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In fluid mechanics, or more generally continuum mechanics, incompressible flow (isochoric flow) refers to a flow in which the material density of each fluid parcel — an infinitesimal volume that moves with the flow velocity — is time-invariant.
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 ]
In the case of an incompressible fluid, Dρ / Dt = 0 (the density following the path of a fluid element is constant) and the equation reduces to: = which is in fact a statement of the conservation of volume.
For constant fluid density, the incompressible equations can be written as a quasilinear advection equation for the fluid velocity together with an elliptic Poisson's equation for the pressure. On the other hand, the compressible Euler equations form a quasilinear hyperbolic system of conservation equations .
This extra equation is the continuity equation for incompressible fluids that describes the conservation of mass of the fluid: ∇ ⋅ v = 0. {\displaystyle \nabla \cdot \mathbf {v} =0.} Due to this last property, the solutions for the Navier–Stokes equations are searched in the set of solenoidal (" divergence -free") functions.
In most flows of liquids, and of gases at low Mach number, the density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. Therefore, the fluid can be considered to be incompressible, and these flows are called incompressible flows. Bernoulli performed his experiments on liquids, so his equation in ...
In fluid dynamics, the Oseen equations (or Oseen flow) describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration. [1]
where is the fluid density and the fluid velocity. To obtain the equations of motion for incompressible flow, it is assumed that the density, ρ {\displaystyle \rho } , 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 ...