A normal form of thin fluid film equations solves the transient paradox

Roberts, A. J. (2006) A normal form of thin fluid film equations solves the transient paradox. Physica D: Nonlinear Phenomena, 223 (1). pp. 69-81. ISSN 0167-2789

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Abstract

Imagine two constant thickness, thin films of fluid colliding together: the transient flow forms a hump where they collide; thereafter they slowly relax. But, apparently reliable lubrication models expressed only in the thickness of the fluid forecast that precisely nothing happens. How can we resolve this paradox? Dynamical systems theory constructs a normal form of the Navier–Stokes equations for the flow of a thin layer of fluid upon a solid substrate. These normal form equations illuminate the fluid dynamics by decoupling the interesting long-term ‘lubrication’ flow from the rapid viscous decay of transient shear modes. The normal form clearly shows the slow manifold of the lubrication model and demonstrates that the initial condition for the fluid thickness of the lubrication model is not the initial physical fluid thickness, but instead is modified by any initial lateral shear flow. With these initial conditions, the lubrication model makes better forecasts. This dynamical systems approach could similarly illuminate other models of complicated dynamics.


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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Author's version deposited in accordance with the copyright policy of the publisher. See also Author's version in Arxiv: http://arxiv.org/PS_cache/nlin/pdf/0408/0408059v1.pdf
Depositing User: ePrints Administrator
Faculty / Department / School: Historic - Faculty of Sciences - Department of Maths and Computing
Date Deposited: 20 Apr 2010 11:14
Last Modified: 02 Jul 2013 23:48
Uncontrolled Keywords: normal form; thin fluid film; lubrication model; nitial conditions; slow manifold
Fields of Research (FOR2008): 02 Physical Sciences > 0203 Classical Physics > 020303 Fluid Physics
09 Engineering > 0915 Interdisciplinary Engineering > 091501 Computational Fluid Dynamics
Socio-Economic Objective (SEO2008): E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
Identification Number or DOI: doi: 10.1016/j.physd.2006.08.018
URI: http://eprints.usq.edu.au/id/eprint/7525

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