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Fluid Mechanics

- Chapter 10

Density

- Recall that the density of an object is its mass

per unit volume (SI unit is kg/m3) - The specific gravity of a substance is its

density expressed in g/cm3

Pressure in Fluids

- Fluids exert a pressure in all directions
- A fluid at rest exerts pressure perpendicular to

any surface it contacts - The pressure at equal depths within a uniform

fluid is the same

SI Unit for Pressure is Pa 1 Pa 1 N/m2 1 atm

101.3 kPa760 mm-Hg

Pressure in Fluids

- Gauge Pressure is a measure of the pressure over

and above the atmospheric pressure - i.e. the pressure measured by a tire gauge is

gauge pressure. If the tire gauge registers 220

kPa then the absolute pressure is 321 kPa because

you have to add the atmosphere pressure (101 kPa) - If you want the absolute pressure at some depth

in a fluid then you have to add atmosphere

pressure

Pressure in Fluids

- Pascals Principle Pressure applied to a fluid

in a closed container is transmitted equally to

every point of the fluid and to the walls of the

container

Buoyancy

- Buoyant force is the force acting on an object

that is immersed in a fluid - Archimedes Principle The buoyant force on a body

immersed in a fluid is equal to the weight of the

fluid displaced by the object - Since the buoyant force acts opposite of gravity,

an object seems to weigh less in a fluid - Apparent Weight Fg-FB

Sinking vs Floating

- Think back to free body diagrams
- If the net external force acting on an object is

zero then it will be in equilibrium

FB

- If FbFg then the object will be in equilibrium

and will FLOAT!

Fg

Cubes floating in a fluid

Density determines depth of submersion

- This equation gives the percent of the objects

volume that is submerged - Vf is the volume of fluid displaced
- Vo is the total volume of the object
- ?o is the density of the object
- ?f is the density of the fluid

Summary of Floating

Continuity Equation

- Continuity tells us that whatever the volume of

fluid in a pipe passing a particular point per

second, the same volume must pass every other

point in a second. - If the cross-sectional area decreases, then

velocity increases

The quantity Av is the volume rate of flow

Bernoullis Principle

- The pressure in a fluid decreases as the fluids

velocity increases. - Fluids in motion have kinetic energy, potential

energy and pressure

How do planes fly?

Bernoullis Equation

The kinetic energy of a fluid element is

The potential energy of a fluid element is

Bernoullis Equation

- This equation is essentially a statement of

conservation of energy in a fluid. Notice that

volume is missing. This is because this equation

is for energy per unit volume.

Applications of Bernoullis Equation

If a hole is punched in the side of an open

container, the outside of the hole and the top of

the fluid are both at atmospheric pressure.

Since the fluid inside the container at the level

of the hole is at higher pressure, the fluid has

a horizontal velocity as it exits.

Applications of Bernoullis Equation

If the fluid is directed upwards instead, it will

reach the height of the surface level of the

fluid in the container.

Sample Problem p. 306 40

- What is the lift (in newtons) due to Bernoullis

principle on a wing of area 80 m2. If the air

passes over the top and bottom surfaces at speeds

of 350 m/s and 290 m/s, respectively. - Lets make point 1 the top of the wing and point

2 the bottom of the wing - The height difference between the top of the wing

and the bottom is negligible

Sample Problem p.306 40

- The net force on the wing is a result of the

difference in pressure between the top and the

bottom. P1 is exerted downward, P2 is exerted

upward - If we know the difference in pressure we can use

that to find the force

Sample Problem p.306 40

- P2-P120318 Pa

P.306 43

- Water at a pressure of 3.8 atm at street level

flows into an office building at a speed of 0.60

m/s through a pipe 5.0 cm in diameter. The pipes

taper down to 2.6 cm in diameter by the top

floor, 20 m above street level. Calculate the

flow velocity and the pressure in such a pipe on

the top floor. Ignore viscosity. Pressures are

gauge pressures.

Find the flow velocity at the top

- A1 is area of first pipe pr2 1.96x10-3 m2
- A2 is area of second pipe pr2 5.31x10-4 m2
- V1 0.6 m/s

Find pressure at the top

P2 1.86 x 105 Pa 1.8 atm

Sample Problem p.305 37

- What gauge pressure in the water mains is

necessary if a fire hose is to spray water to a

height of 12.0 m? - Lets make point 1 as a place in the water main

where the water is not moving and the height is 0 - Point 2 is the top of the spray, so v0 , P

atmospheric pressure, height 12m

Sample Problem p.305 37

Remember that Gauge Pressure is the pressure

above atmospheric pressure. So to get gauge

pressure, we need to subtract atmospheric Pressure

from absolute pressure.

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