling channel. The inlet of the cooling channel is divided into two parts, each with dimensions of 15 mm \(\times\) 8 mm, to provide two separate coolant paths. Both coolant paths are activated for the case when ferrofluid is employed as the coolant. One coolant path is for constant velocity ferrofluid circulation, and the second coolant path is for MHD pump-based ferrofluid circulation. However, in the case of water coolant, only the constant velocity coolant path is activated since water does not offer any magnetic properties for self-circulation like ferrofluid in the second coolant path. For simple understanding, one coolant path acts as conventional pump circulation while the second coolant path is designed for MHD pump circulation. The water coolant cannot be self-circulated using the MHD pump so it only gets circulated using the conventional pump, whereas the ferrofluid coolant can be self-circulated using MHD pump in addition to circulation through the conventional pump.  
Figure 1: Computational domain of MHD pump-based cooling for traction inverter.