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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 system being modelled. Some versions are described below: Incompressible flow: =. This can assume either constant density (strict incompressible) or varying density flow.
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 assumptions for the stream function equation are: The flow is incompressible and Newtonian. Coordinates are orthogonal. Flow is 2D: u 3 = ∂u 1 / ∂x 3 = ∂u 2 / ∂x 3 = 0; The first two scale factors of the coordinate system are independent of the last coordinate: ∂h 1 / ∂x 3 = ∂h 2 / ∂x 3 = 0 ...
Thus for an incompressible inviscid fluid the specific internal energy is constant along the flow lines, also in a time-dependent flow. The pressure in an incompressible flow acts like a Lagrange multiplier, being the multiplier of the incompressible constraint in the energy equation, and consequently in incompressible flows it has no ...
The following assumptions must be met for this Bernoulli equation to apply: [2]: 265 the flow must be steady, that is, the flow parameters (velocity, density, etc.) at any point cannot change with time, the flow must be incompressible—even though pressure varies, the density must remain constant along a streamline;
The assumptions of the equation are that the fluid is incompressible and Newtonian; the flow is laminar through a pipe of constant circular cross-section that is substantially longer than its diameter; and there is no acceleration of fluid in the pipe.
The assumption that a flow is incompressible is a good one and applies to most fluids travelling at most speeds. It is assumptions of this form that will help to simplify the Navier–Stokes equation into differential equations, like Euler's equation, which are easier to work with.
The velocity satisfies the continuity equation for incompressible flow: ∇ ⋅ u = 0. {\displaystyle \quad \nabla \cdot \mathbf {u} =0.} Although in principle the stream function doesn't require the use of a particular coordinate system, for convenience the description presented here uses a right-handed Cartesian coordinate system with ...