A blow-up problem of a class of axisymmetric Navier-Stokes equations with infinite energy

2007 ◽  
Vol 48 (6) ◽  
pp. 065205 ◽  
Author(s):  
K. Ohkitani
Keyword(s):  
Blow Up ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Infinite Energy

Particle dynamics and mixing in a viscously decaying shear layer

1991 ◽  
Vol 227 ◽  
pp. 211-244 ◽  
Author(s):  
E. Meiburg ◽  
P. K. Newton
Keyword(s):  
Exact Solution ◽  
Blow Up ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Core Size ◽  
Mixing Processes ◽  
Navier Stokes Equations ◽  
Interface Generation ◽  
Self Similar ◽  
Detailed Picture

We study the mixing of fluid in a viscously decaying row of point vortices. To this end, we employ a simplified model based on Stuart's (1967) one-parameter family of solutions to the steady Euler equations. Our approach relates the free parameter to a vortex core size, which grows in time according to the exact solution of the Navier-Stokes equations for an isolated vortex. In this way, we approach an exact solution for small values of t/Re. We investigate how the growing core size leads to a shrinking of the cat's eye and hence to fluid leaking out of the trapped region into the free streams. In particular, we observe that particles initially located close to each other in neighbouring intervals along the streamwise direction escape from the cat's eye near opposite ends. The size of these intervals scales with the inverse square root of the Reynolds number. We furthermore examine the particle escape times and observe a self-similar blow-up for the particles near the border between two adjacent intervals. This can be explained on the basis of a simple stagnation-point flow. An investigation of interface generation shows that viscosity leads to an additional factor proportional to time in the growth rates. Numerical simulations confirm the above results and give a detailed picture of the underlying mixing processes.

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