Incompressible flow

In fluid mechanics and continuum mechanics, an incompressible flow is a physical (or hypothetical) flow having an invariant density relative to the time variable. In vector calculus, an incompressible flow velocity has zero divergence. In many physical situations, the flow of a compressible fluid has a good approximation as as incompressible flow.

Quotes

John Canton, in Trans. R. S. 1762, first showed that liquids were compressible, even with the pressure of one atmosphere. His apparatus consisted of a large thermometer which contained the liquid, and the bulb of which was inclosed in an exhausted receiver. The liquid was thus entirely relieved of atmospheric pressure. The height of the liquid was now marked upon the stem, and the end of the sealed thermometer being broken, the air was allowed to enter the stem and press upon the surface of the liquid, and to enter the receiver and press upon the outside of the bulb. The liquid instantly fell in the tube, thus clearly showing a compression produced by the pressure of one atmosphere.
Canton thus found that water was compressed 46/1000000ths of its volume by the pressure of one atmosphere, and this determination is quite exact.
- Alfred M. Mayer, Lecture-notes on Physics: Part 1. Philadelphia: From the Journal of the Franklin institute, 1868. p. 55. (115 pages)

Incompressible flows are those where changes in the fluid density are important. The study of incompressible flow includes such subjects as hydraulics, hydrodynamics, lubrication theory, aerodynamics, and boundary layer theory. It also contains background information for such special subjects as hydrology, stratified flow, turbulence, rotating flows, and biological fluid mechanics. Incompressible flow not only occupies the central position in fluid dynamics, but is also fundamental to the practical subjects of heat and mass transfer.
- Ronald L. Panton, Incompressible Flow (5th ed.). John Wiley & Sons. 2024. p. 1. ISBN 9781119984399. (880 pages)

A method for solving quite general three-dimensional incompressible flow problems, in particular those described by the Navier–Stokes equations, is presented. The essence of the method is the expression of the velocity in terms of scalar and vector potentials, which are the three-dimensional generalizations of the two-dimensional stream function, and which ensure that the equation of continuity is satisfied automatically. Although the method is not new, a correct but simple and unambiguous procedure for using it has not been presented before.
- Stephen Michael Richardson and A. R. H. Cornish, (7 September 1977) "Solution of three-dimensional flow problems". Journal of Fluid Mechanics 82 (2): 309–319. DOI:10.1017/S0022112077000688.

Compressibility
For water (i.e. a liquid) the assumption of incompressibility works very well, meaning if we apply pressure on an element of water, the volume of this element will (approximately) not change. For gases, the assumption of incompressibility (i.e. the volume of an element of air does not change when applying a pressure) works well in most cases of our daily lives (e.g. when walking around, when driving our car). When driving, the air molecules hit your car, but are then quickly displaced around it.
However, in a lot of aerospace applications, the range of speeds encountered is a lot higher. At such speeds, assuming incompressible flow is not correct anymore. Instead of being quickly displaced, as in the car example, the individual molecules will come closer together before getting displaced.
- Laurens Voet, Topic 2: Basic Engine Design: Incompressible Flow. wikis.mit.edu.