We consider the inertially driven, time-dependent biaxial extensional motion of inviscid and viscous thinning liquid sheets. We present an analytic solution describing the base flow and examine its linear stability to varicose (symmetric) perturbations within the framework of a long-wave model where transient growth and long-time asymptotic stability are considered. The stability of the system is characterized in terms of the perturbation wavenumber, Weber number, and Reynolds number. We find that the isotropic nature of the base flow yields stability results that are identical for axisymmetric and general two-dimensional perturbations. Transient growth of short-wave perturbations at early to moderate times can have significant and lasting influence on the long-time sheet thickness. For finite Reynolds numbers, a radially expanding sheet is weakly unstable with bounded growth of all perturbations, whereas in the inviscid and Stokes flow limits sheets are unstable to perturbations in the short-wave limit.
REFERENCES
1.
2.
J. M.
Maerker
and W. R.
Schowalter
, “Biaxial extension of an elastic liquid
,” Rheol. Acta
13
, 627
(1974
).3.
L. A.
Romero
, “The stability of stretching and accelerating plastic sheets. I
,” J. Appl. Phys.
69
, 7474
(1991
).4.
L. A.
Romero
, “The stability of stretching and accelerating plastic sheets. II
,” J. Appl. Phys.
69
, 7487
(1991
).5.
H. A.
Barnes
, J. F.
Hutton
, and K.
Walters
, An Introduction to Rheology
(Elsevier
, Oxford
, 1989
).6.
Y. J.
Jiang
, A.
Umemura
, and C. K.
Law
, “An experimental investigation on the collision behaviour of hydrocarbon droplets
,” J. Fluid Mech.
234
, 171
(1992
).7.
J.
Qian
and C. K.
Law
, “Regimes of coalescence and separation in droplet collision
,” J. Fluid Mech.
331
, 59
(1997
).8.
M.
Orme
, “Experiments on droplet collision, bounce, coalescence and disruption
,” Prog. Energy Combust. Sci.
23
, 65
(1997
).9.
L.
Xu
, W. W.
Zhang
, and S. R.
Nagel
, “Drop splashing on a dry smooth surface
,” Phys. Rev. Lett.
94
, 184505
(2005
).10.
R. D.
Schroll
, C.
Josserand
, S.
Zaleski
, and W. W.
Zhang
, “Impact of a viscous liquid drop
,” Phys. Rev. Lett.
104
, 034504
(2010
).11.
M. M.
Driscoll
, C. S.
Stevens
, and S. R.
Nagel
, “Thin film formation during splashing of viscous liquids
,” Phys. Rev. E
82
, 036302
(2010
).12.
L.
Doubliez
, “The drainage and rupture of a non-foaming liquid film formed upon bubble impact with a free surface
,” Int. J. Multiphase Flow
17
, 783
(1991
).13.
L.
Doubliez
, “Capillary instability of a fast-draining film
,” Colloids Surf.
68
, 17
(1992
).14.
P. D.
Howell
, B.
Scheid
, and H. A.
Stone
, “Newtonian pizza: Spinning a viscous sheet
,” J. Fluid Mech.
659
, 1
(2010
).15.
F.
Savart
, “Memoire sur le choc d'une veine liquide lancee contre un plan circulaire
,” Ann. Chim.
54
, 56
(1833
).16.
F.
Savart
, “Memoire sur le choc de deux veines liquides animees de mouvements directement opposes
,” Ann. Chim.
55
, 257
(1833
).17.
J. C. P.
Huang
, “The break-up of axisymmetric liquid sheets
,” J. Fluid Mech.
43
, 305
(1970
).18.
G. I.
Taylor
, “The dynamics of thin sheets of fluid. I. Water bells
,” Proc. R. Soc. London, Ser. A
253
, 289
(1959
).19.
C.
Clanet
and E.
Villermaux
, “Life of a smooth liquid sheet
,” J. Fluid Mech.
462
, 307
(2002
).20.
E.
Villermaux
and C.
Clanet
, “Life of a flapping liquid sheet
,” J. Fluid Mech.
462
, 341
(2002
).21.
S. P.
Lin
, Breakup of Liquid Sheets and Jets
(Cambridge University Press
, New York
, 2003
).22.
C.
Clanet
, “Waterbells and liquid sheets
,” Annu. Rev. Fluid Mech.
39
, 469
(2007
).23.
R.
Krechetnikov
, “Stability of liquid sheet edges
,” Phys. Fluids
22
, 092101
(2010
).24.
P. D.
Howell
, “Extensional thin layer flows
,” D.Phil. dissertation, University of Oxford
, 1994
.25.
S. S.
Pegler
and M. G.
Worster
, “Dynamics of a viscous layer floating radially over an inviscid ocean
,” J. Fluid Mech.
696
, 152
(2012
).26.
S. S.
Pegler
, J. R.
Lister
, and M. G.
Worster
, “Release of a viscous power-law fluid over an inviscid ocean
,” J. Fluid Mech.
700
, 63
(2012
).27.
P.
Marmottant
and E.
Villermaux
, “Fragmentation of stretched liquid ligaments
,” Phys. Fluids
16
, 2732
(2004
).28.
I.
Frankel
and D.
Weihs
, “Stability of a capillary jet with linearly increasing axial velocity (with applications to shaped charges)
,” J. Fluid Mech.
155
, 289
(1985
).29.
I.
Frankel
and D.
Weihs
, “Influence of viscosity on the capillary instability of a stretching jet
,” J. Fluid Mech.
185
, 361
(1987
).30.
L. A.
Romero
, “The stability of rapidly stretching plastic jets. I
,” J. Appl. Phys.
65
, 3006
(1989
).31.
D.
Henderson
, H.
Segur
, L. B.
Smolka
, and M.
Wadati
, “The motion of a falling liquid filament
,” Phys. Fluids
12
, 550
(2000
).32.
L. B.
Smolka
, A.
Belmonte
, D. M.
Henderson
, and T. P.
Witelski
, “Exact solution for the extensional motion of a viscoelastic filament
,” Eur. J. Appl. Math.
15
, 679
(2004
).33.
L. B.
Smolka
and T. P.
Witelski
, “On the planar extensional motion of an inertially driven liquid sheet
,” Phys. Fluids
21
, 042101
(2009
).34.
N. D.
Di Pietro
and R. G.
Cox
, “The spreading of a very viscous liquid on a quiescent water surface
,” Q. J. Mech. Appl. Math.
32
, 355
(1979
).35.
L. N.
Brush
and S. H.
Davis
, “A new law of thinning in foam dynamics
,” J. Fluid Mech.
534
, 227
(2005
).36.
A. M.
Anderson
, L. N.
Brush
, and S. H.
Davis
, “Foam mechanics: Spontaneous rupture of thinning liquid films with Plateau borders
,” J. Fluid Mech.
658
, 63
(2010
).37.
J.-M.
Chomaz
, “The dynamics of a viscous soap film with soluble surfactant
,” J. Fluid Mech.
442
, 387
(2001
).38.
L. N.
Trefethen
, A. E.
Trefethen
, S. C.
Reddy
, and T. A.
Driscoll
, “Hydrodynamic stability without eigenvalues
,” Science
261
, 578
(1993
).39.
C. M.
Bender
and S. A.
Orszag
, Advanced Mathematical Methods for Scientists and Engineers Systems
(Springer-Verlag
, New York
, 1999
).40.
F.
Doumenc
, T.
Boeck
, B.
Guerrier
, and M.
Rossi
, “Transient Rayleigh-Bénard-Marangoni convection due to evaporation: A linear non-normal stability analysis
,” J. Fluid Mech.
648
, 521
(2010
).© 2013 AIP Publishing LLC.
2013
AIP Publishing LLC
You do not currently have access to this content.
