A Pioneering Pressure Principle
Janel Knepel, Melissa Radke, Nicole Statler, and Beth Wagner

 

             How does an airplane fly?  Why does my shower curtain get “sucked in” when I turn on the shower?  How do perfume bottles work?  Why do windows explode instead of implode during a hurricane?  How do curveballs in baseball work?  Surely, we all have asked at least one of these questions during our lifetime.  Most people, however, would probably not believe that all of these questions lead to the same answer:  Bernoulli’s principle.

Who is Bernoulli?

 

     Daniel Bernoulli was an eighteenth-century scientist.  He was born in the year 1700 in Groningen, Netherlands.  The Bernoullis were a mathematic legacy.  All the Bernoulli mathematicians were fathers of modern calculus.  Daniel’s father, Johann, worked extensively on differential equations, the specialty of the family, and the calculus of variations.  Nicolaus (II), Daniel’s older brother, also worked on differential equations, as well as curves and probability.  Daniel’s uncle, Jacob, worked on differential equations, in addition to infinitesimal geometry, the St. Petersburg paradox, and the development of Bernoulli numbers.  So much talent in one family stirred up bitter rivalries to be the best known of all the Bernoullis. 

            When Daniel was young, his father pushed him toward a business career as a merchant.  At 13, Daniel began attending Basel University in Basel, Switzerland, extensively studying philosophy and logic.  He received his baccalaureate degree in 1715 and his master’s degree the following year.  During this time, he also studied the methods of calculus under his father and his older brother Nicolaus.  After refusing an apprenticeship as a merchant, Daniel’s father sent him back to Basel University to study medicine.  He was determined not to let Daniel enter the field of mathematics.  He claimed there was no money to be made there.  Daniel completed his doctorate in medicine in 1720.  All through the years that he studied medicine, Daniel’s father continued to tutor him in mathematics.  Daniel used his knowledge of mathematics in his work as a doctor, later discovering how to measure blood pressure.

            At age 25, Daniel became a professor of mathematics, along with his brother Nicolaus, at the Imperial Academy in St. Petersburg, Russia.  Nicolaus died from tuberculosis a year later.  While in St. Petersburg, Daniel completed a great wealth of work.  He, like others in his family, worked extensively on differential equations.  However, the most important of his studies was his work on hydrodynamics.  In fact, the term “hydrodynamics” was coined from Daniel’s work that he produced on his findings:  Hydrodynamica. 

            Daniel returned to Basel in 1734.  After entering the same contest at the Paris Academy as his father and being declared a joint winner with him, he became banned from his father’s house.  This shows just how much jealousy festered within the family.  Daniel also continued to work on Hydrodynamica, which was finally published in 1738.  Among the things he discussed in his book Hydrodynamica was the fluid equation, which would come to be known as Bernoulli's principle.  The following year, Johann, Daniel’s father, published his work Hydraulica.  Deceitfully, he predated his book to 1732 instead of the real date, 1739, and claimed that Daniel had stolen all his ideas.  This, too, shows the extreme rivalry within the family.  In 1750, Daniel was appointed chair of physics at Basel University.  Before his death in 1782, Daniel won the Grand Prize of the Paris Academy ten times in the categories of astronomy and nautical topics. 

What is Bernoulli’s principle?

Bernoulli’s principle, also known as Bernoulli’s Law of Pressure Differential, states that as the speed of a fluid increases over a surface, its pressure against the surface decreases, and vice versa.  The equation developed by Daniel Bernoulli to demonstrate this is:

Pressure + (kinetic energy / volume) = constant

In order for this to remain true, the following must conditions must occur:

      Fluid is inviscid (not thick or sticky)

      Fluid is incompressible (traveling at a low velocity, less than the speed of sound)

      Flow is steady

      No heat addition

      Negligible change in height

If any of these do not hold true to the situation, a derivation of Bernoulli’s equation must be used instead.  For example, modified Bernoulli equations exist to encompass energy lost as a result of friction, as well as other energy losses.

            Daniel Bernoulli discovered this principle while studying the conservation of energy in liquids.  He saw that water flowing through a pipe moved faster when the pipe’s diameter was reduced.  He figured that some force was acting upon the water to produce this reaction.  He concluded that this occurred as a result of differences in pressure.  Bernoulli discovered that the pressure of a liquid or gas is lowest when its velocity is the highest and vice versa.  Bernoulli’s principle applies to any fluid, including air.

How does Bernoulli’s principle work?

            To understand how Bernoulli’s principle works, we can look at real world examples.  One such example is the miracle of flight.  For years, humans had tried to duplicate the flight of birds, and many of those who attempted to fly met failure.  It was not until 1903 when Orville and Wilbur Wright first mastered human flight at Kill Devil Hills, North Carolina.  Humans had possessed the knowledge of how flight occurs for over 200 years, but it still took that long to build an ample flight machine.

            So how do airplanes work?  The answer is rather simple.  The curved upper surfaces of airplane wings cause the air flowing on top of the wings to move faster.  The air moves faster because the curved wing shape makes the upper part of the air stream to travel farther in the same amount of time as if it was passing underneath the wing.  According to Bernoulli’s principle, since the air above the wing is moving faster than the air below, the air below has a greater pressure than that above the wing.  As a result, the airplane is able to be lifted off of the ground.  This idea is illustrated in the figure at the left.  This theory applies to ski jumpers as well.  Similarly, it also explains how curveball pitches in baseball are able to curve.  The exception is that, instead of having pressure differences above and below the ball, the pressure differences lay on the sides of the ball.  These pressure differences are created by the rotation of the ball in the air, with air being pulled along by the stitches of the ball.

Another example dealing with the application of Bernoulli’s principle occurs in your very own bathroom.  Ever notice how a shower curtain seems to float inwards while the showerhead is running?  This, too, can be explained by Bernoulli’s principle.  When the water is turned on, the increased velocity of the water traveling through the air creates a pressure drop on the inside of the curtain.  Consequently, the greater pressure on the outside of the curtain pushes it inwards, making it appear as if the shower curtain is being “sucked into” the shower.

The operation of a perfume bottle also applies Bernoulli’s principle.  By squeezing the bulb of the bottle, the speed of the air inside is increased, and a low pressure area now exists within the bottle.  These circumstances cause the perfume in the bottle to be drawn upwards and be sprayed out of the bottle, as illustrated in the figure at the left.

            Bernoulli’s principle is also able to explain why windows explode rather than implode during hurricanes.  The high velocity of the air outside of the window causes the pressure there to be very low.  The air on the other side of the window is still, creating a situation of high pressure.  Because of these pressure differences, the window is pushed outward, exploding in the process.  The same idea occurs in tornadoes as well.

            Airplane pilots not only use Bernoulli’s principle to explain how they take off and fly, but they also use it in another way.  The equation is also used to provide a speedometer called a pitot tube on aircraft.  Named after French scientist Pitot, the device is rather simple, consisting of a tube bent at right angles (seen at the left).  The device computes the velocity of the aircraft by measuring the static and total pressure of the surrounding air flow.  By using the pitot tube, pilots can obtain their velocity with 99% accuracy.

 

 

            Bernoulli’s principle applies in many aspects of our lives.  Through Bernoulli’s principle and equation, we are able to further understand how fluid mechanics and physics work in the real world.  The examples of the curveball, airplane, perfume bottle, and windows in a hurricane are just a few examples of how physics apply to every day life.  Whether we realize it or not, physics is all around us, even if we do not understand exactly what is occurring.

Bibliography

 

Benson, Tom.  “Bernoulli’s Equation.”  4 June 2002.  Online.  http://www.grc.nasa.gov/WWW/K-12/airplane/bern.html.

“Bernoulli’s Equation.”  Online.  http://www.princeton.edu/~asmits/Bicycle_web/Bernoulli.html.

“Bernoulli’s Principle.”  Online.  http://resources.yesican.yorku.ca/lpdd/g06/lp/aero/bernoulli1.html.

“Bernoulli’s Principle.”  29 Sept. 1999.  Online.  http://theory.winnipeg.ca/mod_tech/node68.html.

“Daniel Bernoulli.”  Sept. 1998.  Online.  http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Bernoulli_Daniel.html.

Quinney, Dr. D. A.  “Daniel Bernoulli and the Making of the Fluid Equation.”  Jan. 1997.  Online.  http://plus.maths.org/issue1/bern/index.html.

Suplee, Curt.  Everyday Science Explained.  Washington, D.C.:  National Geographic Society, 1996.