Collapsing BubblesCollapsing Bubbles

Rachel BauerRachel BauerJenna BratzJenna Bratz

Rachel

IntroductionIntroduction

Bubbles have been entertaining children Bubbles have been entertaining children for centuries.for centuries.

Children blew bubbles through clay pipes Children blew bubbles through clay pipes back in the 1700’s.back in the 1700’s.

Today over 200 million bottles of bubbles Today over 200 million bottles of bubbles are sold each year!are sold each year!

Although fun, there is a direct Although fun, there is a direct mathematical reason for which they mathematical reason for which they appear—soap films seek to minimize their appear—soap films seek to minimize their surface energy, which means minimizing surface energy, which means minimizing surface area, making it a sphere.surface area, making it a sphere.

ProcedureProcedure

We suspended a tube off of a lab stand.We suspended a tube off of a lab stand. Grid paper (1cm blocks) was set up behind Grid paper (1cm blocks) was set up behind

the tube.the tube. Three different tubes were used--a Three different tubes were used--a

capillary tube, a straw and a large plastic capillary tube, a straw and a large plastic tubetube

Rachel dabbed the soap solution onto the Rachel dabbed the soap solution onto the tube and blew a bubble and Jenna started tube and blew a bubble and Jenna started and stopped the high speed camera to and stopped the high speed camera to capture the collapse of the bubble.capture the collapse of the bubble.

The camera took 60 frames per second.The camera took 60 frames per second.

Procedure (cont.)Procedure (cont.)

The videos were stored as a sequence of pictures.The videos were stored as a sequence of pictures. Pictures were put into Matlab and a program was Pictures were put into Matlab and a program was

used to find the least squares circle to fit the data used to find the least squares circle to fit the data points around the bubblepoints around the bubble

The average radius was then calculatedThe average radius was then calculated

DataData

Straw Trial 3:

Initial Bubble

After .45 sec

After .7 sec After .85 sec

Theory Theory

We want to begin modeling the We want to begin modeling the deflation of a soap bubble through a deflation of a soap bubble through a narrow tube.narrow tube.

By Poiseuille’s equation we know that By Poiseuille’s equation we know that the change in the gas volume with the change in the gas volume with respect to time is given by , respect to time is given by , where r is the radius, l is the length of where r is the radius, l is the length of the tube, is the viscosity of the air, the tube, is the viscosity of the air, and is the change in pressure. and is the change in pressure.

Theory (cont.)Theory (cont.)

The equation for the change in volume of The equation for the change in volume of the bubble is also given by where the bubble is also given by where R is the radius of the bubble.R is the radius of the bubble.

By the Laplace-Young Law we have By the Laplace-Young Law we have since there are two surfaces of the bubble.since there are two surfaces of the bubble.

Setting the two equations equal and Setting the two equations equal and separating variables we get the following separating variables we get the following equationequation

with initial condition .with initial condition .

Theory (cont.)Theory (cont.)

Solving this differential equation we findSolving this differential equation we find

The radius was then calculated using a The radius was then calculated using a Matlab program that takes points Matlab program that takes points around bubble and finds the least around bubble and finds the least squares circle to fit those points. squares circle to fit those points. (Thanks Derek!)(Thanks Derek!)

Next we wanted to calculate the surface Next we wanted to calculate the surface tension for each of the trials. tension for each of the trials.

AnalysisAnalysis We want to compare the actual radius we We want to compare the actual radius we

found for our data with the expected found for our data with the expected radius given by the theory.radius given by the theory.

Find the least squares curve that Find the least squares curve that approximates our data points.approximates our data points.

t

R

Analysis (cont.)Analysis (cont.)

Minimize the error between the square Minimize the error between the square of the sum of the expected of the sum of the expected (theoretical) radii and the actual radii: (theoretical) radii and the actual radii: , where , where

and n is the number of data and n is the number of data

points we have.points we have. Differentiating E with respect to we Differentiating E with respect to we

have: have:

Analysis (cont.)Analysis (cont.)

We want to find when is equal to 0.We want to find when is equal to 0. We plotted the functions for each trial We plotted the functions for each trial

in Maple and found the x-intercept.in Maple and found the x-intercept. This is the that minimizes the error.This is the that minimizes the error. Capillary Tube:Capillary Tube:

Trial 1: = .0198 N/m = 19.8 dynes/cmTrial 1: = .0198 N/m = 19.8 dynes/cm Trial 2: = .0225 N/m = 22.5 dynes/cmTrial 2: = .0225 N/m = 22.5 dynes/cm Trial 3: = .022 N/m = 22 dynes/cmTrial 3: = .022 N/m = 22 dynes/cm

Analysis (cont.)Analysis (cont.) Plastic TubePlastic Tube

Trial 1: = .00625 N/m = 6.25 dynes/cmTrial 1: = .00625 N/m = 6.25 dynes/cm

StrawStraw Trial 1: = .01335 N/m = 13.35 dynes/cmTrial 1: = .01335 N/m = 13.35 dynes/cm Trial 2: = .01316 N/m = 13.16 dynes/cmTrial 2: = .01316 N/m = 13.16 dynes/cm Trial 3: = .0141 N/m = 14.1 dynes/cmTrial 3: = .0141 N/m = 14.1 dynes/cm

Consistent within the same tube, inconsistent for Consistent within the same tube, inconsistent for different size tubes.different size tubes.

Straw trials seems to be the best, closest to Straw trials seems to be the best, closest to expected value (13-14 dynes/cm).expected value (13-14 dynes/cm).

Two Bubble TheoryTwo Bubble Theory

Extend the model to one with two Extend the model to one with two bubbles, one at each end of the tube:bubbles, one at each end of the tube:

Analysis will begin the same as Analysis will begin the same as above. We now just have two above. We now just have two bubbles with volumes Vbubbles with volumes V11 and V and V22..

Two Bubbles (cont.)Two Bubbles (cont.)

The change in the gas volume is:The change in the gas volume is:

The change in the volume of the two The change in the volume of the two bubbles is:bubbles is:

Two Bubbles (cont.)Two Bubbles (cont.)

The change in pressure will change. The change in pressure will change. We have .We have . Using the Laplace-Young Law we findUsing the Laplace-Young Law we find

Similar to before, set the two equations Similar to before, set the two equations for dVfor dV11 and dV and dV22 equal and plug in the equal and plug in the equations for the change in pressure.equations for the change in pressure.

Two Bubbles (cont.)Two Bubbles (cont.)

Find two coupled nonlinear first order differential Find two coupled nonlinear first order differential equations:equations:

The total volume in this system is The total volume in this system is and therefore, and therefore,

so our system has a conservation law—the so our system has a conservation law—the volume is a constant.volume is a constant.

Two Bubbles (cont.)Two Bubbles (cont.)

Phase Plane analysis of system of Phase Plane analysis of system of equations:equations: Steady-state occurs when RSteady-state occurs when R1 1 = R= R22..

RR1 1 > R> R22 : :

RR1 1 < R< R22 : :

Two Bubbles (cont.)Two Bubbles (cont.)

Directional FieldDirectional Field

x=R1, y=R2

Two Bubbles (cont.)Two Bubbles (cont.)

Since the volume is a constant, the equation Since the volume is a constant, the equation

gives the equation for the trajectories in the gives the equation for the trajectories in the

phase plane:phase plane:

x=R1, y=R2

Spherical CapSpherical Cap

Note as one bubble gets smaller, the shape Note as one bubble gets smaller, the shape changes from a sphere to a spherical cap.changes from a sphere to a spherical cap.

We can modify our model to take this into We can modify our model to take this into account.account.

Assume that RAssume that R11 > R > R22, then R, then R1 1 will increase will increase and Rand R2 2 will decrease as described above.will decrease as described above.

Find equations that model the time after RFind equations that model the time after R2 2

equals the radius of the tube.equals the radius of the tube.

Spherical Cap (cont.)Spherical Cap (cont.)

The equation for RThe equation for R1 1 will stay the will stay the same, since the shape stays same, since the shape stays spherical.spherical.

The volume of the spherical cap is The volume of the spherical cap is

therefore the change in therefore the change in volume of the cap is given by volume of the cap is given by

Spherical Cap (cont.)Spherical Cap (cont.)

We get the following system of We get the following system of equations:equations:

This system also has a conservation This system also has a conservation of volume law. , so we of volume law. , so we have have

Spherical Cap (cont.)Spherical Cap (cont.) The volume is constant, so the equation The volume is constant, so the equation

gives the equation for a gives the equation for a trajectory in the phase plane of this trajectory in the phase plane of this system.system.

a = R1, b = R2

Spherical Cap (cont.)Spherical Cap (cont.) We can plot both phase planes and both trajectories to see We can plot both phase planes and both trajectories to see

the difference after Rthe difference after R22 equals the radius of the tube. equals the radius of the tube.

We start with RWe start with R1 1 > R> R22, R, R11 will increase along the black will increase along the black trajectory until it reaches the point where Rtrajectory until it reaches the point where R22 is equal to the is equal to the radius of the tube, then Rradius of the tube, then R11 will follow the green trajectory and will follow the green trajectory and will not increase as much as it would have if it followed the will not increase as much as it would have if it followed the original trajectory.original trajectory.

x=R1, y=R2

ConclusionConclusion

In general, our surface tension was In general, our surface tension was not consistent throughout our not consistent throughout our different trials. Possible reasons for different trials. Possible reasons for error:error: Air hitting the bubble Air hitting the bubble Bubble not remaining steadyBubble not remaining steady Measurement errorMeasurement error

Conclusion (cont.)Conclusion (cont.)

In the two bubble case, the theory matched In the two bubble case, the theory matched small experimental results from class.small experimental results from class.

Future work:Future work: More experiments to find additional surface More experiments to find additional surface

tension values.tension values. Extend two bubble case to n bubblesExtend two bubble case to n bubbles Attempt to isolate bubble from any disturbances Attempt to isolate bubble from any disturbances

in the labin the lab Complete experiments to further verify two Complete experiments to further verify two

bubble casebubble case