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Biaxial extensional motion of an inertially driven radially expanding liquid sheet
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10.1063/1.4811389
/content/aip/journal/pof2/25/6/10.1063/1.4811389
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/6/10.1063/1.4811389
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Schematic of a radially expanding, thinning liquid sheet with free surfaces located at = ±().

Image of FIG. 2.
FIG. 2.

Evolution of perturbation amplitude in the inviscid short-wave case for > : (a) The oscillatory form of () for early times from initial conditions ( ) = 1, ( ) = 1, and = 1/110 (corresponding to = 110 at We = 1); (b) perturbation magnitude on a log-log plot to illustrate the ( ) transient growth and saturation to the asymptotic amplification factor, for two wavenumbers, α(110) ≈ 14.4 and α(109.9886) ≈ 0.106, obtained from numerical solution of Eqs. (15a) and (15b) with β = 0.

Image of FIG. 3.
FIG. 3.

The amplification factor α() for an inviscid radially expanding sheet subject to axisymmetric perturbations with dashed asymptotes indicating the long- and short-wave limits. For < the sheet is weakly stable, and for < < ∞ the sheet is weakly unstable due to transient growth of the perturbations. The plot for α() uses definition (21) with () determined by solving Eqs. (15a) and (15b) with β = 0, We = 1, and initial conditions ( ) = 1, ( ) = 1.

Image of FIG. 4.
FIG. 4.

The numerically computed amplification factor α() in the low-viscosity limit β → 0 (solid curves) and the estimate given by Eq. (31) (dashed curve). The amplification factor was computed from Eqs. (15a) and (15b) with We = 1, Re = 1000, and initial data ( ) = ( ) = 1.

Image of FIG. 5.
FIG. 5.

Evolution of the magnitude of the surface perturbations, |()|, from Eqs. (15a) and (15b) in the low-viscosity limit, Re = 1000, for (a) = 15 < showing overall growth (α > 1) and (b) = 50 > showing overall decay (α < 1) with ( ) = ( ) = 1 and We = 1. The dashed curve shows the amplitude from Eq. (30) and vertical dotted-dashed lines indicate and ( < ).

Image of FIG. 6.
FIG. 6.

The critical timescale for transition from oscillatory to non-oscillatory evolution for the viscous problem: for β < β and (β) from Eq. (32) for larger β.

Image of FIG. 7.
FIG. 7.

The amplification factor α() for (a) Re = 1 and (b) Re = 2.5 (both with We = 1) compared with Eq. (36) for β > β (dashed curve).

Image of FIG. 8.
FIG. 8.

Evolution of () perturbations in the high viscosity limit with Re = 1: (a) for = 15 > showing decay (α < 1) and (b) for = 8 < showing growth (α > 1). The dashed curves compare Eq. (35) with the results computed from Eqs. (15a) and (15b) with ( ) = ( ) = 1 and We = 1.

Image of FIG. 9.
FIG. 9.

Evolution of || in the moderate viscosity regime with Re = 5 and = 8 > , the threshold given by Eq. (37) . The oscillations, occurring on < ( , indicated by vertical dotted-dashed lines), are well approximated by Eq. (30) (dashed curve). After , there is an intermediate range where Eq. (35) applies leading to the long-time behavior Eq. (19) .

Image of FIG. 10.
FIG. 10.

Maximum amplification factor for axisymmetric perturbations as a function of Reynolds number with asymptotes derived from Eqs. (31) and (36) and with We fixed.

Image of FIG. 11.
FIG. 11.

(a) The growth rates Real(λ) predicted by the frozen-time analysis, and (b) the short-time evolution of the perturbation amplitude from Eqs. (15a) and (15b) (solid curve) compared with the prediction from frozen-time system (38) (dashed) for Re = 5, We = 1, and = 8 (as shown in Fig. 9 ).

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/content/aip/journal/pof2/25/6/10.1063/1.4811389
2013-06-21
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Biaxial extensional motion of an inertially driven radially expanding liquid sheet
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/6/10.1063/1.4811389
10.1063/1.4811389
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