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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, one can greatly simplify the equations governing the flow of the material.

The equation describing an incompressible (isochoric) flow,

${\nabla \cdot \vec u = 0}$ ,

where $\vec u$ is the velocity of the material.

The continuity equation states that,

${\partial \rho \over \partial t} + \nabla \cdot (\rho \vec u) = 0$

This can be expressed via the material derivative as

${\frac{D\rho}{Dt}} = - \rho (\nabla \cdot \vec u)$

Since $ρ > 0$ , we see that a flow is incompressible if and only if,

${\frac{D\rho}{Dt}} = 0$

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

[edit] 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

$Z = {\frac{1}{\rho}} {\frac{d\rho}{dp}}.$

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

[edit] 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.

[edit] Difference between incompressible flow and material

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

$\nabla \cdot \vec u = 0$ .

This is equivalent to saying that

$\tfrac{D\rho}{Dt} = \tfrac{\partial \rho}{\partial t} + \vec u \cdot \nabla \rho = 0$

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 $\tfrac{\partial \rho}{\partial t}$ describes how the density of the material element changes with time. This term is also known as the unsteady term. The second term, $\vec u \cdot \nabla \rho$ 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, $ρ = c o n s t a n t$ . This implies that,

$\tfrac{\partial \rho}{\partial t} = 0$ and

$\nabla \rho = 0$ independently.

From the continuity equation it follows that

$\tfrac{D\rho}{Dt} = \tfrac{\partial \rho}{\partial t} + \vec u \cdot \nabla \rho = 0 \implies \nabla \cdot \vec u = 0$

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 ( $\nu = \tfrac{\mu}{\rho}$ ) 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.

[edit] Related flow constraints

In fluid dynamics, a flow is considered to be incompressible if the divergence of the velocity is zero. However, related formulations can sometimes be used, depending on the flow system to be modelled. Some versions are described below:

Incompressible flow: ${\nabla \cdot \vec u = 0}$ . This can assume either constant density (strict incompressible) or varying density flow. The varying density set accepts solutions involving small perturbations in density, pressure and/or temperature fields, and can allow for pressure stratification in the domain.
Anelastic flow: ${\nabla \cdot \left(\rho_{o}\vec u\right) = 0}$ . Principally used in the field of atmospheric sciences, the anelastic constraint extend incompressible flow validity to stratified density and/or temperature as well as pressure. This allow the thermodynamic variables to relax to an 'atmospheric' base state seen in the lower atmosphere when used in the field of meteorology, for example. This condition can also be used for various astrophysical systems.^[1]
Low Mach-number flow / Pseudo-incompressibility: $\nabla \cdot \left(\alpha \vec u \right) = \beta$ . The low Mach-number constraint can be derived from the compressible Euler equations using scale analysis of non-dimensional quantities. The restraint, like the previous in this section, allows for the removal of acoustic waves, but also allows for large perturbations in density and/or temperature. The assumption is that the flow remains within a Mach number limit (normally less than 0.3) for any solution using such a constraint to be valid. Again, in accordance with all incompressible flows the pressure deviation must be small in comparison to the pressure base state.^[2]

These methods make differing assumptions about the flow, but all take into account the general form of the constraint $\nabla \cdot \left(\alpha \vec u \right) = \beta$ for general flow dependent functions $α$ and $β$ .

[edit] Numerical approximations of incompressible flow

The stringent nature of the incompressible flow equations means that specific mathematical techniques have been devised to solve them. Some of these methods include:

The projection method (both approximate and exact)
Artificial compressibility technique (approximate)
Compressibility pre-conditioning

[edit] References

^ Durran, D.R. (1989). "Improving the Anelastic Approximation". Journal of the Atmospheric Sciences 46 (11): 1453–1461. doi:10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2. http://ams.allenpress.com/archive/1520-0469/46/11/pdf/i1520-0469-46-11-1453.pdf.
^ Almgren, A.S.; Bell, J.B.; Rendleman, C.A.; Zingale, M. (2006). "Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics". Astrophysical Journal 637: 922–936. doi:10.1086/498426. http://seesar.lbl.gov/ccse/Publications/car/LowMachSNIa.pdf.

[edit] See also

[0] Durran, D.R. (1989). "Improving the Anelastic Approximation". Journal of the Atmospheric Sciences 46 (11): 1453–1461. doi:10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2. http://ams.allenpress.com/archive/1520-0469/46/11/pdf/i1520-0469-46-11-1453.pdf.

[1] Almgren, A.S.; Bell, J.B.; Rendleman, C.A.; Zingale, M. (2006). "Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics". Astrophysical Journal 637: 922–936. doi:10.1086/498426. http://seesar.lbl.gov/ccse/Publications/car/LowMachSNIa.pdf.

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