- Lubrication theory
A branch of

fluid dynamics , lubrication theory is used to describe the flow of fluids (liquids or gases) in a geometry in which one dimension is significantly smaller than the others.Interior flows are those where the boundaries of the fluid volume are known, and include those inside

bearing s. Here a key goal of lubrication theory is to determine the pressure distribution in the fluid volume, and hence the forces on the bearing components. The working fluid in this case is often termed alubricant .Free film lubrication theory is concerned with the case in which one of the surfaces containing the fluid is a free surface. In that case the position of the free surface is itself unknown, and one goal of lubrication theory is then to determine this.

Surface tension may then be significant, or even dominant. Issues ofwetting anddewetting then arise. For very thin films (thickness less than onemicrometre ), additional intermolecular forces, such asdisjoining force s, may become significant.**Theoretical basis**Mathematically, lubrication theory can be seen as exploiting the disparity between two length scales. The first is the characteristic film thickness, $H$, and the second is a characteristic substrate length scale $L$. The key requirement for lubrication theory is that the ratio $epsilon\; =\; H/L$ is small, that is, $epsilon\; ll\; 1$.The

Navier-Stokes equations (orStokes equations , when fluid inertia may be neglected) are expanded in this small parameter, and the leading-order equations are then:$frac\{partial\; p\}\{partial\; z\}\; =\; 0$:$frac\{partial\; p\}\{partial\; x\}\; =\; frac\{partial^2\; u\}\{partial\; z^2\}$where $x$ and $z$ are coordinates in the direction of the substrate and perpendicular to it respectively. Here $p$ is the fluid pressure, and $u$ is the fluid velocity component parallel to the substrate.

Further details can be found in the literatureOron, A; Davis S. H., and S. G. Bankoff, " [

*http://prola.aps.org/abstract/RMP/v69/i3/p931_1 Long-scale evolution of thin liquid films*] ", Rev. Mod. Phys. 69, 931 - 980 (1997) ] or in the textbooks given in the bibliography.**Applications**Important application areas include

lubrication of machinery,fluid bearing s,coating (including the preparation ofthin films ,printing , andpainting ), andadhesive s.Biological applications have included studies of liquid flow in the lung and eye.

**Notes****References***Batchelor, G. K. (1976), An introduction to fluid mechanics, Cambridge: CUP. ISBN 978-0-52-109817-5.

*Panton, R. L. (2005), Incompressible Flow (3rd ed.), New York: Wiley . ISBN 978-0-47-126122-3.

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