A fluid is a material that can flow. Liquids - water in a river - molten iron in a steel plant Gases - steam in a heating pipe - air on a windy day The opposite of a fluid is a solid. 2/10/2004
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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.
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Two familiar examples of fluids Water . . . hydrodynamics Air . . . aerodynamics
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Density and Pressure 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
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Buoyancy Why do hot air balloons go up?
How can a great battleship float in water?
Archimedes’ Principle 2/10/2004
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1 3 + 10 71 < π < 3 + 7
Archimedes
This greatest scientist and mathematician of ancient Greece made many discoveries: • accurate value of π • volume and area of a sphere
4 3
πr 3 and 4πr 2
• 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 his book On Floating Bodies
The story of his death 2/10/2004
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Archimedes’ Claw
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. 2/10/2004
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Where have you seen this, and what is it?
Archimedes screw pump 2/10/2004
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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. 2/10/2004
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Why? Imagine the object replaced by a fluid element of the same shape. In 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. 2/10/2004
Buoyancy force
Weight of the fluid element
Q. E. D.
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Floating or sinking? 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.
Fgravity = −Mg = − ρ objectV × g Fbuoyancy = ρ fluidV × g gravity, weight
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) kg/m3
(less than the density of water, so it floats)
What fraction f of the ship is below the water surface? Answer: f = 0.10 2/10/2004
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Generalization:
…because
ρobject ρ fluid
su bm
f =
er ge d
When an object floats in water, the fraction of its volume that lies below the surface is
V
EXAMPLE
ρ = M/V = 0.10 x
103
ρ object V g = ρ fluid f V g weight
buoyancy force
Example: What fraction of an iceberg is below the surface of the ocean? Answer: 92 % 2/10/2004
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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
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Hot air balloons - quantitative At the equilibrium altitude, ρavg = ρatmosphere .
M + ρhot airV = ρ atmosphere V →
ρhot air = ρ atmosphere −
M V
The density of the hot air must be low enough. 2/10/2004
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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
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3 forces act on the fluid element.
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The buoyancy force exists because the pressure at the bottom is greater than the pressure at the top.
h
pbottom = ptop + ρgh Generally, the pressure as a function of depth d in an incompressible fluid in static equilibrium is given by…
p (d ) = p ( 0 ) + ρ gd pressure at depth d
pressure at the surface (d = 0) ρ= density of the fluid
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Pressure versus depth in hydrostatics
p (d ) = p ( 0 ) + ρ gd pressure at depth d
pressure at the surface (d = 0)
This pressure gradient produces the buoyancy force.
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The molecular view of fluid pressure fluid
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
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What is this?
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Distribution of water height h
water depth 0
pground = ptank + ρgh
water depth h
ground level
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Distribution of water
How does the water get up in the tower?
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… by one or more electric pumps. Without electric power, we’d have to pump water out of a well with a hand pump. ISP 209 -- 5A
