Bifurcation Analysis of Fluid Flows


Book Description

A better understanding of the mechanisms leading a fluid system to exhibit turbulent behavior is one of the grand challenges of the physical and mathematical sciences. Over the last few decades, numerical bifurcation methods have been extended and applied to a number of flow problems to identify critical conditions for fluid instabilities to occur. This book provides a state-of-the-art account of these numerical methods, with much attention to modern linear systems solvers and generalized eigenvalue solvers. These methods also have a broad applicability in industrial, environmental and astrophysical flows. The book is a must-have reference for anyone working in scientific fields where fluid flow instabilities play a role. Exercises at the end of each chapter and Python code for the bifurcation analysis of canonical fluid flow problems provide practice material to get to grips with the methods and concepts presented in the book.




Bifurcations in Flow Patterns


Book Description

The main idea of the present study is to demonstrate that the qualitative theory of diffe rential equations, when applied to problems in fluid-and gasdynamics, will contribute to the understanding of qualitative aspects of fluid flows, in particular those concerned with geometrical properties of flow fields such as shape and stability of its streamline patterns. It is obvious that insight into the qualitative structure of flow fields is of great importance and appears as an ultimate aim of flow research. Qualitative insight fashions our know ledge and serves as a good guide for further quantitative investigations. Moreover, quali tative information can become very useful, especially when it is applied in close corres pondence with numerical methods, in order to interpret and value numerical results. A qualitative analysis may be crucial for the investigation of the flow in the neighbourhood of singularities where a numerical method is not reliable anymore due to discretisation er rors being unacceptable. Up till now, familiar research methods -frequently based on rigorous analyses, careful nu merical procedures and sophisticated experimental techniques -have increased considera bly our qualitative knowledge of flows, albeit that the information is often obtained indirectly by a process of a careful but cumbersome examination of quantitative data. In the past decade, new methods are under development that yield the qualitative infor mation more directly. These methods, make use of the knowledge available in the qualitative theory of differen tial equations and in the theory of bifurcations.




Bifurcation Theory


Book Description

In the past three decades, bifurcation theory has matured into a well-established and vibrant branch of mathematics. This book gives a unified presentation in an abstract setting of the main theorems in bifurcation theory, as well as more recent and lesser known results. It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces, and shows how to apply the theory to problems involving partial differential equations. In addition to existence, qualitative properties such as stability and nodal structure of bifurcating solutions are treated in depth. This volume will serve as an important reference for mathematicians, physicists, and theoretically-inclined engineers working in bifurcation theory and its applications to partial differential equations.




Global Bifurcation in Variational Inequalities


Book Description

An up-to-date and unified treatment of bifurcation theory for variational inequalities in reflexive spaces and the use of the theory in a variety of applications, such as: obstacle problems from elasticity theory, unilateral problems; torsion problems; equations from fluid mechanics and quasilinear elliptic partial differential equations. The tools employed are those of modern nonlinear analysis. Accessible to graduate students and researchers who work in nonlinear analysis, nonlinear partial differential equations, and additional research disciplines that use nonlinear mathematics.




Stability Of Gyroscopic Systems


Book Description

The motion of mechanical systems undergoing rotation about a fixed axis has been the subject of extensive studies over a few centuries. These systems are generally subject to gyroscopic forces which are associated with coriolis accelerations or mass transport and render complex dynamics.The unifying theme among topics presented in this book is the gyroscopic nature of the system equations of motion. The book represents comprehensive and detailed reviews of the state of art in four diverse application areas: flow-induced oscillations in structures, oscillations in rotating systems or rotor dynamics, dynamics of axially moving material systems, and dynamics of gyroelastic systems. The book also includes a chapter on dynamics of repetitive structures. These systems feature spatial periodicity and are generally subject to considerable gyroscopic forces. “Gyroelastic systems” and “repetitive structures” are the topics with very recent origins and are still in their infancies compared to the other examples represented in this book. Thus, the contributions on gyroelastic systems and repetitive structures are limited to only modeling, localization and linear stability analysis results.This book covers many important aspects of recent developments in various types of gyroscopic systems. Thus, at last, a comprehensive book is made available to serve as a supplement and resource for any graduate level course on elastic gyroscopic systems, as well as for a course covering the stability of mechanical systems. Moreover, the inclusion of an up-to-date bibliography attached to each chapter will make this book an invaluable text for professional reference.




Practical Bifurcation and Stability Analysis


Book Description

Probably the first book to describe computational methods for numerically computing steady state and Hopf bifurcations. Requiring only a basic knowledge of calculus, and using detailed examples, problems, and figures, this is an ideal textbook for graduate students.




Biomechanical Systems


Book Description

Because of developments in powerful computer technology, computational techniques, advances in a wide spectrum of diverse technologies, and other advances coupled with cross disciplinary pursuits between technology and its greatly significant applied implications in human body processes, the field of biomechanics is evolving as a broadly significant area. The four volumes of Biomechanical Systems: Techniques and Applications presents the advances in widely diverse areas with significant implications for human betterment that occur continuously at a high rate. These include dynamics of musculo-skeletal systems; mechanics of hard and soft tissues; mechanics of muscle; mechanics of bone remodeling; mechanics of implant-tissue interfaces; cardiovascular and respiratory biomechanics; mechanics of blood flow, air flow, flow-prosthesis interfaces; mechanics of impact; dynamics of man machine interaction; and numerous other areas. The great breadth and depth of the field of biomechanics on the international scene requires at least four volumes for adequate treatment. These four volumes constitute a well integrated set that can be utilized as individual volumes. They provide a substantively significant and rather comprehensive, in-depth treatment of biomechanic systems and techniques that is most surely unique on the international scene.




Imperfect Bifurcation in Structures and Materials


Book Description

Most physical systems lose or gain stability through bifurcation behavior. This book explains a series of experimentally found bifurcation phenomena by means of the methods of static bifurcation theory.




Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems


Book Description

The Institute for Mathematics and its Applications (IMA) devoted its 1997-1998 program to Emerging Applications of Dynamical Systems. Dynamical systems theory and related numerical algorithms provide powerful tools for studying the solution behavior of differential equations and mappings. In the past 25 years computational methods have been developed for calculating fixed points, limit cycles, and bifurcation points. A remaining challenge is to develop robust methods for calculating more complicated objects, such as higher- codimension bifurcations of fixed points, periodic orbits, and connecting orbits, as well as the calcuation of invariant manifolds. Another challenge is to extend the applicability of algorithms to the very large systems that result from discretizing partial differential equations. Even the calculation of steady states and their linear stability can be prohibitively expensive for large systems (e.g. 10_3- -10_6 equations) if attempted by simple direct methods. Several of the papers in this volume treat computational methods for low and high dimensional systems and, in some cases, their incorporation into software packages. A few papers treat fundamental theoretical problems, including smooth factorization of matrices, self -organized criticality, and unfolding of singular heteroclinic cycles. Other papers treat applications of dynamical systems computations in various scientific fields, such as biology, chemical engineering, fluid mechanics, and mechanical engineering.




Elements of Applied Bifurcation Theory


Book Description

Providing readers with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems, the focus here is on efficient numerical implementations of the developed techniques. The book is designed for advanced undergraduates or graduates in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the first while updating the context to incorporate recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.