A finite time singularity of the boundary layer equations of natural convection

1985 ◽  
Vol 36 (6) ◽  
pp. 905-911 ◽  
Author(s):  
Graham Wilks ◽  
Roland Hunt
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Finite Time ◽  
Time Singularity ◽  
Boundary Layer Equations ◽  
Finite Time Singularity

scholarly journals The unsteady flow due to an impulsively rotated sphere

2015 ◽  
Vol 471 (2181) ◽  
pp. 20150299 ◽  
Author(s):  
Sophie A. W. Calabretto ◽  
Benjamin Levy ◽  
James P. Denier ◽  
Trent W. Mattner
Keyword(s):  
Boundary Layer ◽  
Angular Momentum ◽  
Classical Problem ◽  
Vortex Pair ◽  
Collision Dynamics ◽  
Time Singularity ◽  
Viscous Boundary Layer ◽  
Boundary Layer Equations ◽  
Viscous Boundary ◽  
Finite Time Singularity

We consider the flow induced by a sphere, contained in an otherwise quiescent body of fluid, that is suddenly imparted with angular momentum. This classical problem is known to exhibit a finite-time singularity in the boundary-layer equations, due to the viscous boundary layer, induced by the sudden rotation, colliding at the sphere's equator. We consider this flow from the perspective of the post-collision dynamics, showing that the collision gives rises to a radial jet headed by a swirling toroidal starting vortex pair. The starting vortex propagates away from the sphere and, in doing so, loses angular momentum. The jet, in turn, develops an absolute instability which propagates back towards the sphere's equator. The starting vortex pair detaches from the jet and expands as a coherent (non-swirling) toroidal vortex pair. We also present results of some new experiments which show good qualitative agreement with our computational results.

scholarly journals Spin-up of stratified rotating flows at large Schmidt number: experiment and theory

1999 ◽  
Vol 389 ◽  
pp. 169-207 ◽  
Author(s):  
R. E. HEWITT ◽  
P. A. DAVIES ◽  
P. W. DUCK ◽  
M. R. FOSTER
Keyword(s):  
Boundary Layer ◽  
Finite Time ◽  
Schmidt Number ◽  
Quantitative Agreement ◽  
Time Singularity ◽  
Boundary Layer Problem ◽  
Similarity Type ◽  
Rotation Rates ◽  
Finite Time Singularity ◽  
Layer Problem

We consider the nonlinear spin-up/down of a rotating stratified fluid in a conical container. An analysis of axisymmetric similarity-type solutions to the relevant boundary-layer problem, Duck, Foster & Hewitt (1997), has revealed three types of behaviour for this geometry. In general, the boundary layer evolves to either a steady state, or a gradually thickening boundary layer, or a finite-time singularity depending on the Schmidt number, the ratio of initial to final rotation rates, and the relative importance of rotation and stratification.In this paper we emphasize the experimental aspects of an investigation into the initial readjustment process. We make comparisons with the previously presented boundary-layer theory, showing good quantitative agreement for positive changes in the rotation rate of the container (relative to the initial rotation sense). The boundary-layer analysis is shown to be less successful in predicting the flow evolution for nonlinear decelerations of the container. We discuss the qualitative features of the spin-down experiments, which, in general, are dominated by non-axisymmetric effects. The experiments are conducted using salt-stratified solutions, which have a Schmidt number of approximately 700.The latter sections of the paper present some stability results for the steady boundary-layer states. A high degree of non-uniqueness is possible for the system of steady governing equations; however the experimental results are repeatable and stability calculations suggest that ‘higher branch’ solutions are, in general, unstable. The eigenvalue spectrum arising from the linear stability analysis is shown to have both continuous and discrete components. Some analytical results concerning the continuous spectrum are presented in an appendix.A brief appendix completes the previous analysis of Duck, Foster & Hewitt (1997), presenting numerical evidence of a different form of finite-time singularity available for a more general boundary-layer problem.

scholarly journals Finite-time singularity versus global regularity for hyper-viscous Hamilton–Jacobi-like equations

Nonlinearity ◽  
2003 ◽  
Vol 16 (6) ◽  
pp. 1967-1989 ◽  
Author(s):  
Hamid Bellout ◽  
Said Benachour ◽  
Edriss S Titi
Keyword(s):  
Finite Time ◽  
Global Regularity ◽  
Time Singularity ◽  
Finite Time Singularity

scholarly journals Interacting vacuum at infinity

2019 ◽  
Vol 16 (04) ◽  
pp. 1950052
Author(s):  
G. Kittou
Keyword(s):  
Finite Time ◽  
Central Extension ◽  
Vacuum Model ◽  
Time Singularity ◽  
Attractor Solution ◽  
The One ◽  
Finite Time Singularity

We apply the central extension technique of Poincaré to dynamics involving an interacting mixture of pressureless matter and vacuum near a finite-time singularity. We show that the only attractor solution on the circle of infinity is the one describing a vanishing matter-vacuum model at early times.

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