Fluid which density is not varying over time:
(1) | \frac{d \rho}{dt} = \frac{\partial \rho}{\partial t} + {\bf u} \cdot \nabla \rho = 0 |
The matter which density is not dependant on pressure p or temperature T:
(2) | \rho(p, T) = \rho_0 = {\rm const} |
which is equivalent to zero compressibility:
(3) | c(p) \equiv 0 \Leftrightarrow \frac{d \rho}{ dp} = 0 |
This also implies that material balance equation:
(4) | \frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) = 0 |
simplifies to:
(5) | \nabla {\bf u} = 0 |
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.
See also
Physics / Fluid (PVT) Analysis / Fluid (PVT) modelling / Compressibility