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Stokes boundary layer due to the sinusoidal oscillation of the far-field flow velocity. The horizontal velocity is the blue line, and the corresponding horizontal particle excursions are the red dots.
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).
In this notation, Stokes' theorem reads as = . In finite element analysis, the first stage is often the approximation of the domain of interest by a triangulation, T. For example, a curve would be approximated as a union of straight line segments; a surface would be approximated by a union of triangles, whose edges are straight line segments ...
It is a special case of the general Stokes theorem (with =) once we identify a vector field with a 1-form using the metric on Euclidean 3-space. The curve of the line integral, ∂ Σ {\displaystyle \partial \Sigma } , must have positive orientation , meaning that ∂ Σ {\displaystyle \partial \Sigma } points counterclockwise when the surface ...
The derivation of the Navier–Stokes equations as well as their application and formulation for different families of fluids, is an important exercise in fluid dynamics with applications in mechanical engineering, physics, chemistry, heat transfer, and electrical engineering.
The concept of a triangulation may also be generalized somewhat to subdivisions into shapes related to triangles. In particular, a pseudotriangulation of a point set is a partition of the convex hull of the points into pseudotriangles—polygons that, like triangles, have exactly three convex vertices. As in point set triangulations ...
The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.
In order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation [20] + + = multiply it for a test function , defined in a suitable space , and integrate both members with respect to the domain : [20] + + = Counter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem ...
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