In fluid dynamics

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, lubrication theory describes 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

Bearing (mechanical)

A bearing is a device to allow constrained relative motion between two or more parts, typically rotation or linear movement. Bearings may be classified broadly according to the motions they allow and according to their principle of operation as well as by the directions of applied loads they can...

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 a lubricant

Lubricant

A lubricant is a substance introduced to reduce friction between moving surfaces. It may also have the function of transporting foreign particles and of distributing heat...

.

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

Surface tension

Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...

 may then be significant, or even dominant. Issues of wetting

Wetting

Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from intermolecular interactions when the two are brought together. The degree of wetting is determined by a force balance between adhesive and cohesive forces.Wetting is important in the bonding or adherence of...

 and dewetting

Dewetting

In fluid mechanics, dewetting is one of the processes that can occur at a solid–liquid or liquid–liquid interface. Generally, dewetting describes the rupture of a thin liquid film on the substrate and the formation of droplets. The opposite process—spreading of a liquid on a substrate—is called...

 then arise. For very thin films (thickness less than one micrometre

Micrometre

A micrometer , is by definition 1×10-6 of a meter .In plain English, it means one-millionth of a meter . Its unit symbol in the International System of Units is μm...

), additional intermolecular forces, such as disjoining forces, 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, , and the second is a characteristic substrate length scale . The key requirement for lubrication theory is that the ratio is small, that is, .
The Navier-Stokes equations

Navier-Stokes equations

In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...

 (or Stokes equations, when fluid inertia may be neglected) are expanded in this small parameter, and the leading-order equations are then

where and are coordinates in the direction of the substrate and perpendicular to it respectively. Here is the fluid pressure, and is the fluid velocity component parallel to the substrate; is the fluid viscosity.

Further details can be found in the literature or in the textbooks given in the bibliography.

Applications

An important application area is lubrication

Lubrication

Lubrication is the process, or technique employed to reduce wear of one or both surfaces in close proximity, and moving relative to each another, by interposing a substance called lubricant between the surfaces to carry or to help carry the load between the opposing surfaces. The interposed...

 of machinery components such as fluid bearing

Fluid bearing

Fluid bearings are bearings which support the bearing's loads solely on a thin layer of liquid or gas.They can be broadly classified as fluid dynamic bearings or hydrostatic bearings. Hydrostatic bearings are externally pressurized fluid bearings, where the fluid is usually oil, water or air, and...

s and mechanical seals

Seal (mechanical)

A mechanical seal is a device which helps join systems or mechanisms together by preventing leakage , containing pressure, or excluding contamination...

. Coating

Coating

Coating is a covering that is applied to the surface of an object, usually referred to as the substrate. In many cases coatings are applied to improve surface properties of the substrate, such as appearance, adhesion, wetability, corrosion resistance, wear resistance, and scratch resistance...

 is another major application area including the preparation of thin films, printing

Printing

Printing is a process for reproducing text and image, typically with ink on paper using a printing press. It is often carried out as a large-scale industrial process, and is an essential part of publishing and transaction printing....

, painting

Painting

Painting is the practice of applying paint, pigment, color or other medium to a surface . The application of the medium is commonly applied to the base with a brush but other objects can be used. In art, the term painting describes both the act and the result of the action. However, painting is...

 and adhesive

Adhesive

An adhesive, or glue, is a mixture in a liquid or semi-liquid state that adheres or bonds items together. Adhesives may come from either natural or synthetic sources. The types of materials that can be bonded are vast but they are especially useful for bonding thin materials...

s.

Biological applications have included studies of red blood cells in narrow capillaries and of liquid flow in the lung and eye.