Incompressible Fluid

Website Links For
Flow

Information About

Incompressible Fluid

The equation describing an incompressible (isochoric) flow,

:

{
abla \cdot ec u = 0}

,

where

ec u

is the velocity of the material.

The Continuity Equation states that,

:

{\partial 
ho \over \partial t} + 
abla \cdot (
ho ec u) = 0

This can be expressed via the Material Derivative as
:

{rac{D
ho}{Dt}} = - 
ho (
abla \cdot ec u)

Since

{
ho > 0}

, we see that a flow is incompressible if and only if,
:

{rac{D
ho}{Dt}} = 0

that is, the mass density is constant following the material element.

RELATION TO COMPRESSIBILITY FACTOR

In some fields, a measure of the incompressibility of a flow is the change in density as a result of the pressure variations. This is best expressed in terms of the Compressibility Factor

:

Z = {rac{1}{
ho}} {rac{d
ho}{dp}}.

If the compressibility factor is acceptably small, the flow is considered to be incompressible.

RELATION TO SOLENOIDAL FIELD

An incompressible flow is described by a velocity field which is Solenoidal . But a solenoidal field, besides having a zero Divergence , also has the additional connotation of having non-zero Curl (i.e., rotational component).

Otherwise, if an incompressible flow also has a curl of zero, so that it is also Irrotational , then the velocity field is actually Laplacian .

DIFFERENCE BETWEEN INCOMPRESSIBLE FLOW AND MATERIAL

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

abla \cdot ec u = 0

.
This is equivalent to saying that
:

frac{D
ho}{Dt} = 	frac{\partial 
ho}{\partial t} + ec u \cdot 
abla 
ho = 0

i.e. the Material Derivative of the density is zero. Thus if we follow a material element, it's mass density will remain constant. Note that the material derivative consists of two terms. The first term

frac{\partial 
ho}{\partial t}

is the unsteady term and describes how the density of the material element changes with time. This term is also know as the ''unsteady term''. The second term,

ec u \cdot 
abla 
ho

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,

ho = constant

. This implies that,
:

frac{\partial 
ho}{\partial t} = 0

and
:

abla 
ho = 0

''independently''.
From the continuity equation it follows that
:

frac{D
ho}{Dt} = 	frac{\partial 
ho}{\partial t} + ec u \cdot 
abla 
ho = 0 \implies 
abla \cdot ec u = 0

Thus incompressible 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 (

u = 	frac{\mu}{
ho}

) 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.

SEE ALSO

Compressible Flow

Navier-Stokes Equations