The Mathematics of Bubbles
Payal Desai
Lucy Chen
Ishika Arora
3*
The Shape of Bubbles
• Ideally: bubbles are spherical
• Shape of bubbles determined by
“constraints”
–
–
–
–
Shape of the bubble wand (not always flat)
Volume of the air in the soap bubble
Hard surface
Gravity
• Bubbles always adjust to the constraints to
minimize SA
• Minimal surface
– If the soap film does not enclose any air
– Has a total curvature of 0 at every point
• Minimal construction
– If the soap film encloses air -> bubble
– Total curvature at each point is larger than 0
– Bubble will minimize surface area given the
volume
Catenoid – A Special Shape
• Leonhard Euler
proved that the true
minimal surface
whose boundary
comprises two
identical parallel
circles is a catenoid
• Not as common as the
more spherical shaped
bubbles
How to make a catenoid
• Revolve f(x) = cos(x) from -.1 to .1 around
the x-axis
Toroidal Bubble
• Observed by Walters and Davidson in 1963
• Created by mass release of gas in water
• Found in nature
The Double Bubble Conjecture
Says…
• When two bubbles join together
(forming a double bubble)
– Because bubbles minimize SA, this
is the optimal shape for enclosing
and separating two chambers of air
(two bubbles)
– Will form a boundary between the
two bubbles
• Same size bubbles  inner curve
will be flat
• Different size bubbles  inner
curve will be curved
– Why?: Internal pressure
Pressure
• If pressures are even within the two bubbles
(same size bubbles), then the pressures will
create a flat wall
• The bigger the bubble, the lower the pressure
– Smaller bubble will have more pressure
– Wall will be pushed towards the larger bubble
• Laplace Equation: equation of pressure
inside a bubble
• R = radius
• Sigma = surface tension
Bubble Pressure
• pressure will be lower in a region with moving air, relative
to a region with slower movements
– Demonstrated by Bernoulli’s equation:
•
– For a small bubble the height across the bubble is
negligible and since the air under the bubble is not
moving (vunder = 0), then we can reduce the Bernoulli
equation to:
•
Electricity
• Soap bubbles can be
attracted by electrical
fields
• possible to make soap
bubbles float using static
electricity
• In an electric field all
water molecules will be
aligned with their positive
pole in direction towards
any negative charges
nearby
Angles
• All bubbles meet at 120 degrees
Rate of Thickness
• Suppose you are blowing a spherical
bubble. When the radius is 10 cm it is
expanding at 1cm/sec.
• At this same time the thickness is 0.1 cm.
At what rate is the thickness decreasing at
this time?
Why Your Bubbles Just Don’t Last
Long…
• The bubble mixture
– Regular water has a harsher, and rougher mineral
called calcium carbonate which makes it easier to
pop
– Impurities in the container decrease quality
• Weather
– Dry air (vaporization of water)
– Too much sunlight bakes them
• Foam
– Foam has many little bubbles, so if a little bubble
breaks, the big one will break too
Long-Lasting Bubbles
• Decrease mixture evaporation
• Jar method
– Put bubble in a jar on a ring of wire
– Close jar
– Humidity inside the jar will increase until the air is
saturated with vapor
• Flat bubbles are more resistant to popping
• If kept below 0 degrees C, bubbles can be
frozen
Color of Bubbles
• Soap bubbles have a
rainbow color because of
the reflection and refraction
of light
– Reflection: light is reflected
off of the surface at the same
angle that it hit the surface
(the angle of incidence equals
the angle of reflection)
– Refraction: as light passes
from one transparent medium
to another, it changes speed
and bends.
Color of Bubbles Cont.
• In a bubble:
– An inner and outer surface form the thin film creating
the bubble
• Various densities in the soap bubble will produce particular
wavelengths of reflected and refracted light.
• When white light travels through the bubble, certain colors will
be absorbed (refracted) and other colors will be reflected
according to the density of the medium through which the light
travels.
– We see the reflected light as color
– The shifting of colors in the bubble is caused by
changes in density, or thickness, in different parts of the
bubble
Color of Bubbles Cont.
The result?
Works Cited
"Air pressure in bubbles." Soapbubble.dk. 28 May 2009
<http://www.soapbubble.dk/en/bubbles/airpressure.php>.
"Bubbles - Color and Light." Whitaker Center. Whitaker Center. 28 May 2009
<http://www.whitakercenter.org/Education/Curriculum%20Guides/Grades%2
03-5/3-5%20Color%20and%20Light/Bubbles%20Pre.pdf>.
"Floating soap bubbles." Soapbubble.dk. 28 May 2009
<http://www.soapbubble.dk/en/bubbles/floating.php>.
"Frank Morgan's Math Chat - DOUBLE BUBBLE CONJECTURE PROVED."
Mathematical Association of America: MAA Online. MAA. 28 May 2009
<http://www.maa.org/features/mathchat/mathchat_3_18_00.html>.
"The geometry of soap films and soap bubbles []." Soapbubble.dk. 28 May 2009
<http://www.soapbubble.dk/en/bubbles/geometry.php>.
"HowStuffWorks "How Light Works"" Howstuffworks "Science Channel"
Science Channel. 28 May 2009
<http://science.howstuffworks.com/light13.htm>.
"Make your own soap bubble mixture." Soapbubble.dk. 28 May 2009
<http://www.soapbubble.dk/en/bubbles/bubblemixture.php>.
"Proof of the Double Bubble Conjecture." AIMS Journals. AIMS. 28 May 2009
<http://aimsciences.org/journals/pdfs.jsp?paperID=2394&mode=full>.