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Von Kármán swirling flow is a flow created by a uniformly rotating infinitely long plane disk, named after Theodore von Kármán who solved the problem in 1921. [1] The rotating disk acts as a fluid pump and is used as a model for centrifugal fans or compressors.
Example of a measure that is invariant under the action of the (unrotated) baker's map: an invariant measure. Applying the baker's map to this image always results in exactly the same image. In dynamical systems theory, the baker's map is a chaotic map from the unit square into itself.
Here the fluid is subject to the Taylor-Proudman theorem which says that small motions will tend to produce purely two-dimensional perturbations to the overall rotational flow. However, in this case the effects of rotation and viscosity are usually characterized by the Ekman number and the Rossby number rather than by the Taylor number.
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.
For example, in the laminar flow within a pipe with constant cross section, all particles travel parallel to the axis of the pipe; but faster near that axis, and practically stationary next to the walls. The vorticity will be zero on the axis, and maximum near the walls, where the shear is largest.
The term (ω ∙ ∇) u on the right-hand side describes the stretching or tilting of vorticity due to the flow velocity gradients. Note that (ω ∙ ∇) u is a vector quantity, as ω ∙ ∇ is a scalar differential operator, while ∇u is a nine-element tensor quantity. The term ω(∇ ∙ u) describes stretching of vorticity due to flow ...
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time.
Potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications.