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An illustration of Stokes' theorem, with surface Σ, its boundary ∂Σ and the normal vector n.The direction of positive circulation of the bounding contour ∂Σ, and the direction n of positive flux through the surface Σ, are related by a right-hand-rule (i.e., the right hand the fingers circulate along ∂Σ and the thumb is directed along n).
The cross differentiated Navier–Stokes equation becomes two 0 = 0 equations and one meaningful equation. The remaining component ψ 3 = ψ is called the stream function. The equation for ψ can simplify since a variety of quantities will now equal zero, for example:
By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, [4] = = = Here, the closed integration path ∂S is the boundary or perimeter of an open surface S , whose infinitesimal element normal d S = n dS is oriented according to the right-hand rule .
The right-hand side of the equation is often used to formulate integral laws; the left-hand side then leads to equivalent differential formulations (see below). The theorem is often used in situations where Ω {\displaystyle \Omega } is an embedded oriented submanifold of some bigger manifold, often R k {\displaystyle \mathbb {R} ^{k}} , on ...
The Navier–Stokes equations (/ n æ v ˈ j eɪ s t oʊ k s / nav-YAY STOHKS) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades ...
In fluid dynamics, Stokes' law gives the frictional force – also called drag force – exerted on spherical objects moving at very small Reynolds numbers in a viscous fluid. [1] It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations. [2]
Discretization of the Navier–Stokes equations of fluid dynamics is a reformulation of the equations in such a way that they can be applied to computational fluid dynamics. Several methods of discretization can be applied: Finite volume method; Finite elements method; Finite difference method
In mathematics, the Navier–Stokes equations are a system of nonlinear partial differential equations for abstract vector fields of any size. In physics and engineering, they are a system of equations that model the motion of liquids or non-rarefied gases (in which the mean free path is short enough so that it can be thought of as a continuum mean instead of a collection of particles) using ...