Reynolds equation

The Reynolds Equation is a partial differential equation governing the pressure distribution of thin viscous fluid films in Lubrication theory. It should not be confused with Osborne Reynolds' other namesakes, Reynolds number and Reynolds-averaged Navier–Stokes equations. It was first derived by Osborne Reynolds in 1886.[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.

General usage

The general Reynolds equation is:

${\displaystyle {\frac {\partial }{\partial x}}\left({\frac {\rho h^{3}}{12\mu }}{\frac {\partial p}{\partial x}}\right)+{\frac {\partial }{\partial y}}\left({\frac {\rho h^{3}}{12\mu }}{\frac {\partial p}{\partial y}}\right)={\frac {\partial }{\partial x}}\left({\frac {\rho h\left(u_{a}+u_{b}\right)}{2}}\right)+{\frac {\partial }{\partial y}}\left({\frac {\rho h\left(v_{a}+v_{b}\right)}{2}}\right)+\rho \left(w_{a}-w_{b}\right)-\rho u_{a}{\frac {\partial h}{\partial x}}-\rho v_{a}{\frac {\partial h}{\partial y}}+h{\frac {\partial \rho }{\partial t}}}$

Where:

• ${\displaystyle p}$ is fluid film pressure.
• ${\displaystyle x}$ and ${\displaystyle y}$ are the bearing width and length coordinates.
• ${\displaystyle z}$ is fluid film thickness coordinate.
• ${\displaystyle h}$ is fluid film thickness.
• ${\displaystyle \mu }$ is fluid viscosity.
• ${\displaystyle \rho }$ is fluid density.
• ${\displaystyle u,v,w}$ are the bounding body velocities in ${\displaystyle x,y,z}$ respectively.
• ${\displaystyle a,b}$ are subscripts denoting the top and bottom bounding bodies respectively.

The equation can either be used with consistent units or nondimensionalized.

The Reynolds Equation assumes:

• The fluid is Newtonian.
• Fluid viscous forces dominate over fluid inertia forces. This is the principle of the Reynolds number.
• Fluid body forces are negligible.
• The variation of pressure across the fluid film is negligibly small (i.e. ${\displaystyle {\frac {\partial p}{\partial z}}=0}$)
• The fluid film thickness is much less than the width and length and thus curvature effects are negligible. (i.e. ${\displaystyle h\ll l}$ and ${\displaystyle h\ll w}$).

For some simple bearing geometries and boundary conditions, the Reynolds equation can be solved analytically. Often however, the equation must be solved numerically. Frequently this involves discretizing the geometric domain, and then applying a finite technique - often FDM, FVM, or FEM.

Derivation from Navier-Stokes

A full derivation of the Reynolds Equation from the Navier-Stokes equation can be found in numerous lubrication text books.[2][3]

Solution of Reynolds Equation

In general, Reynolds equation has to be solved using numerical methods such as finite difference, or finite element. 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. In this solution it was assumed that the pressure profile follows Hertz solution. The model is therefore accurate at high loads, when the hydrodynamic pressure tends to be close to the Hertz contact pressure.[7]

Applications

The Reynolds equation is used to model the pressure in many applications. For example:

• Ball bearings
• Air bearings
• Journal bearings
• Squeeze film dampers in aircraft gas turbines
• Human hip and knee joints
• Lubricated gear contacts

Reynolds Equation adaptations

In 1978 Patir and Cheng introduced an average flow model[8] which modifies the Reynolds equation to consider the effects of surface roughness on partially lubricated contacts.

References

1. Reynolds, O. (1886). "On the Theory of Lubrication and Its Application to Mr. Beauchamp Tower's Experiments, Including an Experimental Determination of the Viscosity of Olive Oil". Philosophical Transactions of the Royal Society of London. Royal Society. 177: 157–234. doi:10.1098/rstl.1886.0005. JSTOR 109480.
2. Hamrock, Bernard J.; Schmid, Steven R.; Jacobson, Bo O. (2004). Fundamentals of Fluid Film Lubrication. Taylor & Francis. ISBN 978-0-8247-5371-9.
3. Szeri, Andras Z. (2010). Fluid Film Lubrication. Cambridge University Press. ISBN 978-0-521-89823-2.
4. "Reynolds Equation: Derivation and Solution". tribonet.org. 12 November 2016. Retrieved 10 September 2019.
5. Akchurin, Aydar (18 February 2016). "Analytical Solution of 1D Reynolds Equation". tribonet.org. Retrieved 10 September 2019.
6. Akchurin, Aydar (22 February 2016). "Semi-Analytical Solution of 1D Transient Reynolds Equation(Grubin's Approximation)". tribonet.org. Retrieved 10 September 2019.
7. Akchurin, Aydar (4 January 2017). "Hertz Contact Calculator". tribonet.org. Retrieved 10 September 2019.
8. Patir, Nadir; Cheng, H. S. (1978). "An Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication". Journal of Lubrication Technology. 100 (1): 12. doi:10.1115/1.3453103. ISSN 0022-2305.