Dear colleagues! My name is NN. I will present the team of Russia with a problem Honey coils.
The problem says: A thin, downward flow of viscous liquid, such as honey, often turns itself into circular coils. We are proposed to study and explain this phenomenon.
The phenomenon of a jet coiling was investigated by various authors. Our work continues their studies.
Let’s start with basic observations and experiments.
This is our experimental setup. With its design we can observe coiling of a thin liquid jet.
We made our first experiment with honey. Colliding the ground, the honey jet really formsregular coils. Unfortunately, honey we used was slightly inhomogeneous due to its candying. Actually this is common for any honey, worsening the reproduction of experimental results. Therefore we decided to use the silicon oil with high viscosity.
This silicon oil is very similar to honey in its viscous properties.
We observe the phenomenon of coiling with different heights of the liquid supply. The lowest height allows the coiling regime was about 2 cm.
For the 8 cm height, the filament is thinner, the coils are smaller, the coiling occurs faster. You can see that the coils create a pile.
With the height is about 15 cm the coils are very small and the coiling is very fast.
We found how the coiling frequency depends on the fall height. This dependence is close to quadratic.
Now I will explain the origin of viscous coils: qualitatively first, and then with a simple quantitative model.
Let’s start with the model of the falling viscous sheet. This slide shows our experimental setup to obtain this flow. The sheet is produced by a long slot in a liquid supply. The surface tension causes the lateral edges to come together.
In this video you can see how the sheet folds on itself in a regular manner.
Let me explain how this folding occurs. When the sheet reaches the ground, its lower part is under the stress of the upper moving parts. Thus the buckling instability develops, creating a bend.An identical scenario leads to the formation of the second half of the periodic fold. Asa result, the sheet folds on itself in a regular manner.
Next step is to consider model of a round jet. Here you can see the computer simulation made by Batty and Houston. Reaching the ground, the liquid column is bent in a random plane. Then it loses stability in the other direction, so the coiling begins.
Now let me discuss the theoretical model which gives a scaling estimation for the coiling frequency.
The physical parameters governing the coiling phenomenon include fluid density ρ, dynamic viscosity η, the flow rate Q, gravity g and the height H over which the jet falls. Near the ground surface the jet has the radius a and velocity U. Coils are laid at a frequency Ω and radius R.
The flow rate, the radius of the jet and the jet velocity are linked through the continuity of flow. The circular frequency of coiling is expressed through the jet velocity and the radius of the coil.
The inertial forces due to the whirling are balanced by the viscous forces due to the differential velocities. So we can write the second Newton’s law for a shortsegment of a filament. The inertial force is a product of a mass and an acceleration. The viscous force acting in the cross-section of the filament is estimated as a product of the dynamic viscosity, the cross-section and the velocity gradient. To get the difference between the viscous forces acting in the two end sections of the segment, it is necessary to introduce this differential factor. As a result of the subsequent calculations we obtain that the coiling frequency is proportional to the square of the jet velocity.
In this model of coiling we neglect surface tension and gravity.The surface tension has only a small effect on the viscous jet and the coiling frequency. This is evident from the fact that even a very thin jet does not break up into droplets.Neglect of gravity means that the jet is coiled because it hits the surface with a certain velocity.We explain this neglect below.
Let’s summarize our current results. The experiment demonstrates that the coiling frequency is proportional to the square of the fall height. The theory predicts that the coiling frequency is proportional to the square of the jet velocity. To have a consistency, the jet velocity should be proportional to the fall height.
To find the jet velocity consider the shape of the jet.
We made this photo and then measured the jet diameter.
The dependence of the jet diameter on the height is shown on this diagram.
Using the continuity of the flow, we find how the jet velocity depends on the fall height. This diagram shows that the viscous jet falls down sufficiently slowly than in the free fall. From this experiment we obtain that the jet velocity is proportional to the fall height in the range from 2 to 20 cm (as used in our coiling experiments).
Now we can compare the theory with the experiment.
The theory predicts that the coiling frequency is proportional to the square of the jet velocity. The experiment shows that the jet velocity is proportional to the fall height. Hence the coiling frequency is proportional to the square of the fall height. On the other hand, our coiling experiment also shows that the coiling frequency is proportional to the square of the fall height. Consequently, our theoretical model is in a good agreement with the experimental results.
Now we can calculate coiling centrifugal acceleration as the function of the fall height. It is seen that at heights more then 5 cm centrifugal acceleration is large then gravitational, so that the influence of gravity on the coiling process can be neglected.
This boundary is visible, if we present the results of our main experiment on a logarithmic scale.However, our scaling theory is also applicable well below this limit.
Finally we compare our results with theoretical predictions of other authors. Our theory gives the scaling Ω ~ H2, which is in good agreement with our experiments. The theory of Mahadevan gives slightly different scaling Ω~H5/3, which is not consistent with our experimental results.
Let’s summarise our investigation.
Coiling of a viscous jet is explained by the interplay of viscous and inertial forces.
The coiling frequency of a viscous jet is proportional to the square of the jet velocity.
There are our references.