CHAPTER 2
Fluid Mechanics
FLUID MECHANICS
• Fluid statics – study of fluids at rest in
equilibrium situations.
• Fluid dynamics – study of fluids in motion
DENSITY
• Important property of any material
• Mass per unit volume
• A homogeneous material has the same
density throughout
• SI unit: 1 kg/m3
• CGS unit: 1 g/cm3
DENSITY
• Two objects made of the same material
have the same density even though they
may have different masses and different
volumes
DENSITY
𝑚
𝜌=
𝑉
• ρ = density
• m = mass of homogeneous material
• V = volume
DENSITY
Densities of Some Common Substances
SPECIFIC GRAVITY
• Ratio of a material’s density to the density
of water at 4.0°C (1000 kg/m3)
• Pure number without units
AVERAGE DENSITY
• The density of some materials varies from
point to point within the material.
• Density of a material depends on
environmental factors such as temperature
and pressure.
EXAMPLE 1
Find the mass and weight of the air in a
living room at 20°C with a 4.0 m x 5.0 m
floor and a ceiling 3.0 m high. What are the
mass and weight of an equal volume of
water?
PRESSURE IN A FLUID
• Defined as the normal force per unit area at
a point
• SI unit: pascal (1 pascal = 1 Pa = N/m2)
PRESSURE IN A FLUID
𝑑𝐹⊥
𝑝=
𝑑𝐴
• If the pressure is the same at all points of a
finite plane surface:
𝐹⊥
𝑝=
𝐴
• p = pressure
• F┴ = net normal force
• A = area
ATMOSPHERIC PRESSURE
• pa
• Pressure of the earth’s atmosphere
• Varies with weather changes and with
elevation
• 1 atmosphere (atm): normal atmospheric
pressure at sea level (average value) =
101,325 Pa
EXAMPLE 2
In the room described in Example 1, what is
the total downward force on the surface of
the floor due to air pressure of 1.00 atm?
PRESSURE, DEPTH, AND PASCAL’S
LAW
𝑝2 − 𝑝1 = −𝜌𝑔 𝑦2 − 𝑦1
(pressure in a fluid of uniform density)
• p1 and p2 are the pressures at elevations y1
and y2 (ρ and g are constant)
• This equation shows that when y increases,
p decreases.
PRESSURE, DEPTH, AND PASCAL’S
LAW
𝑝 = 𝑝0 + 𝜌𝑔ℎ
• The pressure p at a depth h is greater than
the pressure p0 at the surface by an
amount ρgh.
• Pressure is the same at any two points at
the same level in the fluid.
• The shape of the container does not matter.
PRESSURE, DEPTH, AND PASCAL’S
LAW
PASCAL’S LAW
Pressure applied to an enclosed fluid is
transmitted undiminished to every portion of
the fluid and the walls of the containing
vessel.
ABSOLUTE PRESSURE AND GAUGE
PRESSURE
• Gauge pressure – excess pressure above
atmospheric pressure
• Absolute pressure – total pressure
EXAMPLE 3
A storage tank 12.0 m deep is filled with
water. The top of the tank is open to the air.
What is the absolute pressure at the bottom
of the tank? The gauge pressure?
PRESSURE GAUGES
• Mercury barometer – common pressure
gauge
- consists of a long glass
tube, closed at one end, that has been filled
with mercury and then inverted in a dish of
mercury.
BUOYANCY
• Archimedes’s principle states: When a
body is completely or partially immersed in
a fluid, the fluid exerts an upward force on
the body equal to the weight of the fluid
displaced by the body.
EXAMPLE 4
A 15.0-kg solid gold statue is being raised
from a sunken ship. What is the tension in
the hoisting cable when the statue is (a) at
rest and completely immersed; and (b) at
rest and out of the water?
SURFACE TENSION
• The surface of a liquid behaves like a
membrane under tension.
• Arises because the molecules of the liquid
exert attractive forces on each other.
FLUID FLOW
• Ideal fluid – fluid that is incompressible (density cannot
•
•
•
•
change) and has no internal friction (viscosity)
Flow line – the path of an individual particle in a moving
fluid.
Steady flow – if the overall flow pattern does not change
with time.
Streamline – a curve whose tangent at any point is in the
direction of the fluid velocity at that point.
Flow tube – tube formed by flow lines passing through the
edge of an imaginary element of area.
FLUID FLOW
• Laminar flow – adjacent layers of fluid slide
smoothly past each other and the flow is
steady.
• Turbulent flow – no steady-state pattern;
the flow pattern changes continuously.
THE CONTINUITY EQUATION
• The mass of a moving fluid doesn’t change
as it flows.
𝐴1 𝑣1 = 𝐴2 𝑣2 (continuity equation,
incompressible fluid)
𝑑𝑉
𝑑𝑡
= 𝐴𝑣 (volume flow rate)
𝜌1 𝐴1 𝑣1 = 𝜌2 𝐴2 𝑣2 (continuity equation,
compressible fluid)
CONTINUITY EQUATION
EXAMPLE 5
As part of a lubricating system for heavy
machinery, oil of density 850 kg/m3 is
pumped through a cylindrical pipe of
diameter 8.0 cm at a rate of 9.5 liters per
second. (a) What is the speed of the oil?
What is the mass flow rate? (b) If the pipe
diameter is reduced to 4.0 cm, what are the
new values of the speed and volume flow
rate? Assume that the oil is incompressible.
BERNOULLI’S EQUATION
1
1
𝑝1 + 𝜌𝑔𝑦1 + 𝜌𝑣1 2 = 𝑝2 + 𝜌𝑔𝑦2 + 𝜌𝑣2 2
2
2
1 2
𝑝 + 𝜌𝑔𝑦 + 𝜌𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
2
• Relates the pressure p, flow speed v, and
elevation y for any two points, assuming
steady flow in an ideal fluid.
BERNOULLI’S EQUATION
EXAMPLE 6
Water enters a house through a pipe with an
inside diameter of 2.0 cm at an absolute
pressure of 4.0 x 105 Pa (about 4 atm). A
1.0-cm-diameter pipe leads to a secondfloor bathroom 5.0 m above. When the flow
speed at the inlet pipe is 1.5 m/s, find the
flow speed, pressure, and volume flow rate
in the bathroom.