Switzerland (where Bernoulli spent much of his life), 1738. The total energy of fluid pressure, gravitational potential energy, and kinetic energy of a moving fluid remains constant. For liquid flow- ing in a pipe, an increase in velocity occurs simultaneously with decrease in pressure.
Cross Reference: Bernoulli-Euler Law.
In 1738, German clockmaker Franz Ketterer invented the cuckoo clock, and Joseph Guillotin (the French physician who invented the guillotine) was born. When the French government persisted in naming the execution machine after Guillotin, his relatives decided to change their family name.
Imagine a fluid flowing steadily through a pipe that carries the liquid from the top to the bottom of a hillside. The pressure of the liquid will change along the pipe. Daniel Bernoulli (1700–1782) discovered the law that relates pressure, flow speed, and height for a fluid flowing in a pipe.
Today, we write Bernoulli’s Law as v2
2 +gz+ p ρ =C,
wherevis the fluid velocity,gthe acceleration due to gravity,zthe eleva- tion (height) of a point in the fluid, pthe pressure,ρthe fluid density, and Cis a constant. Scientists prior to Bernoulli had understood that a moving body exchanges its kinetic energy for potential energy when the body gains height. Bernoulli realized that, in a similar way, a moving fluid exchanges its kinetic energy for pressure.
As with most laws in this book, Bernoulli’s Law holds for an idealized situation. For example, the formula assumes a steady (nonturbulent) fluid flow in a closed pipe. The fluid must be incompressible. Because most fluids are only slightly compressible, Bernoulli’s Law is often a useful approximation. The fluid should not be viscous, which means that the fluid should not have internal friction. Although no real fluid meets all these criteria, Bernoulli’s relationship is generally very accurate for free-flowing regions of fluids that are away from the walls of pipes or containers, and especially useful for gases and light liquids. The equation can be general- ized to a steadycompressibleflow (in which changes in density play a role) by adding the internal energy per unit mass to the left-hand side of the formula.
Bernoulli’s Law often makes reference to a subset of the informa- tion included in the above equation, namely, that the decrease in pres- sure occurs simultaneously with an increase in velocity. The idea that an increase in the speed of a fluid results in a decrease in the pressure is at the core of many everyday phenomena. Bernoulli’s Law predicts correctly that a shower curtain is pulled inward when the water first comes out of the shower head because the increase in water and air velocity inside the shower causes a pressure drop. The pressure difference between the outside and inside of the curtain causes a net force on the shower curtain that sucks the curtain inward.
Bernoulli’s formula has numerous practical applications in the fields of aerodynamics, where it is considered when studying flow over airfoils—
such as wings, propeller blades, rudders (whose shapes control stability or propulsion)—and flow in supersonic nozzles. Bernoulli’s Law is used when designing aVenturi throat—a constricted region in the air passage of a carburetor that causes a reduction in pressure, which in turn causes fuel vapor to be drawn out of the carburetor bowl. The term “venturi” is also applied to a short tube with a constricted region that facilitates the measurement of fluid pressures and velocities as a fluid flows through the tube. The fluid increases speed in the smaller diameter region, reducing its pressure and producing a partial vacuum via Bernoulli’s Law. This Venturi effect is named after the Italian physicist Giovanni Battista Venturi (1746–
1822). In carburetors, the Venturi effect sucks gasoline into an engine’s intake air stream. Additionally, Bernoulli’s Law plays a role in the func- tioning of Pitot tubes used for aircraft speedometers.
I have frequently seen the Venturi effect in action when squeezing a flexible hose through which water flows. If the flow is sufficiently strong, the constriction I put in the hose remains in the hose, even when I remove my hand, because the partial vacuum produced in the constriction is suffi- cient to keep the hose collapsed.
The fact that pressure falls with increasing velocity is exploited by an airplane wing, which is designed to create an area of fast flowing air on its upper surface. The pressure near this area is lower; thus, the wing tends to be pulled upward.
Daniel Bernoulli (1700–1782), Dutch-born Swiss mathematician, physi- cist, and medical doctor famous for his wide variety of work in mathemat- ics, hydrodynamics, vibrating systems, probability, and statistics.
CURIOSITY FILE: Both Daniel Bernoulli and his father, Johann Bernoulli, had to secretly study mathematics against their respective fathers’ strict
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orders to forget about mathematics and to pursue more prosperous careers.
• Bernoulli wrote on whatever subjects struck his fancy—one of his papers discussed formulas for computing the relationship between the number of oarsmen on a ship and the resultant ship velocity. • In 1738, Bernoulli published “Exposition of a New Theory on the Measurement of Risk,”
dealing with the economic theory of risk aversion and overall happiness gained from a good or service.
There is no philosophy which is not founded upon knowl- edge of the phenomena, but to get any profit from this knowledge it is absolutely necessary to be a mathemati- cian.
—Daniel Bernoulli, Letter to John Bernoulli III, January 7, 1763
Just as a great river is fed by small streams, some even barely noticeable . . . , so science and technology proceeds from small individual contributions until it becomes an ever-increasing flow of knowledge and techniques. This big river of fluid mechanics is closely associated with Daniel Bernoulli, the author of the first textbook in this field.
—G. A. Tokaty, History and Philosophy of Fluid Mechanics
Daniel Bernoulli was a member of a truly remarkable family which produced no fewer than eight mathemati- cians of ability within three generations, three of whom—
James I (1654–1705), John I (1667–1748), and Daniel—
were luminaries of the first magnitude.
—S. L. Zabell, in John Eatwell et al.’sUtility and Proba- bility
Daniel Bernoulli is one of the most versatile scientists presented in this book and comes from a family of extraordinary Swiss mathematicians. Not only did he study fluid flow, as discussed in the preceding section, but he also investigated a variety of topics in mathematics, biology, physics, and astronomy.
Before discussing Daniel, let me discuss his famous father, Johann (1667–1748). Mathematical genius seemed to be the very essence of the Bernoulli brain. Swiss mathematician Johann conducted pioneering work with his brother Jacob Bernoulli (1654–1705) in calculus and many areas of applied mathematics. Johann’s father tried hard to push Johann into
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a business career, but Johann preferred mathematics and the academic life. Johann and Jacob’s relationship gradually deteriorated. In particular, Jacob was jealous of Johann’s friendship with Leibniz and worried that his younger brother was a better mathematician than he. Jacob did all he could to slow down the rise of his brother. For example, in 1695, Jacob convinced the members of the University of Basel’s academic senate to reject Johann’s application for professorship. When Johann learned of his brother’s betrayal, he was furious.
Both the meanness and mathematics that were so apparent in Johann’s generation progressed into the next. Daniel Bernoulli was born in Gronin- gen, the Netherlands, while his father taught mathematics at the University of Groningen, and the Bernoulli family returned to their native city of Basel, Switzerland, when Daniel was five years old. The poor relationship between Daniel and his dad started from an early age. Just as Johann’s father had tried to force Johann into a merchant career, Johann did the same to his son. Johann virtually forbade his son to become educated in mathematics, which Johann said did not make sufficient money. More- over, Johann mapped out Daniel’s entire future and even selected the woman that Daniel must eventually marry. However, Daniel was stubborn.
According to Michael Guillen, author ofFive Equations That Changed the Word:
Daniel Bernoulli decided to stop altogether his pretending to go along with his father’s astrology-like notions of what God expected of him—that included the business of becoming a merchant, marry- ing some preselected girlfriend,and, the nonmathematical charade he had been carrying on for several years now. Consequently, the young man broke the bad news to his father and begged for permis- sion to pursue his love of mathematics.
Johann finally told Daniel that he did not have to become a merchant, but instead Daniel now must become a medical doctor—and no mathematics would be allowed. Time passed, and finally Johann relented and agreed to personally teach Daniel mathematics, as long as Daniel continued his medical education.
While growing up in Basel, Switzerland, Daniel had managed to study philosophy and logic, obtaining his master’s degree in 1716. In 1721, he obtained his doctorate in medicine. His dissertation was on the mechanics of breathing.
In the meantime, he did pursue mathematics, and his various math- ematics papers, published in 1724, caused the St. Petersburg Academy, along with Empress Catherine I of Russia, to invite him for a visit. Daniel’s 128 | a r c h i m e d e s t o h a w k i n g
St. Petersburg years (1725–1733) were his most creative, leading to famous papers on hydrodynamics, oscillations, and probability.
Johann Bernoulli sent the Swiss mathematician Leonhard Euler (1707–
1783) to St. Petersburg in order to work with Daniel, which led to fruitful and creative discussions. Euler would turn out to be one of the greatest and most prolific mathematicians who ever lived.
While in Russia, Daniel Bernoulli invented ways for measuring the pressure flowing through pipes by punching a hole in the wall of a pipe and attaching a small glass tube to the hole. As the water flowed through the pipe, it would also enter the glass tube, and the height that it rose in the upward-pointing tube was a measure of the pressure of the flowing water. Bernoulli recognized that this approach might be used to measure blood pressure, and he told his friend Christian Goldbach, “I made a new discovery that will be very useful in the design of the water supply, but mainly, it will open a new era in physiology.” Physicians throughout Europe began to measure blood pressure by sticking pointed-end glass tubes directly into patients’ arteries. Ouch!
Bernoulli’s fascinating paper on probability, which was finally pub- lished in 1738, described a paradox now known as the “St. Petersburg Para- dox.” The puzzle involves coin flips and money that a gambler is to receive depending on the outcome of the flips. Philosophers and mathematicians have wondered: What is the fair price for joining this game? How much would you be ready to pay for joining this game?
Here’s one way to view the St. Petersburg scenario: Flip a penny until it lands tails. The total number of flips,n, determines the prize, which equals
$2n. Thus, if the penny lands tails the first time, the prize is $21= $2, and the game ends. If the penny comes up heads the first time, it is flipped again. If it comes up tails the second time, the prize is $22=$4, and the game ends.
And so on. A detailed discussion on the paradox of this game is beyond the scope of this book, but a rational gambler would enter a game if and only if the price of entry was less than the expected value of the financial payoff. According to some analyses of the St. Petersburg game, any finite price of entry is smaller than the expected value of the game, and a rational gambler might desire to play the game no matter how large we set a finite entry price to play the game!
Peter L. Bernstein inAgainst the Gods: The Remarkable Story of Risk comments on the mystery and profundity of Bernoulli’s St. Petersburg Paradox:
His paper is one of the most profound documents ever written, not just on the subject of risk but on human behavior as well. Bernoulli’s emphasis on the complex relationships between measurement and gut touches on almost every aspect of life.
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Daniel’s primary work,Hydrodynamica, was completed around 1734 but not published until 1738. In this book, he discussed fluid pressure and velocity and presented his famous law stating that fluid pressure increases as its velocity decreases. The word “hydrodynamics”—which refers to the branch of modern science that deals with the motion of fluids—derives from the title of Daniel’s work. His father, jealous of Daniel’s work, published his ownHydraulica, which may have been predated to 1732 to make it appear that his work was written before his son’sHydrodynamica.
Johann had even persuaded Euler to write in the preface ofHydraulica, “I was thoroughly astounded by the very fluent application of Your principles to the solution of the most intricate Problems, because of which . . . Your very distinguished Name will forever be revered among future genera- tions.” According to Guillen,
Daniel Bernoulli could never prove it, but he would always suspect his father of plagiarism and his alleged friend Euler of duplicity.
“Of my entireHydrodynamica, of which indeed I in truth need not credit one iota to my father,” Bernoulli lamented, “I am robbed all of a sudden, and therefore in one hour I lose the fruits of work of ten years.”
Daniel shared the 1735 prize from the Paris Academy of Sciences for his work on planetary orbits with his father, who promptly threw Daniel out of the house for obtaining a prize he felt should be his alone. According to John J. O’Connor and Edmund F. Robertson’s entry on Bernoulli inThe MacTutor History of Mathematics Archive:
Daniel’s father was furious to think that his son had been rated as his equal, and this resulted in a breakdown in relationships between the two. The outcome was that Daniel found himself back in Basel but banned from his father’s house. Whether this caused Daniel to become less interested in mathematics or whether it was the fact that his academic position was a non-mathematical one, certainly Daniel never regained the vigor for mathematical research that he showed in St. Petersburg.
In 1750, Daniel was appointed chair of physics at Basel, where he taught until his death in 1782. In many ways, Daniel turned out to be the “Carl Sagan” of his era, as he enjoyed clarifying science for a general public and did not confine himself to a single field of knowledge. Ten of his essays entered in competitions of the Paris Academy won awards. Essay topics included marine navigation and technology, magnetism, astronomy, 130 | a r c h i m e d e s t o h a w k i n g
Frontispiece for Chapter 1 of Daniel Bernoulli’sHydrodynamica, published in 1738.
planetary orbits, and the optimal shapes for sand-filled hourglasses and boat anchors.
His hourglass was special in that it was designed to keep good time even when a ship was rocking in heavy seas. To achieve this, Bernoulli’s award-winning invention involved mounting an hourglass atop an iron slab floating in mercury. Even when a ship was in a storm, the density of the
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mercury would help keep the timepiece from becoming too agitated by the movements of the ship.
Sadly, the precise content of many of his popular science lectures is not known today. His papers covered so many areas of science that I give a flavor of their diversity by presenting a sampling of topics:
Medicine
r
Mechanics of breathing and muscular contractionr
An elucidation of the shape of the optic nerve at its attachment point to the retinar
Computing mechanical work done by the heartMathematics
r
The game of faro (a card game in which the players lay wagers on the top card of the dealer’s pack)r
Riccati’s differential equations [named after Count JacopoFrancesco Riccati (1676–1754), these mathematical equations have the formy=q0(x)+q1(x)y+q2(x)y2and are not easily solved using standard elementary techniques]
r
Lunulae (a class of crescent shapes)r
Divergent sine and cosine seriesr
Infinite continued fractionsr
Probability and statistics applied in the areas of economics, disease spread, and population statisticsMechanics
r
Theory of rotating bodiesr
Frictionr
Fluid pressure on pipe wallsr
Fluid flow from container holesr
Oscillations of fluids in tubes immersed in water tanksr
Investigations of pumps, windmill sails, and the Archimedean screwr
Atmospheric pressuresr
Refraction of lightr
Air flow from small openingsr
Theories of ocean tidesr
The actions of sails and oarsr
The mechanics of flexible bodiesr
Velaria, lintearia, and catenaria (geometrical curves sometimes exhibited by natural processes)r
Oscillations of ropes loaded with weightsr
Vibrations of threads of uneven thicknesses, plates in water and organ pipes, and musical instrument strings132 | a r c h i m e d e s t o h a w k i n g
Another less famous law that bears Bernoulli’s name is the Bernoulli- Euler Law, which is useful in studying, for example, a horizontal beam that supports a vertical load that causes the beam to bend. The law states that an elastic beam of thicknesst, bent to a radius of curvatureR(R>t), has a bending momentMgiven byM=EI/R, whereEis Young’s modulus for the material of the beam, andIis the second moment of area of the cross- section of the beam about an axis that is normal to the plane of bending.
Young’s modulus is a measure of the stiffness of a given material. Bernoulli suggested the law in 1742 and Leonhard Euler derived it in 1744.
In his 1738 textbook on hydrodynamics, Bernoulli suggested his famous Bernoulli’s Law of Fluid Dynamics, in which he states that “proportional- ity” existed between pressure and velocity. He wrote, “It is clearly very amazing that this very simple rule, which nature affects, could remain unknown up to this time.” It was not until 1755 that Euler derived the fuller expression that relates pressure, velocity, density, and height.
I think of both Bernoulli and Euler whenever I squeeze a bulb that sits atop a perfume bottle. Squeezing the bulb over the liquid perfume creates a low pressure area due to the higher speed of the air, which then sucks the perfume up to the opening of the bottle. Similarly, Bernoulli’s Law helps us understand why house windows often explode outward in a hurricane.
The high speed of the air outside the window pane results in lower pressure outside than inside, thus pulling the glass outward. If you believe that a hurricane is approaching, it may wise to open a few windows to equalize the pressure.
Graham Cleverley, my colleague and a historian of science, in a per- sonal communication to me, comments on Bernoulli’s Law as it relates to airplane wings:
Conventional airplanes fly more economically because their design takes Bernoulli’s principle into account. Without the principle . . . , sailing ships wouldn’t be able to sail against the wind. Europeans would not have discovered America until the nineteenth century.
And when we got there and played baseball, we wouldn’t be able to throw curveballs or sliders. . . . You wouldn’t be able to bend it like Beckham. Or serve like Sampras. And table tennis would be very dull.
Ray Kurzweil writes in John Brockman’s What We Believe but Cannot Prove:
It is the nature of engineering to take a natural, often subtle effect and control it, with a view toward greatly leveraging and magnifying it. . . . Consider, for example, how we have focused and amplified the
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