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The normal self-similar solution is also referred to as a self-similar solution of the first kind, since another type of self-similar exists for finite-sized problems, which cannot be derived from dimensional analysis, known as a self-similar solution of the second kind.
Self-similar processes are stochastic processes satisfying a mathematically precise version of the self-similarity property. Several related properties have this name, and some are defined here. A self-similar phenomenon behaves the same when viewed at different degrees of magnification, or different scales on a dimension.
Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. [2] Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to ...
Originally constructed as a curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński but appeared as a decorative pattern many centuries before the work of Sierpiński.
The self-similar solution exists because the equations and the boundary conditions are invariant under the transformation ,,, where is any positive constant. He introduced the self-similar variables Developing Blasius boundary layer (not to scale). The velocity profile ′ is shown in red at selected positions along the plate.
Self-Similar solution. The problem on the whole is similar to the one dimensional heat conduction problem. Hence a self-similar variable can be introduced [4]
Many of these systems evolve in a self-similar fashion in the sense that data obtained from the snapshot at any fixed time is similar to the respective data taken from the snapshot of any earlier or later time. That is, the system is similar to itself at different times. The litmus test of such self-similarity is provided by the dynamic scaling.
The self-similar solution to be described corresponds to the region , that is to say, the shock wave has travelled enough to forget about the initial condition. Since the shock wave in the self-similar region is strong, the pressure behind the wave p 1 {\displaystyle p_{1}} is very large in comparison with the pressure ahead of the wave p 0 ...
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