Continuity example – falling water

Why does the stream of water from a tap get narrower as it falls? Hint: Is the water accelerating as it falls?

Answer: The fluid accelerates under gravity so the velocity is higher down lower. Since Av = const (i.e. A is inversely proportional to v), the cross sectional area, and the radius, is smaller down lower Bernoulli’s principle

Bernoulli conducted experiments like this: https://youtu.be/9DYyGYSUhIc (from 2:06)

A fluid accelerates when entering a narrow section of tube, increasing its kinetic energy (=½mv2) Something must be doing work on the parcel. But what? Bernoulli’s principle resolves this dilemma. Daniel Bernoulli “An increase in the speed of an ideal fluid is accompanied 1700-1782 by a drop in its pressure.”

The fluid is pushed from behind Examples of Bernoulli’s principle

Aeroplane wings high v, low P Compressible fluids? Bernoulli’s equation OK if not too much compression.

Venturi meters Measure velocity from pressure difference between two points of known cross section. Bernoulli’s equation

New Concept: A fluid parcel of volume V at pressure P has a pressure potential energy

Epot-p = PV. It also has gravitational potential energy:

Epot-g = mgh Ignoring friction, the energy of the parcel as it flows must be constant:

Ek + Epot = const 1 mv 2 + PV + mgh = const ) 2 For incompressible fluids, we can divide by volume, showing conservation of energy 1 density: ⇢v 2 + P + ⇢gh = const 2 This is Bernoulli’s equation. Bernoulli equation example – leak in water tank A full water tank 2m tall has a hole 5mm diameter near the base. What velocity is the stream of water that emerges from the hole? State and justify all assumptions.

College physics p409 Bernoulli equation example 1 – snoring

Estimate the drop in pressure associated with normal breathing through an airway of diameter 1cm