J. Fluids in Motion
Chapter Objectives
• Study fluid dynamics.
• Understanding Bernoulli’s Equation.
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J. Fluids in Motion
Chapter Outline
1. Fluid Flow
2. Bernoulli’s Equation
3. Viscosity and Turbulence
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J. Fluids in Motion
1. Fluid Flow
• An ideal fluid is a fluid that is incompressible, that is
density do not change, and has no internal friction
(viscosity).
• The path of an individual particle in a moving fluid is
called a flow line.
• The overall flow pattern does not change with time,
the flow is called steady flow.
• A streamline is a curve whose tangent at any point is
in the direction of the fluid velocity at that point.
• In the figure, the flow lines passing through the edge
of an imaginary element area and form a tube called
a flow tube.
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J. Fluids in Motion
1. Fluid Flow
Steady flow
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J. Fluids in Motion
1. Fluid Flow
• The figure shows pattern of fluid flow from left to right
round a number of shapes.
• These patterns are typical of laminar flow.
• At sufficient high flow rates, the flow can become
irregular and chaotic and is called turbulent flow.
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J. Fluids in Motion
Turbulent flow
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J. Fluids in Motion
1. Fluid Flow
The continuity equation
• The mass of a moving fluid doesn’t change as it flows.
• This leads to a quantitative relationship called
continuity equation.
• The figure shows a flow tube with changing cross
sectional area. If the fluid is incompressible, the
product Av has the same value at all points along the
tube.
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J. Fluids in Motion
1. Fluid Flow
The continuity equation
• In steady flow the total mass in the tube is constant,
so ∆m = ∆m
ρA v ∆t = ρA v ∆t
1
2
1
1
2
2
Av = A v
1
1
2
2
• The product Av is the volume flow rate ∆V/∆t, the
rate which volume crosses a section of the tube
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J. Fluids in Motion
14.4 Fluid Flow
The continuity equation
• The mass flow rate is the mass flow per unit time
through a cross section. This is equal to the density
times the volume flow rate.
• We can generalize Eq. (14.10) for the case in which
fluid is not incompressible.
• If ρ1 and ρ 2 are the densities at section 1 and 2, then
ρ Av = ρ A v
1
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1
1
2
2
2
J. Fluids in Motion
Example 1. Incompressible fluid flow
As part of a lubricating system for heavy
machinery; oil of density 850kg/m3 is pumped
through a cylindrical pipe of diameter 8.0cm at a
rate of 9.5 liters per second. (1L = 0.001 m3)
A) What is the speed of the oil?
B) If the pipe diameter is reduced to 4.0cm, what
are the new values of the speed and volume
flow rate? Assume that the oil is incompressible.
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J. Fluids in Motion
Example 1. (SOLN)
A) The volume flow rate is equal the product A1v1
where A1 is the cross-sectional are of the pipe of
diameter 8.0cm and radius 4.0cm. Hence
volume flow rate 9.5 × 10
v =
=
A
π (4 × 10
−2
1
1
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−3
)
= 1.9 m/s
J. Fluids in Motion
Example 1. (SOLN)
B) Since the oil is incompressible, the volume flow rate
has the same value of (8.5L/s) in both sections of pipe.
(
(
π 4.0 × 10
A1
v2 = v1 =
A2
−2
π 2.0 × 10
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) (1.9) = 7.6m / s
2
)
−2 2
J. Fluids in Motion
2. Bernoulli’s Equation
• Bernoulli’s equation states that the relationship of
pressure, flow speed, and height for flow of an ideal,
incompressible fluid.
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J. Fluids in Motion
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J. Fluids in Motion
2. Bernoulli’s Equation
• The subscript 1 and 2 refer to any point along the flow
tube,
1
p + ρgy + ρv = constant
2
2
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J. Fluids in Motion
Example 2. Bernoulli’s equation
A water tank has a spigot near its bottom. If the top
of the tank is open to the atmosphere, determine
the speed at which the water leaves the spigot
when the water level is 0.500 m above the spigot.
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J. Fluids in Motion
Example 2. (SOLN)
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J. Fluids in Motion
Example 2. (SOLN)
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J. Fluids in Motion
Example 2. (SOLN)
Torricelli’s
theorem.
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That is the speed of efflux
from an opening at a distance
h before the top surface of
liquid is the same as the
speed a body would acquire
in falling freely through a
height h.