# Daniel Bernoulli Establishes the Field of Hydrodynamics

# Daniel Bernoulli Establishes the Field of Hydrodynamics

*Overview*

Swiss mathematician and physicist Daniel Bernoulli (1700-1782) brought a high level of mathematical rigor to the study of natural phenomena, particularly phenomena associated with liquids and gases. He was especially fascinated with the behavior of fluids. Having been trained as a physician, Bernoulli combined his medical knowledge with his mathematical and analytical skills to study the speed with which a liquid flows through an enclosed space such as a blood vessel. Bernoulli's observations resulted in the first effective method of measuring blood pressure but, more importantly, put Bernoulli on the path that would lead to his derivation of the fluid equation. That equation showed that a fluid's pressure is determined by its velocity, the pressure decreasing as the velocity increases. Bernoulli's formulation related pressure to the law of conservation of energy, placing the behavior of fluids (and by extension gases) squarely within the realm of mathematical and mechanical physics. Bernoulli's insight would affect all studies of fluids and gases for centuries, guiding studies of the behavior of water under pressure, the hull designs of ships, problems of turbulence in turbines, and explaining, for example, the behavior of air traveling over and beneath an airplane's wing.

*Background*

Daniel Bernoulli was born into a distinguished family of mathematicians and scientists. His father, Jean (Johann) Bernoulli (1667-1748), was a mathematician, chemist, and professor; his uncle, Jakob Bernoulli (1654-1705), was a mathematician and natural philosopher. Both of the elder Bernoullis made substantial contributions to calculus and to probability theory.

Despite this family history, Daniel Bernoulli faced obstacles to becoming a mathematician. His father, in particular, aware that mathematics offered little in the way of financial reward, was opposed to Daniel's interest, and sought to steer his son into a career as a merchant. Daniel raised strenuous objections and his father finally relented, permitting the boy to study medicine but continuing to prohibit mathematics. Daniel entered medical school and obtained his degree in 1721. But he never gave up his fascination with mathematics, and ultimately even his father could not hold the young man back. Jean Bernoulli himself began at last to tutor Daniel in advanced mathematics. Daniel began applying the mathematical lessons to his medical observations and studies, specifically the flow of blood through the body's circulatory system.

Unable to find the teaching post he desired, Daniel Bernoulli began to travel around Europe. By 1723 Bernoulli was in Italy, where he was taken ill. As he recuperated he turned his attention to the design of an hourglass whose flow of sand would remain constant even aboard a ship in storm-tossed waters. The French Academy awarded Bernoulli's hourglass first prize in a design competition. Bernoulli also published his first book, *Some Mathematical Exercises*.

Upon turning 25, Bernoulli at last received a teaching commission—from no less an employer than Empress Catherine I of Russia, who offered Bernoulli the position of mathematics professor at the Imperial Academy of St. Petersburg. Bernoulli accepted the position, and his brother Nikolas was given the Academy's second mathematics chair, although Nikolas died within a year of their arrival in Russia.

Working with Leonhard Euler (1707-1783), Bernoulli tackled questions of fluid behavior. Studies of the properties of water were nearly as old as science—around 250 b.c. the great Greek scientist Archimedes (c. 287-212 b.c.) supposedly noticed the displacement of water by his body when he entered his bath. Whether or not Archimedes actually exclaimed "Eureka!" upon observing the displacement, his observation did found the science of studying water (or other fluids) at rest, or hydrostatics.

Over the ensuing centuries hydrostatics remained a relatively active field of study and inquiry. Displacement of fluids by solids, and particularly different displacement properties of different materials—density—received much scientific attention. Properties of surface tension also attracted scientific study. But the careful, systematic study of fluids in motion—hydrodynamics—awaited the arrival of Daniel Bernoulli. Applying himself, along with Euler, first to the behavior of blood in veins, Bernoulli essentially created the field of hydrodynamics in a very few years.

Using a thin-walled pipe to simulate a blood vessel and running water through it at various speeds, Bernoulli discovered that a hollow straw, inserted through the wall of the pipe, drew water to different levels depending upon the pressure of the fluid flowing through the pipe. Blood pressure could be measured in similar fashion, using a hollow glass needle inserted directly into a vein. (Not until 1896 was a less painful method of measuring blood pressure discovered!)

With the lessons learned from his blood studies, Bernoulli turned to the larger question of the relationship of fluid pressure to the laws of conservation of energy. Understanding that kinetic, or moving, energy becomes potential energy when a body is raised in height, Bernoulli achieved his great insight into the behavior of fluids: when a fluid moves, its kinetic energy is transferred to pressure. If the fluid's velocity increases, the pressure decreases; if the velocity decreases, the pressure rises.

Bernoulli captured his insight in an equation: where p is pressure and u is its velocity. This equation still bears the name Bernoulli's principle.

Quickly seeing the potential for his insights, Bernoulli set to work cataloging and extending his observations, applying his mathematical skills to his studies. Equally important, he extended those studies to the properties of gases.

In doing so, Bernoulli assumed that gases are composed of tiny particles. By viewing gases in this way, Bernoulli was able to speculate, mathematically, about their properties, much as he had about liquids. Bernoulli's research and studies of gas properties laid the foundation for the kinetic theory of gases that would be developed in the next century. On a practical level, Bernoulli's exploration of gas properties enabled him to create a mathematical means for measuring atmospheric pressure at different altitudes, showing that the pressure changed as altitude changed.

Bernoulli's ability to combine careful scientific observation and measurement with a high level of mathematical analysis enabled him to create, by 1738, his treatise *Hydrodynamica*, which is considered to be one of the great achievements of eighteenth-century science. In this book Bernoulli distilled his insights and mathematical analyses of the properties of fluids and gases and in one stroke, as it were, created a major branch of physics. Ironically and sadly, Bernoulli's father grew obsessively jealous of his son's accomplishments and published his own book, *Hydraulics*, which many felt plagiarized Daniel's work. The two remained estranged, and that estrangement seems to have drained Daniel of interest in further pursuing hydrodynamics.

Bernoulli devoted the rest of his life to anatomy, botany, and pure mathematics, making contributions to each field. But it is his establishment of hydrodynamics as a field of mathematical inquiry that remains the greatest of Daniel Bernoulli's many accomplishments, a field that remains vigorous and important to this day.

*Impact*

Bernoulli's creation of hydrodynamics (or, as it is more commonly referred to today, *fluid dynamics*) is a scientific contribution of great importance. First, he applied himself to the mechanical properties of fluids in motion, a relatively unexplored but increasingly important field. Next, he quantified the study of fluids by bringing mathematical attention to bear on their behavior in various circumstances. That in itself was a major contribution to establishing the field as a pure science rather than an area of natural philosophy. Equally important, Bernoulli extended the study of fluid behavior to include the behavior of gases, thus establishing principles on which further inquiry and study could be built.

As a result of the foundation Bernoulli established, hydrodynamics was able to investigate and explain not only the behavior of fluids and gases in motion, but also the effects of variables on that motion. Turbulence, which occurs when pressure and velocity undergo irregular fluctuations, is perhaps the most important of the areas of inquiry that derived from Bernoulli's starting points. Most natural flows—rivers, air currents—are, obviously, irregular or turbulent. By creating the mathematical and scientific framework for studying turbulence, scientists were able to create a more accurate picture of the ways in which physical laws are revealed in the natural world.

Bernoulli's contributions also extended to technology. In medicine, of course, he contributed to the understanding of blood flow and blood pressure. But as the Industrial Revolution gathered force and advances in metallurgy made possible the creation of high-pressure steam (and water) vessels, the ability to predict the behavior of liquids and gases flowing under pressure became critical. In turn, studies of turbulence became
crucial to the design of turbine blades turned by steam. *Hydrodynamica* itself contained Bernoulli's insights into the effect of fluid dynamics on ship propulsion and hull, lessons that were put to work by the naval industries.

In the twentieth century Bernoulli's contribution found applications that he would have perhaps considered pure fantasy, as aerodynamics was created as a sub-discipline of hydrodynamics. Aerodynamics uses mathematical tools to study the effects of air flow over wings—to study the physics of flight, and to use those studies to create aircraft that take advantage of fluid dynamic principles in order to lift themselves from the ground and into the skies. Aerodynamic studies also affect the design of automobile bodies.

**KEITH FERRELL**

*Further Reading*

Bernoulli, Daniel. *Hydrodynamica. Argentorati, sumptibus*. J. R. Dulseckeri, 1738.

Quinney, D. A. "Daniel Bernoulli and the Making of the Fluid Equation." *PLUS *1 (January 1997).

Wilson, Derek Henry. *Hydrodynamics*. London: E. Arnold, 1959.

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