Extensional flow at low Reynolds number with surface tension
Tóm tắt
The extensional flow and break-up of fluids has long interested many authors. A slender viscous fluid drop falling under gravity from beneath a horizontal surface is examined. After reviewing previous work which has neglected surface tension, a one-dimensional model which describes the evolution of such a drop, beginning with a prescribed initial drop shape and including the effects of gravity and surface tension, is investigated. Inertial effects are ignored due to the high viscosity of the fluid. Particular attention is paid to the boundary condition near the bottom of the drop where the one-dimensional approximation is no longer valid. The evolving shape of the drop is calculated up to a crisis time at which the cross-sectional area at some location goes to zero. Results are compared with those obtained when surface tension is neglected. Near to the crisis time, as the Reynolds number increases and inertia becomes non-negligible, the model assumptions are invalid so that the model does not describe actual pinch-off of a fluid drop.
Tài liệu tham khảo
Eggers J (1997) Nonlinear dynamics and breakup of free surface flows. Rev Mod Phys 69: 865–929
Eggers J, Villermaux E (2008) Physics of liquid jets. Rep Prog Phys 71: 036601
Rayleigh L (1892) On the instability of a cylinder of viscous liquid under capillary force. Phil Mag 34: 145–154
Eggers J (1993) Universal pinching of 3D axisymmetric free surface flow. Phys Rev Lett 71: 3458–3460
García FJ, Castellanos A (1994) One-dimensional models for slender axisymmetric viscous liquid jets. Phys Fluids 6: 2676–2689
Papageorgiou DT (1995) Analytical description of the breakup of liquid jets. J Fluid Mech 301: 109–132
Henderson D, Segur H, Smolka LB, Wadati M (2000) The motion of a falling liquid filament. Phys Fluids 12: 550–565
Fontelos MA (2004) On the evolution of thin viscous jets: filament formation. Math Methods Appl Sci 27: 1197–1220
Xu Q, Basaran OA (2007) Computational analysis of drop-on-demand drop formation. Phys Fluids 19: 102–111
Decent SP (2009) Asymptotic solution of slender viscous jet break-up. IMA J Appl Math 74: 741–781
Wilson SDR (1988) The slow dripping of a viscous fluid. J Fluid Mech 190: 561–570
Zhang DF, Stone HA (1997) Drop formation in viscous flows at a vertical capillary tube. Phys Fluids 9: 2234–2242
Wilkes ED, Phillips SD, Basaran OA (1999) Computational and experimental analysis of dynamics of drop formation. Phys Fluids 11: 3577–3598
Trouton FT (1906) On the coefficient of viscoustraction and its relation to that of viscosity. Proc R Soc Lond A 77: 426–440
Matovich MA, Pearson JRA (1969) Spinning a molten threadline. Steady-state isothermal viscous flows. Ind Eng Chem Fundam 8: 512–520
Peregrine DH, Shoker G, Symon A (1990) The bifurcation of liquid bridges. J Fluid Mech 212: 25–39
Ting L, Keller JB (1990) Slender jets and thin sheets with surface-tension. SIAM J Appl Math 50: 1533–1546
Schulkes RMSM (1994) The evolution and bifurcation of a pendant drop. J Fluid Mech 278: 83–100
Shi X, Brenner MP, Nagel SR (1994) A cascade of structure in a drop falling from a faucet. Science 265: 219–222
Stokes YM, Tuck EO, Schwartz LW (2000) Extensional fall of a very viscous fluid drop. Q J Mech Appl Math 53: 565–582
Stokes YM, Tuck EO (2004) The role of inertia in extensional fall of a viscous drop. J Fluid Mech 498: 205–225
Bradshaw-Hajek BH, Stokes YM, Tuck EO (2007) Computation of extensional fall of slender viscous drops by a one-dimensional Eulerian method. SIAM J Appl Math 67: 1166–1182
Senchenko S, Bohr T (2005) Shape and stability of a viscous thread. Phys Rev E 71: 56301
Stone HA, Leal LG (1989) Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J Fluid Mech 198: 399–427
Hiemenz PC, Rajagopalan R (1997) Principles of colloid and surface chemistry, 3rd edn. Marcel Dekker, Inc., New York