Fluids

Physics, Chapter 10

by Giancoli

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First, a few reminders:

• atomic mass unit (u) - based on setting carbon-12 equal to 12.000 00 u
• Avogadro’s number
• = 6.022 x 10²³ particles/mol
• STP = 0°C at 1 atm
• D = m/V (in Physics we use ρ = m/V)
• 1 g/cm³ = 1000 kg/m³ = density of H₂O
• atoms of every kind are nearly the same size (diameter = ~0.1-0.3 nm)

Liquids as fluids

• fluid = able to flow
• viscosity - internal resistance or friction offered to an object moving through a fluid
• ideal liquids are incompressible and nonviscous
• real liquids, of course, are slightly compressible and their viscosity varies significantly

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• water is considered “nonviscous” because its particles are so small

Gases as fluids

• the density of our atmosphere at sea level is around 1/800th that of H₂O
• each square inch of exposed surface at sea level is struck by 2 x 10²⁴ air molecules per second!

• at STP, there are approx. 3 x 10¹⁹ molecules of air per cubic centimeter
• that’s around 3 nm per air particle (10 times its radius)
• each air particle travels at ~450 m/s and has ~6 x 10⁹ collisions per second
• actual speed depends on air temperature

Pressure

• pressure is defined as the scalar value of the force acting perpendicular to, and distributed over, a surface divided by the area of that surface
• in other words:
• P = F / A

• pressure is measured in Pascals
• 1 Pa = 1 N/m²
• Pascals are so small that we often use kiloPascals (kPa)
• “standard pressure”:
• 1 atmosphere
• 101,300 Pa
• 101.3 kPa

Direction of pressure

• the force exerted by a fluid at rest acting on any rigid surface is always perpendicular to that surface
• fluids exert pressure equally in all directions
• gravity is the ultimate cause of hydrostatic pressure
• imagine a column of fluid over an object in that fluid

Pressure equations

or, how we get from P = F / A to P_l = ρgh

F_w = mg

and

ρ = m / V

so:

m = ρV

which means:

F_w = ρVg

which we can plug into:

P = F / A

to make:

P = ρVg / A

since

V = area * height

this means A & V cancel, so:

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P_l = ρgh

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(because the density of ideal liquids is nearly uniform, use subscript l)

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and taking surface pressure into account:

P = P_s + P_l

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“The Hero of Haarlem”

• a Dutch folk tale tells of a boy named Hans who held back the North Sea by sticking his finger in a hole in a dike that was leaking
• he really could have done it because P_l = ρgh only relies on depth and density of liquid, not volume
• pressure is independent of shape of vessel

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• pressure at every point at a given horizontal level in a single body of fluid at rest is the same!