In fluid mechanics (specifically lubrication theory), the Reynolds equation is a partial differential equation governing the pressure distribution of thin viscous fluid films.
[1] The classical Reynolds Equation can be used to describe the pressure distribution in nearly any type of fluid film bearing; a bearing type in which the bounding bodies are fully separated by a thin layer of liquid or gas.
The Reynolds Equation assumes: For some simple bearing geometries and boundary conditions, the Reynolds equation can be solved analytically.
Frequently this involves discretizing the geometric domain, and then applying a finite technique - often FDM, FVM, or FEM.
In certain simplified cases, however, analytical or approximate solutions can be obtained.
[4] For the case of rigid sphere on flat geometry, steady-state case and half-Sommerfeld cavitation boundary condition, the 2-D Reynolds equation can be solved analytically.
This solution was proposed by a Nobel Prize winner Pyotr Kapitsa.
Half-Sommerfeld boundary condition was shown to be inaccurate and this solution has to be used with care.
In case of 1-D Reynolds equation several analytical or semi-analytical solutions are available.
In 1916 Martin obtained a closed form solution[5] for a minimum film thickness and pressure for a rigid cylinder and plane geometry.
This solution is not accurate for the cases when the elastic deformation of the surfaces contributes considerably to the film thickness.
In 1949, Grubin obtained an approximate solution[6] for so called elasto-hydrodynamic lubrication (EHL) line contact problem, where he combined both elastic deformation and lubricant hydrodynamic flow.
For example: In 1978 Patir and Cheng introduced an average flow model,[8][9] which modifies the Reynolds equation to consider the effects of surface roughness on lubricated contacts.
The average flow model spans the regimes of lubrication where the surfaces are close together and/or touching.
They also presented terms for adjusting the contact shear calculation.
In these regimes, the surface topography acts to direct the lubricant flow, which has been demonstrated to affect the lubricant pressure and thus the surface separation and contact friction.
Leighton et al.[10] presented a method for determining the flow factors needed for the average flow model from any measured surface.
Harp and Salent[11] extended the average flow model by considering the inter-asperity cavitation.
Chengwei and Linqing[12] used an analysis of the surface height probability distribution to remove one of the more complex terms from the average Reynolds equation,
Knoll et al. calculated flow factors, taking into account the elastic deformation of the surfaces.
Meng et al.[13] also considered the elastic deformation of the contacting surfaces.
The work of Patir and Cheng was a precursor to the investigations of surface texturing in lubricated contacts.
Demonstrating how large scale surface features generated micro-hydrodynamic lift to separate films and reduce friction, but only when the contact conditions support this.