Bifurcations in Flow Patterns: Some Applications of the Qualitative Theory of Differential Equations

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.