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Physics 101: Lecture 25 Fluids in Motion: Bernoulli’s Equation

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Note: Everything we do assumes fluid is non-viscous and incompressible.

Physics 101: Lecture 25Fluids in Motion: Bernoulli’s Equation- Today’s lecture will cover Textbook Sections 11.7-11.10
- Fluids in motion: Continuity & Bernoulli’s equation

Physics 101: Lecture 24Archimedes Principle (summary)

- Buoyant Force (FB)
- FB=weight of fluid displaced
- FB = fluid Vdispl g
- W = Mg = object Vobject g
- If object floats….
- FB=W
- Therefore fluid gVdispl. = object gVobject
- Therefore Vdispl./Vobject = object / fluid

Concept Question

Suppose you float a large ice-cube in a glass of water, and that after you place the ice in the glass the level of the water is at the very brim. When the ice melts, the level of the water in the glass will:

1. Go up, causing the water to spill out of the glass.

2. Go down.

3. Stay the same.

FB =W gVdisplaced

W = ice gVice Vdisplaced = Vice under water =Vicerice/rW

Fluids in Motion

- Consider an ideal fluid (incompressible and nonviscous) that flows steadily.
- Steady Flow:

Every fluid particle passing trough the same point in the stream has the same velocity.

Streamlines are used to visualize the trajectory of fluid

particles in motion. The velocity vector of the fluid

particle is tangent to the streamline.

The fluid velocity can vary from point to point along a

streamline but at a given point the velocity is constant in

time.

Equation of Continuity

- Mass is conserved as the fluid flows.

If a certain mass of fluid enters a pipe at one end at a

certain rate, the same mass exits at the same rate

at the other end of the tube (if nothing gets lost in

between through holes, for instance).

Mass flow rate at position 1 = Mass flow rate at position 2

r1 A1 v1 = r2 A2 v2

r A v = constant along a tube that has a single entry

and a single exit point for fluid flow.

Concept Question

A stream of water gets narrower as it falls from a faucet (try it & see).

This phenomenon can be explained using the equation of continuity

A1

V1

The water\'s velocity is increasing as it flows down, so in order to compensate for the increase in velocity, the area must be decreased because the density*area*speed must be conserved

A2

V2

Bernoulli’s Equation

- Work-Energy Theorem : Wnc = change of total mechanical energy

applied to fluid flow :

Difference in pressure => net force is not zero => fluid accelerates

Pressure is due to collisional forces which is a nonconservative force:

Wnc = (P2-P1) V

Consider a fluid moving from height h1 to h2. Its total mechanical

energy is given by the sum of kinetic and potential energy. Thus,

Wnc = Etot,1 –Etot,2 = ½ m v12+m g h1 –( ½ m v22+m g h2)

v1

A2 r2

v2

Fluid Flow (summary)- Mass flow rate: Av (kg/s)
- Continuity: 1A1 v1 = 2A2 v2
- i.e., mass flow rate the same everywhere
- e.g., flow of river
- For fluid flow without friction (nonviscous):
- Bernoulli: P1 + 1/2v12 + gh1 = P2 + 1/2v22 + gh2

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