Dimensionless Parameters

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An Introduction to Dimensionless Parameters in the Study of Viscous FluidFlowsDavid Guerra , Kevin Corley , Paolo Giacometti , Eric Holland , Michael Humphreys et al.

Citation: Phys. Teach. 49 , 175 (2011); doi: 10.1119/1.3555507 View online: http://dx.doi.org/10.1119/1.3555507 View Table of Contents: http://tpt.aapt.org/resource/1/PHTEAH/v49/i3 Published by the American Association of Physics Teachers

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DOI: 10.1119/1.3555507 THE PHYSICS TEACHER V ol. 49, M ARCH 2011 175

dratic fit function, included with the data gathering softwis used to fit a curve to the section of data enclosed in brets. The measured drain time is found by following the cfit to a point where it crosses the location on the graph thequal to the original height of the water in the bottle.

In the original laboratory exercise,3 water was used as thefluid and its relatively low viscosity allowed the use of Bnoullis law and the continuity equation to analyze the sWith these expressions, an expression for the time (t d) it takeswater to drain out of a hole drilled in a bottle cap screwe

onto the inverted bottle was developed. The expression drain time (t d),

(1)

4

4

2* 1 ,d

H Dt

g d

=

depends on the parameters of the experiment, such as theight of the liquidH , the diameter of the bottleD, thediameter of the hole in the capd , and the local accelerationdue to gravity g as demonstrated in Fig. 3. This expressiowas derived with no consideration of viscosity.3

An example of the data is presented in Fig. 2. The ori

height of the water in the bottle was 20 cm; therefore, thtance on the vertical axis from the starting point of 0.18 the blue dashed line at 0.38 cm is the 20-cm span. The pwhich the best-fit curve, in black, intersects the blue-da

An Introduction to DimensionlessParameters in the Study of ViscousFluid FlowsDavid Guerra, Kevin Corley, Paolo Giacometti, Eric Holland, Michael Humphreys, and Michael Nicotera,Saint Anselm College, Manchester, NH

It has been suggested that there is a need to deepen theunderstanding of fluid dynamics in the introductoryphysics course and to offer interesting experiments todo so.1To address this need we have developed a labora-tory experiment and the supporting analysis to demonstratethe role of viscosity and the interestingly mysterious use ofdimensionless parameters in fluid dynamics.2Since viscosityindicates the frictional dependence between the layers of aflowing fluid, a thoughtful student may ask why or when vis-cosity can be neglected. The laboratory experiment presented

here uses common fluids to provide a concrete answer to thisquestion and an easily understandable example of the role ofdimensionless parameters in fluid dynamics.

ExperimentThe experimental arrangement utilized in this laboratory

exercise, shown in Fig. 1, was previously presented in this jour-nal.3As previously described, a motion sensor is set up abovean inverted 2-L plastic bottle with the bottom cut off. Bottlecaps with holes of different diameters are screwed on the bottleso that the size of the exit hole can easily be adjusted. As thefluid in the bottle drains into a bucket positioned below the

bottle, the motion sensor detects the position of the top of thefluid and graphs a position-versus-time graph of the watersmotion. An example of the data taken with the device, withwater as the fluid, is displayed in Fig. 2 (red curve). The qua-

Fig. 1. The experimental set-up consists of an invertedsoda bottle with a set of bottle caps with different sizedholes and a motion sensor to monitor the water level.

Fig. 2. Plot of the position of the top of the water draining out of aninverted bottle with a cap hole size of d = 0.89 cm, an initial height ofwater of 20 cm, and a bottle diameter D = 11 cm.



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176 THE PHY SICS T EACHER V ol. 49, M ARCH 2011

H

D

z

d v 2

v 1

Fig. 3. Diagram of the experimental setup.

presented in Fig. 2, it is clear that viscosity is insignifica

since an analysis that ignores viscous effects correctly pthe drain time of the system. If liquids with higher viscoand/or bottle caps with smaller holes are used, it eventubecomes impossible to match the drain time computed wEq. (1) and the measured drain time. Even with the marfreedom in adjusting the part of the data curve in whichcurve fit is set, the match must still fail for a negative rebe recorded. For example, the predicted drain time with(1) for a cap hole size ofd = 0.2 cm, an initial height of fluid16 cm, and a bottle diameter D = 11 cm is 546.62 s = 9.11 mAs demonstrated in Fig. 4, with the computed drain tim9.11 min, indicated by the orange dashed line, and the lo

tion of the original height of 16 cm, indicated by the dasblue line, it is clear that no adjustment of the line fit segmwill produce a match of the computed and measured dratimes. This failure to match the drain times in the experindicates that viscosity is too large to ignore, and conserBernoulli-based derivations of the drain time are not appriate. This result provides an answer to the question, w viscosity too large to ignore? It is interesting to note thanot simply a value of viscosity at which it becomes signbut that a combination of factors, which includes the visand density of the fluid and the size of the exit hole, detemines when viscosity is significant.

Upon studying the list of the failures and successes omatching the computed versus measured drain time for fluids used in the lab with different cap hole diameters (d )in Table II, it is clear that as the viscosity increases the sthe exit hole at which viscosity is significant increases. water, which has a relatively small viscosity compared tother liquids used, we were able to match the computedmeasured drain times for cap hole sizes down to the smawe could drill. For the other, more viscous fluids, failurmatch the drain times occurred at different size caps thanot seem to follow the scaling indicated by the viscosityNotice that both the corn oil and the pancake syrup-wat

line indicates the drain time, when the orange dashed line isfollowed down to the horizontal time axis. With slight adjust-ments to the portion of the curve chosen for the curve fit, thecomputed drain time matches well with the measured drain

time at the level of precision proper for an introductory lab.In the case displayed in Fig. 2, the computed and experimentaldrain times are 30.86 s and 30.75 s, respectively. This is lessthan a 1% difference.

In the laboratory exercise presented in this paper, we usethe ability to match the predicted and experimental draintime as a measure of the significance of the viscosity of thefluid since viscosity is omitted from Bernoulli-based deriva-tions.4 The first step in the new exercise is to measure viscosi-ties and densities of the fluids to be used in this experiment.Thus, the viscosities and densities of maple syrup, corn oil,and a mixture of pancake syrup and water are listed in Table I.

The densities were measured by finding the mass of a known volume of the fluid. We found that the most consistent andsimple method to measure the viscosities of these fluids iswith an Ostwald viscometer. Following a standard procedure5based on Poiseulles equation,6 we obtained the viscosities ofthe common fluids used in the lab. The measured values fallwithin ranges of viscosities given in charts found on the Inter-net;7 thus, viscosity values from reliable sources could be usedif no method for measuring viscosity is readily available. Inaddition, there are several papers that discuss other methodsto measure8 and demonstrate9 viscosity that can be used.

In the position-versus-time graph of the draining water

Water Corn oil Pancakesyrup +

H2O

Maplesyrup

Viscosity(Pa*s)

0.001 0.075 0.099 0.175

Density(kg/m 3 )

998 903 1230 1300

Table I. Viscosities and densities of the fluids used in the labora-tory exercise.

Cap holed (cm)

Water Corn oil Pancakesyrup +

H2O

Maplesyrup

0.1 P F F F

0.146 P F F F

0.2 P P(close) F(close) F0.278 P P P F(close)

0.355 P P P P

0.45 P P P P

0.87 P P P P

1.61 P P P P

Table II. The failures (F) and successes (P) of matching the com-puted and measured drain-time for the fluids used in the lab withdifferent cap-hole diameters are presented. The additional labelof (close) indicates some indecision in the judgment of success-ful matching of the data to the predicted drain-time.



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the resulting value has no dimensions. So if the densitywater is entered in kg/m3, the velocity of the water flowinthrough the pipe in m/s, the length of pipe in m, and th viscosity of the water in Pa*s = kg/(m*s), the combinaof units in the Reynolds yields a number with no unitsThrough experiment, values of the Reynolds number lthan 2000 (Re < 2000) predict that the flow of a fluid wil

laminar, and for values of the Reynolds number over 300(Re > 3000) the flow will be turbulent. Thus, a transitioregion exists for Reynolds numbers between 2000 and(2000


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178 THE PHY SICS T EACHER V ol. 49, M ARCH 2011

students could try several different fluids to prove that thN value at which viscosity can no longer be considered innificant depends not only on the viscosity, but the param

of the experimental arrangement. Either way, this exercilustrates the predictive powers of dimensionless paramefluid dynamics in a simple laboratory arrangement.

Appendix: Buckingham Theory

Another way to find the dimensionless parameter forexperiment is the Buckingham theory.11 The basic idea ofthis mathematical system is to find the combination of vables that generates a dimensionless parameter. The keychoosing one base quantity; in the case presented belowcosity () is chosen since it is the focus of the study prese

in this paper. The other three quantities chosen were den( ) the diameter of the exit holes (d ), and the acceleration dueto gravity ( g ). This choice of quantities, which will be raisto the powers , , and and combined in a way such thatthe result of the product of viscosity, density, diameter, aacceleration due to gravity result in a unit-less quantity, resented in the Buckingham theory as the product of thesequantities set equal to the product of the parameters of le(L), time (T ), and mass ( M ) all raised to the zero (0) power:

( * *d * g g )=L0T 0 M 0. (A.1)

The first step in the procedure is to substitute for each othe chosen quantities, , d , and g the combination of theparameters of length (L), time (T ), and mass ( M ), which rep-resent the units of the quantity. For example, since densis mass per volume, the quantity ( M /L3) is substituted in theplace ofr in Eq. (A.1):

( M /LT )( M /L3) (L) (L/T ) = L0T 0 M 0. (A.2)

The next step is to write equations that represent the exnents of the parameters of length, time, and mass for E

fluid, which are the forces we believe control the flow of fluidin our system. Substituting in the expression forFr andRe,the expression ofN in terms of the parameters of the experi-

ment isN =Fr /Re =v /( gL)/ ( vL)/ =/(rL3/2 g 1/2). (10)

Another way to develop this same dimensionless parameterfor this system is with the Buckingham theory, which isoutlined in the appendix of this paper. This provides a math-ematically more rigorous technique for finding a dimension-less parameter.

ConclusionBased on our assertion of the controlling forces,N should

be a predictor for the significance of viscosity in the flow offluid through the system in the exercise. It is this relation-ship between a dimensionless parameter and the behavior ofa liquid that is mysteriously interesting. As can be seen forcorn oil, maple syrup, and the watered down pancake syrup inTable III, the three fluids agree onN values of approximately0.46 when the viscosity of the fluid flowing through the spe-cific hole size is definitely significant and cannot be neglected.The transition from insignificant viscosity to significant vis-cosity occurs atN values of approximately 0.29. ForN valuesless than 0.25, it is clear that viscosity is insignificant. Thisbracketing behavior of the dimensionless parameter is not

surprising since it was developed in part from the Reynoldsnumber, which also exhibits a transition section betweenlaminar and turbulent flow.

There are several options for implementing the laboratoryexercises presented in this paper. For a given fluid the studentscould first calculate at what cap size the data will no longerfit a drain time predicted by a Bernoulli-based calculation.They would then run several trials, starting with larger caphole sizes going down to smaller cap hole sizes, to find the ex-perimentalN at which the data can no longer be fit to a curvethat intersections the theoretical drain time. For a longer lab,

Cap holed (cm)

Water Corn oil Pancake syrup +H2O

Maple syrup

N P/F N P/F N P/F N P/F0.1 0.0102 P 0.805 F 0.813 F 1.35 F

0.146 0.0058 P 0.46 F 0.46 F 0.77 F0.2 0.0036 P 0.29 P(close) 0.29 F(close) 0.48 F

0.278 0.0022 P 0.17 P 0.18 P 0.29 F(close)

0.355 0.0015 P 0.12 P 0.12 P 0.20 P

0.45 0.011 P 0.084 P 0.085 P 0.14 P

0.87 0.0004 P 0.031 P 0.032 P 0.053 P

1.61 0.00016 P 0.012 P 0.013 P 0.021 P

Table III. A list of the failures (F) and successes (P) of matching the computed vs measured drain times forthe fluid used in the laboratory exercise with different cap-hole diameters and the value of the dimension-less parameter N .



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THE PHYS ICS TEA CHER V ol. 49, M ARCH 20 11 179

5 A. Halpern, Experimental Physical Chemistry: A LaboratoryTextbook, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 1pp. 295305.

6 Ref. 4, pp. 350-351.7. www.research-equipment.com/viscosity%20chart.html.8. R. Digilov and M. Reiner, Weight-controlled capillary v

eter, Am. J. Phys. 73 , 10201022 (Nov. 2005); L. Courbin etDesign of a low cost Zimm-Crothers viscometer: From tto experiment, Am. J. Phys. 73 , 851855 (Sept. 2005).

9. J. Libii, Demonstration of viscous damping in the underate laboratory, Am. J. Phys. 68 , 195198 (Feb. 2000).

10. R. Serway, Physics for Scientists and Engineers,3rd ed. (SaundersCollege Publishing, Philadelphia, 1992), pp. 421413.

11. A. Alexandrou,Principles of Fluid Mechanics (Prentice Hall,Upper Saddle River, NJ 2001), Chap. 6, pp. 214232.

David V. Guerra is a professor of physics at Saint Anselm College inManchester, NH. Along with his work involving students in the development of laboratory exercises, he conducts research in laser developmentand applications of lasers, such as lidar (laser radar).

Department of Physics, Saint Anselm College, 100 Saint Anselm

Drive, Manchester, NH 03102; [email protected]

Kevin Corley graduated from Reading Memorial High School in 2003 andreceived his BA in applied physics from Saint Anselm College in 2007.Kevin has been working at Covidien in Mansfield, MA, since 2007 as aresearch engineer in Advanced Wound Care Research and Development.

Paolo Giacometti graduated from Saint Anselm College in 2009 with aBA in applied physics and a certificate in computational physical science.He is currently at Dartmouth College pursuing his PhD in mechanicalengineering.

Eric Holland graduated from Saint Anselm College in 2009 with a BA inapplied physics. He is now a graduate student at Yale University pursuinga PhD in physics.

Mike Humphreys is a mathematics and physics teacher at Xavier HighSchool in Middletown, CT. He received his BA in applied physics fromSaint Anselm College in 2008. Along with teaching at Xavier, Mike is thecoach of the JV hockey team, as well as JV lacrosse and moderates theEngineering Club.

Michael Nicotera graduated from Revere High School in 2003 andreceived his BA in applied physics from Saint Anselm College in 2007.Michael is currently a field engineer for William A. Berry & Son, Inc., aconstruction management company based in Danvers, MA.

(A.2). For example, in the first term of Eq. (A.2), mass israised to the 1 and in the second term it is raised to , andit appears in no other terms on the left-hand side and has apower of zero on the right-hand side of the equation for theexponents, or M is

M : 1+ = 0. (A.3)

The equations for the other to two parameters are found ina similar manner, with care taken to include the sign of theexponents in the summation.

T : 1 + 2 = 0 (A.4)L: 1 +3 + + = 0. (A.5)

These three equations are solved for the three unknowns of , , and in the following steps.

From Eq. (A.3): = 1.

From Eq. (A.4): = 1/2.

With and known, Eq. (A.5) can be solved to give : = 3/2, which results in the same dimensionless parameter:N = -1d -3/2 g -1/2 =/(rL3/2 g 1/2).

AcknowledgmentsWe would like to thank Mrs. Kathy Shartzer for her insightinto and help with the laboratory equipment and ProfessorJeffrey Schnick for his careful review of the manuscript.

References

1. V. Jokinen et al.,Apparatus for an easy demonstration of the ba-sic phenomena in fluid flow, Am. J. Phys. 64 , 12071209 (Sept.1996).

2. See V. Thomsen, Estimating Reynolds number in the kitchensink,Phys. Teach. 31 , 410 (Oct. 1993) for an interesting experi-ment.

3. D. Guerra, A. Plainstaid, and M. Smith, A Bernoullis law lab ina bottle,Phys. Teach. 43 , 456459 (Oct. 2005).

4. D. Giancoli,Physics for Scientists and Engineers with ModernPhysics, 3rd ed. (Prentice Hall, Upper Saddle River, NJ, 2000),pp. 345347.

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Dimensionless Parameters