Chapter 14

Fluid Mechanics

Fluids

• Fluids (Ch. 6) – substances that can flow (gases, liquids)

• Fluids conform with the boundaries of any container in which they are placed

• Fluids lack orderly long-range arrangement of atoms and molecules they consist of

• Fluids can be compressible and incompressible

Blaise Pascal(1623 - 1662)

Density and pressure

• Density

• SI unit of density: kg/m3

• Pressure (cf. Ch. 12)

• SI unit of pressure: N/m2 = Pa (pascal)

• Pressure is a scalar – at a given point in a fluid the measured force is the same in all directions

• For a uniform force on a flat area

V

mV

0lim

A

FP

A

0lim

A

FP

Atmospheric pressure

• Atmospheric pressure:

• P0 = 1.00 atm = 1.013 x 105 Pa

Fluids at rest

• For a fluid at rest (static equilibrium) the pressure is called hydrostatic

• For a horizontal-base cylindrical water sample in a container

mgFF 12 gyyAAPAP )( 2112 gyyPP )( 2112

hgPP 0

Fluids at rest

• The hydrostatic pressure at a point in a fluid depends on the depth of that point but not on any horizontal dimension of the fluid or its container

• Difference between an absolute pressure and an atmospheric pressure is called the gauge pressure

hgPPPg 0

hgPP 0

Chapter 14Problem 12

The tank is filled with water 2.00 m deep. At the bottom of one sidewall is a rectangular hatch 1.00 m high and 2.00 m wide that is hinged at the top of the hatch. (a) Determine the force the water causes on the hatch. (b) Find the torque caused by the water about the hinges.

Measuring pressure

• Mercury barometer

• Open-tube manometer

hgP 0

0;

;0

22

011

Phy

PPygyyPP )( 2112

PPhy

PPy

22

011

;

;0gyyPP )( 2112

hgPPPg 0

Pascal’s principle

• Pascal’s principle: A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container

• Hydraulic lever

• With a hydraulic lever, a given force applied over a given distance can be transformed to a greater force applied over a smaller distance

2

2

1

1

A

F

A

FP

2

121 A

AFF

2211 xAxAV

1

221 A

Axx

11 xFW 22 xF

Archimedes’ principle

• Buoyant force: For imaginary void in a fluid

p at the bottom > p at the top

• Archimedes’ principle: when a body is submerged in a fluid, a buoyant force from the surrounding fluid acts on the body. The force is directed upward and has a magnitude equal to the weight of the fluid that has been displaced by the body

gmB f

Archimedes of Syracuse

(287-212 BCE)

Archimedes’ principle

• Sinking:

• Floating:

• Apparent weight:

• If the object is floating at the surface of a fluid, the magnitude of the buoyant force (equal to the weight of the fluid displaced by the body) is equal to the magnitude of the gravitational force on the body

Bmg

Bmg

Bmgweightapparent

Chapter 14Problem 28

A spherical aluminum ball of mass 1.26 kg contains an empty spherical cavity that is concentric with the ball. The ball barely floats in water. Calculate (a) the outer radius of the ball and (b) the radius of the cavity.

Motion of ideal fluids

Flow of an ideal fluid:

• Steady (laminar) – the velocity of the moving fluid at any fixed point does not change with time (either in magnitude or direction)

• Incompressible – density is constant and uniform

• Nonviscous – the fluid experiences no drag force

• Irrotational – in this flow the test body will not rotate about its center of mass

Equation of continuity

• For a steady flow of an ideal fluid through a tube with varying cross-section

xAV tAv tvAtvA 2211

2211 vAvA

constAv

Equation of continuity

Bernoulli’s equation

• For a steady flow of an ideal fluid:

• Kinetic energy

• Gravitational potential energy

• Internal (“pressure”) energy

Daniel Bernoulli(1700 - 1782)

intEUKE gtot

2

2mvK

mgyU g

VPE int

2

2vV

gyV

Bernoulli’s equation

• Total energy

intEUKE gtot

VPgyVvV

2

2

Pgyv

V

Etot 2

2

const

22

22

11

21

22Pgy

vPgy

v

Chapter 14Problem 49

A hypodermic syringe contains a medicine having the density of water. The barrel of the syringe has a cross-sectional area A = 2.50 × 10-5 m2, and the needle has a cross-sectional area a = 1.00 × 10-8 m2. In the absence of a force on the plunger, the pressure everywhere is 1 atm. A force F of magnitude 2.00 N acts on the plunger, making medicine squirt horizontally from the needle. Determine the speed of the medicine as it leaves the needle’s tip.

Questions?

Answers to the even-numbered problems

Chapter 14

Problem 6(a) 1.01 × 107 Pa(b) 7.09 × 105 N outward

Answers to the even-numbered problems

Chapter 14

Problem 8225 N

Answers to the even-numbered problems

Chapter 14

Problem 223.33 × 103 kg/m3

Answers to the even-numbered problems

Chapter 14

Problem 38(a) 0.825 m/s (b) 3.30 m/s (c) 4.15 L/s