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

Kunihiko Kakoi

doi:10.3390/lubricants9050056

Abstract

This paper presents a formulation of point contact elastohydrodynamic lubrication analysis for an isothermal, non-Newtonian flow. A coordinate system of the pressure gradient was employed herein. A Couette flow and a Poiseuille flow were considered along the directions of the zero and non-zero pressure gradients, respectively. The Poiseuille flow velocity was assumed to be represented by a 4th-order polynomial of z along the film thickness direction. The Couette flow velocity was assumed to be represented by a linear function of z. Subsequently, the modified Reynolds equation, which contains an equivalent viscosity, was obtained. Using Bauer’s rheological model, the formulation presented in this study was applied to a grease that has been previously experimented upon. The results of previous studies were compared with those of the present study and a reasonable agreement was noted. The distribution of the equivalent viscosity showed a notable difference from that of Newtonian flow. The formulation can be incorporated easily to the usual elastohydrodynamic lubrication calculation procedure for Newtonian flow. The method can be easily applied to other non-Newtonian rheological models. The equivalent viscosity can be calculated using the one-parameter Newton-Raphson’s method; as a result, the calculation can be performed within a reasonable time.

Highlights

Performing experiments on non-Newtonian flows, including grease flows, is considerably time-consuming and costly
It is important to numerically analyze the phenomena corresponding to non-Newtonian flows
It is convenient that the non-Newtonian elastohydrodynamic lubrication (EHL) calculation can be executed within a reasonable calculation time and without large modification to the usual Newtonian EHL calculation procedure

Summary

Introduction

Performing experiments on non-Newtonian flows, including grease flows, is considerably time-consuming and costly. Grease flows can be well defined using Bauer’s model; 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. The numerical methods for isothermal, non-Newtonian EHL analyses can be classified in terms of the accurac2yofo1f4 u(z) and v(z), as follows: Method 1: Exact solution of u(z) and v(z) is obtained. The authors obtained an exact solution of line contact EHL problems considering the power law model and demonstrated the effectiveness of their proposed method. Bordenet et al [12] obtained an exact solution of pure rolling point contact EHL problems considering Bauer’s model for n = 1/2 and applied the approach to grease

Overview of the Proposed Method

Calculation of Velocity Distribution as a Function of z

Application to a Grease