In
fluid mechanicsFluid mechanics is the study of how fluids move and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, the study of fluids in motion. It is a branch of continuum mechanics, a subject which models matter without using the...
or more generally
continuum mechanicsContinuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and mechanical behavior of materials modeled as a continuum, i.e., solids and fluids...
, an
incompressible flow is
solidMatter is generally found in three different forms: solid, liquid, and gas . The solid state of matter is characterized by a distinct structural rigidity and resistance to deformation . Most solids have high values both of Young's modulus and of the shear modulus of elasticity...
or
fluidA fluid is a substance that continually deforms under an applied shear stress. All gases are fluids, but not all liquids are fluids. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
flow in which the
divergenceIn 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 scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the...
of velocity is zero. This is more precisely termed
isochoricAn isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a thermodynamic process during which the volume of the closed system undergoing such process remains constant...
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,
,
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.
In some fields, a measure of the incompressibility of a flow is the change in density as a result of the pressure variations.
In
fluid mechanicsFluid mechanics is the study of how fluids move and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, the study of fluids in motion. It is a branch of continuum mechanics, a subject which models matter without using the...
or more generally
continuum mechanicsContinuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and mechanical behavior of materials modeled as a continuum, i.e., solids and fluids...
, an
incompressible flow is
solidMatter is generally found in three different forms: solid, liquid, and gas . The solid state of matter is characterized by a distinct structural rigidity and resistance to deformation . Most solids have high values both of Young's modulus and of the shear modulus of elasticity...
or
fluidA fluid is a substance that continually deforms under an applied shear stress. All gases are fluids, but not all liquids are fluids. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
flow in which the
divergenceIn 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 scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the...
of velocity is zero. This is more precisely termed
isochoricAn isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a thermodynamic process during which the volume of the closed system undergoing such process remains constant...
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,
,
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 factorThe compressibility factor is a useful thermodynamic property for modifying the ideal gas law to account for the real gas behaviour. In general, deviations from ideal behavior become more significant the closer a gas is to a phase change, the lower the temperature or the larger the pressure...
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
divergenceIn 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 scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the...
, 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.
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.
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: . This can assume either constant density (strict incompressible) or varying density flow. The varying density set accepts solutions involving small perturbations in density
The density of a material is defined as its mass per unit volume. The symbol of density is ρ .- Formula :Mathematically:where: is the density, is the mass, is the volume....
, pressure and/or temperature fields, and can allow for pressure stratificationAtmospheric stratification is the division of the atmosphere into distinct layers, each with specific characteristics such as temperature or composition....
in the domain.
- Anelastic flow: . Principally used in the field of atmospheric sciences
Atmospheric sciences is an umbrella term for the study of the atmosphere, its processes, the effects other systems have on the atmosphere, and the effects of the atmosphere on these other systems. Meteorology includes atmospheric chemistry and atmospheric physics with a major focus on weather...
, 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.
- Low Mach-number flow / Pseudo-incompressibility: . The low Mach-number
Mach number is the speed of an object moving through air, or any fluid substance, divided by the speed of sound as it is in that substance...
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.
These methods make differing assumptions about the flow, but all take into account the general form of the constraint for general flow dependent functions and .
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
The projection method is an effective means of numerically solving time-dependent incompressible fluid-flow problems. It was originally introduced by Alexandre Chorin and independently by Roger Temam as an efficient means of solving the incompressible Navier-Stokes equations...
(both approximate and exact)
- Artificial compressibility technique (approximate)
- Compressibility pre-conditioning