Information about Incompressible Flow
In fluid mechanics or more generally continuum mechanics, an incompressible flow is solid or fluid flow in which the divergence of velocity is zero. This is more precisely termed isochoric flow. It is an idealization used to simplify analysis. In reality, all materials are compressible to some extent. Note that isochoric refers to flow, not the material property. This means that under certain circumstances, a compressible material can undergo (nearly) incompressible flow. However, by making the 'incompressible' assumption, the governing equations of material flow can be simplified significantly.
The equation describing an incompressible (isochoric) flow,
where
is the velocity of the material.
The continuity equation states that,
This can be expressed via the material derivative as
Since
, we see that a flow is incompressible if and only if,
that is, the mass density is constant following the material element.
If the compressibility factor is acceptably small, the flow is considered to be incompressible.
Otherwise, if an incompressible flow also has a curl of zero, so that it is also irrotational, then the velocity field is actually Laplacian.
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, 
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,
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.
The equation describing an incompressible (isochoric) flow,
,
where
is the velocity of the material.
The continuity equation states that,
This can be expressed via the material derivative as
Since
, we see that a flow is incompressible if and only if,
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
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
.
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, 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.
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.
See also
Fluid mechanics is the study of how fluids move and the forces on them. (Fluids include liquids and gases.) Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, the study of fluids in motion.
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Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i.e., liquids and gases).
The fact that matter is made of atoms and that it commonly has some sort of heterogeneous
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The fact that matter is made of atoms and that it commonly has some sort of heterogeneous
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A solid object is in the states of matter characterized by resistance to deformation and changes of volume. At the microscopic scale, a solid has these properties :
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- The atoms or molecules that comprise the solid are packed closely together.
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FLUID (Fast Light User Interface Designer) is a graphical editor that is used to produce FLTK source code. FLUID edits and saves its state in text .fl files, which can be edited in a text editor for finer control over display and behavior.
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In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a (signed) scalar.
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Isochoric may refer to:
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- For geometry, cell-transitive.
- For chemistry, isochoric process.
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The convective derivative (also commonly known as the advective derivative, substantive derivative, or the material derivative) is a derivative taken with respect to a coordinate system moving with velocity u
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The compressibility factor (Z) is used to alter the ideal gas equation to account for the real gas behaviour.[1] The compressibility factor is usually obtained from the compressibility chart.
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In vector calculus a solenoidal vector field is a vector field v with divergence zero:
This condition is satisfied whenever v has a vector potential, because if
then
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This condition is satisfied whenever v has a vector potential, because if
then
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In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a (signed) scalar.
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In vector calculus a conservative vector field is a vector field which is the gradient of a scalar potential. There are two closely related concepts: path independence and irrotational vector fields.
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In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following differential equations:
Since the curl of v
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Since the curl of v
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The convective derivative (also commonly known as the advective derivative, substantive derivative, or the material derivative) is a derivative taken with respect to a coordinate system moving with velocity u
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The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances such as liquids and gases. These equations establish that changes in momentum in infinitesimal volumes of fluid are simply the sum of dissipative
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