The matter which density is not dependant on pressure $\begin{array}{l}p\end{array}$ or temperature $\begin{array}{l}T\end{array}$ :

(1)	$\begin{array}{l}\displaystyle \rho(p, T) = \rho_0 = {\rm const}\end{array}$

which is equivalent to zero compressibility:

(2)	$\begin{array}{l}\displaystyle c(p) \equiv 0 \Leftrightarrow \frac{d \rho}{ dp} = 0\end{array}$

This also implies that material balance equation:

(3)	$\begin{array}{l}\displaystyle \frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) = 0\end{array}$

simplifies to:

(4)	$\begin{array}{l}\displaystyle \nabla {\bf u} = 0\end{array}$

The incompressible matter does not exist but in many practical applications the pressure and temperature variations stay within the sufficiently narrow range where a given matter can be considered as incompressible.

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Incompressible matter

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