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Bifurcations in Flow Patterns : Some Applications of the Qualitative Theory of Differential Equations in Fluid Dynamics - P.G. Bakker

Bifurcations in Flow Patterns

Some Applications of the Qualitative Theory of Differential Equations in Fluid Dynamics


Published: 31st October 1991
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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.

'This is an interesting albeit specialized book...In short, Bakker has made an important contribution to the literature of fluid mechanics and this book should be a part of the library at every engineering school as well the specialists mentioned...' Appl. Mech. Rev. 45:8 1992

Some Elements of the Qualitative Theory of Differential Equations
Phase space representation of a dynamical systemp. 1
Phase portraits near singular pointsp. 5
Topological structure of phase portraits, structural stability, bifurcationp. 9
Higher-order singularities in R[superscript 2]p. 16
Bifurcation of vector fields, unfoldingsp. 24
Center manifoldsp. 30
An approach to physical unfoldings in flow patternsp. 39
Topology of Conical Flow Patterns
Local conical stagnation point solutions in irrotational flowp. 52
Classification of conical stagnation points in conical flowsp. 59
Analytical unfoldings in conical flowsp. 76
External corner flow; a nonanalytical unfolding of a starlike nodep. 95
Topological Aspects of Steady Viscous Flows Near Plane Walls
A way to obtain local solutions of the Navier-Stokes equationsp. 123
Steady viscous flow near a plane wall, elementary singular points in the flow patternsp. 126
Higher-order singularities in the flow patternp. 132
Unfolding of the topological saddle point of the third orderp. 145
Unfolding of a topological saddle point of the fifth orderp. 163
Unfolding of a saddle point with three hyperbolic sectors in a half plane, [Tau][subscript xx] [actual symbol not reproducible] 0p. 169
Unfolding of a saddle point with two or four hyperbolic sectors in a half plane, [Tau][subscript xx] [actual symbol not reproducible] 0p. 178
Viscous flow near a circular cylinder at low Reynolds numbersp. 194
Index of subjectsp. 205
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792314288
ISBN-10: 079231428X
Series: Nonlinear Topics in the Mathematical Sciences
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 211
Published: 31st October 1991
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.5 x 15.5  x 1.4
Weight (kg): 1.1