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Analytical Fluid Dynamics in Lagrangian Variables
This book is devoted to the problems of the analytical description of incompressible fluid flows in Lagrangian variables. Commonly, specialists in fluid dynamics prefer to use the Euler approach. For example, even classical textbooks by J. Batchelor, L.G. Loitsyansky, L.D. Landau, and E.M. Lifshitz do not provide the Lagrange equations of f luid dynamics. Exceptions are the fundamental work by G. Lamb and the two-volume work by N.E. Kochin, I.A. Kibel, and N.V. Rose, but even there, only a few paragraphs are devoted to Lagrangian variables. An extreme point of view on this issue was expressed by D. Kuchemann in his book “The Aerodynamic Designof Aircraft”. In his opinion, using the Lagrangian description in practice is inconvenient and there is usually no need for it. This observation needs to be commented upon. D. Kuchemann was engaged in aircraft design. Indeed, it is inconvenient to use Lagrangian variables to solve flow problems, but they turn out to be preferable to Eulerian variables in some cases when other types of flows are described. This applies, in particular, to the problems of vortex dynamics and flows with a free boundary. I am inclined to consider the Lagrange method as an additional (to the Euler method) technique for studying fluid motion. It is well known that for most exact solutions, the transition from one description method to another is simply impossible, and each method is a different mathematical reflection of the real flow. Moreover, the natural (deep) essence of fluid dynamics is related to tracing the trajectories of the motion of specific fluid particles. This type of analysis is exactly what the Lagrange method offers.
The purpose of this publication is to reflect as fully as possible on the achievements of analytical fluid dynamics in Lagrangian variables. A significant part of it consists of my original articles. Considerable attention has been paid to the intricacies of the Lagrangian description and its popularization among fluid dynamics specialists, undergraduates, graduate students, and anyone interested in prob lems of fluid mechanics. This book can be considered, among other things, a tutorial on Lagrangian fluid dynamics. In 2006, Cambridge University Press published A. Bennett’s book Lagrangian Fluid Dynamics. This was the first book published in Western countries devoted exclusively to the Lagrangian description of fluid motion. Although the title is similar, it differs from the present book in both scope and presentation. While Bennett’s book devotes considerable attention to the statistical description of fluids (more than half the book), I focus exclusively on problems of classical fluid dynamics. This publication can be a useful addition to well-known fluid dynamics courses that use only Eulerian variables. I would like to express my gratitude to Anastasiya Volodina and Alexey Krayev for translating this book.
