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Irrational rotation on a 2-torus For a linear flow on the torus, all orbits are either periodic or dense on a subset of the n {\displaystyle n} -torus, which is a k {\displaystyle k} -torus. When the components of ω {\displaystyle \omega } are rationally independent all the orbits are dense on the whole space.
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.
In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to the body. They are named in honour of Leonhard Euler. Their general vector form is
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.
The problem has a cylindrical symmetry and can be treated in two dimensions on the orthogonal plane. Line sources and line sinks (below) are important elementary flows because they play the role of monopole for incompressible fluids (which can also be considered examples of solenoidal fields i.e. divergence free fields).
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 flow regions where vorticity is known to be important, such as wakes and boundary layers, potential flow theory is not able to provide reasonable predictions of the flow. [1] Fortunately, there are often large regions of a flow where the assumption of irrotationality is valid which is why potential flow is used for various applications.