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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.
The self-similar solution tries to describe the flow when the shock wave has moved through a distance that is extremely large when compared to the size of the explosive. At these large distances, the information about the size and duration of the explosion will be forgotten; only the energy released E {\displaystyle E} will have influence on ...
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.
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.
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 ...
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]
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A Ricci soliton (,) yields a self-similar solution to the Ricci flow equation = (). In particular, letting ():=and integrating the time-dependent vector field ():= to give a family of diffeomorphisms , with the identity, yields a Ricci flow solution (,) by taking