Fluid which density is not varying over time:

(1)	$\begin{array}{l}\displaystyle \frac{d \rho}{dt} = \frac{\partial \rho}{\partial t} + {\bf u} \cdot \nabla \rho = 0\end{array}$

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

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

which is equivalent to zero compressibility:

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

This also implies that material balance equation:

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

simplifies to:

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

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.

Page tree

See also

Page tree

Incompressible matter

See also