Collapsing%20Bubbles

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Collapsing%20Bubbles

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Rachel Collapsing Bubbles Rachel Bauer Jenna Bratz Introduction Bubbles have been entertaining children for centuries. Children blew bubbles through clay pipes back ... – PowerPoint PPT presentation

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Title: Collapsing%20Bubbles

1
Collapsing Bubbles
Rachel

Rachel Bauer
Jenna Bratz

2
Introduction

Bubbles have been entertaining children for
centuries.
Children blew bubbles through clay pipes back in
the 1700s.
Today over 200 million bottles of bubbles are
sold each year!
Although fun, there is a direct mathematical
reason for which they appearsoap films seek to
minimize their surface energy, which means
minimizing surface area, making it a sphere.

3
Procedure

We suspended a tube off of a lab stand.
Grid paper (1cm blocks) was set up behind the
tube.
Three different tubes were used--a capillary
tube, a straw and a large plastic tube
Rachel dabbed the soap solution onto the tube and
blew a bubble and Jenna started and stopped the
high speed camera to capture the collapse of the
bubble.
The camera took 60 frames per second.

4
Procedure (cont.)

The videos were stored as a sequence of pictures.
Pictures were put into Matlab and a program was
used to find the least squares circle to fit the
data points around the bubble
The average radius was then calculated

5
Data
Straw Trial 3
Initial Bubble
After .45 sec
After .7 sec
After .85 sec
6
Theory

We want to begin modeling the deflation of a soap
bubble through a narrow tube.
By Poiseuilles equation we know that the change
in the gas volume with respect to time is given
by , where r is the radius, l is
the length of the tube, is the viscosity of
the air, and is the change in pressure.

7
Theory (cont.)

The equation for the change in volume of the
bubble is also given by where R
is the radius of the bubble.
By the Laplace-Young Law we have since
there are two surfaces of the bubble.
Setting the two equations equal and separating
variables we get the following equation
with initial condition .

8
Theory (cont.)

Solving this differential equation we find
The radius was then calculated using a Matlab
program that takes points around bubble and finds
the least squares circle to fit those points.
(Thanks Derek!)
Next we wanted to calculate the surface tension
for each of the trials.

9
Analysis

We want to compare the actual radius we found for
our data with the expected radius given by the
theory.
Find the least squares curve that approximates
our data points.

R
t
10
Analysis (cont.)

Minimize the error between the square of the sum
of the expected (theoretical) radii and the
actual radii , where
and n is the number of data
points we have.
Differentiating E with respect to we have

11
Analysis (cont.)

We want to find when is equal to 0.
We plotted the functions for each trial in Maple
and found the x-intercept.
This is the that minimizes the error.
Capillary Tube
Trial 1 .0198 N/m 19.8 dynes/cm
Trial 2 .0225 N/m 22.5 dynes/cm
Trial 3 .022 N/m 22 dynes/cm

12
Analysis (cont.)

Plastic Tube
Trial 1 .00625 N/m 6.25 dynes/cm
Straw
Trial 1 .01335 N/m 13.35 dynes/cm
Trial 2 .01316 N/m 13.16 dynes/cm
Trial 3 .0141 N/m 14.1 dynes/cm
Consistent within the same tube, inconsistent for
different size tubes.
Straw trials seems to be the best, closest to
expected value (13-14 dynes/cm).

13
Two Bubble Theory

Extend the model to one with two bubbles, one at
each end of the tube
Analysis will begin the same as above. We now
just have two bubbles with volumes V1 and V2.

14
Two Bubbles (cont.)

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

15
Two Bubbles (cont.)

The change in pressure will change.
We have .
Using the Laplace-Young Law we find
Similar to before, set the two equations for dV1
and dV2 equal and plug in the equations for the
change in pressure.

16
Two Bubbles (cont.)

Find two coupled nonlinear first order
differential equations
The total volume in this system is
and therefore,
so our system has a conservation lawthe
volume is a constant.

17
Two Bubbles (cont.)

Phase Plane analysis of system of equations
Steady-state occurs when R1 R2.
R1 gt R2
R1 lt R2

18
Two Bubbles (cont.)

Directional Field

xR1, yR2
19
Two Bubbles (cont.)

Since the volume is a constant, the equation
gives the equation for the trajectories in the
phase plane

xR1, yR2
20
Spherical Cap

Note as one bubble gets smaller, the shape
changes from a sphere to a spherical cap.
We can modify our model to take this into
account.
Assume that R1 gt R2, then R1 will increase and R2
will decrease as described above.
Find equations that model the time after R2
equals the radius of the tube.

21
Spherical Cap (cont.)

The equation for R1 will stay the same, since the
shape stays spherical.
The volume of the spherical cap is
therefore the change in volume
of the cap is given by

22
Spherical Cap (cont.)

We get the following system of equations
This system also has a conservation of volume
law. , so we have

23
Spherical Cap (cont.)

The volume is constant, so the equation
gives the equation for a
trajectory in the phase plane of this system.

a R1, b R2
24
Spherical Cap (cont.)

We can plot both phase planes and both
trajectories to see the difference after R2
equals the radius of the tube.
We start with R1 gt R2, R1 will increase along the
black trajectory until it reaches the point where
R2 is equal to the radius of the tube, then R1
will follow the green trajectory and will not
increase as much as it would have if it followed
the original trajectory.

xR1, yR2
25
Conclusion

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

26
Conclusion (cont.)

In the two bubble case, the theory matched small
experimental results from class.
Future work
More experiments to find additional surface
tension values.
Extend two bubble case to n bubbles
Attempt to isolate bubble from any disturbances
in the lab
Complete experiments to further verify two bubble
case