Bernoulli
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The Bernoulli Approach

Using The Bernoulli Principle to Explain Lift

"It would be better for the true physics

if there were no mathematicians on earth."

Daniel Bernoulli (Feb 8, 1700 - Mar 17, 1782)

Daniel Bernoulli was the son and nephew of the respected mathematicians Johann Bernoulli and Jakob Bernoulli. He was influenced by Sir Issac Newton (1642-1727) and by Leonhard Euler (1707-83). From 1725-1749 he won prizes for work in astronomy, gravity, tides, magnetism, ocean currents, and the behavior of ships at sea. Around 1738 he wrote Hydrodynamica, established the basis for the kinetic theory of gases, discovered how to measure blood pressure, considered the basic properties of fluid flow, pressure, density, and velocity, and presented the fundamental relationship now known as the Bernoulli Principle. He also proposed the Saint Petersburg paradox, concerning a strategy for geometrical betting.

Accounting for lift with the Bernoulli Principle

Thousands of Internet sites, many by educational institutions and by students, account for lift with the Bernoulli Principle. Typically, they interpret Bernoulli's Principle as: high velocity fluid causes low pressure. They use this relationship to account for a variety of lift phenomena, such as airplanes, curve balls in baseball, ping-pong balls sticking in funnels, perfume atomizers, levitators and other such devices.

A standard Bernoulli Principle diagram, such as this, appears in numerous textbooks and on countless web sites.

 

Origin of the Bernoulli Principle

The Bernoulli Principle derives from the physics of a closed hydraulic loop of constant density, with varying cross sections and with no internal flow motivation. From these constraints, since the total energy in such a system must be everywhere equal, a decrease in pressure (thermal) energy must balance an increase in kinetic energy (given constant gravitational energy). Since fluid flowing through a narrow cross section flows faster, it has higher kinetic energy and must have lower pressure. From this, the popular interpretation of Bernoulli's Principle: fast flow means low pressure.

Standard Bernoulli Principle Diagram

A standard pipe diagram, such as this, appears in textbooks and on web sites. It assumes steady, incompressible (constant density), friction-free, unmotivated flow.

It seems unclear from this diagram (1) how to motivate the fluid flow and (2) that the fluid flow is actually part of the same constant-density closed loop.

 

The Fundamental Bernoulli Principle Assumptions

The Bernoulli Principle assumes (1) a closed-loop system of varying cross section, (2) constant density of the fluid and (3) no flow motivation. While this situation might obtain in a theory, in practice there is simply no such thing as a closed hydraulic loop in which fluid circulates with no flow motivation. The velocity in such a loop would have to be zero, so the equation would only be valid at  v = 0, at which point lift would also be zero.

Scientists interested in understanding lift have had no alternative simple scientific rule to relate lift to fluid flow. So despite the fundamental assumptions, the Bernoulli Principle came to fill the conceptual vacuum and became the standard explanation for lift.

 

The Mathematics of the Bernoulli Principle

The Bernoulli Principle is a statement of energy balance. It correctly states that for certain conditions (steady flow, no friction and incompressible fluid), the sum of the thermal, kinetic and gravitational energies is constant at all locations. Thus, for fluid moving through a system of pipes of varying cross section, the fluid that moves through narrow pipes, moves fast and has low pressure while the fluid that moves through wide pipes moves slowly and has high pressure. The formula for this principle is:

ET + EK + EG = k

For systems that do not involve gravity change, this becomes

ET + EK = k

Thermal energy + kinetic energy is everywhere constant.

Now since  ET = PV  and  EK = Mv²/2 

PV + Mv²/2 = k

The product of Pressure and Volume plus 1/2 the product of Mass and the square of velocity is everywhere constant.

So for flow from a large diameter pipe to a small diameter pipe within a closed system, velocity (v) increases so pressure (P) falls. Students often interpret Bernoulli's Principle to say that that fast flow "causes" low pressure. In practice, the opposite may be true since it takes a pressure gradient to "cause" fluids to flow. Continuing,

Dividing both sides by volume,

P + dv2/2 = k

and, rearranging,

P = k - dv2/2

So, an increase in velocity indicates a decrease in pressure.

Bernoulli's Law, by Jakob Bernoulli is an altogether different concept; it states that a large number of items chosen at random from a population will, on the average, have the characteristics of the population. Bernoulli's Principle, by Daniel Bernoulli has to do with fluid flow.

 

Text Book Examples

The examples below, from standard textbooks, show examples of using the Bernoulli Principle to explain lift. (Click on the problem number, or the tab at the top of the page, to view the problem and solution.)

 

 

1. Halliday and Resnick

Problem # HR-74-P

 

2. Halliday and Resnick

Problem # HR-75-P

No diagram.

3.  Potter and Foss

Problem # 2.32

 

Conclusions and Observations

The Bernoulli Principle appears widely in explanations for lift, even in those which seem to compromise the qualifying assumptions of incompressibility (constant density), no flow motivation, and a closed fluid loop.

Proponents of the application of Bernoulli's Principle claim that the problem with the assumptions are mostly minor artifacts and point out that the density assumption becomes moot if the velocity is low enough to assure incompressible flow. However, at low velocity, approaching zero, as the mathematical problems disappear, so does the lift.

Other factors seem to favor choosing the Bernoulli Principle to address lift problems. There has been no easily comprehensible theory of lift which acknowledges the importance of density effects. A closed-form derivation of the Navier-Stokes equation for compressible flow presents a formidable and daunting mathematical challenge. Numerical solutions seem to require substantial computational power and re-formulations of basic fluid dynamics equations into unfamiliar forms compatible with integral calculus.

Above all, the Bernoulli Principle to explain lift has stood the test of time and has gained wide popularity and acceptance.