1

2/10/2004 ISP 209 -- 5A 1

FluidsFluids

i.e., gases and liquidsi.e., gases and liquids

2/10/2004 ISP 209 -- 5A 2

A fluid is a material that can flow.

LiquidsLiquids- water in a river- molten iron in a steel plant

GasesGases- steam in a heating pipe- air on a windy day

The opposite of a fluid is a solid.

2

2/10/2004 ISP 209 -- 5A 3

Flow requires the ability to change shape.

In a gas or liquid, the molecules are free to slide around – not held at fixed positions.

In a solid, the molecules are held in fixed relative positions.

2/10/2004 ISP 209 -- 5A 4

Two familiar examples of fluidsWater . . . hydrodynamicsAir . . . aerodynamics

3

2/10/2004 ISP 209 -- 5A 5

Two important parameters of any fluid are• density ρ = mass per unit volume [kg/m3]• pressure p = force per unit area on a surface in contact with the fluid [N/m2 = Pa for pascal]

Useful numbers for the CAPA

ρ (water) = 1.00 x 103 kg/m3

p (air at STP) = 1 atm = 1.013 x 105 Pa = 14.7 psi

Density and PressureDensity and Pressure

2/10/2004 ISP 209 -- 5A 6

Buoyancy

Why do hot air balloons go up?

How can a great battleship float in water?

Archimedes’ Principle

4

2/10/2004 ISP 209 -- 5A 7

ArchimedesArchimedes

This greatest scientist and mathematician of ancient Greece made many discoveries:

• accurate value of π

• volume and area of a sphere

• levers and machines: “Give me a lever long enough and a place to stand and I will move the Earth.”

• density and buoyancy: The story of the King’s crown

The story of his death

2334 4 rr ππ and

71

7110 33 +<<+ π

his book On Floating Bodies

2/10/2004 ISP 209 -- 5A 8

Archimedes lived in the Greek kingdom of Syracuse on the island of Sicily. The Roman army and navy laid siege to Syracuse. Archimedes, an old man at the time, was put in charge of the defense of the city. He held off the Romans for about a year with military inventions like the Claw.

Archimedes’ Claw

5

2/10/2004 ISP 209 -- 5A 9

Archimedes screw pump

Where have you seen this, and what is it?Where have you seen this, and what is it?

2/10/2004 ISP 209 -- 5A 10

Archimedes’ Principle of Buoyancy

An object immersed in a fluid experiences an upward force equal in strength to the weight of the displaced fluid.

This force is called the buoyancy force.

6

2/10/2004 ISP 209 -- 5A 11

Why?Imagine the object replaced by a fluid element of the same shape.

Weight of the fluid element

Buoyancy forceIn static equilibrium the buoyancy force must balance the weight of the equivalent fluid element.

The buoyancy force on the original object is the same as the buoyancy force on the equivalent fluid element.

Q. E. D.

2/10/2004 ISP 209 -- 5A 12

Floating or sinking?

gVFgVMgF

×=

×−=−=

fluidbuoyancy

objectgravity

ρρ

gravity, weight

buoyancy Suppose an object of mass M and volume V is completely immersed in a fluid. Will it rise or sink?

Let upward be the positive direction.

VgFFF )( objectfluidgravitybuoyancynet ρρ −=+=

Compare the densities:fluid > object ⇒ positive Fnet ⇒ risesfluid < object ⇒ negative Fnet ⇒ sinks

7

2/10/2004 ISP 209 -- 5A 13

Example: Imagine a battleship with the parameters listed below.

M = 3,000 metric tons = 3 x 106 kg

V = 3 x 104 m3 (mostly air)

ρ = M/V = 0.10 x 103 kg/m3

What fraction f of the ship is below the water surface? Answer: f = 0.10

(less than the density of water, so it floats)

EXAM

PLE

2/10/2004 ISP 209 -- 5A 14

Generalization:

When an object floats in water, the fraction of its volume that lies below the surface is

fluid

object

ρρ

=f

Example: What fraction of an iceberg is below the surface of the ocean?

…because f V g V g fluidobject ρρ =weight buoyancy force

V subm

erge

d

Answer: 92 %

8

2/10/2004 ISP 209 -- 5A 15

Hot air balloons - qualitative

This is tricky. We need to compare the density of the atmosphere (the fluid) to that of the hot air and balloon (the floating object).

1- Atmospheric density decreases with altitude.

2- Density of an ideal gas is inversely proportional to its temperature (because the gas expands with temperature and ρ = M/V).

Thus, the balloon rises until the net force is 0, i.e., to the altitude where ρavg = ρatmosphere .

2/10/2004 ISP 209 -- 5A 16

Hot air balloons - quantitative

At the equilibrium altitude,ρavg = ρatmosphere .

atmosphereair hot

atmosphereair hot

VM

VVM

−=→

=+

ρρ

ρρ

The density of the hot air must be low enough.

9

2/10/2004 ISP 209 -- 5A 17

What physical entity exerts the buoyancy force?

• The fluid – by pressure.

• More precisely, the molecules of the fluid that are adjacent to the surfaces of the immersed object.

Imagine a rectangular volume element, and consider the net force on the parcel of fluid.

Pressure = force / area; i.e., p = F/A 3 forces act on

the fluid element.

2/10/2004 ISP 209 -- 5A 18

ρ= density of the fluid

The buoyancy force exists because the pressure at the bottom is greater than the pressure at the top.

ghpp ρ+= topbottom

Generally, the pressure as a function of depth d in an incompressible fluid in static equilibrium is given by…

0 gdpdp ρ+= )()(pressure at depth d

pressure at the surface (d = 0)

h

10

2/10/2004 ISP 209 -- 5A 19

Pressure versus depth in hydrostatics

0 gdpdp ρ+= )()(pressure at depth d

pressure at the surface (d = 0)

This pressure gradient produces the buoyancy force.

2/10/2004 ISP 209 -- 5A 20

The molecular view of fluid pressureThe molecular view of fluid pressure

A molecule strikes the surface, and so exerts a sharp impulsive force on the surface.

Molecules are so incredibly small that these collisions occur almost continuously; i.e., they produce a steady force.

Pressure = force / area

fluid

11

2/10/2004 ISP 209 -- 5A 21

What is this?

2/10/2004 ISP 209 -- 5A 22

Distribution of water Distribution of water

height h

ground level

water depth 0

water depth h

tankground ghpp ρ+=

12

2/10/2004 ISP 209 -- 5A 23

Distribution of waterDistribution of water

How does the water get up in the tower?

… by one or more electric pumps. Without electric power, we’d have to pump water out of a well with a hand pump.