Incompressible Flow - Difference Between Incompressible Flow and Material

Difference Between Incompressible Flow and Material

As defined earlier, an incompressible (isochoric) flow is the one in which

This is equivalent to saying that

i.e. the material derivative of the density is zero. Thus if we follow a material element, its mass density will remain constant. Note that the material derivative consists of two terms. The first term describes how the density of the material element changes with time. This term is also known as the unsteady term. The second term, describes the changes in the density as the material element moves from one point to another. This is the convection or the advection term. For a flow to be incompressible the sum of these terms should be zero.

On the other hand, a homogeneous, incompressible material is defined as one which has constant density throughout. For such a material, . This implies that,

and

independently.

From the continuity equation it follows that

Thus homogeneous materials always undergo flow that is incompressible, but the converse is not true.

It is common to find references where the author mentions incompressible flow and assumes that density is constant. Even though this is technically incorrect, it is an accepted practice. One of the advantages of using the incompressible material assumption over the incompressible flow assumption is in the momentum equation where the kinematic viscosity can be assumed to be constant. The subtlety above is frequently a source of confusion. Therefore many people prefer to refer explicitly to incompressible materials or isochoric flow when being descriptive about the mechanics.