Daniel Bernoulli

Claimed by Brynn McFarland, this is a work in progress!

Personal Life

Daniel Bernoulli was born on February 9, 1700 in Groningen, the Netherlands. He came from a family of distinguished mathematicians, but reportedly was the brightest of three children of the family. While growing up in Switzerland, his relationship with his father gradually deteriorated due to competition in academia.

First studying at the University in Basel, he was later invited to study at the Saint Petersburg Academy of Sciences. He studied mathematics, business, medicine, anatomy, and botany, eventually earning his Ph.D. in the latter two. Eventually he returned to Basel as a professor of anatomy, botany, and physics.

He made significant contributions to Physics, most notably the Bernoulli Principle, which is used extensively in aerodynamics and fluid dynamics. His contributions were not limited to only Physics though, he has also been called the founder of mathematical economics and was one of the first people to begin work on a kinetic theory of gases.

Contributions to Physics

Hydrodynamica

Bernoulli published Hydrodynamica in 1738, and the title eventually gave rise to the name "hydrodynamics" for the field of fluid dynamics. The full title of the text was "Hydrodynamics, or commentaries on the forces and motions of fluids". In the book, Bernoulli characterizes fluid mechanics according to conservation of energy by examining several examples such as fluid coming out of an opening or fluid flowing through a tube. These examples gave the first adequate theory of the motion of incompressible fluids and also hydrodynamic pressure. In the text he also describes work and efficiency of hydraulic machines, the kinetic theory of gases, and several ideas that would eventually be formulated into the Bernoulli Principle.

The Bernoulli Principle

The Bernoulli Principle was developed in Hydrodynamica when he asked the following question: 'Let a very wide vessel ACEB, which is to be kept constantly full of water, be fitted with a horizontal cylindrical tube ED; and at the extremity of the tube let there be an orifice o emitting water at a uniform velocity; the pressure of the water against the walls of the tube ED is sought.' To solve this problem, Bernoulli employs his principle of equality between the actual descent and potential ascent, which states that 'If any number of weights begin to move in any way by the force of their own gravity, the velocities of the individual weights will be everywhere such that the products of the squares of these velocities multiplied by the appropriate masses, gathered together, are proportional to the vertical height through which the centre of gravity of the composite of the bodies descends, multiplied by the masses of all of them.' In only considering quasi-one-dimensional fluid motion and by using the principle of conservation of energy, Bernoulli develops the following equation:

[math]\displaystyle{ {v}{dv \over dx}={(a-v^2) \over 2c} }[/math]

where:

[math]\displaystyle{ v\, }[/math] is the velocity of the fluid

[math]\displaystyle{ dx\, }[/math] is the length of the cylindrical tube

[math]\displaystyle{ a\, }[/math] is the height of the tube

[math]\displaystyle{ c\, }[/math] is the distance from the entrance of the tube

After making several assumptions such as the pressure is proportional to the ratio of the increase in velocity over time, and that the hole o is infinitesimal, thus the fluid is virtually motionless which makes the pressure determined by the height of the column, he develops this equation:

[math]\displaystyle{ {p_1 \over \gamma} + {V_1^2 \over 2g}={p_2 \over \gamma} + {V_2^2 \over 2g} }[/math]

where:

[math]\displaystyle{ p_1\, }[/math] is the pressure in the tube

[math]\displaystyle{ p_2\, }[/math] is the pressure at the exit of the tube

[math]\displaystyle{ V_1\, }[/math] is the velocity of the fluid in the tube

[math]\displaystyle{ V_2\, }[/math] is the velocity of the fluid exiting the tube

[math]\displaystyle{ \gamma\, }[/math] is the specific weight of the fluid

[math]\displaystyle{ g\, }[/math] is the acceleration due to gravity

A simpler version of Bernoulli's Principle for an incompressible fluid where frictional forces is negligible is:

[math]\displaystyle{ {v^2 \over 2}+gz+{p \over \rho}=\text{constant} }[/math]

where:

[math]\displaystyle{ v\, }[/math] is the velocity of the fluid in the tube

[math]\displaystyle{ g\, }[/math] is the acceleration due to gravity

[math]\displaystyle{ z\, }[/math] is the height of the tube

[math]\displaystyle{ p\, }[/math] is the pressure

[math]\displaystyle{ \rho\, }[/math] is the density of the fluid

The main idea of the Bernoulli Principle is that for a steady flow, the sum of all the energies involved will be the same at any point in the tube, therefore, the velocity of the fluid through the tube will increase coinciding with a decrease in pressure or a decrease in the potential energy.

Euler-Bernoulli Beam Equation

The Euler-Bernoulli Beam Equation is based on the theory of elasticity, which states that elasticity is the property of solid materials to deform under the application of an external force and to regain their original shape after the force is removed.

The equation is:

[math]\displaystyle{ {d^2 \over dx^2}[{EI}{d^2w \over dx^2}]=p }[/math]

where:

[math]\displaystyle{ w\, }[/math] is the displacement of the beam

[math]\displaystyle{ p\, }[/math] is the force per unit length acting in the direction of w

[math]\displaystyle{ E\, }[/math] is the Young's Modulus for the beam

[math]\displaystyle{ I\, }[/math] is the moment of inertia of the beam's cross section

[math]\displaystyle{ x\, }[/math] is the position along the beam

The Euler-Bernoulli Beam Equation relates the displacement, or deflection of the beam (w) to its pressure loading (p). The equation is widely used in engineering applications, such as civil or mechanical to determine the strength of beams when stress is applied.

References

Wilkins, D.R. "The Bernoullis." Web. 30 Nov. 2015. <http://www.maths.tcd.ie/pub/HistMath/People/Bernoullis/RouseBall/RB_Bernoullis.html#DanielBernoulli>.

Rothbard, Murray. "Daniel Bernoulli and the Founding of Mathematical Economics." Mises Institute. Mises Institute, 10 Feb. 2011. Web. 30 Nov. 2015. <https://mises.org/library/daniel-bernoulli-and-founding-mathematical-economics>.

Guinness, I. "Daniel Bernoulli, Hydrodynamica (1738)." Landmark Writings in Western Mathematics 1640-1940. Amsterdam: Elsevier, 2005. Print.

Dahr, S. "3.1 Theory of Elasticity." Web. 30 Nov. 2015. <http://www.iue.tuwien.ac.at/phd/dhar/node17.html>.

"Euler-Bernoulli Beam Equation." EFunda, 2015. Web. 30 Nov. 2015. <http://www.efunda.com/formulae/solid_mechanics/beams/theory.cfm>.