Chapter Eleven - The Reynolds Equation

Abstract

At the foundation of the lubrication field is the Reynolds equation, a differential equation which relates the pressure in a thin liquid film to the viscosity, geometry, and kinematics. The Reynolds equation is an approximation to the Navier–Stokes equations, employing a thin-film assumption. Although it was derived with the assumption of constant viscosity, it has been used for decades as the basis of elastohydrodynamic lubrication (EHL) where the viscosity is anything but constant, varying orders-of-magnitude in the space of 0.1mm. To predict the film thickness in EHL, a form of Reynolds equation is required which allows for the shear dependence of viscosity. In one of the oddest quirks of classical EHL, the only nonNewtonian Reynolds derivations assumed the sinh-law, Eq. (7.7), and the solutions obtained have been incapable of predicting the effect of shear-thinning on film thickness without altering the assumed piezoviscous response. The Reynolds equation is derived here for some of the viscosity functions of Table 2.1.