## 2 Simulation Analysis
### Flow-Field Simulation  
The tool base used in this study was a tube electrode with a cut depth of 15 mm, a wall thickness of 1 mm, and a closed end. The outer and inner diameters were 10 mm and 8 mm, respectively. The electrolyte outlet channel on its side wall was in the form of a slit, and the slit height and slit width were 13.6 mm and 0.4 mm, respectively. Figure 2a shows the tool base of two different outlet slit arrangements, where the outlet slit arrangements of tools A and B were linear and spiral, respectively. The number of electrolyte outlet slits that distributed uniformly along the circumference of each tool base was 4. The distance between the bottom end of the outlet slit and the end surface of the cathode tool was 1.1 mm. The pitch of tool B was 54.4 mm, and the two ends of the adjacent outlet slits were on the same bus. In order to clarify the influence of the layout of the outlet slit on the flow velocity distribution of the electrolyte in the side gap, a flow field simulation geometric model of tools A and B was established. Considering the flow field simulation model of tool A, as an example, the boundary conditions were set as shown in Figure 2b. Section A was selected at a distance of 0.05 mm  
Figure 1: Schematic diagram of electrochemical mill-grinding (ECMG). from the machined surface to observe the flow velocity distribution of the electrolyte in the side gap. The simulation parameters are shown in Table 1. The flow field simulation was performed in Fluent 17.0. The four electrolyte outlet slits of the cathode tool were evenly distributed on the circumferential surface, which means that each time the cathode tool rotated, the same flow field state repeated 4 times. Therefore, we analyzed the flow field between the 0\({}^{\circ}\) and 90\({}^{\circ}\) rotation angle. Figure 3 presents the cloud diagram of the electrolyte flow velocity distribution in the side gap at different rotation angles of Tool A. When the cathode tool was rotated to 30\({}^{\circ}\) and 45\({}^{\circ}\) (see Figure 3a,b), the non-electrolyte-outlet-slit area was directly opposite section A, and the high-velocity area was distributed on a vertical straight line. The flow velocity decreased from the middle to the upper and lower ends. When the cathode tool was rotated to 90\({}^{\circ}\) (see Figure 3c), the electrolyte outlet slit was directly opposite section A. At this time, the area of high flow velocity was distributed on the parallel side of the electrolyte outlet slit, and the flow velocity in the middle was higher than that in the upper and lower ends. Analysis of the simulation results shows that the distribution of the flow velocity was directly related to the layout of the outlet slit. The flow field of tool A was relatively concentrated. Even if the cathode tool was rotated through different angles, the high-velocity area always acted in the middle of the side gap. In the process of electrolytic processing, the flow field has a great influence on the conductivity. Electrolytic products accumulate at the locations with lower flow rates, resulting in a significant change in the composition and concentration of local electrolytes in the processing zone, thereby reducing the processing reaction rate [23]. The difference in the high and low flow rates under a non-uniform flow field for a long time resulted in different rates of material removal, and it was difficult for the tool A to ensure the flatness of the machined surface. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Inlet pressure (MPa) & 0.6 \\ Outlet pressure (MPa) & 0 \\ Side and bottom gap (mm) & 0.2 \\ Rotational speed (rpm) & 1000 \\ Step times (s) & \(2.5\times 10^{-3}\) \\ Total simulation times (s) & \(6.25\times 10^{-2}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Flow field simulation parameters. Figure 2: Schematic diagram of (**a**) cathode tool electrolyte outlet slit layout (**b**) flow field simulation geometric model of tool A. Flow field simulation was performed on tool B.