Self-Similar Hydrodynamics with Heat Conduction

1.1 Dimensional analysis and self-similarity Dimensional and similarity theory provides one with the possibility of prior qualitativetheoretical analysis and the choice of a set for characteristic dimensionless parameters. The theory can be applied to the consideration of quite complicated phenomena and makes the processing of experiments much easier. What is more, at present, the competent setting and processing of experiments is inconceivable without taking into account dimensional and similarity reasoning. Sometimes at the initial stage of investigation of certain complicated phenomena, dimensional and similarity theory is the only possible theoretical method, though the possibilities of this method should not be overestimated. The combination of similarity theory with considerations resulting from experiments or mathematical operations can sometimes lead to significant results. Most often dimensional and similarity theory is very useful for theoretical as well for practical use. All the results obtained with the help of this theory can be obtained quite easily and without much trouble. A phenomenon is called self-similar if the spatial distributions of its properties at various moments of time can be obtained from one another by a similarity transformation. Establishing self-similarity has always represented progress for a researcher: selfsimilarity has simplified computations and the representation of the characteristics of phenomena under investigation. In handling experimental data, self-similarity has reduced what would seem to be a random cloud of empirical points so as to lie on a single curve of surface, constructed using self-similar variables chosen in some special way. Selfsimilarity enables us to reduce its partial differential equations to ordinary differential equations, which substantially simplifies the research. Therefore with the help of selfsimilar solutions researchers have attempted to find the underlying physics. Self-similar solutions also serve as standards in evaluating approximate methods for solving more complicated problems. Scaling laws, which are obtained as a result of the dimensional analysis and other methods, play an important role for understanding the underlying physics and applying them to practical systems. When constructing a full-scale system in engineering, numerical simulations will be first made in most cases. Its feasibility should be then demonstrated experimentally with a reduced-scale system. For astrophysical studies, for instance, such scaling considerations are indispensable and play a decisive role in designing laboratory experiments. Then one should know how to design such a miniature system and how to