Chapter 10 Notes – Fluids
There are three common states of matter – solid, liquid, and gas.
A solid maintains a fixed shape and fixed size. When a strong force is applied to that
solid, in most cases, it keeps its shape and volume. A liquid does not have a fixed shape
but takes the shape of its container. Its shape changes, but it takes a large force per area
to significantly change its volume. Finally, a gas does not have a fixed shape or volume.
Pressure significantly affects the volume and shape of a gas. Liquids and gases do not
have a fixed shape, so therefore, they have the ability to flow – aka fluids.
10.1 Statics – Density and Specific Gravity
We often mix the terms weight and density, but there is a significant difference – volume.
It’s often said that iron weighs more than wood, but something is missing from that
statement. They must be the same volume to compare. What one must really mean is
that iron has a higher density than wood.
Density – mass per unit volume (variable: ρ: greek symbol rho and units: kg/m3)
Density equation: ρ=
𝑚
𝑉
(m = mass and V = volume)
The densities of pure substances such as copper are constant throughout the object.
There is a table 10-1 on page 276 of your book which gives a list of density values for
pure substances at 0ºC and 1 atm. The pressures and temperatures are listed as constant
because they do affect the density of a substance.
Specific Gravity (SG) – (of a substance) is the ratio of the density of that substance to
the density of water at 4.0ºC. There are no units for SG because it is a ratio.
The density of water is 1.00 g/cm3 or 1.00 x 103 kg/m3. In order to calculate the SG for
any substance, simply take the value of the density and divide by the density of water
(matching the units). For example, the density of lead is 11.3 x 103 kg/m3, so the specific
gravity for lead is 11.3.
10.2 Pressure in Fluids
Pressure – force per unit area (variable: P and units: N/m2 or Pascal (Pa))
𝐹
Pressure equation: P= 𝐴 (F = force and A = area)
The units of N/m2 can be interchanged with Pascal, which is named after Blaise Pascal.
A 60-kg person standing on a 500 cm2 area exerts a
(60kg)(9.8m/s2)/(0.050m2) = 12 x 103 Pa or N/m2.
A fluid exerts pressure in all directions. If the pressure in the fluid is not the same in
all directions, then the fluid must be flowing in a particular direction. Think about scuba
diving. When you dive down a certain depth, you can feel the pressure of the water all
around you equally.
Another important fact about a fluid at rest is that the force due to fluid pressure is
always perpendicular to any surface it is in contact with. This concept is much like a
normal force that we studied in physics 1. Due to Newton’s 3rd Law, the container of the
liquid can only push back with and equal and opposite force that is perpendicular to the
force of the fluid.
Have you ever wondered how pressure changes as you go deeper into the ocean? I
recently spent time snorkeling in Grand Cayman and the farther I dove down, the more
my ear felt the effects of the pressure. When we dive down, we have the weight of the
water above our head. W = mg – if we substitute the density equation in for mass, we are
left with W = ρVg. The volume is simply the area (A) of water times the depth (h). We
can substitute this in for volume – W = ρAhg. Now plug this into the pressure equation
and we’re left with:
P=ρgh (ρ = density of fluid, g = acceleration of gravity, h = depth of fluid or pressure
head)
10.3 Atmospheric Pressure and Gauge Pressure
If we look at the Earth’s atmosphere like a giant swimming pool, then the pressure is
least at the top of the atmosphere and greatest at the surface of the atmosphere (or at the
bottom of the swimming pool). The issue is that it’s very hard to measure from the top of
the atmosphere, so we instead measure the difference in depth to compare pressures in
the atmosphere.
Atmosphere – unit of pressure on the Earth’s surface at sea level (1 atm = 1.013E5 N/m2
= 101.3 kPa)
Another pressure commonly used in meteorology is the bar. 1 bar = 100 kPa
How come humans aren’t crushed by the enormous pressure on the surface? Living cells
have an internal pressure that matches the atmospheric pressure. An balloon exceeds
atmospheric pressure so it can press against the atmosphere and keep its shape.
Pressure Gauge – A device used to measure the pressure above the atmospheric
pressure. i.e. – tire gauge
In order to measure the absolute pressure, one must add the reading on the pressure
gauge plus the atmospheric pressure. i.e. – 220 kPa tire gauge reading + 101 kPa = 321
kPa of absolute pressure.
10.4 Pascal’s Principle
The Earth’s atmosphere exerts a pressure on all objects that it is contact with. Because of
this, it’s important not to forget the Earth’s atmospheric pressure when calculating the
pressure on a submerged object. If we calculate the water pressure on an object at 100m
depth, P = ρgh = (1000 kg/m3)(9.8 m/s2)(100m) = 9.8E5 N/m2 or 9.7 atm
This is the pressure due to the water, but the atmosphere sits on top of the water.
Therefore we need to add the atmospheric pressure to calculate the total pressure on that
object. Therefore the pressure on the object is really 10.7 atm (9.7 atm + 1 atm). Blaise
Pascal, a French philosopher, noticed this and created a principle.
Pascal’s Principle – pressure applied to a confined fluid increases the pressure
throughout by the same amount.
This principle is applied many places, but two examples are hydraulic brakes in a car and
a hydraulic lift.
The drawing above shows that the confined fluid (green) has the same pressure
throughout. Therefore, when the driver pushes the brakes (Pressure In), the fluid presses
against the brake cylinders (calipers) with equal pressure (Pressure Out). This causes the
brake pads to clamp down on the disk (rotor) attached to the wheel.
The trick used is that Pin = Pout so therefore Fout/Aout = Fin/Ain. If there is different
area between the master cylinder and the calipers, then a little force on the brake pedal
can put a great deal of force on the calipers.
10.5 Measurement of Pressure: Gauges and the Barometer
There are many devices used to measure pressure.
Manometer – U-shaped tube partially filled with liquid, usually mercury or water.
The equation to describe the manometer is P = Po + ρgh where Po is the atmospheric
pressure ρ is the density of the liquid, h is difference in level of liquid, and P is the
pressure being measured. An example would be a Radon detector system.
Other measurements of pressure include: 1 mm-Hg = 133 N/m2 = 1 torr (named after the
scientist Evangelista Torricelli who invented the barometer). So many units of pressure
can be a nuisance, so N/m2 = Pa can always be used as the standard units.
Barometer – a manometer closed at one end, filled with mercury, and inverted into a
bowl of mercury.
If the pressure of the atmosphere drops, then the mercury inside the bowl will not have as
much pressure on it and mercury inside the tube will drop. If the pressure increases,
mercury will be pushed down in the bowl and up into the tube. The atmosphere can
support a tube of Mercury 76 cm tall. If the mercury in the tube drops, it creates a
vacuum in the top of the tube.
10.6 Buoyancy and Archimedes’ Principle
Submerged objects in a liquid often appear to weigh less than they do when outside the
fluid. You may be able to lift something heavy like a rock under the ocean that you
wouldn’t be able to lift on land. If you lift the rock up above the surface of the water, it
appears to become much heavier. NASA uses underwater diving to simulate spacewalks
or gravity conditions on the moon.
These are examples of buoyancy. The weight of the submerged object does not change.
Instead there is an upward buoyancy force exerted by the liquid on the object. Divers or
fish nearly have a buoyancy force equal to the gravitational force so they end up
balancing. Wood has a greater buoyancy force than its weight, so it floats on the fluid.
Since pressure increases as an object is submerged deeper, the pressure on the bottom of
the object is greater than the pressure on the top of the object. This causes an upward
buoyancy force (FB).
If F1 is a fluid force at the top of a submerged object and F2 is a fluid force at the bottom
of the same object then F2 – F1 = FB. If we substitute pressure P = F/A = ρgh into the
equation, we get FB = ρgA(h2-h1) = ρgV since volume is equal to Ah. If we substitute in
mass for density * volume, then we get FB = mfg where mf is the mass of the fluid. This
means that the buoyant force is equal to the displaced mass of the fluid times gravity.
Archimedes’ Principle – the buoyant force on a body immersed in a fluid is equal to the
weight of the fluid displaced by that object.
Archimedes apparently thought of this in his bathtub while thinking he could determine if
the king’s crown was made of gold or not. See examples 10-5 and 10-6.
Any object that floats is less dense than water. This means that the weight of the object
can’t displace a fluid that has more weight (assuming same volume). When an object
floats, we can say FB = Fg (weight of object) – from this we can write a ratio of volume
of object displaced/volume of full object = density of object/density of fluid. This will
give a percentage or fraction of how much of the object is submerged under the fluid.
𝑉𝑑𝑖𝑠𝑝 𝜌𝑜
=
𝑉𝑜
𝜌𝑓
where Vdisp is the volume of fluid displaced by object and Vo is the entire volume of the
object.
10.7 Fluids in Motion; Flow Rate and the Equation of Continuity
Fluid Dynamics (hydrodynamics) – Fluids in motion
Streamline or laminar flow – flow of fluid is smooth, so that neighboring layers of the
fluid slide by each other smoothly.
Turbulent Flow – Erratic, small, whirlpool-like circles that are called eddy currents
Viscosity – Internal friction of a fluid while flowing – laminar flow has a low viscosity
and turbulent has a high viscosity.
A drop of dye can quickly show whether the flow is laminar or turbulent.
Flow Rate – Mass of fluid that passes a given point per unit time
We can substitute in density and volume for mass, and then break down volume into area
and length. If water has a laminar flow through a pipe, we get
𝛥𝑚 𝜌𝛥𝑉 𝜌𝛥𝐴𝛥𝑙
=
=
= 𝜌𝐴𝑣
𝛥𝑡
𝛥𝑡
𝛥𝑡
The flow rate into the pipe is equal to the flow rate out. This means that if the pipe
changes diameter, then ρ1A1v1 = ρ2A2v2 (density * Area of pipe * velocity of fluid) This
is called the equation of continuity.
If the fluid is incompressible (most liquids), then the density remains constant through the
line. If this is the case, then, A1v1 = A2v2
Av (area * velocity) is a flow rate of volume per time or m3/s. This system is used for
blood flow in the human body.
Also, wheezing is a symptom of this principle of flow rate and Bernoulli’s Principle. The
airways get so tiny that the velocity of air through them increases at a great rate. The
velocity gets so high that it creates a very low pressure in the bronchioles. This causes
the bronchioles to temporarily collapse, or wheeze.
10.8 Bernoulli’s Equation
Bernoulli’s Principle (worked out by Daniel Bernoulli) – where the velocity of a fluid is
high, the pressure is low, and where the velocity is low, the pressure is high.
High pressure slows the fluid down and low pressure allows the fluid to accelerate and
speed up.
Bernoulli dervived his equation from the above scenario. He asked, how much work
would be required to move fluid from area 1 to area 2? It turns out there are three
magnitudes of work acting on the fluid. First: Fluid to left exerts a pressure on area 1
Second: Fluid to the right exerts a pressure on area 2 Third: Work is done by gravity
since the fluid is moving uphill.
W1 = F1Δl1 = P1A1Δl1 where P = Pressure, A = Area, and l = length of pipe
W2 = F2Δl2 = -P2A2Δl2 (Note that there is a negative sign because the force points in the
opposite direction of the motion of the fluid).
W3 = -mg (h2-h1) This is the amount of work done due to gravity. The negative sign is
there because if the pipe flows uphill, gravity does negative work on the system.
The net work done on the fluid is W = W1 + W2 + W3
Now if we remember back to the work – energy principle from last year, the change in
kinetic energy is equal to the net work done on the system. Now let’s put it all together.
ΔKE = W1 + W2 + W3
½ mv22 – ½ mv12 = P1A1Δl1 - P2A2Δl2 - mg (h2-h1)
Divide through by volume and rearrange to get the equation below known as Bernoulli’s
Equation: