# Lubrication theory

From Soft-Matter

Lubrication theory refers to a simplification of the Navier-Stokes equations which assumes that one dimension of the problem is significantly smaller than the others.

It is in most cases formulated for two dimensions, where the governing equations to first order then become

- <math>\frac{\partial p}{\partial z} = 0</math>
- <math>\frac{\partial p}{\partial x} = \frac{\partial^2 u}{\partial z^2}</math>

where *p* is pressure, *u* is the velocity component in the x direction and *z* is the small dimension in the problem.

The term derives from the tremendous importance the areas of lubrication of machinery and fluid bearings had when these equations where first formally developed.

Today these equations finds application in a very wide range of fields: from free films over biological flows to the study of elastohydrodynamic interactions or the splashing of water drops.