# Incompressible fluid

A fluid whose density does not depend on the pressure is called incompressible - in contrast to compressible fluids.

Conversely, this means that fluids whose density changes, for example due to thermal influences, can be incompressible. Since these effects are usually considerably smaller in practice than changes in density due to changes in pressure, a fluid is regarded as incompressible if the density is constant along each trajectory. However, a constant overall density is not a criterion for incompressibility.

Incompressible fluids do not exist in reality; they represent an idealization that enormously simplifies many calculations with negligible errors, e.g. B. Water in water pipes under normal conditions . In certain applications of hydraulics or fluid technology, however , the low compressibility of a hydraulic fluid must be taken into account.

Flows of principally compressible fluids (e.g. gases ) can be regarded as incompressible if the Mach number is small.

The incompressibility of a fluid is synonymous with the disappearance of compressibility , which is defined as the relative change in volume with a change in pressure and constant temperature: ${\ displaystyle \ kappa}$

{\ displaystyle {\ begin {aligned} \ kappa & = 0 \\\ Leftrightarrow - {\ frac {1} {V}} \ left ({\ frac {\ partial V} {\ partial p}} \ right) _ {T} & = 0 \\\ Leftrightarrow \ left ({\ frac {\ partial V} {\ partial p}} \ right) _ {T} & = 0 \ end {aligned}}}

This formulation is derived from the continuity equation as the freedom from divergence of the flow, neglecting any temperature dependence:

${\ displaystyle {\ vec {\ nabla}} \ cdot {\ vec {v}} = 0 \ Leftrightarrow {\ text {div}} \; {\ vec {v}} = 0}$

The underlying mathematical model are the Navier-Stokes equations .

## literature

• Batchelor, GK: An introduction to fluid dynamics, Cambridge University Press, 2000, ISBN 0-521-66396-2 .
• Lexicon of Physics, Spectrum Akademischer Verlag GmbH, Heidelberg, 1999, ISBN 3-86025-293-3 .