Good afternoon! My name is NN. I will present the team of Russia with a problem «Jet and Film».
The problem says: A thin liquid jet collides with a soap film. Depending on some appropriate parameters, the jet can either pass through the film, or flow into it creating interesting shapes. We a proposed to explore and explain this interaction and the shapes being created.
This phenomenon was recently investigated by Kirstetter, Raufaste and Celestini. In our work we reproduce some of their results and obtain some new ones.
This is our experimental setup. With its design we can change an incident angle of the jet. Also we can change velocity of the jet, varying the level of the liquid supply.
We observed three regimes of interaction between the jet and the film. Firstly, the jet can pass through the film and refract. Secondly, the jet can reflect from the film. Thirdly, the jet can be absorbed by the film and undulate along it. Regime of reflection is transitional between refraction and absorption regimes.
In our work we study both regimes of refraction and absorption and a boundary between them.
The incident jet is characterized by incident angle θi, velocity v and radius r.
The incident angle of is designated as θi, and the refraction angle as θr.
Regime of interaction between the jet and the film is determined by the Weber number.
The Weber number is a ratio of characteristic inertial and capillary forces, acting in the system. This dimensionless parameter quantifies the relative importance of inertia and capillarity.
In order to calculate the Weber number, we measure the surface tension of the liquid with a help of capillary waves.
Here are values of the Weber number for different nozzles and supply levels, realized in our studies.
Let’s begin with the refraction regime.
Here you can see the jet refracting by the film. Note to the meniscus-like shape of the film. The meniscus and the jet interact with each other. As a result, the jet drags the film and the film slows the jet.
It can be noticed that the film acts on the jet only in the vertical direction. So the horizontal velocity of the jet is conserved. As a result, the ratio of sines is equal to the inverse ratio of the jet velocities.
Let us assume that the film is tangential to the jet in the meniscus on the line of its contact. So the balance of a longitudinal momentum can be written. The angle between the directions of the incident and the refracted jet is small. Thus changing of the longitudinal momentum is approximately equal to the surface tension force, projected on the initial direction of the jet. To find this force, one can multiply the surface tension coefficient on the double perimeter section of the jet. Double, because the film has two surfaces. Using this equation, we obtain the ratio of the jet velocities.
As a result, we get the law of refraction which is similar to Snell’s law in optics.
To check this theory we studied the refraction experimentally. Experimental points lie on straight lines, passing through the origin of coordinates. However the slope coefficients for experimental results are slightly different from the theoretical predictions. This is due to the roughness of the theory. The agreement between experiment and theory can be remarkably improved by replacing factor 4 by a factor 3.
Then we studied a boundary between refraction and absorption regimes.
We measured a critical angle of incidence, when the transition occurs. The critical angle should be always measured with the same transition from refraction to absorption, because of the phenomenon of hysteresis which occurs in this transition.
Due to the refraction law, the refraction angle increases with increasing of the incident angle.
When the incident angle reaches the critical value, the refracted jet goes parallel to the film. This is a condition in which the jet is absorbed by the film.
This diagram represents our experimental results. The pink line corresponds to our theoretical prediction. We see that all experimental points lie below this line. A better agreement can be reached replacing factor 4 in this formula by a factor 5.
Let me notice some details we can see on this experimental video. If there is the absorption and the velocity of the jet increases, the wavelength grows and the jet may go to reflection. We also see that the refraction and reflection are similar, only in the refraction the jet separates from the film immediately, but upon reflection it first passes the lower half-wave.
Finally let’s consider the absorption regime.
On this slide you can see the wave-like shape of the jet. No doubt that the film acts on the jet with a capillary force.
To calculate the value of this force, we need firstly to find out the angle α between the film and the vertical. Using the Laplace equation, we obtain that this angle at the top of the wave is equal to the incident angle. Then by averaging this angle along the half-wave, we obtain the average value of cos α.
Then we consider a half-wave and calculate its momentum balance. The horizontal momentum is conserved. The vertical momentum is reversing. Thus we can equate the vertical momentum changing per time to the force of surface tension averaged along the half-wave.
Substituting the average cosine from the previous slide, we obtain the ratio of the wavelength to the radius of the jet. This ratio is proportional to the Weber number.
This diagram represents the experimental dependence of the relative wavelength on the Weber number. There is a perfect agreement of experimental results with our theory.
This diagram represents experimental results of measuring the wavelength versus the incident angle. They are in good agreement with our theory too.
Let me summarize our researches.
The nature of observed jet-film interaction is determined by two parameters: the incident angle of the jet and the Weber number.
There are three regimes of interaction.
First is absorption. It occurs when incident angle is large and Weber number is low.
Second is refraction. Vice versa, it can be observed for small incident angles and high Weber numbers.
Third is reflection. It appears to be transient between two previous modes.
There are the papers used in our studies.