Chapter 10: Fluids
Don’t Be Dense
• Fluid = any state of matter with the ability to
flow. Mostly liquids and gasses (also plasma
and colloids, but we will stick to liquid and
gas)
• Density (ρ) = a ratio of mass per volume
• ρ = m/v
• Specific gravity (SG) the ratio of the density of
a substance to that of water at 4.0oC (which is
1.00g/cm3)
Under Pressure
• Pressure = force per unit area
pressure = P = F/A
• The units: N/m2 named the Pascal (Pa)
1Pa = 1N/m2
• Example: A 60kg person whose feet have a total
area of 500cm2 exerts a pressure on the Earth of
F/A = mg/A = (60kg)(9.8m/s2) / (0.050m2) =
12,000N/m2. If he stood on one foot, the force is
the same, but the area is cut in half, so the
pressure is doubled to 24,000N/m2.
Pressure Cont.
• A fluid exerts a pressure in all directions.
• The force due to fluid pressure always acts
perpendicular to any surface it is in contact
with.
• Pressure depends on depth.
Imagine a point in the ocean at a depth h. The
pressure on this point is caused by the force of
the weight of the column of water above it.
20,000 Leagues
• Here’s the math:
• F = mg = ρAhg, coming from if ρ = m/v, then
m = ρv and v = Ah.
• The pressure, P = F/A = (ρAhg)/A = ρgh
• P = ρgh
• This means that the pressure at equal depths
in a uniform fluid are equal.
Incompressible Fluid
• The last equation can be used for a fluid that
does not change in density with depth.
• This is actually quite an appropriate
approximation for liquids (except at really high
pressures) but this is not very true for gasses.
• ΔP = ρgΔh
Atmospheric Pressure
•
•
•
•
The pressure at sea level is 1.013 x 105 N/m2
This is referred to as one atmosphere (atm)
1 atm = 1.013E5N/m2 = 101.3kPa
Another unit of pressure is the bar (often used in
meteorology)
• 1 bar = 100kPa
• How on Earth does a human body survive
100,000 N of force on every square meter of its
body?
Gauge Pressure
• The answer is that every cell in the human
body has an internal pressure close to that of
1 atm, so the forces balance out.
• A tire gauge measures the pressure above that
of atmospheric, which is called the gauge
pressure.
• If one wants the absolute pressure, P, one
must add atmospheric pressure and gauge
pressure together.
Pascal’s Principal
• “Pressure applied to a confined fluid increases
the pressure throughout by the same
amount”.
• Basically Pascal says that liquids in confined
spaces are socialist, equally distributing the
pressure.
• This fact is incredibly useful.
In and Out
• Pascal’s Principal means that the pressure put
into the system at one end is equal to the
pressure experienced at the other end or
• Pout = Pin or
• Fout /Aout = Fin/Ain or
• Fout/Fin = Aout /Ain
• Where Fout/Fin is called the mechanical advantage.
• For example, of the output piston is 20 times the
in put piston, then force gets multiplied by 20.
Bouyancy
• Why do submerged objects appear to weigh less?
• Remember, weight is a force. So to diminish
weight is to diminish the force by applying a
counter force.
• To place an object into a fluid would be to
provide it another surface, which will supply a
force perpendicular to the surface. Newton’s
third then dictates that the fluid will exert an
opposite perpendicular force (i.e in the upward
direction.)
Buoyant Force
• For a submerged cylinder, the fluid will exert a
pressure down on the top of the cylinder and
a pressure up on the bottom of the cylinder.
• The downward pressure is Pdown = ρfgh1, where
h1 is the depth at which the top of the cylinder
resides and ρf is the density of the fluid.
• Using P = F/A, we get Fdown = PdownA = ρfgh1A
• Similarly, Fup = PupA = ρfgh2A
Buoyant Force Cont
• The buoyant force, FB, is the difference
between F2 and F1.
• F B = F 2 – F1
= ρfgA(h2 – h1)
= ρfgAh
= ρfgv, where v is the volume of the cylinder.
Archimedes‘ Principle
• From the last slide we had ρfgv
• But wait, ρ = m/v and m = ρv, therefore,
ρfgv = mfg, which is the weight of the fluid, which
takes up the same volume as the object.
• The cool thing is this works regardless of the
shape of the object. This was first found by
Archimedes and is thus named the Archimedes’
Principle: the buoyant force on a body immersed
in a fluid is equal to the weight of the fluid
displaced by that object.
Apparent weight
• As we stated earlier, submerged objects
appear to weigh less.
• w‘ = the apparent weight of the object
• w ‘ = F ’T
= w – FB
= ρOgv – ρfgv
• This yields a useful proportion
O
w

'
ww
f
Fluid dynamics
• Everything we have done thus far would be called
fluid statics.
• Now we will explore a fluid that is moving.
• To begin with, there are two types of fluid
motion:
• Laminar flow = smooth flow
• Turbulent flow = has erratic, small whirlpools
called “eddy” currents. Much energy is removed
from the system in these eddy currents.
Inertia analog
• Not every fluid flows at the same rate. Water
and ketchup do not flow the same.
• The difference is internal friction, or resistance
to flow, called viscosity.
• We usually think of viscous fluids as being
“thick”
• Several factors affect viscosity: density,
pressure, temperature, electrostatic factors,
etc.
Flow rate
• The mass flow rate is defined at the Δm of
fluid that passes a given point per unit time
Δt, or Δm/Δt.
• Picture a fluid flowing in a pipe that reaches a
choke point.
• A1v1 = A2v2 this is called the equation of
continuity
Bernoulli’s Principle
• Bernoulli’s Principle: where the velocity of a
fluid is high, the pressure is low, and where
the velocity is low, the pressure is high.