Dimensional Analysis Beyond the Pi Theorem

Although the Dimensional Analysis and Physical Similar is a well understood subject and the general concepts of dynamical similarity are explained in this book. Our exposition is essentially different from those available in the literature, although it follows in its general ideas that are known as Pi Theorem and there are many excellent books by the different authors are published, which one can refer to, however dimensional analysis goes way beyond Pi theorem or namely known as Buckingham's Pi Theorem. Many techniques via self similar solutions can bound these solutions to problems that seem to be intractable.§§A time-developing phenomenon is called self-similarity if the spatial distributions of its properties at various different moments of time can be obtained from one another by a similarity transformation and the fact that we identify one of the independent variables of dimension with time is nothing new from subject of Dimensional Analysis point of view. However this is where the boundary of dimensional analysis goes beyond Pi Theorem and steps into a new arena that is known as self-similarity, which has always represents progress for researchers§§In recent years there has been a surge of interest in self-similar solutions of the First and Second kind. Such solutions are not newly discovered; they had been identified and in fact so named by Zel'dovich famous Russian Mathematician in 1956, in the context of a variety of problems, such as shock waves in gas dynamics, and filtration through elasto-plastic materials.§§Self-Similarity has simplified computations and the representation of the properties of phenomena under investigation and it handles experimental data, and reduces what would be a random cloud of empirical points so as to lie on a single curve or surface, construct procedure that is known to us as self-similar where variables could be chosen in some special way.§§