# Formulation to Calculate Isothermal, Non-Newtonian Elastohydrodynamic Lubrication Problems Using a Pressure Gradient Coordinate System and Its Verification by an Experimental Grease

## Abstract

**:**

## 1. Introduction

#### 1.1. Classification of Calculation Methods

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

- Method 2-A: Local X-direction is the sliding direction.
- Method 2-B: Local X-direction is the direction of the pressure gradient.

#### 1.2. Previous Works

## 2. Overview of the Proposed Method

## 3. Calculation of Velocity Distribution as a Function of z

## 4. Calculation of Equivalent Viscosity, Flow, and Surface Force

## 5. Application to a Grease

_{min}=100.0 s

^{−1}, then ${\eta}^{*}$ varies linearly with the gradient $\mathrm{d}{\eta}^{*}/\mathrm{d}\dot{\gamma}$ at c

_{min}. Bauer’s parameter ${k}_{1}$ was assumed to be the base oil ambient viscosity ${\eta}_{0}$. The values of Bauer’s parameters ${\tau}_{0}$, ${k}_{2}$, and $n$ were determined from the apparent viscosity curve of grease A, as shown in Figure 9 of Kochi et al. [1]. When $P=0$, the curve was assumed to pass through the following three points: 100 mm/s, 6.46475 Pa·s; 10,000 mm/s, 0.16712 Pa·s; and 1,000,000 mm/s, 0.06573 Pa·s.

## 6. Conclusions

- (1)
- A modified Reynolds equation, which contains an equivalent viscosity, was obtained.
- (2)
- The EHL calculation procedure for Newtonian flows can be applied to non-Newtonian flows by simply replacing the Newtonian viscosity with the equivalent viscosity.
- (3)
- If rheological equations are incorporated, any isothermal, non-Newtonian EHL calculation can be performed easily.
- (4)
- As the equivalent viscosity is calculated using the one-variable Newton–Raphson method, the EHL calculation can be performed within a reasonable calculation time.
- (5)
- Using Bauer’s model, the formulation was applied to a grease that was evaluated experimentally by Kochi et al. [1]. The results obtained using the proposed method and the experimental results were compared, and reasonable agreement was noted.
- (6)
- In the case of sliding velocity, the equivalent viscosity shows a figure-eight-shaped distribution in the vicinity of the contact point.

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

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

**AMA Style**

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(5):56.
https://doi.org/10.3390/lubricants9050056

**Chicago/Turabian Style**

Kakoi, Kunihiko.
2021. "Formulation to Calculate Isothermal, Non-Newtonian Elastohydrodynamic Lubrication Problems Using a Pressure Gradient Coordinate System and Its Verification by an Experimental Grease" *Lubricants* 9, no. 5: 56.
https://doi.org/10.3390/lubricants9050056