Parameters Affecting Whether Water Will Flow Out from an Inverted Open Bottle Tsunetaka Sasaki' 2-15-21 Yagumo, Meguro, Tokyo 152 We are quite familiar with the fact that water does not flow out from a bottle when i t is inverted with its mouth left open if the mouth is small enough in size, as in the case of a pipet or a buret. The question then arises as to what is the maximum size of the mouth that will still prevent the outflow of water. However, a precise study concerning such phenomena cannot he found to date as far as the present author is aware, except for our preliminary paper concerning such phenomena that does not contain detailed calculation~.~ The present paper describes the calculation of such a maximum size based upon the equilihrium between gravity and surface tension acting upon water when it flows out. Experimental confirmation of the results of the calculation was also attempted by using a series of test tubes varying in diameter. Mode of Inverting Test Tube Two different modes of inverting the test tuhe containing water were used. Test Tube with Cover Plate. A test tube containing water is first covered with a plastic plate and turned upside down, with the plate held in place. The plate is them removed horizontally, and the observation of whether water flows out is made. Test Tube without Cover Plate. The test tube containing water up to the brim is slowly tilted without a cover plate from an upright position through a horizontal t o an upsidedown position, and the similar observation is made. ~he.calculationsareconducted under the assumption that the thirkness of the wall of the test rube is nealiaible com. pared with the diameter of the tube. Test Tube wlth Cover Plate Figure 1shows the sideview of the lower end of the inverted test tuhe containing water. The straight line AB indicates the circular surface of water just after the removal of the cover plate. The outflow of water begins from one half of the water surface, while air enters the tube from the other half. Nowwe assume, for the sake of simplicity, that initial plane of the water surface, AB, becomes S-shaped as indicated by APOQB, P and Q being the lowest and highest points of the surface and Po and Qo being the respective points in the initial plane surface. For such a deformation or the outflow of water, only the hydrostatic pressure, F h , due to gravity, which impels the water downward, and Laplace pressure, Fa,due to the surface tension which counteracts the outflow, are considered. Laplace Pressure. Laplace pressure acting on the water surface is expressed by
Figure 1. Side view of inverted test tuba containing water. AP&IQoB. initial plane water surlaface: APOQB, outflowing water surface: x, maximum distance of surface flow; R radius of the tube: 0, center of the circle.
Figure 2. Bottom view of the test tube shown in Figure 1. CP&, straight line vertical to diameter AOB; other notations are the same as those in Figure 1.
where a denotes surface tension of water, and r and r' the principal radius of curvature a t the point P in Figure 1. A similar pressure hut opposite in direction acts a t the point Q. ~ h e " a l u e s r a n d+;in be estimated with the aid f; ~ i ~ u i e 2, whichis the bottom view of the tubeshown in Fimre 1: the notations used are the same, except for CD whichdenotes a straight line passing through Puand is vertical to the line AB. I t can readily he seen that r a n d r' at point P are the values for the curves APO and CPD, respectively. Under the assumption that these twocurves are arcs of different radiiand have a middle point P in common, r and r' are given by the following equations,
with ~ = R / 2 , ~ = & 7 R /andx 2,
=m==
Putting the values of eqs 2 and 3 in eq 1yields Emeritus Professor of Tokyo Metropolitan University. Sasaki. T. Chem. Educ. (Chem. Soc. Japan) 1987, 35.464-465. (In Japanese.) Volume 66 Number 12 December 1989
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assumed to proceed in an S shape similar to that of Figure 1, and x increases with the angle of tilt. Then the maximum hydrostatic pressure
Figure 3. Hydmstatic and Laplace pressures. 5, hydrostatic pressure as a function of outflow x: F,, Laplace pressure as a function of x and radius R increasing hwn A Mraugh D.
The value of 2 s in eq 4 instead of s in eq 1 comes from the fact that the pressures at P and Q both acting to resist the outflow of water have been summed. HydrostaticPressure. As is evident in Figure 1, hydrostatic pressure Fh is expressed by Fh = 2xpg
(5)
where p denotes the density of water and g the gravity constant. It is pertinent to add that the atmospheric pressure which is often stated to counteract the outflow of water3 can be disregarded in this case, since it acts both to resist the outflow of water a t the point P and to aid the entrance of air at the point Q; thus, two effects manually cancel. Calculation of Maximum Diameter. Figure 3 shows eq 4 as functions of x and R, and eq 5 as a function of x . In this figure, eq 5 is a straight line passing through the origin with the slope of 2pg, while curves A, B, C, and D show eq 4 in the increasine order of R arbitrarilv chosen. It is co&rmed in Figure 3 that in the range of small x , F, is larger than Fh for smaller value of R as shown bv curves A and B, while F, is smaller than Fh for larger val"es of R as shown by the curve D. This means that water does not flow out of the inverted tube in the cases of A and B, while the outflow occurs in the case of D. Thus the critical condition of the outflow or the maximum diameter Dmof the test tube hindering the outflow of water can be determined from the radius R of the curve C, which has the same slope at the origin as that of eq 5, namely Introducing the condition of x2 D m> 1.7 cm could be confirmed, which is in agreement with the result of calculation. More precise confirmation of Dm may be possible, but considering the assumptions and approximations adopted in the calculation, the confirmation described above may be considered sufficient. A similar measurement in the case of the test tube without the cover plate gave the result of 1.4 cm > Dmf> 1.3 cm which is also considered satisfactory as the confirmation of the calculation results. Summary The maximum diameter Dm of the open test tube from which water does not flow out when the tube is inverted was calculated. In the case of the test tube with acover plate that was removed after inversion. calculation cave D- = 1.77 cm: while in the case of the test tube withoit the Gate, Dm' = 1.33 cm was obtained. Both of these values were ex~erimentally confirmed as 1.8 cm > Dm> 1.7 cm and 1.4 c k > D d > 1.3 cm, respectively.
