In this paper, based on the Navier–Stokes equation (N–S equation), a mathematical model of hydrodynamic lubrication of different surface textures was established, and the solution method of the optimal surface texture model was determined. The purpose is to study how to choose the optimal solution for different surface textures under different working conditions, and provide theoretical guidance for subsequent research.
2.1 Introduction to Basic Theory
2.1.1 Derivation of N–S Equation
The N–S equation is obtained by substituting and sorting the components of the viscous fluid motion momentum equation in the form of stress with the expression given by the generalized Newton's internal friction law. If the fluid is incompressible, the dynamic viscosity µ is constant. The N–S equation can be simplified as:
$$\begin{gathered} \rho \frac{{{\text{d}}\nu_{x} }}{{{\text{d}}t}} = \rho f_{x}  \frac{\partial p}{{\partial x}} + 2\mu \frac{{\partial^{2} \nu_{x} }}{{\partial x^{2} }} + \mu \frac{\partial }{\partial y}(\frac{{\partial \nu_{x} }}{\partial x} + \frac{{\partial \nu_{x} }}{\partial y}) + \mu \frac{\partial }{\partial z}(\frac{{\partial \nu_{x} }}{\partial z} + \frac{{\partial \nu_{z} }}{\partial x}) \hfill \\ = \rho f_{x}  \frac{\partial p}{{\partial y}} + \mu \nabla^{2} \nu_{x} \hfill \\ \nabla^{2} = \frac{{\partial^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} }}{{\partial y^{2} }} + \frac{{\partial^{2} }}{{\partial z^{2} }}. \hfill \\ \end{gathered}$$
(1)
Similarly, equations in the form of projections on the y and z axes can be derived. The two ends of the equation are divided by the density at the same time, and the three projection forms of the N–S equation of incompressible viscous flow can be obtained from the relationship of dynamic viscosity µ, kinematic viscosity and density.
$$\begin{gathered} \frac{{{\text{d}}v_{x} }}{{{\text{d}}t}} = f_{x}  \frac{1}{\rho }\frac{\partial p}{{\partial x}} + \nu \nabla^{2} v_{x} , \hfill \\ \frac{{{\text{d}}v_{y} }}{{{\text{d}}t}} = f_{y}  \frac{1}{\rho }\frac{\partial p}{{\partial y}} + \nu \nabla^{2} v_{y} , \hfill \\ \frac{{{\text{d}}v_{z} }}{{{\text{d}}t}} = f_{z}  \frac{1}{\rho }\frac{\partial p}{{\partial z}} + \nu \nabla^{2} v_{z} . \hfill \\ \end{gathered}$$
(2)
2.1.2 Couette Flow
The N–S equation of viscous fluid motion is a secondorder nonlinear partial differential equation, which is difficult to be solved. In addition, in practical engineering, the viscous flow with complex flow boundaries is encountered, and with the change of time and space, the flow parameters in the flow field are also constantly affected and changed. But when the flow boundary is relatively simple and the flow parameters are mostly constant, the solution of the flow can be obtained. As a kind of viscous flow, Couette flow is of great significance in engineering, and accurate analytical solutions can be obtained under some conditions [32].
The typical form of Couette flow is two infinite planes with a distance of h_{0}, in which an incompressible fluid with a dynamic viscosity of µ flows in a fixed direction. Couette flow also includes two other forms. The first is that there is a speed difference between the two planes, so that the two planes produce relative motion but the pressure in the direction of flow does not change. The second is that there is not only a speed difference between the two planes, but also a pressure gradient in the flow direction. At this time, the solution of the linear equation is additive, and the independent solutions of velocity and pressure can be added.
2.2 Basic Derivation of Theory
Each surface texture corresponds to one or more pressure values and velocity values that can give full play to its tribological properties. This matching mechanism is the core of theory.
Before deriving the basic theory, the applicable premise must be determined first:

1.
The effects of volume force such as gravity and magnetic force are ignored.

2.
Assume that there is no relative sliding between the fluid and solid interface.

3.
The fluid belongs to a kind of Couette flow, but it is not exactly the same.

4.
The fluid is Newtonian fluid, and the flow mode is turbulent flow. There may be vortex and turbulent flow in the oil film.

5.
Compared with the viscous force of lubricating oil, the inertial force is smaller, so its effect is ignored, including the inertial force when the fluid accelerates and the centrifugal force when the fluid moves in a circular motion.
2.2.1 Setting of Fluid Boundary Conditions Based on Couette Flow
As shown in Figure 1, the thickness of the oil film between the two moving friction pairs is h_{0}. The lubricating oil is an incompressible fluid with a dynamic viscosity of µ, which is driven by the upper plate to flow positively towards the X axis, and the moving speed of the upper plate is U. The depth of texture is h_{p}, and the crosssectional width is 2r_{p}. The upper plate moves to the right at speed U, while the lubricating oil has a pressure gradient in the x direction. The red line represents the speed gradient change. The purple line represents the velocity distribution under different pressures. When P > 0, the velocity distribution is the rightmost purple line, the pressure promotes the flow of lubricating oil, and the average velocity is greater than the velocity without pressure difference. When P < 0, the velocity distribution is the leftmost purple line. At this time, the flow caused by the upper plate is not enough to overcome the flow caused by the reverse pressure difference, so reverse flow occurs.
The N–S equation can be reduced to:
$$\nu_{x} (y) = U\frac{y}{{h_{0} }}  \frac{1}{2\mu }\frac{{{\text{d}}p}}{{{\text{d}}x}}y(h_{{0}}  y).$$
(3)
By dividing both sides of Eq. (3) by y, the equation becomes:
$$\frac{{\nu_{x} (y)}}{y} = U\frac{{1}}{{h_{0} }}  \frac{1}{2\mu }\frac{{{\text{d}}p}}{{{\text{d}}x}}(h_{{0}}  y).$$
(4)
Eq. (4) is the relationship between the fluid velocity and pressure in the Couette flow form. This equation is a simplified N–S equation based on Couette flow.
2.2.2 Summary of Theory Formula
Substituting Eq. (4) into the pressurefriction conversion equation (Eq. (5)), Eq. (6) is obtained.
$$f = \frac{{\frac{{{\text{d}}p}}{{{\text{d}}x \cdot (D_{1} + D_{2} )}}(D_{1}^{2}  D_{2}^{2} )\pi }}{n} = \frac{{\frac{{{\text{d}}p}}{{{\text{d}}x}}}}{n}(D_{1}  D_{2} )\pi ,$$
(5)
$$f = \frac{{\frac{U}{{h_{0} }}  \frac{{\nu_{x} (y)}}{y}}}{{(h_{{0}}  y)}} \cdot \frac{2\mu }{n} \cdot (D_{1}  D_{2} )\pi .$$
(6)
The conversion Eq. (5) is derived from the actual experimental conditions in this paper. dp/dx·(D_{1}−D_{2})/n is the pressure value in the unit area, n is the number of texture distribution on the surface of the test piece, D_{1} is the inner diameter of the grinding piece, D_{2} is the outer diameter of the grinding piece. Eq. (5) can be used to combine the hydrodynamic equations with the surface texture parameters, and fully fit the actual working conditions of this article, which improves the accuracy and credibility of the equation. However, there is a certain error in the solution of the pressure in the cell texture area, and this equation also has a certain error. Eq. (6) is the relationship between friction force and fluid flow characteristics, it is necessary to continue to derive the equation. The final Eq. (7) is obtained.
$$\frac{F}{f} = \frac{P \cdot S}{f} = \frac{{P \cdot S^{{}} (h_{0}  y) \cdot n}}{{(\frac{U}{{h_{0} }}  \frac{{v_{x} (y)}}{y}) \cdot 2\mu (D_{1}  D_{2} )\pi }}.$$
(7)
2.3 Simplification of Theory Formula
According to the experimental conditions in this paper, the substitution amount is simplified. The outer diameter, inner diameter and contact area of the grinding piece are fixed. The lubricating oil is selected from Mobil Series No. 1 lubricating oil, and the parameters are fixed. In Eq. (7), the value of h_{0} is critical. Since there is an independent variable y in the film thickness direction, if h_{0} is retained, it will increase the difficulty of solving and make the equation have multiple solutions. The value of h_{0} is set to 1, and the range of the independent variable y is [0, 1], and Eq. (8) is obtained.
$$\begin{aligned}u &= \frac{P \cdot 0.0000885 \cdot (1  y) \cdot 24}{{(\frac{U}{1}  \frac{{v_{x} (y)}}{y}) \times 2 \times 0.055 \times (0.01 + 0.014)\pi }} \\ &= \frac{P \cdot (1  y) \cdot 0.002124}{{0.0083 \cdot (\frac{U}{1}  \frac{{v_{x} (y)}}{y})}}.\end{aligned}$$
(8)
In Eq. (8), the pressure P and speed U need to be set according to the actual working conditions. In this model, the velocity v at any point has only the xaxis component, v_{y }= v_{z }= 0, so it can be considered that v_{x }= v. This part needs to obtain data through simulation to solve, the solution process is explained in detail in the simulation analysis section.