**1. Introduction**

Performing experiments on non-Newtonian flows, including grease flows, is considerably time-consuming and costly. Therefore, it is important to numerically analyze the phenomena corresponding to non-Newtonian flows. Numerical approaches can help obtain a variety of data that cannot be determined experimentally. Grease flows can be well defined using Bauer's model; however, owing to the extreme complexity of this model, it is difficult to determine the exact solution as well as approximate solutions for point contact, isothermal, non-Newtonian elastohydrodynamic lubrication (EHL) analyses. Therefore, it is convenient that the non-Newtonian EHL calculation can be executed within a reasonable calculation time and without large modification to the usual Newtonian EHL calculation procedure. Kochi et al. [1] performed experiments on grease under soft EHL conditions and measured the film thickness and traction forces. The method proposed in the present study can be applied to the grease considered in Kochi et al. [1] to validate this theoretical approach.

#### *1.1. Classification of Calculation Methods*

As shown in Figure 1, the Z-direction is considered to be the film thickness direction. The flow velocities along the X- and Y-directions are denoted by *u* and *v*, respectively, which are functions of *x*, *y*, and *z*; however, when considering only *z* dependency, the velocities can be expressed as *u*(*z*) and *v*(*z*), respectively. The numerical methods for isothermal,

**Citation:** Kakoi, K. Formulation to Calculate Isothermal, Non-Newtonian Elastohydrodynamic Lubrication Problems Using a Pressure Gradient Coordinate System and Its Verification by an Experimental Grease. *Lubricants* **2021**, *9*, 56. https://doi.org/10.3390/ lubricants9050056

Received: 16 February 2021 Accepted: 11 May 2021 Published: 14 May 2021

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non-Newtonian EHL analyses can be classified in terms of the accuracy of *u*(*z*) and *v*(*z*), as follows:

Method 1: Exact solution of *u*(*z*) and *v*(*z*) is obtained. Method 2: Approximate solution of *u*(*z*) and *v*(*z*) is obtained.

**Figure 1.** Global coordinate system O, X, Y, Z.

Method 2 can be further classified in terms of the employed coordinate system. Method 2-A: Local X-direction is the sliding direction. Method2-B:LocalX-directionisthedirectionofthepressuregradient.
