A Simple Low Deborah Number Model for Unsteady Hydrodynamic Lubrication, Including Fluid Inertia

In unsteady hydrodynamic lubrication the film thickness varies with time, producing the squeeze film term in the Reynolds equation, where the fluid is assumed to be Newtonian and fluid inertia effects are neglected. A Deborah and Reynolds number perturbation solution correction to lubrication theory is presented which can accommodate arbitrary smooth two‐dimensional surface geometry and arbitrary time variation in film thickness. The constitutive equation used is one recently proposed by Harnoy in which a new type of material derivative is used allowing separate consideration of relaxation and normal stress effects, each governed by a separate material parameter. A low Deborah number expansion is developed to obtain a simple two‐parameter (viscosity and relaxation time) stress‐explicit relation. Corrections to the lubrication theory for inertia and viscoelastic effects in terms of Reynolds and Deborah numbers are presented. Solutions to several problems are presented: (1) parallel surfaces with constant approach velocity, (2) parallel surfaces with constant load, and (3) the squeeze film damper bearing. Important phase shifting effects between the squeezing rate and the load are now exhibited, whereas lubrication theory requires that they be exactly in phase.