Fluids

Fluids are materials that can flow, gases and liquids.

Air is the most common gas, and moves from place to place as wind.

Water is the most familiar liquid.

New Symbol“Rho”

The mass density is the mass m of a substance divided by its volume V:

SI Unit of Mass Density: kg/m3

Density

• Density of water is – 1, 000 kg /m3

– 1 g/ml– 1 pound/pint

Example 1 Blood as a Fraction of Body Weight

The body of a man whose weight is 690 N contains about5.2x10-3 m3 of blood. The density of human blood is 1.060 g/ml

(a) Find the blood’s mass and (b) express it as a percentage of the body weight.

Example 1 Blood as a Fraction of Body Weight

The body of a man whose weight is 690 N contains about5.2x10-3 m3 of blood. The density of human blood is 1.060 g/ml

(a) Find the blood’s mass and (b) express it as a percentage of the body weight.

kg 5.5mkg1060m102.5 333 Vm

11.1 Mass Density

N 54sm80.9kg 5.5 2 mgW(a)

(b) %8.7%100N 690

N 54Percentage

New Symbol “P”The pressure P exerted by a fluid is defined as the magnitude F of the force acting perpendicular to a surface divided by the area A over which the force acts:

The SI unit for pressure: newton/meter2 = (N/m2) = pascal (Pa).

11.2 Pressure

Example 2 The Force on a Swimmer

Suppose the pressure acting on the backof a swimmer’s hand is 1.2x105 Pa. Thesurface area of the back of the hand is 8.4x10-3m2.

(a)Determine the magnitude of the forcethat acts on it.(b) Discuss the direction of the force.

11.2 Pressure

A

FP

N 100.1

m104.8mN102.13

2325

PAF

Since the water pushes perpendicularly against the back of the hand, the forceis directed downward in the drawing.

Other units of pressure

• 1 atm = the pressure at sea level at 0 c.

• 1 atm = 1.013*105 Pa

• Pressure – Not a vector. Can push any direction.

• Forces result from pressure differentials.

New Terms

• Absolute Pressure – The pressure at a point relative to the vacuum of space.

• Gauge Pressure – The pressure at a point relative to Earth’s surface.

New Terms

• Absolute Pressure – The pressure at a point relative to the vacuum of space.

• Gauge Pressure – The pressure at a point relative to Earth’s surface.– What Pressure gauges read.

Absolute and gauge pressure

• Pabs = Pgauge + 1 atm

• Pabs = Pgauge + 1.013* 105 Pa

Pressure in a liquid

• In a liquid, the force pressing down on a surface is the weight of all the fluid above the surface(fluid column)

Pressure in a liquid

• Weight = mg

= ρVg

Volume of the column is the area times height.

= ρAhg

Pressure in a liquid

• Weight = ρAhg

P = Weight/Area

= ρAhg/A

Pgauge = ρgh

Pressure Difference

• The pressure difference between different depths reduces to

• ΔP = ρgΔh

ΔP

• ΔP will be same between Gauge pressure and absolute pressure.

• The two are just offset by a constant, so differences are preserved.

Solar water heater.

• A solar water heater works by pumping water through panels on the roof where the sun warms it, and then storing and circulating it as needed. The storage tank is in the basement 12 m below.

• What is the gauge pressure and absolute pressure in the water tank?

Pressure in an irregular container

• The shape of the container doesn’t matter. Only depth.

Pressure in an irregular container

• The shape of the container doesn’t matter. Only depth.

11.3 Pressure and Depth in a Static Fluid

Example 4 The Swimming Hole

Points A and B are located a distance of 5.50 m beneath the surface of the water. Find the pressure at each of these two locations.

Pascal’s Law

• Pressure applied to an enclosed fluid is transmitted undiminished to every surface of the fluid.

Hydraulics

• Pressure at points depend only on elevation. So A & B have same pressure.

1

1

2

2

A

F

A

F

Hydraulics

• If you’re dealing with a liquid (incompressable), there is no change in Volume.

2211 AXAX

Hydraulics

The input piston has a radius of 0.0120 mand the output plunger has a radius of 0.150 m.

The combined weight of the car and the plunger is 20500 N. Suppose that the inputpiston has a negligible weight and the bottomsurfaces of the piston and plunger are atthe same level. What is the required inputforce?

Hydraulics

The input piston has a radius of 0.0120 mand the output plunger has a radius of 0.150 m.

If the input piston is pushed 6 m, how far is the Output plunger lifted?

Pressure Measurements:Manometer

• One end of the U-shaped tube is open to the atmosphere

• The other end is connected to the pressure to be measured

• If P in the system is greater than atmospheric pressure, h is positive– If less, then h is negative

Manometer Problem

• A sample of gas connected to a water filled manometer, and Δh between the sides is + 127 millimeters. What is the absolute pressure of the sample?

Pressure Measurements: Barometer

• Invented by Torricelli (1608 – 1647)

• A long closed tube is filled with mercury and inverted in a dish of mercury

• Measures atmospheric pressure as ρgh

Pressure Measurements: Barometer

• A mercury barometer is used to measure the atmospheric pressure on an unidentified planet. A Hg (ρ = 13.534 g·cm−3) barometer shows a column height of 92 mm. What is the atmospheric pressure?

Quick Question

• How deep would a scuba diver need to go for the absolute pressure to be 2 atms?

Question

• A person is standing in the water breathing through a snorkel. The water level is just over the tip of his head. Estimate the force his diaphragm would need exert in order for him to breath.

Question

• A wooden cube (ρ = .8) 1 meter to a side is submerged in water.

• List all the forces acting on it.

• What is the net force?

Buoyancy

• ARCHIMEDES’ PRINCIPLE

• Any fluid applies a buoyant force to an object that is partially or completely immersed in it;

• the magnitude of the buoyant force equals the weight of the fluid that the object displaces:

11.6 Archimedes’ Principle

APPAPAPFB 1212

ghPP 12

ghAFB

hAV

gVFB

fluiddisplaced

of mass

Aluminum!

• A 1 kg block of aluminum (ρ= 2700 kg /m3) is under water. What will it’s free “fall acceleration” be?

If the object is less dense than the fluid, it will float. It will displace as it’s own weight in fluid, with the rest above the surface.

The magnitude of the buoyant forceis equal to the magnitude of itsweight.

11.6 Archimedes’ Principle

Example 9 A Swimming Raft

The raft is made of solid squarepinewood. Determine whetherthe raft floats in water and ifso, how much of the raft is beneaththe surface.

11.6 Archimedes’ Principle

N 47000

sm80.9m8.4mkg1000 233

max

gVVgF waterwaterB

m 8.4m 30.0m 0.4m 0.4 raftV

11.6 Archimedes’ Principle

N 47000N 26000

sm80.9m8.4mkg550 233

gVgmW raftpineraftraft

The raft floats!

11.6 Archimedes’ Principle

gVwaterwaterN 26000

Braft FW

If the raft is floating:

23 sm80.9m 0.4m 0.4mkg1000N 26000 h

m 17.0sm80.9m 0.4m 0.4mkg1000

N 2600023

h

An ideal fluid is incompressible and has zero viscosity. Water is really close to an ideal fluid. We have “steady-state” flow.

When the flow is steady, streamlines are often used to representthe trajectories of the fluid particles.

New Term

“Mass Flow Rate” How much mass of a fluid passes through a given cross section of area per second.

For a well behaved fluid, it is constant.

11.8 The Equation of Continuity

2222 vAt

m

111

1 vAt

m

distance

tvAVm

11.8 The Equation of Continuity

222111 vAvA

EQUATION OF CONTINUITY

The mass flow rate has the same value at every position along a tube that has a single entry and a single exit for fluid flow.

SI Unit of Mass Flow Rate: kg/s

11.8 The Equation of Continuity

Example 12 A Garden Hose

A garden hose has an unobstructed openingwith a cross sectional area of 2.85x10-4m2. It fills a bucket with a volume of 8.00x10-3m3

in 30 seconds.

Find the speed of the water that leaves the hosethrough (a) the unobstructed opening and (b) an obstructedopening with half as much area.

11.8 The Equation of Continuity

AvQ

sm936.0

m102.85

s 30.0m1000.824-

33

A

Qv

(a)

(b) 2211 vAvA

sm87.1sm936.0212

12 v

A

Av

BERNOULLI’S EQUATION

In steady flow of a nonviscous, incompressible fluid, the pressure, the fluid speed, and the elevation at two points are related by:

2222

121

212

11 gyvPgyvP

Blood flow

Blood flows out of the heart into the Aorta (r = .009 m) where the speed is .33 m/s. From there it flows into a capillary where the speed is .00034 m/s. Find the cross sectional area of the capillary.

11.10 Applications of Bernoulli’s Equation

Conceptual Example 14 Tarpaulins and Bernoulli’s Equation

When the truck is stationary, the tarpaulin lies flat, but it bulges outwardwhen the truck is speeding downthe highway.

Account for this behavior.

11.10 Applications of Bernoulli’s Equation

11.10 Applications of Bernoulli’s Equation

Example 16 Efflux Speed

The tank is open to the atmosphere atthe top. Find and expression for the speed of the liquid leaving the pipe atthe bottom.

2222

121

212

11 gyvPgyvP

atmPPP 2102 v

hyy 12

ghv 212

1

ghv 21

Flow of an ideal fluid.

Flow of a viscous fluid.

Problem

• The oil in a lubrication system has a density of 850 kg/m3. It flows through a cylindrical pipe of 8 cm diameter at 9.5 liters/second.

• Find the velocity of the oil.

• Find the velocity if the pipe narrows to 4cm diameter.

Question• The watermain going into a home is 2cm in

diameter, and the influx speed is 1.5 m/s. The pipe going to the shower in on the 2nd floor (5m up) is 1cm in diameter.

• Find the pressure on the second floor if it is the only thing using water in the home.

• Find the volume flow rate of the showerhead.

• A water tank has a radius R, is attached to a piston with radius r.

• How far (∆x) does the piston need to be pushed to raise the water level an amount (∆y)

• If the piston is height H below the water level, how much force and how much pressure need to be applied by the piston to move the piston by the same ∆x as before?

• A spigot is two meters below water level. IT starts off with a radius of 5 cm, then tapers to 2 cm.

• How long will it take for 1 liter to drain? (assume waterlevel doesn’t change much)

• Find the velocity of water at the efflux.

Question

• An icecube is floating in a drinking glass of water. As it melts, what happens to the water level?

• ρ = 917 kg/m3