Same fluid can behave as compressible and incompressible depending upon flow conditions. Flows in which variations in density are negligible are termed as . “Area de Mecanica de Fluidos. Centro Politecnico Superior. continuous interpolations. both for compressible and incompressible flows. A comparative study of. Departamento de Mecánica de Fluidos, Centro Politécnico Superior, C/Maria de Luna 3, . A unified approach to compressible and incompressible flows.
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In fluid mechanics or more generally continuum mechanicsincompressible flow isochoric flow refers to a flow in which the material density is constant within a fluid parcel —an infinitesimal volume that moves with the flow velocity.
An equivalent statement that implies incompressibility is that the divergence of the flow velocity is zero see the derivation below, which illustrates why these conditions are equivalent. Incompressible flow does not imply that the fluid itself is incompressible. It is shown in the derivation below that under the right conditions even compressible fluids can — to a good approximation — be modelled as an incompressible flow. Incompressible flow implies that the density remains constant within a parcel of fluid that moves with the flow velocity.
Mathematically, this constraint implies that the material derivative discussed below of the density must vanish to ensure incompressible flow. Before introducing this constraint, we must apply the conservation of mass to generate the necessary relations. The conservation of mass requires that the time derivative of the mass inside a control volume be equal to the mass flux, Jacross its boundaries.
Mathematically, we can represent this constraint in terms of a surface integral:. The negative sign in the above expression ensures that outward flow results in a decrease in the mass with respect to time, using the convention that the surface area vector points comoresible. Now, using the divergence theorem we can derive the relationship between the flux and the partial time derivative of the density:. The partial derivative of the density with incomoresible to time need not vanish to ensure incompressible flow.
When we speak of the partial derivative of the density with respect to time, we refer to this rate of change within a control volume of fixed position.
By letting the partial time derivative of the density be non-zero, we are not restricting ourselves to incompressible fluidsbecause the density can change as observed from a fixed position as fluid flows through the control volume.
This approach maintains generality, and not requiring that the partial time derivative of the density vanish illustrates that compressible fluids can still undergo incompressible flow. What interests us is the change in density of a control volume that moves along with the flow velocity, u.
The flux is related to the flow velocity through the following function:. The previous relation where we have used the appropriate product rule is known as the continuity equation.
Now, we need the following relation about the total derivative of the density where we apply the chain rule:. So if we choose a control volume that is moving at the same rate as the fluid i. A change in the density over lfuido would imply that the fluid had either compressed or expanded or that the mass contained in our constant volume, dVhad changedwhich we have lfuido.
We must then require that the material derivative of the density vanishes, and equivalently for non-zero density so must the divergence of the flow velocity:. And so beginning with the conservation of mass and the constraint that the density within a moving volume of fluid remains constant, it has been shown that an equivalent condition required compresuble incompressible flow is that the divergence of the flow velocity vanishes.
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. An incompressible flow is described by a solenoidal flow velocity field.
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But a solenoidal field, besides having a zero divergencealso has the additional connotation of having non-zero curl i. Otherwise, if an incompressible flow also has comprwsible curl of zero, so that it is also irrotationalthen the flow velocity field is actually Laplacian. Thus fluivo we follow a material element, its mass density remains constant. Note that the material derivative consists of two terms. This term is also known as the unsteady term. This is the advection term convection term for scalar field.
For a flow to be incompressible the sum of these terms should be zero. On the other hand, a homogeneous, incompressible material is rluido that has constant density throughout. 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. 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. In fluid dynamics, a flow is considered incompressible if the divergence of the flow velocity is zero. However, related formulations can sometimes be used, depending on the flow uncompresible being modelled.
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Some versions are described below:. The stringent nature of the incompressible flow equations means that specific mathematical techniques have been devised to solve them.
Some of these methods include:.
From Wikipedia, the free encyclopedia. For the property of vector fields, see Solenoidal vector field. For the topological property, see Incompressible surface. Journal of the Atmospheric Sciences.