We study the detailed process of bifurcation in the flow’s topological structure for a two-dimensional (2-D) incompressible flow subject to no-slip boundary conditions and its connection with boundary-layer separation. The boundary-layer separation theory of M. Ghil, T. Ma and S. Wang, based on the structural-bifurcation concept, is translated into vorticity form. The vorticity formulation of the theory shows that structural bifurcation occurs whenever a degenerate singular point for the vorticity appears on the boundary; this singular point is characterized by nonzero tangential second-order derivative and nonzero time derivative of the vorticity. Furthermore, we prove the presence of an adverse pressure gradient at the critical point, due to reversal in the direction of the pressure force with respect to the basic shear flow at this point. A numerical example of 2-D driven-cavity flow, governed by the Navier Stokes equations, is presented; boundary-layer separation occurs, the bifurcation criterion is satisfied, and an adverse pressure gradient is shown to be present.
If an (n + 2)-dimensional Lorentzian manifold is indecomposable, but non-irreducible, then its holonomy algebra is contained
in the parabolic algebra (R⊕so(n)).Rn. We show that its projection onto so(n) is the holonomy algebra of a Riemannian manifold.
This leads to a classification of Lorentzian holonomy groups and implies that the holonomy group of an indecomposable Lorentzian
spin manifold with parallel spinor equals to G . Rn where G is a product of SU(p), Sp(q), G2 or Spin(7).