Incompressibility

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Possible deformation of a volume element of an incompressible substance

Incompressibility refers to the property of a substance not to change its volume under the influence of pressure at constant temperature, i.e. not to be compressed :

${\displaystyle \left({\frac {\partial V}{\partial p}}\right)_{T}=0}$

The volume always consists of the same number of particles (i.e. the mass remains constant).

Complete incompressibility does not occur in reality, all real materials are compressible, even if z. T. only to a very small extent. The following is usually given as the size :

• for fluids , the isothermal compressibility
• for solids the reciprocal value , the isothermal compression modulus .

Incompressibility therefore stands for the approximation of an infinitely low compressibility or an infinitely high compression modulus. Rubber is often considered incompressible because its compression modulus is very large compared to its shear modulus .

Compared to gases , liquids and solids are often viewed as incompressible. At normal pressure, most liquids are 1000 to 10,000 times less compressible than gases, and solids are usually ten times less compressible.

Incompressible bodies do not experience any change in volume due to a change in pressure, but they can experience a change in shape .

In hydrodynamics , the  following simplified mathematical formulation is used for incompressibility - assuming an incompressible fluid :

${\displaystyle {\begin{aligned}{\vec {\nabla }}\cdot {\vec {v}}&=0\\\Leftrightarrow {\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}&=0\qquad \qquad (1)\end{aligned}}}$

Where is the flow velocity . ${\displaystyle {\vec {v}}=(v_{x},v_{y},v_{z})}$

This relationship is called freedom from divergence , since the divergence represents the flow velocity. ${\displaystyle {\vec {\nabla }}\cdot {\vec {v}}}$

The picture tries to clarify this connection: e.g. a bilateral horizontal flow with the velocity components and into the volume element causes a simultaneous vertical flow with the velocity component out of the volume element. ${\displaystyle v_{x}}$ ${\displaystyle v_{y}}$ ${\displaystyle v_{z}}$

Equation (1) was derived using the continuity equation :

${\displaystyle {\partial \rho \over \partial t}=-\nabla \cdot (\rho {\vec {v}})=-{\vec {v}}\cdot \nabla \rho -\rho \nabla \cdot {\vec {v}}\qquad \qquad (2)}$

For incompressible flow, the definition of a particle means that its density does not change:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\rho (t,\mathbf {x} (t))={\partial \rho \over \partial t}+{\vec {v}}\cdot \nabla \rho =0\qquad \qquad (3)}$

The comparison of (2) and (3) leads directly to equation (1).