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In fluid mechanics, a two-dimensional flow is a form of fluid flow where the flow velocity at every point is parallel to a fixed plane. The velocity at any point on a ...
The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange in 1781, [1] is defined for incompressible (divergence-free), two-dimensional flows. The Stokes stream function , named after George Gabriel Stokes , [ 2 ] is defined for incompressible, three-dimensional flows with axisymmetry .
The first two scale factors of the coordinate system are independent of the last coordinate: ∂h 1 / ∂x 3 = ∂h 2 / ∂x 3 = 0, otherwise extra terms appear. The stream function has some useful properties: Since −∇ 2 ψ = ∇ × (∇ × ψ) = ∇ × u, the vorticity of the flow is just the negative of the Laplacian of ...
Flux F through a surface, dS is the differential vector area element, n is the unit normal to the surface. Left: No flux passes in the surface, the maximum amount flows normal to the surface.
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
The one-dimensional (1-D) Saint-Venant equations were derived by Adhémar Jean Claude Barré de Saint-Venant, and are commonly used to model transient open-channel flow and surface runoff. They can be viewed as a contraction of the two-dimensional (2-D) shallow-water equations, which are also known as the two-dimensional Saint-Venant equations.
Rayleigh's equation only concerns two-dimensional perturbations to the flow. From Squire's theorem it follows that the two-dimensional perturbations are less stable than three-dimensional perturbations. Kelvin's cat's eye pattern of streamlines near a critical layer.
Shown is a sphere in Stokes flow, at very low Reynolds number. Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, [1] is a type of fluid flow where advective inertial forces are small compared with viscous forces. [2] The Reynolds number is low, i.e. . This is a typical situation in flows where the ...