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A viscous buoyancy-driven flow exhibiting finite-time blow-up

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posted on 2014-12-12, 13:21 authored by Anthony Kay
We consider 2D viscous flow driven by buoyancy forces resulting from a quadratic horizontal density variation in an unbounded domain between horizontal walls. The density is a quadratic function of the concentration of a tracer, so we solve the vorticity equation under the Boussinesq approximation, together with an advection–diffusion equation for the tracer. Stagnation-point similitude eliminates dependence on the horizontal coordinate. For the case of small Grashof number (large viscosity), the flow passes through three stages. A transient adjusts from the initial condition of static fluid to a regime in which buoyancy and viscous forces are in balance. The flow and temperature gradient slowly intensify until eventually the non-linear advection terms become dominant. The flow then enters its final stage, in which a more rapid intensification leads to a singularity in finite time. While no rigorous proof has been found that this blow-up occurs, the combination of asymptotic analysis and numerical computation provides strong support for its occurrence.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

IMA JOURNAL OF APPLIED MATHEMATICS

Volume

78

Issue

6

Pages

1214 - 1236 (23)

Citation

KAY, A., 2013. A viscous buoyancy-driven flow exhibiting finite-time blow-up. IMA Journal of Applied Mathematics, 78 (6), pp. 1214 - 1236.

Publisher

Oxford University Press on behalf of the Institute of Mathematics and its Applications / © The Author

Version

  • AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2013

Notes

This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The version of record KAY, A., 2013. A viscous buoyancy-driven flow exhibiting finite-time blow-up. IMA Journal of Applied Mathematics, 78 (6), pp. 1214 - 1236 is available online at: http://dx.doi.org/10.1093/imamat/hxs018

ISSN

0272-4960

Language

  • en

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