Editing Incompressible flow (section)

== Difference from material ==
As defined earlier, an incompressible (isochoric) flow is the one in which 
:<math> \nabla \cdot \mathbf u = 0. \, </math>
This is equivalent to saying that
:<math> \frac{D\rho}{Dt} = \frac{\partial \rho}{\partial t} + \mathbf u \cdot \nabla \rho = 0</math>
i.e. the [[Substantive derivative|material derivative]] of the density is zero. Thus if one follows a material element, its mass density remains constant. Note that the material derivative consists of two terms. The first term <math> \tfrac{\partial \rho}{\partial t} </math> describes how the density of the material element changes with time. This term is also known as the ''unsteady term''. The second term, <math>\mathbf u \cdot \nabla \rho</math> describes the changes in the density as the material element moves from one point to another. This is the ''advection term'' (convection term for scalar field). For a flow to be accounted as bearing incompressibility, the accretion sum of these terms should vanish.

On the other hand, a '''homogeneous, incompressible material''' is one that has constant density throughout. For such a material, <math>\rho = \text{constant} </math>. This implies that, 
:<math> \frac{\partial \rho}{\partial t} = 0 </math> and
:<math>\nabla \rho = 0</math> ''independently''.

From the continuity equation it follows that 
:<math> \frac{D\rho}{Dt} = \frac{\partial \rho}{\partial t} + \mathbf u \cdot \nabla \rho = 0 \ \Rightarrow\  \nabla \cdot \mathbf u = 0 </math>

Thus homogeneous materials always undergo flow that is incompressible, but the converse is not true. That is, compressible materials might not experience compression in the flow.