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J. Fluids in Motion

Chapter Objectives

• Study fluid dynamics.• Understanding Bernoulli’s Equation.

2005 Pearson Education South Asia Pte Ltd

J. Fluids in Motion

Chapter Outline

1. Fluid Flow2. Bernoulli’s Equation3. Viscosity and Turbulence


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 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 .

J. Fluids in Motion

1. Fluid Flow

Steady flow


Steady flow

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 .


J. Fluids in Motion


Turbulent flow

J. Fluids in Motion

1. Fluid Flow

• 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

The continuity equation


sectional area. If the fluid is incompressible, the product Av has the same value at all points along the tube.

J. Fluids in Motion

1. Fluid Flow

• In steady flow the total mass in the tube is constant, so


21mm ∆=∆

tvAtvA ∆=∆2211

ρρ

2211vAvA =


• The product Av is the volume flow rate ∆V/∆t, the rate which volume crosses a section of the tube

2211

J. Fluids in Motion

14.4 Fluid Flow

• 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.



fluid is not incompressible.• If and are the densities at section 1 and 2, then 1ρ 2ρ

222111vAvA ρρ =

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?


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.

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

( ) m/s 9.1104105.9rate flow volume 3

1=

××==

−

−

πAv


( ) m/s 9.1104 2

1

1=

×==

−πAv

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.

( )224.0 10π −×


( )( )

( )22

12 1 222

4.0 101.9 7.6 /

2.0 10

Av v m s

A

π

π

−

−

×= = =

×

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.


J. Fluids in Motion


J. Fluids in Motion

2. Bernoulli’s Equation

• The subscript 1 and 2 refer to any point along the flow tube,

constant 21 2 =++ vgyp ρρ


2

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.


J. Fluids in Motion

Example 2. (SOLN)


J. Fluids in Motion

Example 2. (SOLN)

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

Torricelli’s theorem.


in falling freely through a height h.

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