Fluid Mechanics

Fluid Mechanics

• Liquids and gases have the ability to flow

• They are called fluids

• There are a variety of “LAWS” that fluids

obey

Density

• Regardless of form (solid, liquid, gas) we

can define how much mass is squeezed into

a particular space

density mass

volume

Mass Density

2 kg, 4000 cm3

Wood

177 cm3

45.2 kg

; mass m

Densityvolume V

Lead: 11,300 kg/m3

Wood: 500 kg/m3

4000 cm3

Lead

Same volume

2 kg Lead

Same mass

Gases

• The primary difference between a liquid

and a gas is the distance between the

molecules

• In a gas, the molecules are so widely

separated, that there is little interaction

between the individual molecules

Boyle’s Law

Boyle’s Law • Pressure depends on density of the gas

• Pressure is just the force per unit area exerted by

the molecules as they collide with the walls of the

container

• Remember:

• Pressure is measured in pascal units (Pa)

• 1Pa = 1 Newton / m2 (force/area)

• At sea level, 1atm = 101.3 kPa or 101,300 N per

square meter

• Double the density, double the number of collisions

with the wall and this doubles the pressure

Boyle’s Law

Density is mass

divided by

volume.

Cut the volume

in half and you

double the

density and thus

the pressure.

Boyle’s Law

• At a given temperature for a given quantity

of gas, the product of the pressure and the

volume is a constant

P1V1 P2V2

Pressure

• A measure of the amount of force exerted

on a surface area

pressure force

area

Pressure

Pressure is the ratio of a force F to the area A over

which it is applied:

Pressure ; Force F

PArea A

A = 2 cm2

1.5 kg

2

-4 2

(1.5 kg)(9.8 m/s )

2 x 10 m

FP

A

P = 73,500 N/m2

The Unit of Pressure (Pascal):

A pressure of one pascal (1 Pa) is defined as a force of

one newton (1 N) applied to an area of one square meter

(1 m2).

21 Pa = 1 N/mPascal:

In the previous example the pressure was 73,500 N/m2.

This should be expressed as:

P = 73,500 Pa

Pressure / Density Example

Tofu

Cookbook

Schmedrick uses his 6 lb tofu recipe book to teach his little brother

Poindexter about density and pressure. He sets the book on the table

and calculates the pressure on the table, which depends on the book’s

orientation. The book’s density is 6 lb / (9” · 14” · 3”) = 0.0159 lb / in 3.

Tofu Cookbook

14”

3” 9”

P = 6 lb / (9” · 14” )

= 0.0476 lb / in 2

P = 6 lb / (9” · 3” )

= 0.222 lb / in 2

P = 6 lb / (3” · 14” )

= 0.143 lb / in 2

Pressure in a Fluid

• The pressure is just the weight of all the

fluid around the object

• Atmospheric pressure is just the weight of

all the air above on an area on the surface of

the earth

• In a swimming pool the pressure on your

body surface is just the weight of the water

above you (plus the air pressure above the

water)

Fluid Pressure

Fluid exerts forces in many directions. Try to submerse a rubber

ball in water to see that an upward force acts on the float.

• Fluids exert pressure in

all directions. F

Pressure in a Fluid

• So, the only thing that counts in fluid pressure is the

gravitational force acting on the mass ABOVE you

• The deeper you go, the more weight above you and

the more pressure

• Go to a mountaintop and the air pressure is lower

• Pressure in a fluid is the result of the forces exerted

by molecules as they bounce off each other in all

directions. Therefore, at a given depth in a liquid or

gas, the pressure is the same and acts in every

direction

Pressure in a Fluid

Pressure acts

perpendicular

to the surface

and increases

at greater

depth.

Pressure vs. Depth in Fluid

Pressure = force/area

; ; mg

P m V V AhA

Vg AhgP

A A

h

mg Area

• Pressure at any point in a

fluid is directly proportional

to the density of the fluid

and to the depth in the fluid. P = gh

Fluid Pressure:

Independence of Shape and Area.

Water seeks its own level,

indicating that fluid pressure

is independent of area and

shape of its container.

• At any depth h below the surface of the water

in any column, the pressure P is the same.

The shape and area are not factors.

*Properties of Fluid Pressure*

• The forces exerted by a fluid on the walls of its

container are always perpendicular.

• The fluid pressure is directly proportional to the

depth of the fluid and to its density.

• At any particular depth, the fluid pressure is the

same in all directions.

• Fluid pressure is independent of the shape or area

of its container.

Pressure in a Fluid

Barometers

• The height of the mercury

column in a barometer directly

measures air pressure.

• The weight of the column of

mercury is balanced by the

force exerted at the bottom due

to the air pressure.

• Normal air pressure is 760mm

or 760 torr

• Since mercury is 13.6 times

heavier than water, a water

barometer would have to be

13.6 times longer.

Pascal’s Principle

• Pressure applied to a fluid is transmitted

throughout the fluid. • Ex) squeezing tube of toothpaste

• Hydraulic machines work using Pascal’s

principle.

Pascal’s Law

Pascal’s Law: An external pressure applied

to an enclosed fluid is transmitted uniformly

throughout the volume of the liquid.

Fout Fin Aout Ain Pressure in = Pressure out

in out

in out

F F

A A

Hydraulic Press

oil

A2 F1

A1

F2

A force F1 is applied to a hydraulic press. This increases the pressure

throughout the oil, lifting the car--Pascal’s principle. This would not

work with air, since air is compressible. The pressure is the same

throughout the oil. The volume of oil pushed down on the left is the

same as the increase on the right. The distance pushed on the left is the

trade off.

h1

h2

Example 3. The smaller and larger pistons of a

hydraulic press have diameters of 4 cm and 12 cm.

What input force is required to lift a 4000 N weight

with the output piston?

Fout Fin Aout Ain ; in out out in

in

in out out

F F F AF

A A A

2

2

(4000 N)( )(2 cm)

(6 cm)inF

2; 2

DR Area R

F = 444 N

Rin= 2 cm; Rout = 6 cm

Floating in Fluids We all know that dense objects sink in fluids of lower density. A

rock sinks in air or water, and oil floats on top of water.

Basements stay cool in the summer because cool air is denser

than warm air. The USS Eisenhower is a 95 000 ton nuclear

powered aircraft carrier made of dense materials like steel, yet it

floats. If you weigh yourself under water, the scale would say

you are lighter than your true weight. All of these facts can be

explained thanks one of the greatest scientists of all time--the

Greek scientist, mathematician, and engineer--Archimedes.

USS Eisenhower Archimedes

Archimedes’ Principle

• An object that is completely or partially submerged in

a fluid experiences an upward buoyant force equal to

the weight of the fluid displaced.

2 lb

2 lb

• The buoyant force is due to the

displaced fluid. The block material

doesn’t matter.

• If the buoyant force on an object is

greater than the force of gravity acting

on the object, the object will float.

• The apparent weight of an object in a

liquid is gravitational force (weight)

minus the buoyant force

Buoyancy Net upward

force is

called the

buoyant

force!!!

Displacement of Water

The amount of

water displaced is

equal to the

volume of the

rock.

Flotation

Flotation

• A floating object displaces a weight of fluid equal

to its own weight. An object floats if its density is

less than the density of the fluid it is placed in.

Submarines & Blimps A sub is submerged in water, while a

blimp is submerged in air. In each a

buoyant force must balance the weight

of the vessel. Blimps and hot air

balloons must displace huge amounts

of air because air isn’t very dense. The weight of the air a blimp

displaces is equal to the blimp’s weight. Likewise, the weight of

the water a sub displaces is equal to the sub’s weight.

Buoyancy in a Gas

• An object surrounded by air is buoyed up by

a force equal to the weight of the air

displace.

• Exactly the same concept as buoyancy in

water. Just substitute air for water in the

statement

• If the buoyant force is greater than the

weight of the object, it will rise in the air

Buoyancy in a Gas

Since air gets less

dense with altitude,

the buoyant force

decreases with

altitude. So helium

balloons don’t rise

forever!!!

Atmospheric Pressure

• Just the weight of the air above you

• Unlike water, the density of the air

decreases with altitude since air is

compressible and liquids are only very

slightly compressible

• Air pressure at sea level is about 105

newtons/meter2

Bernoulli’s Principle

• When the speed of a fluid increases, the

pressure exerted by the fluid decreases.

Bernoulli’s Principle • Uses: airplanes, hose-end sprayers

• Energy conservation requires that the

pressure be lower in a fluid that is moving

faster

Bernoulli’s Principle

Air is not incompressible, but the Bernoulli principle can

explain, in part, why an airplane flies. The upper surface of

the wing has a smaller radius of curvature than the bottom

surface. Air on top must travel farther, so it moves faster, and

the pressure there is lower, creating lift. Also, because of the

wing’s upward tilt, air is pushed downward. So, the air

pushes back on the wing in the direction of F.

Viscosity

• The resistance to flow by a fluid

• When a container of liquid is tilted to allow

flow, the flowing particles will transfer

energy to the particles that are stationary.

• Increasing temperature of a fluid will

decrease viscosity

Viscosity Different kinds of fluids flow more easily than others. Oil, for

example, flows more easily than molasses. This is because molasses

has a higher viscosity, which is a measure of resistance to fluid flow.

Inside a pipe or tube a very thin layer of fluid right near the walls of

the tube are motionless because they get caught up in the microscopic

ridges of the tube, or microwelds. Layers closer to the center move

faster and the fluid sheers. The middle layer moves the fastest.

The more viscous a fluid is, the more the layers want to cling together,

and the more it resists this shearing. The resistance is due the frictional

forces between the layers as the slides past one another. Note, there is

no friction occurring at the tube’s surface since the fluid there is

essentially still. The friction happens in the fluid and generates heat.

The Bernoulli equation applies to fluids with negligible viscosity.

v = 0