Abstract
Most of the problems in lubrication are studied within the context of Reynolds’ equation, which can be derived by writing the incompressible Navier-Stokes equation in a dimensionless form and neglecting terms which are small under the assumption that the lubricant film is very thin. Unfortunately, the Reynolds equation is often used even though the basic assumptions under which it is derived are not satisfied. One example is in the mathematical modelling of elastohydrodynamic lubrication (EHL). In the EHL regime, the pressure is so high that the viscosity changes by several orders of magnitude. This is taken into account by just replacing the constant viscosity in either the incompressible Navier-Stokes equation or the Reynolds equation by a viscosity-pressure relation. However, there are no available rigorous arguments which justify such an assumption. The main purpose of this two-part work is to investigate if such arguments exist or not. In Part A, we formulate a generalised form of the Navier-Stokes equation for piezo-viscous incompressible fluids. By dimensional analysis of this equation we, thereafter, show that it is not possible to obtain the Reynolds equation, where the constant viscosity is replaced with a viscosity-pressure relation, by just neglecting terms which are small under the assumption that the lubricant film is very thin. The reason is that the lone assumption that the fluid film is very thin is not enough to neglect the terms, in the generalised Navier-Stokes equation, which are related to the body forces and the inertia. However, we analysed the coefficients in front of these (remaining) terms and provided arguments for when they may be neglected. In Part B, we present an alternative method to derive a lower-dimensional model, which is based on asymptotic analysis of the generalised Navier-Stokes equation as the film thickness goes to zero.
Highlights
It is well-known that the consequences of lubrication in e.g. bearings and gears, in terms of energy loss, materials wastage, premature failures and environmental impact are substantial, see e.g. literature.[1]
We show that it is only possible to formulate a generalisation of the Navier-Stokes equation for incompressible piezo-viscous fluids, by using implicit constitutive relations.[2,3]
One apparent example is the modelling of elastohydrodynamic lubrication
Summary
It is well-known that the consequences of lubrication in e.g. bearings and gears, in terms of energy loss, materials wastage, premature failures and environmental impact are substantial, see e.g. literature.[1]. We are interested in the class of lubrication problems wherein the lubricant can be considered as incompressible and piezo-viscous This means that the Navier-Stokes consititutive relation,a presented here in (4), is not applicable. Apply the theory for implicit consititutive relations developed in literature,[2,3] and combine the implicit relation given in (10) for the lubricant rheology with conservation of mass, balance of linear and angular momentum, to model the flow. This type of system is very complex, but the fact that the fluid domain in lubrication is
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