I checked the literature for a night, and I felt that one thing was smashed into another thing. This topic is a big pitfall.
- I found the analytical solution of the various shapes of rigid bodies calculated by the Soviets and smashed into the liquid under the linear approximation (the linear approximation of this kind of problem can be applied as a ghost)
- Found the cavity shape brought out after the research was dropped, and The movement of the things that fell at this time.
- I found the resistance to the head of the rigid body in contact with water for hundreds of microseconds, and the exciting beam velocity (the water droplets excited at this time are too small and will evaporate in the air later)
The above picture reflects the relationship between the speed of the splashed water and the Reynolds number…very terrifying speed, but the tiny water droplets splashed during this time period have nothing to do with our problem…)
- Consider the relationship between the cross-sectional shape and the amount of splashing water (but what we want to find is the relationship with the incident velocity lol)
- There is also a claim that a drop of water dropped into a shallow pool, which was solved perfectly in experiments, calculations, or theory (but what I want is to fall into the water…)
- It is something that is directly related to altitude (or water entry speed) without the question asked.
So… Let’s clarify the physics problem first
We consider a solid ball with a radius of D falling into the water at a speed v (why is it just a ball? We do physics but only study the spherical chicken in a vacuum). This ball is considered to be a static-free fall from a height of h. Naturally, As a fluid-related problem, we need to consider these parameters:
That is, the density, viscosity, and surface tension coefficient of water.
After getting these five parameters, we can get two dimensionless numbers
, Reflects the influence of viscous force and is often used to measure the parameters of turbulence
, Reflects the ratio of kinetic energy to surface tension energy. In addition, as a solid-fluid interaction problem, we also have a dimensionless parameter, The Contact angle between ball and water
, Reflect whether it is a hydrophobic material or a hydrophilic material.
According to the general idea of dimensionless processing, these three dimensionless numbers can uniquely determine the result we need: the ratio of the maximum height of splashed water to the initial height of the ball in free fall.
Physical Images
(The pictures in this part are taken from Kubota Y, Mochizuki O. Splash formation by a spherical body plunging into water[J]. Journal of visualization, 2009, 12(4): 339-346.)
According to these dimensionless numbers, they are different The entire physical process exhibits different characteristics. (In this part of the discussion, the balls are all acrylic balls with a polished surface, and the contact angle is fixed and does not change.)
When Re and We are small, the effect of viscosity and surface tension is stronger. A typical process is as follows Figure
(The time in the figure is D/v as a unit, the same below)
It can be seen that when the ball enters the water, the stronger surface tension maintains a relatively stable cavity behind the ball. When the ball continues to sink, the cavity is replaced by water pouring in at both ends. The influx of water filled the cavity and collided with each other, igniting a cone-shaped spray directly above, without splashing.
I’m sure you all know what it’s like to be hit directly in the asshole by this kind of spray.
With the increase of Re and We, the physical process has undergone significant changes. Two typical processes are shown in the following figure
a, b The two sets of pictures are scenarios of two different parameters under the same parameter area.
Carefully observe the two pictures with T=0.12, pay attention to the lower part of the ball, you can see a layer of water (film) climbs up along the surface of the ball, and forms wavy patterns (fingers)
At the forefront; at T=0.25 At that time, the water film in picture a is still climbing up, while the water film in picture b has already shown signs of splashing out. Whether the water film splashes depends on the strength of the centrifugal force and the adsorption force, and the rising speed of the water film is generally between 2 to 5 times the descending speed of the sphere (James Oliver, Water entry and related problems, PhD Thesis, Oxford University (2002) ), so the faster the sphere drops, the sooner the water film splashes. Pay attention to the wavy pattern at the front of The water film. The number of patterns can be calculated theoretically. have Allen. R. F., The role of surface tension in splashing., Journal of Colloid and Interface Science, 51 (1975), 350-351.）
At T=0.75, the foremost segment of the water film meets at the top, collides and splashes outward. Obviously, the more wavy patterns in the front section of the water film, the more water droplets will be excited after the collision;
When T=1.14, the subsequent water collides more violently, forming larger splashes.
In this case, we found that no gas cavity could be formed.
As the parameters continue to change, the falling process presents a new face, as shown in the figure below
At T=0.55, we can observe the splashing water, and then the rapid drop of the ball will cause irregular air bubbles on the surface. As the liquids on both sides quickly come closer and collide, a fairly large, broken splash will be excited. An article published on Nature Physics (Duez C, Albert C, Clanet C, et al. Making a splash with water repellency[J]. Nature Physics, 2007, 3(3): 180-183.) pointed out that for The contact angle with liquid is less than hydrophilic solid, the dividing point between the second type of spray and the third type of spray is placed
(That is, when this parameter is greater than 0.1, it will become the third type of splash), Calculating with water at room temperature, this critical speed is roughly between [
].
I believe that the friends who have used the toilet in the countryside have a deep understanding of this scene (at least 0.5m+ from the top of the pit to the bottom)
Back to the problem itself
Given other parameters, what is the relationship between the maximum height of the splashed water and the incident velocity of the ball? Unfortunately, I did not find any articles that are interested in the maximum height of splashing water (Of course, I found a lot of articles about how to suppress splashes in diving, but these people who do sports physics only know one thing. Simulate with a computer, and do not analyze the analysis results) The only article is the source of the previous part. He studied the maximum height of water splashing only by changing the incident speed in the case of the second type of water spray.
His result is very simple:
The ratio of the maximum height of the splashed water to the height of the free fall of the ball is almost constant, and less than one.
However, This conclusion cannot simply be generalized to general.
The result of the problem is directly related to the two dimensionless parameters Re, We (let’s omit the contact angle for now). This article is equivalent to drawing a curve on this two-dimensional plane.
And we got a conclusion on the sky blue curve in the middle.
So, when we changed the ball, changed the liquid, and considered the same problem, it was equivalent to drawing a new curve on this two-dimensional plane.
Can the conclusion of the original paragraph be generalized?
We don’t know [-0-]
Of course, we can guess, but we need to experiment to verify it.
Let me talk about other things.
The easiest and most effective way to prevent water splashing is also mentioned by the title of the subject.
Of course, we can also open up other brains with a piece of paper.
For example, this paper Bergeron, V., Bonn, D., Martin, J.-Y., Vovelle, L., Controlling droplet deposition with polymer additives, Nature 405 772-775 (2000) by adding a small amount of polymer to the water to change the properties of water to reduce splashes. It’s the same article on Nature Physics mentioned above, considering the water splash caused by the change of contact angle.
The balls in the left and right pictures have almost no difference except that the contact angle with water on the left is 15 degrees (hydrophilic) and the contact angle with water on the right is 100 degrees (hydrophobic), but the splashes and air they stimulate Bubbles are different.
From this we can give a new suggestion: change the diet structure so that the pulled out Xiang has good hydrophilicity.
So the conclusion is…
Not too much. Actually: a tuft of something falling into the water, under different parameters, the splashes that are stirred up have three different forms. These three forms may be encountered when we defecate.
There is insufficient evidence to show that in the second form, the maximum height of the water splash is proportional to the release height of the object in free fall, and this proportional coefficient is naturally related to other parameters.
Finally, I wish you all the luck to avoid every drop of water when you shit.