Evolution of solitary three- dimensional waves on vertically falling liquid films S. Alekseenko, V. Antipin, V. Guzanov, S. Kharlamov, D. Markovich Institute of Thermophysics Russian Academy of Sciences Siberian Branch

Introduction The first theoretical solution: Petviashvili V. I., Tsvelodub O. Yu., Horseshoe-shaped solitons on an inclined viscous liquid film, Dokl. Acad. Nauk. SSSR (in Rusian), V. 238, pp (1978) Problems: slow natural evolution, chaotic interaction. Solution: short impact by working liquid in upper part of waveless region

The case 2 gives low level of optical distortion Laser beam makes easy calibration procedure Laser Induced Fluorescence technique

Experimental setup Doubled Nd:YaG laser (3 – 10 ms, 7.5 Hz) 532 nm Rhodamine 6G (~0.01%) Low-pass filter (>550 nm) Water-alcohol solution: ρ=931 kg/m 3, ν =2.72·10 -6 m 2 /s, σ=0.03 kg/s 2 Water-glycerol solution: ρ=1070 kg/m 3, ν =2.15·10 -6 m 2 /s, σ=0.065 kg/s 2 Mass of drops: 0.3 – 10 mg, velocity: 0.4 – 1.5 m/s Re=1 – 25 (Re=q/v, where q – volumetric flow rate per unit area, v – kinematic viscosity)

Experimental errors Main errors: spatial redistribution of intensity of laser beam from flash to flash and CCD matrix noise (2 – 3% under experimental condition) Additional errors: spatial intensity redistribution under free curvilinear boundary (in presented later cases these errors are sufficiently less then 1%)

Preliminary results Fast formation of wave train occurs when excitation energy is low Evolution as one-humped solitary wave occurs for higher excitation energy

Evolution. Wave train Water-glycerol solution. Re=12.

Evolution. Solitary wave Water-alcohol solution. Re=25.

Parameters Section is passed via center of main peak in downstream direction mm Dy Velocity C is defined by cross-correlation function

Evolution Re=10 water- glycerol solution Two energy of excitation: E < E

Evolution Re=5 water- alcohol solution Two energy of excitation: E < E

Parameters of stationary solitary waves Water-alcohol solution Re=2.5 C, cm/s X, mm A, mm X, mm Dy, mm X, mm Dx, mm X, mm

Parameters of stationary solitary waves Water-glycerol solution Re=10 X, mm A, mm X, mm Dx, mm X, mm Dy, mm X, mm C, cm/s

Shape of stationary solitary waves Water-alcohol solution Re=2.2

Shape of stationary solitary waves Water-alcohol solution Re=2.5 mm Re ~ 1 KS Petviashvili, Tsvelodub, 1978

Shape of stationary solitary waves Water-alcohol solution Re=3.9

Shape of stationary solitary waves Water-alcohol solution Re=4.7 mm

Shape of stationary solitary waves Water-glycerol solution Re=10

Theory mm Demekhin E. A., Kalaydin E. N., Shapar S. M., Shelistov V. S., Stability of three-dimensional solitons in vertically falling liquid films, Dokl. Akad. Nauk, 2007, vol. 413, 2, pp. 193 – 197. Comparison with theory (gKS) Experiment Water-alcohol solution Re=3.9

Experiment Comparison with theory (gKS) Theory Water-alcohol solution Re=3.9 mm

Comparison with theory (KS) Petviashvili V. I., Tsvelodub O. Yu., Horseshoe-shaped solitons on an inclined viscous liquid film, Dokl. Acad. Nauk. SSSR (in Rusian), V. 238, pp (1978) - undisturbed film thickness - mean flow rate velocity

Conclusions Experimental results on evolution of 3-D solitary waves on falling liquid films for Reynolds numbers of film flow Re < 25 are presented. LIF method is applied to measure the shape and instant velocity of the waves. Generation of the wave train as well as evolution in the form of one- humped solitary wave are main scenarios of the wave evolution. Stationary 3-D solitary waves were registered for several experimental conditions. For Re<5 their amplitudes and velocities agree well with theoretically predicted values based both on KS and gKS equation. The shape of registered stationary 3-D waves agree well with the shape of stationary KS wave for Re<3 and with stationary gKS waves for 3<Re<5. For moderate Re~10 there exists essential difference between experiment and theories.