This book formulates the kinematical conservation laws (KCL), analyses them and presents their applications to various problems in physics. Finally, it addresses one of the most challenging problems in fluid dynamics: finding successive positions of a curved shock front. The topics discussed are the outcome of collaborative work that was carried out mainly at the Indian Institute of Science, Bengaluru, India. The theory presented in the book is supported by referring to extensive numerical results. The book is organised into ten chapters. Chapters 1-4 offer a summary of and briefly discuss the theory of hyperbolic partial differential equations and conservation laws. Formulation of equations of a weakly nonlinear wavefront and those of a shock front are briefly explained in Chapter 5, while Chapter 6 addresses KCL theory in space of arbitrary dimensions. The remaining chapters examine various analyses and applications of KCL equations ending in the ultimate goal-propagation of a three-dimensional curved shock front and formation, propagation and interaction of kink lines on it.

Autorentext

PHOOLAN PRASAD worked in the Department of Mathematics, Indian Institute of Science (IISc), Bengaluru, India. With over 52 years of experience, he is an Indian mathematician who specialised in partial differential equations and fluid mechanics. He held the distinguished Chair of Mysore Sales International Limited (MSIL) in the area of Physical Science, was the chairman of the department and a professor of super-time-scales before his retirement. Later, he was made honorary professor at the Department of Atomic Energy (DAE)-Raja Ramanna Fellow; Senior Scientist at the Indian National Science Academy (INSA); and Senior Scientist Platinum Jubilee Fellow at the National Academy of Sciences (NASI), India. In 1983, he was awarded the Shanti Swarup Bhatnagar Prize for Science and Technology-India's highest science award in the field of mathematical sciences. He is fellow of the National Academy of Sciences, India; the Indian Academy of Sciences and the Indian National Science Academy. Professor Prasad started his career as a research fellow at the Department of Applied Mathematics at IISc, in 1965. Thereafter, he became a professor at IISc in 1977. At the same time, he was a postdoctoral fellow at the University of Leeds in the U.K. (1970-72), the Alexander von Humboldt Fellow (1980-81) and visiting professor at many other universities including Cambridge University. He has also played an important role in the development of quality education in the field of mathematics from school education to research. He succeeded in assessing the basic properties of the equations of various physical phenomena, generalised these mathematical properties and then developed general theories to explain new results in the field of nonlinear waves. He gave a proof of the existence of a new type of wave on the interface of a clear liquid and a mixture in a sedimentation process, which was verified experimentally much later.

Inhalt

Introduction.- Single first-order PDE.- Single conservation law.- Systems of hyperbolic PDE and conservation laws.- Propagation of nonlinear wave and shock.- Kinematical conservation laws (KCL).- Conservation forms of energy transport equations.- 2-D KCL, WNLRT and SRT.- 2-D WNLRT and SRT - some applications.- 3-D WNLRT and SRT: Theory and Applications.- Appendix and Notations.