Free surface of a liquid in a rotating frame with time-depend velocity
FFree surface of a liquid in a rotating frame with time-depend velocity
Mart´ın Monteiro ∗ Universidad ORT Uruguay
Fernando Tornar´ıa † CES-ANEP, Montevideo, Uruguay
Arturo C. Mart´ı ‡ Instituto de F´ısica, Facultad de Ciencias,Universidad de la Rep´ublica, Igu´a 4225, Montevideo, 11200, Uruguay (Dated: August 14, 2019)The shape of liquid surface in a rotating frame depends on the angular velocity. In thisexperiment, a fluid in a rectangular container with a small width is placed on a rotatingtable. A smartphone fixed to the rotating frame simultaneously records the fluid surfacewith the camera and also, thanks to the built-in gyroscope, the angular velocity. Whenthe table starts rotating the surface evolves and develops a parabolic shape. Using videoanalysis we obtain the surface’s shape: concavity of the parabole and height of the vertex.Experimental results are compared with theoretical predictions. This problem contributesto improve the understanding of relevant concepts in fluid dynamics. ∗ [email protected] † [email protected] ‡ marti@fisica.edu.uy a r X i v : . [ phy s i c s . e d - ph ] A ug I. STATEMENT OF THE PROBLEM
When we gently stir a cup of coffee or tea the free surface develops a familiar parabolic shape.This in fact an expression of an ubiquitous phenomenon called vortex. In general, in fluid mechan-ics, vortex motion, is characterized by fluid elements moving along circular streamlines [1]. Thereare two basic types of vortex flows: irrotational vortex, that occurs tipically around a sink, androtational vortex or solid body rotation, that occurs in the aforementioned example. In particular,the free surface of a rotating liquid develops the well-known parabolic profile whose characteristicsdepends on the angular velocity. This last is an usual problem in introductory courses when dealingwith fluid mechanics. Although it is not difficult from the theoretical point of view, it is not easyto address it experimentally. Here we propose an experiment to analyze the parabolic shape of aliquid surface in a rotating table as a function of the time-dependent angular velocity.Rotating fluids were studied in several experiments. To mention a few, the profile of circularuniform motion of liquid surface was determined using a vertical laser beam reflected from thecurved surface [2] and determine the acceleration of gravity. More recently [3], this experimentalsetup was improved using the fact that a rotating liquid surface will form a parabolic reflector whichwill focus light into a unique focal point. Another interesting experiment is the Newton’s bucketwhich provides a simple demonstration that simulates Mach’s principle allowing to observe theconcave shape of the liquid [4]. In other experiments rotating fluids were studied in the frameworkof the equivalence principle and non-inertial frames [5, 6].In the present experiment a narrow container is placed on a rotating table whose angularvelocity can be manually controlled. As shown in the next Section the liquid surface developsa parabolic shape whose concavity and the location of the vertex can be related to the angularvelocity of the table and to the gravitational acceleration. The experimental setup, described inSection III, in additon to the container on the rotating table, includes a smartphone, also fixedto the rotating table, allows us to register the shape of the liquid surface with the camera andthe angular velocity with the gyroscope. This ability to measure simultaneously with more thanone sensor is a great advantage of smartphones since it allows us to perform a great variety ofexperiments, even outdoors, avoiding the dependence on fragile or unavailable instruments (see forexample Refs. [7–11]). Thanks to the analysis of the digital video it is possible to readily obtainthe characteristic of the parabolic shape. The results are presented in Section IV and, finally, theconclusion is given in Section V.
II. SHAPE OF A LIQUID SURFACE IN A ROTATING FRAME
The free surface of a liquid in a rotating frame is obtained from the points where the pressureis equal to the atmospheric pressure. Let us consider the pressure field in a fluid, p ( (cid:126)r ), subjectedto a constant acceleration (cid:126)a and a gravitational field (cid:126)g . After transients, when a fluid is rotatingas a rigid body, i.e. the fluid elements follow circular streamlines without deforming and viscousstresses are null [1]. Under these hypothesis, pressure gradient, gravitational field and particleacceleration are related by the simple expression ∇ p = ρ ( (cid:126)g − (cid:126)a ) . (1)In the present experiment, we consider a fluid in a narrow prismatic container, as shown inFig. 1, whose basis is L × d where L (cid:29) d and its height is large enough so that the fluid does notoverflow. When the system is at rest, the fluid, with density ρ and negligible viscosity, reaches aheight H . The container is placed on a rotating table whose angular velocity, ω , around the verticalaxis passing through the geometrical center can be externally controlled. In this experiment theangular velocity is slowly varied so that the transient effects can be neglectd. Figure 1 also displaysthe cylindrical polar coordinates with unitary vectors (ˆ r, ˆ θ, ˆ z ), where ˆ r coincides with the basis ofthe container and ˆ z is a vertical axis through the center of the container. FIG. 1. A liquid in a prismatic container with a free surface, z (cid:48) ( r ), mounted on a rotating table with angularvelocity ω displays a parabolic shape. The figure also indicates the definition of the coordinate axes in therelative system. Under these assumptions, the velocity field is that of a rigid body and can be expressed as (cid:126)u = ωr ˆ θ while the acceleration (cid:126)a = − ω r ˆ r . To obtain the pressure field, after substituting theseexpressions in the Eq. (1) we obtain − ω r ˆ r = − ∇ pρ − g ˆ z. (2)where in the case of an axisymmetric field the gradient can be written as ∇ p = ∂p∂r ˆ r + ∂p∂z ˆ z. (3)The pressure field can be easily integrated to obtain p ( r, z ) = − ρgz + ρω r C (4)where C is a constant of integration with dimensions of pressure. The equation of the free surface, z (cid:48) ( r ), is obtained using the constraint that the pressure corresponds to the atmospheric pressure p atm and results z (cid:48) ( r ) = r ω g − p atm ρg + Cρg . (5)The constant C can be obtained using the mass conservation and the fact that the fluid is incom-pressible, HL = 2 (cid:90) L/ z (cid:48) ( r ) dr = 2 (cid:90) L/ (cid:18) r ω g − p atm ρg + Cρg (cid:19) dr. (6)Performing the integral we get the expression for CC = p atm + ρgH − ρω L . (7)Finally, the pressure field inside the fluid can be expressed as p ( r, z ) = p atm + ρg ( H − z ) + ρω (cid:18) r − L (cid:19) (8)where we can appreciate static and dynamics contributions. The free surface can be finally ex-pressed as z (cid:48) ( r ) = H − ω g (cid:18) L − r (cid:19) (9)We notice that the concavity and the location of the vertex of the parabole depend on theangular velocity. The vertex of the parabole, given by r = 0 is located at z v = H − ω L g . (10)When the angular velocity is ω ≥ √ gH/L the parabole vertex reaches the bottom of the containerand these expressions are not longer valid. It is also interesting that there are two nodal pointsgiven by z (cid:48) ( r ) = H with r = ± L/ √
12 that always belong to the free surface.
III. EXPERIMENTAL SET-UP AND DATA PROCESSING
The experimental setup shown in Fig. 2 consists of a prismatic container and a smartphone,model LG-G2 (with digital camera and built-in gyroscope), both of them fixed to a rotating table.The dimensions of the container containing dyed water were 25 cm width, 15 cm height and 2cm thickness. The rotating table was powered by a DC motor so the rotational speed could beadjusted by varying the voltage applied to the motor.Initially, with the rotating table at rest we turn on the video camera and we start recordingthe measures provided with the gyroscope and proximity sensors with the Androsensor app . Tosynchronize the video and the and data provided by the app , we cover a few seconds simultaneouslythe lens of the camera and the proximity sensor. In this way we obtain a common time referencefor the video and sensor data.
FIG. 2. Experimental set-up composed of a narrow container mounted on a rotatory table. The smartphone,also fixed to the rotating system, supplies both the video of the of the time-evolving surface and the angularvelocity obtained with the gyroscope.
Then, we turn on the power supply and the rotating table starts rotating. Throughout theexperiment the power supply is slowly increased by small jumps and so does the angular velocity.Figure 3 shows the temporal evolution of the angular velocity. The blue arrow indicates the timeused to synchronize the video and the sensor data. By the end of the experiment both files, videoand sensor data, are transferred to a computer. The accompanying video abstract shows a synopsisof the experimental video. The video is available at https://youtu.be/6SsX16rNoVE.
FIG. 3. Temporal evolution of the angular velocity. The appreciable jumps are produced by the operatorregulating the DC power supply. The blue arrow indicates the instant in which the hand uncover the cameraand the proximity sensor to register a mark to synchronize the video and the gyroscope sensor.
To analyze the characteristics of the time-evolving surface, firstly, we extract the individualframes from the digital video. Next, we select 15 frames, corresponding to different values of theangular velocities displayed in Fig. 3. Each frame was analyzed with the video analysis softwareTracker [12]. Several points, tipically 8, on the interface were manually labeled and, then, weperform a parabolic fit, y = Ax + Bx + C . The coefficient A corresponds to the concavity of theparabole. From the coefficients B and C the height of the vertex, H = − B / (4 A ), can be readilyobtained. IV. RESULTS
After the data were processed, we compared the results for the concavity and the height of theparabole with the model prediction. Figure 5 shows the experimental relationship between theconcavity of the parabole and the angular velocity and the model prediction. The slope of thelinear fit 20 . · rad /s , according to Eq. 9, is with good agreement 2 g .The height of the parabole vertex as a function angular velocity squared is plotted in Figure 6.The slope of the linear fit results − . · s /rad which is very similar to the value, according FIG. 4. Tracker screen-shot showing one frame of the digital video with the free surface and the pointsselected (left). The right panel shows the parabolic shape and the coordinates of the selected points.FIG. 5. Relationship between the angular velocity and the concavity of the fluid surface fitted to a parabole.The points indicate the experimental results and the solid line the model prediction. The slope of the linearfit shown in inset is 20 . · rad /s . to Eq. 10, given by the model − L / g = − . · s /rad . In addition, the interceptcorresponds to the water level with the rotating table at rest. In the experiment, the value obtainedis − . − . FIG. 6. Height of the parabole vertex as a function of the angular velocity squared. The red line indicatesa linear fit (slope − . · s /rad and intercept − . V. CONCLUSIONS AND PROSPECTS
The present proposal aims at experimenting with the free surface of a liquid rotating with a timedependent angular velocity. Thanks to a smartphone both the shape of the surface and the angularvelocity are simultaneously measured. Using video analysis software we obtain the coefficients ofthe parabolic profile which can be related to the angular velocity, the gravitational acceleration andthe water level with the rotating table at rest. The present experiment yields very good agreementwith the theoretical model. This simple and inexpensive proposal provides an opportunity forstudents to engage with challenging aspects of fluid dynamics without sophisticated or expensiveequipment.
ACKNOWLEDGMENTS
We are very grateful to Cecilia Cabeza for productive discussions. This work was partiallysupported by the program
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