Chapter 11

Fluids

11.1 Mass Density

• Fluids are materials that can flow, and they include both gases and liquids.

The mass density of a liquid or gas is an important factor that determines its important factor that determines its behavior as a fluid.

The mass density ρ is the mass m of a substance divided by its volume V:

The SI unit for mass density is kg/m³.

• Equal volumes of different substances generally

have different masses, so the density depends

on the nature of the material.

• Gases have the smallest densities because gas • Gases have the smallest densities because gas

molecules are relatively far apart and contain

large fraction of empty space.

• In liquids and solids the molecules are much

more tightly packed and the lighter packing

leads to larger densities.

• The densities of gases are very sensitive to changes in temperature and pressure.

• It is the mass of a substance , not its weight that enters into the definition of weight that enters into the definition of density

Example: Blood as a fraction of

bodyweight• The body of a man whose weight is 690 N

contains about 5.2x10⁻³m³ (5 qt) of blood.

• (A) Find the blood’s weight

• (B) Express it as a percentage of the body

⁻

• (B) Express it as a percentage of the body weight.

•

Compare Densities

• A convenient way to compare densities is to use the concept of specific gravity. The specific gravity of a substance is its density divided by the density of a density divided by the density of a standard reference material, usually chosen to be water at 40C.

3

/100.1

tan

4

tan

3mkgx

cesubsofdensity

Cwateratofdensity

cesubsaofdensitygravityspecific

o==

Pressure

• The pressure P exerted by a fluid is defined as the magnitude F of the force acting perpendicular to a surface divided by the area A over which the force acts:by the area A over which the force acts:

• The SI unit for pressure is Pascal

• (1 Pa = 1 N/m²).

Example: The Force on a

Swimmer• Suppose the pressure acting on the back

of a swimmer’s hand is 1.2x105 Pa. The surface area of the back of the hand is 8.4x10-3m2.is 8.4x10-3m2.

(a)Determine the magnitude of the force that acts on it.

(b) Discuss the direction of the force.

A

FP =

(b)The hand (palm downward) is oriented parallel to the bottom

(a)

(b)The hand (palm downward) is oriented parallel to the bottomof the pool . Since the water pushes perpendicularly against theback of the hand, the force f is directed downward. This downwardforce is balance by an upward acting force on the palm, so that thehand is in equilibrium.If the hand were rotated 900, the directions of these forceswould also be rotated by 900, always being perpendicular tothe hand.

Atmospheric pressure at sea level 1.013 x 10⁵ Pa = 1 atmosphere

Pressure and Depth in a Static

Fluid

• P₂₂₂₂=P₁₁₁₁+ρgh

• In the presence of gravity, the upper layers of a

fluid push downward on the layers beneath, with

the result that fluid pressure is related to depth.

₁₁₁₁

₂₂₂₂

the result that fluid pressure is related to depth.

• The equation shows that the relation is where P₁₁₁₁

is the pressure at one level, P₂₂₂₂ is the pressure at

a level that is h meters deeper, and g is the

magnitude of the acceleration due to gravity

(9.80 m/s²)

(b)

Example: The Swimming Hole

• Points A and B are both located at a distance of h = 5.50 m below the surface of the water. Find the pressure at each of these two points. these two points.

Solution:•P₂=P₁+ρgh•P₁ = 1.01 x 10⁵ Pa } atmospheric pressure•ρ = 1.000 x 10³ kg/m³ in table 11.1 is the density of water •h = 5.50 m •

Example: Blood Pressure

• Blood in the arteries is flowing, but as a first approximation, the effects of this flow can be ignored and blood treated as a static fluid. Estimate the amount by which static fluid. Estimate the amount by which the blood pressure P2 in the anterior tibial artery at the foot exceeds the blood pressure P1 in the aorta at the heart when the person is (a) reclining horizontally and (b) standing.

11.4 Pressure Gauges

• Two basic types of pressure gauges are the mercury barometer and the open-tube manometer.

• The gauge pressure is the amount by • The gauge pressure is the amount by which a pressure P differs from atmospheric pressure. The absolutepressure is the actual value for P.

ghPP ρ+= 12

ghP ρ= ghPatm ρ=

( )( )( )

mm 760m 760.0

sm80.9mkg1013.6

Pa 1001.1233

5

==

×

×==

g

Ph atm

ρ

AB PPP ==2

ghPPA ρ+= 1

ghPP atm ρ=−43421

pressure gauge

2

Pop Quiz

• 1. The ice on a lake is 0.010m thick. The lake is circular, with a radius of 480m. Find the mass of the ice

• 2. If the area of a hammer is 2 m2 and a • 2. If the area of a hammer is 2 m2 and a wall is hit with a force of 10 Newtons, what is the pressure the hammer puts on the wall?

Pascal’s Principle

• Any change in the pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and the enclosing walls. the enclosing walls.

Pascal’s Principle

• Pascal's principle states that when pressure is applied to a confined liquid, this pressure is transmitted, without loss, throughout the entire liquid and to the walls of the container.

• For example your eyeballs contain a liquid.• For example your eyeballs contain a liquid.

• A sharp blow to the front of an eyeball will produce a higher pressure which is transmitted to the opposite side.

• This large pressure may cause the optic nerve to be damaged

(b)

• Automobile Hydraulic Lift

• A hydraulic lift for automobiles is an example of a force multiplied by hydraulic press, based on Pascal’s Principle. The fluid in the small cylinder must be moved fluid in the small cylinder must be moved much further than the distance the car is lifted.

Hydraulic Lift

Archimedes Principle

• Any fluid applies a buoyant force to an object that is partially or completely immersed in it; the magnitude of the buoyant force equals the weight of the buoyant force equals the weight of the fluid that the object displaces:

• FB = Wfluid

• Magnitude of Weight of

• buoyant force displaced fluid

SOLUTION:

• Buoyancy

• Buoyancy arises from the fact that fluid pressure increases with depth and from the fact that the increased pressure is the fact that the increased pressure is exerted in all directions (Pascal's principle) so that there is an unbalanced upward force on the bottom of a submerged object

{ {ofWeight

fluid

of Magnitude

WFB =

ARCHIMEDE’S PRINCIPLE

fluid displacedofWeight

forcebuoyant of Magnitude

( )APPAPAPFB 1212 −=−=

ghAF B ρ=

{gVFB

fluiddisplaced

of mass

ρ=

Example: A Swimming Raft

• A solid, square pinewood raft measures 4.0 m on a side and is 0.30 m thick. (a) Determine whether the raft floats in water, and (b) if so, how much of the raft is and (b) if so, how much of the raft is beneath the surface.

( )( )( ) m 8.4m 30.0m 0.4m 0.4 ==raftV

max == gVVgF waterwaterB ρρ

( )( )( )

N 47000

sm80.9m8.4mkg1000 233

=

=

( )( )( )sm80.9m8.4mkg550 233=

== gVgmW raftpineraftraft ρ

The raft floats

( )( )( )

N 47000N 26000

sm80.9m8.4mkg550

<=

=

• (B) If the raft is

floating

Braft FW = Braft FW =gVwaterwaterρ=N 26000

( )( )( ) ( )23 sm80.9m 0.4m 0.4mkg1000N 26000 h=

( )( )( )( )m 17.0

sm80.9m 0.4m 0.4mkg1000

N 2600023

==h

11.7 Fluids in Motion

• Fluids can move or flow in many ways. Water may flow smoothly and slowly in a quiet stream or violently over a waterfall. The air may form a gentle breeze or a The air may form a gentle breeze or a raging tornado. To deal with such diversity, we will identify basic types of fluid flow

= Change in Volume

t = time

Bernoulli's Equation

• The Bernoulli equation states that,

• In the steady flow of non viscous, incompressible fluid of density ρ, the pressure P, the fluid speed v, and the elevation y at any two points (1 and 2) are related by

• P1 + ½ ρ v12 + ρ gy1 = P2 + ½ ρ v2

2 + ρ gy2

• where

• points 1 and 2 lie on a streamline,

• the fluid has constant density,

• the flow is steady, and

• there is no friction

y1 = y2 Then,

insertinsert

Basic Types of Fluid Flow

• 1. Steady Flow VS Unsteady flow

• Steady flow - the velocity of the fluid particles at any point is constant as time passes.passes.

• - every particle passing through this point has the same velocity. Example if the velocity that that point is 5m/s, then every particles passing here will be with a velocity of 5m/s.

In the river, water flows faster near

the center and slower near its bank

• Unsteady flow exists whenever the velocity at a point in a fluid changes as time passes.

• Turbulent flow is an extreme example of • Turbulent flow is an extreme example of unsteady flow. It occurs when there are sharp obstacles or bends in the path of fast moving fluid.

In a turbulent flow the velocity at a

point changes erratically from

• moment to moment both in magnitude and direction.

• 2. Compressible VS Incompressible

• Incompressible – the density of a liquid • Incompressible – the density of a liquid remains almost constant as the pressure changes.

• Liquid are incompressible while gases are compressible.

However, there are situations in

which the density of a flowing

• gas remains constant enough that the flow can be considered incompressible.

• 3. Viscous VS Non-viscous

• Viscous fluid does not flow readily like • Viscous fluid does not flow readily like honey. Honey have a large viscosity.

• Water is less viscous and flow more steadily.

• The flow of viscous fluid is an energy dissipating process.

The viscosity hinders neighboring

layers of fluid from sliding freely

past one another.

A fluid with zero viscosity flows in an unhindered

manner with no dissipation of energy.

Although no real fluid has zero viscosity at normal Although no real fluid has zero viscosity at normal

temperature, some fluids have negligibly small

viscosities.

An incompressible, non viscous fluid is called an

Ideal fluid.

• There are many ways to measure viscosity, including attaching a torque wrench to a paddle and twisting it in a fluid, using a spring to push a rod into a fluid, and seeing how fast a fluid pours through a hole. This exercise uses one of the oldest and easiest ways: we will simply see how oldest and easiest ways: we will simply see how fast a sphere falls through a fluid. The faster the sphere falls, the lower the viscosity. This makes sense: if the fluid has a high viscosity it strongly resists flow, so the sphere falls slowly. If the fluid has a low viscosity, it offers less resistance to flow, so the ball falls faster

Units of η

• Laminar shear of fluid between two plates. Friction between the fluid and the moving boundaries causes the fluid to shear. The force required for this action is a measure force required for this action is a measure of the fluid's viscosity

• Gases

• Viscosity in gases arises principally from the

molecular diffusion that transports momentum

between layers of flow. The kinetic theory of between layers of flow. The kinetic theory of

gases allows accurate prediction of the

behaviour of gaseous viscosity, in particular that,

within the regime where the theory is applicable:

• Viscosity is independent of pressure; and

• Viscosity increases as temperature increases

• Liquids• In liquids, the additional forces between molecules become

important. This leads to an additional contribution to the shear stress though the exact mechanics of this are still controversial. Thus, in liquids:

• Viscosity is independent of pressure (except at very high pressure); and and

• Viscosity tends to fall as temperature increases (for example, water viscosity goes from 1.79 cP to 0.28 cP in the temperature range from 0 °C to 100 °C); see temperature dependence of liquid viscosity for more details.

• The dynamic viscosities of liquids are typically several orders of magnitude higher than dynamic viscosities of gases

Poiseuille’s Law

• A fluid whose viscosity is η, flowing through a pipe of radius R and length L, has a volume flow rate Q given by:

Q= ππππR4(P2-P1)/ 8ηηηηLQ= ππππR4(P2-P1)/ 8ηηηηL