Business Cards and Paper Airplanes

 

By Scott Little

 

In addition to being a member of SAS, I am also a member of the Society for Industrial

 

and Applied Mathematics (SIAM). While reading this month’s newsletter, I came across

 

an interesting article on the properties of falling paper using business cards (1). The

 

article is based on the research of mathematician Z. Jane Wang of Cornell University. Dr.

 

Wang has done extensive research on how insects actually fly. She takes a standard sized

 

business card, 2-1/2”x3” and drops it from an elevation to demonstrate the random

 

movements of an insect wing. By using a single card, she is, in her own words “setting

 

the wing free”. She then can study it’s aerodynamic properties.

 

Most objects that can fly, including airplanes, do so by taking advantage of the Bernoulli

 

Principle. The principle states, in simplified terms, that a fall in pressure of a fluid will

 

always be accompanied by an increase in speed (2). In the case of an airplane, the fluid is

 

the air, and the drop in pressure is caused by the difference in speed of the air moving

 

over the top and bottom of the wing. The air takes the same amount of time for it to

 

travel over the top or bottom of the wing, but the distance is longer for the top due the

 

wing’s curvature. This creates a drop in the pressure over the top of the wing, and the

 

higher pressure underneath pushes up on the bottom of the wing, causing lift. (Please see

 

figure 3 for an illustration of this effect.)

 

Mathematically, the Bernoulli Principle is represented by the following equation:

 

P+(KE/V)=C

 

Where:

P=pressure

KE= kinetic(moving) energy of the object

V= volume of fluid

C=constant

 

In the case of an airplane wing, the volume is normally calculated to be the cubic feet

 

of air under the wings. Although this can explain many types of flight, it does not account

 

for the random flight of certain insects. Dr. Wang believes that the same physical forces

 

that explain how creatures such as the Bumble Bee are able to fly also explains the

 

behavior of falling business cards. She uses a set of equations known as the Napier-

 

Stokes Equations to explain the movements of these fluttering objects. The Napier-Stokes

 

Equations are a set of differential equations that are used to calculate the flow of a fluid

 

around an object. In it’s most basic form, the general equation is (3):

 

 

r Dv/Dt = -▼p +▼· T+ f

 

The left side is the pressure of the fluid times the change in its’ speed over time.

 

The right side of the equation is the summation of all the forces acting on the body.

-▼p= the pressure gradient, that is, the sum of the dimensional pressure forces from the normal stresses in the fluid flow.

These dimensional forces are in the X, Y, Z directions and sometimes include time “t”.

▼· T= the shear forces of the fluid. These are forces acting at an angle to the pressure and could include cross winds, etc.

f= the other forces, including gravity.

 

As can be seen from the above explanation, the pressure is effected by the speed, time,

 

shear forces, and other forces, such as gravity, to explain lift. To do more precise

 

measurements, Dr. Wang developed some specialized aluminum plates the same size as a

 

business card. She then dropped these plates in a tank full of water and filmed their

 

movements with a high-speed camera.

 

For my own experiment, I went back to the basics and used the standard paper business

 

card and dropped them off my stairway. I then tracked their approximate movements and

 

distance traveled and plotted them onto a diagram.

 

Fig. 1 Courtesy of Scott Little

 

 

 

I held the card in two different positions, one as flat and the other as sideways from my

 

position. Both positions were tested 5 times to obtain some uniformity. All doors and

 

windows near the stairway were closed to minimize cross air currents.

 

In addition to the business cards, I decided to test the flight patterns of a flying wing. I

 

first saw this design when I was around ten. It was in a paper airplane book that my

 

cousin, who happens to be an engineer, had bought me. I was interested in the fact that

 

this simple folded paper had won a contest in staying aloft over so many more

 

complicated paper airplane designs. It seemed a good random variable to test, because

 

although it was almost flat, the wing utilized the Bernoulli Principle in it’s design.

 

All 3 test subjects were held 6 feet above the stairway and let go. The flying wing

 

traveled  the farthest at 17’-7”, and the flat card came in second with 7’.  The side card

 

came in last with 6’. The wing could have probably flown farther, but my stairway is only

 

4’ wide and did not allow for much lateral movement. As can be seen, the flat surface has

 

more area to be pushed against, and will stay aloft slightly longer.  The position of the

 

cards are shown here in figure 2.

 

 

 

 

 

 

Fig. 2 Courtesy of Scott Little

 

 

 

 

 

 

 

 

 

 

The flying wing design, as well as an illustration of the Bernoulli Principle, are shown

 

below.

 

 

Fig. 3 Courtesy of Scott Little

 

 

Although the wing did exhibit the more classical principles of flight, the business card

 

can also stay aloft, but for a shorter period. Perhaps it is combination of all the Bernoulli

 

and Napier-Stokes forces that give the flying wing it’s extra lift. But, as Dr. Wang had

 

concluded, a Bumble Bee could not fly by using the same principles a plane uses, so

 

there must other things at work.

 

By using the Napier-Stokes equations, she concluded that a fluttering piece of paper uses

 

the same physics to stay aloft as a dragonfly’s wings. These equations can also explain

 

how a falling leaf can momentarily rise with no wind to carry it aloft. She found a simple

 

way to test her theory; just drop a piece of paper.

 

In conclusion, there are more forces at work than just one to create lift in a piece of paper.

 

In my personal interpretation of the physics involved, there appears to be a more efficient

 

use of these forces in the flat business card. This could be due to the larger surface area

 

being exposed in the direction of descent, sort of a “parachute” effect.  It could also be

 

that as the flat card fell, it spun in a direction perpendicular to the falling, like a helicopter

 

rotor. The side card spun in the same direction, creating a slight down force.  I would like

 

to do more research on these theories, and perhaps plumb the Napier-Stokes Equations

 

for answers.

 

 

 

 

 

 

 

Sources

 

(1) Finn, David L ” Falling Paper and Flying Business Cards”, SIAM News, Vol. 40/ Number 4, May 2007.

 

(2) http://theory.uwinnipeg.ca/mod_tech/node68.html

 

(3) http://en.wikipedia.org/wiki/Navier-Stokes_equations