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Weakly nonlinear instability of planar viscous sheets

Published online by Cambridge University Press:  23 October 2013

Lijun Yang ,
Chen Wang ,
Qingfei Fu ,
Minglong Du  and
Mingxi Tong
Lijun Yang*
Affiliation:
School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China
Chen Wang
Affiliation:
School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China
Qingfei Fu
Affiliation:
School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China
Minglong Du
Affiliation:
School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China
Mingxi Tong
Affiliation:
School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China
*
†Email address for correspondence: [email protected]
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Abstract

A second-order instability analysis has been performed for sinuous disturbances on two-dimensional planar viscous sheets moving in a stationary gas medium using a perturbation technique. The solutions of second-order interface disturbances have been derived for both temporal instability and spatial instability. It has been found that the second-order interface deformation of the fundamental sinuous wave is varicose or dilational, causing disintegration and resulting in ligaments which are interspaced by half a wavelength. The interface deformation has been presented; the breakup time for temporal instability and breakup length for spatial instability have been calculated. An increase in Weber number and gas-to-liquid density ratio extensively increases both the temporal or spatial growth rate and the second-order initial disturbance amplitude, resulting in a shorter breakup time or length, and a more distorted surface deformation. Under normal conditions, viscosity has a stabilizing effect on the first-order temporal or spatial growth rate, but it plays a dual role in the second-order disturbance amplitude. The overall effect of viscosity is minor and complicated. In the typical condition, in which the Weber number is 400 and the gas-to-liquid density ratio is 0.001, viscosity has a weak stabilizing effect when the Reynolds number is larger than 150 or smaller than 10; when the Reynolds number is between 150 and 10, viscosity has a weak destabilizing effect.

JFM classification

Instability: Nonlinear instability Interfacial Flows (free surface): Thin films Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 735 , 25 November 2013 , pp. 249 - 287
Copyright
©2013 Cambridge University Press 

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