Daniel Bernoulli

David ApplegateCassandra Diamond

Erin RyanTiffany Liang

Background• Born on February 8th,

1700o Groningen, Netherlands

• Swiss mathematician and physicist

• Leonhard Euler• Received Bachelor’s

degree at 15 and Master’s degree at 16

Background• Bernoulli Family

o 8 family members were mathematicians

• University of Baselo Medicine, metaphysics, and natural philosophy

Studies• Bernoulli’s Equation• Bernoulli’s Principle• Kinetic Theory of Gases

Bernoulli’s Equation

Bernoulli’s Equation• Mathematical model of fluid flow through a

conduit• Based on the conservation of energy law applied

to the fluid• This equation can be applied to incompressible

fluids as well as compressible gases or fluids moving at low Mach Numbers

• Bernoulli first published this equation in Hydrodynamica (1783).

Energy Applied to the Fluid

Energy Type Formula

o Enthalpic

o Gravitational

o Kinetic

o Friction Loss

o Work

The Bernoulli equation concerns itself with incompressible (constant density) and adiabatic (no heat applied/removed) fluid flow.

Combining Terms results in the Bernoulli Equation

Derivation of Bernoulli’s Equation

• Differential energy balance:

• Plugging in equations for entropy and enthalpy:

• Gives:

• Integrate:

The Incompressible Fluid Bernoulli Balance

Note: P2, P1, V2,V1 terms are initial (1) and final (2) states.

Bernoulli’s Principle

What is Inviscid Flow?

-Idealized form of fluid-Supposes that fluid has a viscosity of 0-Looks like laminar, but drag has no effect

Standard Laminar Inviscid Flow

Images courtesy of me spending way too long in MS Paint

Inviscid Flow Around Objects

Around a sphere Around a wing

Images courtesy of http://www.allstar.fiu.edu

So are you going to talk about Bernoulli?

Bernoulli's Principle

-For inviscid flows, kinetic and potential energy are exchangeable

-That means as pressure changes, velocity changes to compensate (and vice versa)

-Forms the backbone of the Bernoulli Equation

Bernoulli’s Kinetic Theory

KINETIC THEORY

• In his most famous work,

Hydrodynamica , Bernoulli was

the first to postulate on the

kinetic theory of gases.

• This included • the idea that pressure is a result

of the collisions between gas molecules and the walls of a container

• the theorem that temperature is related to the velocity, or the kinetic energy of the molecules in a substance.

P1 = atmospheric pressure

PS = pressure when piston at height s

𝑃𝑆𝑃1

=𝑉 2

𝑠

There was not enough experimental evidence at the time to quantify a relationship between temperature and average molecular velocity, but an equation would eventually be developed. This idea is also the basis of the identification of an absolute temperature scale.

Maxwell and Boltzmann later expanded on Bernoulli’s theory and used statistical methods to determine a more qualitative relationship between temperature and average translational kinetic energy of molecules.

Aerodynamics Application

• An airfoil on the wing of an airplane forces the air along the upper surface to travel a longer distance, increasing its velocity.

• According to Bernoulli’s principle, this increase in velocity causes a decrease in pressure and creates lift.

• The flaps on the wings of planes perform a similar function.

Source Citation• Levermore, Dave. (2001):

<http://www2.math.umd.edu/~lvrmr/History/EarlyTheories.html>.

• Whitaker, Robert D. "University of South Florida." University of South Florida. 56. (1979): 315-318. <http://pubs.acs.org/doi/pdfplus/10.1021/ed056p315>.

• Friedman, Erich. (2005): <http://www2.stetson.edu/~efriedma/periodictable/html/B.html>.