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Incompressible flow does not imply that the fluid itself is incompressible. It is shown in the derivation below that under the right conditions even the flow of compressible fluids can, to a good approximation, be modelled as incompressible flow.
The liquid is modeled as an incompressible fluid (i.e. with constant density), ... Oseen's approximation sums up to neglecting the term non-linear in ...
In the steady incompressible case the mass equation is simply: =, that is the mass conservation for a steady incompressible flow states that the density along a streamline is constant. Then the Euler momentum equation in the steady incompressible case becomes:
The incompressible flow assumption typically holds well with all fluids at low Mach numbers (say up to about Mach 0.3), such as for modelling air winds at normal temperatures. [16] the incompressible Navier–Stokes equations are best visualized by dividing for the density: [17]
The upstream flow is uniform and has no vorticity. The flow is inviscid, incompressible and has constant mass density ρ. The flow therefore remains without vorticity, or is said to be irrotational, with ∇ × V = 0 everywhere. Being irrotational, there must exist a velocity potential φ: =.
In fluid dynamics, the Boussinesq approximation (pronounced, named for Joseph Valentin Boussinesq) is used in the field of buoyancy-driven flow (also known as natural convection). It ignores density differences except where they appear in terms multiplied by g , the acceleration due to gravity .
To obtain the equations of motion for incompressible flow, it is assumed that the density, , is a constant. Furthermore, occasionally one might consider the unsteady Stokes equations, in which the term ρ ∂ u ∂ t {\displaystyle \rho {\frac {\partial \mathbf {u} }{\partial t}}} is added to the left hand side of the momentum balance equation.
In computational fluid dynamics, the projection method, also called Chorin's projection method, is an effective means of numerically solving time-dependent incompressible fluid-flow problems. It was originally introduced by Alexandre Chorin in 1967 [1] [2] as an efficient means of solving the incompressible Navier-Stokes equations.