# Bifurcations in Flow Patterns

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Oktober 1991

## Beschreibung

### Beschreibung

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.### Inhaltsverzeichnis

I Some Elements Of The Qualitative Theory Of Differential Equations.- 1. Phase space representation of a dynamical system.

- 2. Phase portraits near singular points.

- 3. Topological structure of phase portraits, structural stability, bifurcation.

- 3.1. Topological structure of phase portraits.

- 3.2. Structural stability, bifurcation.

- 3.3. Saddle connections.

- 3.4. Multiple limit cycle.

- 4. Higher-order singularities in R2.

- 4.1. Higher-order singular points with one zero eigenvalue.

- 4.2. Higher-order singular points having both eigenvalues zero; p=0.

- 5. Bifurcation of vector fields, unfoldings.

- 6. Center manifolds.

- 7. An approach to physical unfoldings in flow patterns.

- 8. References.- II Topology Of Conical Flow Patterns.

- 1. Introduction.

- 1.1. Concepts and definitions.

- 1.2. A survey of conical flow theory.

- 1.3. Conical streamlines, conical stagnation points.

- 1.4. Transition phenomena in conical flow patterns.

- 2. Local conical stagnation point solutions in irrotational flow.

- 2.1. Conical potential equation.

- 2.2. Conical stagnation point solutions.

- 3. Classification of conical stagnation points in conical flows.

- 3.1. First-order conical stagnation points.

- 3.2. Irrotational attachments and separations.

- 3.3. Higher-order conical stagnation points.

- 4. Analytical unfoldings in conical flows.

- 4.1. Bifurcation parameters.

- 4.2. Approximate solutions near regular points.

- 4.3. Saddle-node bifurcation.

- 4.4. Bifurcation of topological saddle point.

- 4.5. Bifurcation of topological node.

- 5. External corner flow; a nonanalytical unfolding of a starlike node.

- 5.1. The flow around an external corner.

- 5.2. Boundary conditions and bifurcation modes.

- 5.3. Bifurcations of the starlike node.

- 5.4. Symmetrical external corners.

- 5.5. Transition of oblique saddle to starlike node.

- 6. References.- III Topological Aspects Of Steady Viscous Flows Near Plane Walls.

- 1. A way to obtain local solutions of the Navier-Stokes equations.

- 2. Steady viscous flow near a plane wall, elementary singular points in the flow patterns.

- 2.1. Approximate solutions near a plane wall.

- 2.2. Elementary singular points located at the wall (on-wall singularities).

- 2.3. Elementary singular points in the flow (free singularities).

- 3. Higher-order singularities in the flow pattern.

- 3.1. Higher-order singular points in the flow field.

- 3.2. Higher-order singular points on the wall.

- 4. Unfolding of the topological saddle point of the third order.

- 4.1. Local phase portraits of the unfolding.

- 4.2.Incipient bubble separation.

- 4.3. Separation along a moving wall.

- 4.4. MRS-criterion for separations in flows along a moving wall.

- 4.5. Unfolding model for moving wall separations.

- 5. Unfolding of a topological saddle point of the fifth order.

- 5.1. Description of the unfolding.

- 5.2. Bubble capturing by a secondary separation.

- 6. Unfolding of a saddle point with three hyperbolic sectors in a half plane, ?xx? 0.

- 6.1.Universal physical unfolding.

- 6.2. Movement principle.

- 6.3.Bifurcation sets, flow patterns.

- 7. Unfolding of a saddle point with two or four hyperbolic sectors in a half plane, ?xx? 0.

- 7.1. Universal physical unfolding.

- 7.2. Determination of codimension.

- 7.3. Neighbouring singular points, local bifurcation sets Bsand Bc.

- 7.4. Flow patterns and global bifurcation sets Bg1 and Bg2.

- 8. Viscous flow near a circular cylinder at low Reynolds numbers.

- 8.1. Description of flow topology.

- 8.2. Symmetrical bifurcations.

- 8.3. Asymmetrical bifurcations, transition scenario's.

- 9. References.- Index of subjects.

### Pressestimmen

'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 1992EAN: 9780792314288

ISBN: 079231428X

Untertitel: Some Applications of the Qualitative Theory of Differential Equations in Fluid Dynamics.
1991. Auflage.
Book.
Sprache: Englisch.

Verlag: Springer

Erscheinungsdatum: Oktober 1991

Seitenanzahl: 228 Seiten

Format: gebunden

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