The Mathematics of BubblesPayal DesaiPayal DesaiLucy ChenLucy Chen

Ishika AroraIshika Arora3*3*

The Shape of Bubbles

• Ideally: bubbles are sphericalIdeally: bubbles are spherical

• Shape of bubbles determined by “constraints”Shape of bubbles determined by “constraints”– Shape of the bubble wand (not always flat)Shape of the bubble wand (not always flat)

– Volume of the air in the soap bubbleVolume of the air in the soap bubble

– Hard surfaceHard surface

– GravityGravity

• Bubbles always adjust to the constraints to Bubbles always adjust to the constraints to minimize SAminimize 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%203-5/3-5%20Color%20and%20Light/Bubbles%20Pre.pdf>.

"Floating soap bubbles." Soapbubble.dk. 28 May 2009 <http://www.soapbubble.dk/en/bubbles/floating.php>.

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"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>.