# Analysis of Grinding Flow Field under Minimum Quantity Lubrication Condition
## 2 Flow Field Analysis of the Grinding Wheel  
### Theoretical Modeling of the Grinding Wheel Flow Field  
When the grinding wheel rotates, it will drive the air micro-cluster on its surface to move. Because the air is viscous, the air micro-clusters on the surface of the grinding wheel will drive its adjacent air micro-clusters to move, forming a circular air flow around the grinding wheel, that is, circumferential flow. The air on the side surface of the grinding wheel flows to the edge of the grinding wheel under the action of centrifugal force, and new air is constantly replenished to the side of the grinding wheel, forming a radial flow from the center of the grinding wheel to the edge of the grinding wheel. For the grinding wheels with pores, osmotic flow and internal flow will also be formed. The circumferential flow, the radial flow, the osmotic flow and the internal flow jointly constitute the flow field of the grinding wheel [20]. When simulating the flow field of grinding wheel, the compressibility of air has a great influence on the simulation results. The compressibility of air can be judged according to its Mach number, which is recorded as _Ma. Ma_ can be calculated using Equation (1). \[Ma=\frac{v_{a}}{a} \tag{1}\]\[a=20.05\sqrt{T} \tag{2}\]  
Here, \(v_{a}\) is the flow rate of air, \(a\) is the sound velocity in the environment, and \(T\) is the ambient temperature. The flow rate of air can be substituted by the maximum velocity of the air, that is, the rotational velocity of the grinding wheel. According to Equations (1) and (2), the \(Ma\) is 0.1156, and is smaller than 0.3 if the maximum velocity of grinding wheel is 40 m/s and the ambient temperature \(T\) is 298 K. In this case, the air can be considered incompressible gas when analyzing the flow field of grinding wheel. According to the theory of fluid mechanics, the flow field of grinding wheel meets the continuity equation, momentum conservation equation and energy conservation equation [21]. The following is the continuity equation:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial\rho u_{i}}{\partial x_{i}}=0 \tag{3}\]  
Since the air is considered an incompressible gas, the continuity equation can be reduced to Equation (4). \[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{4}\]  
The following equations, (5) and (6), are the momentum conservation equation and the energy conservation equation, respectively. \[\rho\frac{Du}{Dt}=\rho f_{x}+\frac{\partial\tau_{xx}}{\partial x}+\frac{ \partial\tau_{xx}}{\partial y}+\frac{\partial\tau_{xx}}{\partial z}-\frac{ \partial p}{\partial x} \tag{5}\]  
\[\rho\frac{Dv}{Dt}=\rho f_{y}+\frac{\partial\tau_{yy}}{\partial y}+\frac{ \partial\tau_{xy}}{\partial x}+\frac{\partial\tau_{xy}}{\partial z}-\frac{ \partial p}{\partial y} \tag{6}\]  
\[\rho\frac{D}{Dt}\bigg{(}e+\frac{1}{2}u_{i}u\bigg{)}=\rho f_{i}u_{i}-u_{i}\frac {\partial P}{\partial x_{i}}+\frac{\partial}{\partial x_{i}}\bigg{(}u_{i}u \bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x _{i}}\bigg{)}\bigg{)}-\frac{\partial}{\partial x_{i}}\bigg{(}\lambda\frac{ \partial T}{\partial x_{i}}\bigg{)} \tag{7}\]  
Here, \(\rho\) is the fluid density; \(f\) is the physical force; \(\tau\) is the shear force; \(u_{i}\) is the velocities \(u\), \(v\) and \(w\) in the \(x\) and \(y\) directions; \(\lambda\) is the thermal conductivity of the fluid; \(T\) is the temperature of the flow field; and \(e\) is the internal energy. As the number of unknowns in the equation is redundant to the number of equations, the \(k\)-\(\epsilon\) turbulence model is introduced to make the equation closed. Equation (7) is the \(k\)-equation and Equation (8) is the \(\epsilon\)-equation.