The breaking of water stream under the light of total internal reflection of laser Jiho Yang Colegio Retamar Abstract In this paper, we report the circumstances that produce breaking of water stream. The usual way to visualize this is by using a strobe, as how they do in Science museums. Experimental methodology that we have used to investigate this topic is based on an experiment carried out by two scientists in 1850 that contributed to studies about total reflections. By using a laser, we could observe that the laser light is scattered due to refraction when water stream breaks.
Taking into account this act, we recorded videos and then analyzed them computationally. This phenomenon is considered to be very complex and they cannot be studied in the same way as other physical phenomenon. Our initial hypothesis about this phenomenon was that water stream breaks when its flow turns from laminar state to turbulent state. Then we extended this study including the influence of surface tension in this phenomenon. We expected that as surface tension decreases, the stream will break with higher velocities then before. l.
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Introduction We often see in science museums, using a strobe, how a water stream breaks and orms drops while performing a parabolic movement. The object of this paper is to study the conditions in which a “water stream” breaks. The motivation for choosing this topic was the first demonstration of total internal reflection carried out by Colladon and Babinet in Paris in 1840s1. Titled as “On the reflections of a ray of light inside a parabolic liquid stream “, it was published in 1842. The paper was based on the description of “light fountain”.
Using optical characteristic of water, they observed the behaviour of the light when it propagates inside the water stream 2 . They ound out that the light could be seen on the end of the water stream where the light enters and the other end where the water stream starts to form drops. This phenomenon shows that inside of the continuous water stream, total internal reflection of light is produced. Then when it comes to the moment that this “continuity’ breaks, forming various water drops, the light refracts. In this study, we have used this behaviour of light to find Figure 1.
The experiment out the breaking point of water stream. We set our Colladon and Babinet hypothesis and then extended it. Our hypothesis was based on the Reynolds number. In order to perform this study, we observed the variation of the breaking point of water stream by varying its outflow velocity. Later on, we realised that surface tension may be related as well. For these reason, we extended this study including the influence of surface tension. This time, we observed the variation of the breaking point of the mixture of water and alcohol depending on its concentration.
All experiments were filmed by a digital camera, stacked into one picture and then analyzed with a computer program. Daniel Colladon y Jacques Babinet, Compte Rendu des S?©ances de L’Acad?©mie des Sciences, S?©ance du lundi, octobre 5, 1842. pg. 800. http://www. biodiversitylibrary. org/item/20656#page/814/mode/1 up 2 The image belongs to La Nature, Paris: Masson, 1884. It can also be found in the digital edition: http://cnum. cnam. fr/SYN/4KY28. 23. html II. Experimental setup 1 . Apparatus In order to carry out the experiments, we made a similar apparatus as the Figure 1 . We attached a cork on one side of a transparent plastic box.
The cork had a small hole on one side to make it possible for a light to pass and a bigger hole on the other side for a small green laser. The small hole was then covered with a piece of ransparent plastic to avoid the contact between water and the green laser. On the other side of the plastic box, we attached a water outlet with a small crystal cylinder. Then this side was covered with black coloured insulating tape. Finally, on the top of the plastic box, a rubber hose was connected with a big tub to supply the water. The length of the water outlet stream designed to be lcm to make is easier to measure the breaking point with the ruler.
Water tub Power supply Plastic box Figure 2a. Apparatus for experiments Green laser Figure 2b. Apparatus in detail Rubber hose Water outlet Figure 2c. Apparatus in detail Figure 2d. Apparatus in detail This apparatus were then vertically fixed, as we only studied the vertical movement of water stream. A ruler was vertically fixed as well to measure the position of breaking point. The hose was fixed with clamps, so that the water could pass easily. 2 Figure 3. (a) Apparatus vertically fixed (b) Hose fixed with clamps 2. Software and procedure of image processing All the experiments were filmed by a digital camera.
Then we had to stack frames of the video, so that we could observe more clearly the position of water stream where the light intensity is the highest. Moreover, by doing this, we could obtain very recise data Just with one experiment. This process was done using a computer program -“HandyAvi” demo version. This way, we obtained one photo for each video. Figure 4. An example of frame stacked photo Using the computer program, “GIMP 2” we turned the photo so that the ruler is parallel to the pixel measurement shown on the program, in order to measure the breaking point precisely. 3 Figure 5. a) Figure 3 turned with “GIMP 2” (b) Stacked photo cut into a line The measurement on the left of Figure 5 (a) is the number of pixel of the picture. Then, every stacked picture was cut down from the lcm line, as marked on Figure 5 a). The result was the Figure 5 (b). Figure 5 (b) was then analyzed with “Matlab” with the object to obtain a light intensity graph respect to its position. We used the algorithm of Travis which is very simple to program in Matlab. With this algorithm, we can obtain relative light intensities of each pixel of the image. This is the code used: a = name. ormat”) c = b(: , l)/x + b(: ,2)/x+ + , x)/x where ???x” is the number of pixels of width of the imported image. Ford, Adrian y Roberts, Alan en Colour Space Conversions, digital edition: http://www. poynton. com/PDFs/coloureq. pdf 4 Figure 6. Image processing with the “Matlab” (a) Defined functions; we can see the number of rows of the original matrix. (b) Files imported. (c) Codes used. (d) Final matrix which shows the average intensity of Figure 4(b). (e) Final matrix represented in a table; we can copy these data to plot a graph with excel. This way, we could obtain the graph of light intensity respect to the picture”s number of pixel.
However, as we have cut every stacked photo from lcm line, by observing the relation between the number of pixel and the measurement of ruler, we could convert the number of pixel into cm unit. The relation between them was: 1 cm = 15 ixels And considering lcm line as the start point, we obtained a graph of light intensity respect to the position of the water stream, drawn with excel. 5 Ill. Theoretical background 1. Bernoulli’s equation The Bernoulli”s principle states that for a continuous incompressible flow, the sum of all forms of energy is constant.
This equation relates the fluid”s pressure, velocity and the potential energy, when there is no external work done to it. pvl 2 + pgyl = p + pv22+ pgy2 @ @ ??? This is the Bernoulli”s equation. From this equation, if we apply several circumstances we can obtain other expression useful to carry out the experiments. In our experiment, the pressure in the water outlet and the tub is atmospheric pressure. The water outlet is much smaller than the size of the tub. According to the continuity equation, the discharge is constant and it is expressed as: Q = A1 vl = A2 where A is the area of the tube, v is the velocity of water and Q is the discharge.
Therefore, the outflow velocity is much greater than the velocity of tub”s decreasing water level, making it negligible Applying all those circumstances to obtain other expression: , we p atm + p(O) + pgyl = p atm + pv2 2 + pgy2 2 Now, let Y2 = O and yl = h Thus, pvl 2 + pgh pv22 Therefore, 2gh This expression shows that the velocity of the outflow velocity is greater if the height between the water outlet and the tub”s water level is higher. 6 2. Reynolds number The Reynolds number4 (N R ), is a number that represents the ratio between the fluid”s inertial and viscous force by a relation between some important parameters for a fluid.
The Reynolds number shows numerically whether inertial or viscous force is more influential on the fluid”s form. In this way, it indicates whether a phase of a fluid is a laminar or turbulent flow. A turbulent flow is a chaotic and disordered hase of a fluid, meanwhile a laminar flow is formed by an ordered stream lines. The numerical range of laminar and turbulent flow is: If N R < 2000, the flow is laminar If N R > 4000, the flow is turbulent If 2000 < N R < 4000, the flow is in the transition between laminar and turbulent flow In this transition state, the fluid has both laminar and turbulent flow.
These ranges, however, are only valid when a fluid is passing through a pipe or channel. The mathematical expression for the Reynolds number6 is: VDP Where v is the average velocity of a fluid, D is the diameter of the pipe, p is the fluid”s ensity and p is the dynamic viscosity (unit: The Reynolds number is dimensionless: kg x mx 3x MOTT, Robert L, Mec?¤nica de fluidos, Pearson Education Inc. M?©xico 2006. Pg 230 MOTT. RObert L. op. Cit. pg 231 MOTT. RObert L. op. Cit. pg230 7 3. Tate’s law Let”s imagine that we have a dropper full of a liquid inside of it. This liquid drop will fall when its weight equals to its surface tension. Tate”s law states that the weight of falling liquid drop is proportional to tube”s radius and liquid”s surface tension. The application of this law allows us to relatively measure a liquid”s surface tension. This applied formula is: m here m, y are mass and surface tension of a liquid, and m’ , y’ are mass and surface tension of another liquid. As the distilled water”s surface tension at certain temperature is know, by measuring the mass of determined number of water drops and other liquid drops, we can calculate this other liquid”s surface tension. ‘V. Hypothesis, experiments and analysis 1 .
Hypothesis Our hypothesis to explain why a water stream breaks was based on the Reynolds number. As seen previously, the range of the Reynolds number for a flow to be turbulent was not valid in our case because we are studying the behaviour of a water stream in open air. However, those ranges clearly show that if the Reynolds number increases, it is more probable for a flow to be turbulent. This turbulent phase is what makes the fluid to be disordered and then form various drops. Therefore, our hypothesis was that breaking of water stream is related to the Reynolds number.
And that if the Reynolds number increases, breaking point must get closer to water outlet. 2. Experiments Experiments were carried out by varying the height between the water outlet and the tub”s water level in order to observe the variation of the breaking point depending on the water stream”s initial velocity. Obviously, the increase of this height will cause the increase of outflow velocity, and then the increase of the Reynolds number. Discharge for each height was measured to calculate the water stream”s initial outflow velocity.
As we were only varying its initial velocity, other conditions of the water had to be constant. Therefore, we used distilled water for the experiments as its dynamic viscosity and density is known. Also, water”s temperature for each experiment was measured as its dynamics viscosity and density vary depending on 8 Information of Tate”s law found from: http://www. sc. ehu. es/sbweb/fisica/fluidos/ ension/tate/tate. htm temperature. The position of the water stream with the highest intensity of green light was considered as its breaking point. This, however, cannot be measured in exact value.
Therefore, we set the possible range of breaking point and calculated its average value to plot a graph. In total, eight experiments were carried out for eight different heights. 3. Analysis We obtained 8 graphs of light intensity respect to water stream ‘s position. Then we set ranges for possible breaking points. In order to observe their variation, we calculated its average values, and then drew another 8 graphs of breaking point espect to the initial outflow velocity. Next graph is one example of light intensity graph respect to water stream’s position.
Graph 1. Intensity-h(cm) Intensity 20 10 15 h(cm) Graph 1. Light intensity graph of the Figure 4(b) The high intensity shown at first of the Graph 1 is due to the light refraction caused by the crystal water outlet. The h (cm) is the breaking point. We could see easily that the light intensity suddenly starts to increase later on, and decreases back. By observing the range of the most intense position, excluding initial part of the graph and by calculating the average value, we obtained a new graph of water stream”s reaking point respect to its outflow velocity. Graph 2. h(cm)-outflow velocity(cm/s) 14 13 12 11 50 70 Outflow velocity(cm/s) Graph 2. Graph of breaking point of water stream respect to its initial velocity Graph 2 clearly showed that the h gets less as its initial velocity increases. The breaking points of water stream gets closer to water outlet as its initial velocity increases. By calculating the Reynolds number of each experiment, we obtained next table: Range of v (m/s) 1 . 55-1 . 68 Average v (m/s) 1 . 61 Range of N R 0. 00280-0. 00296 (m) 0. 00288 4130-4750 4430