General Physics IGeneral Physics I

Lecture 16: Fluid MechanicsLecture 16: Fluid Mechanics

Prof. WAN, Xin （万歆）

xinwan@zju.edu.cnhttp://zimp.zju.edu.cn/~xinwan/

MotivationsMotivations

Newton’s laws for fluid statics?– Force → pressure

– Mass → density

How to treat the flow of fluid?– Difficulties:

– Continuous medium– Change of the shape (still incompressible!)

– Frame of reference

DensityDensity

Density (scalar): mass of a small element of material m divided by its volume V

– For infinitesimally small V:

= m / V

– For homogeneous material:

= m / V

PressurePressure

Under static condition, force acts normal or perpendicular to the surface – vector.

Pressure: the magnitude of the normal force per unit surface area – scalar.

A

Δ A⃗

Δ F⃗

Δ F⃗ = pΔ A⃗

p =Δ FΔ A

Common UnitsCommon Units

2

5

5

2

1 1 / [SI]

1 1.01325 10

1 10 1

1 760 760

1 14.7 / .

pascal N m

atmosphere Pa

bar Pa atm

atm mm of Hg torr

atm lb in

Origin of PressureOrigin of Pressure

Microscopic origin: collisions of molecules of the fluid with the surface– Action and reaction: Newton’s third law

liquid

WallΔ v⃗

f⃗ reaction

Pressure in a Fluid at RestPressure in a Fluid at Rest

( ) ( )

( ) 0

dm g Ady g gAdy

pA gAdy p dp A

yy+dy

A

pA

(p+dp)A

(dm)g

dpg

dy

Pressure in a Fluid at RestPressure in a Fluid at Rest

Assumption: , g independent of y

0 0

0 0

0 0 0

' '

( )

( )

p y

p y

dpg dp g dy

dy

p p g y y

p p g y y p gh

Mercury BarometerMercury Barometer

How to measure atmospheric pressure? E. Torricelli (1608-47)

For mercury, h = 760 mm.

How high will water rise?

patm

p=0

atmatm

PP gh h

g

Vacuum!

h

U-Tube ManometerU-Tube Manometer

1 1 2 2A atmp gh p gh

SnorkelingSnorkeling

Can I use a longer snorkel to look closer at the underwater life?

Atmospheric PressureAtmospheric Pressure

Gases are compressible. Thus, varies!

50 sea level

0 0

00

1.013 10

, for constant

p p Pa

ppV nRT T

p

dp pg g

dx p

dy

0 0

0 0

00

0 0

00 0

0

( / )0

''

'

ln ln ( )

p y

p y

g p h

gdp p dpg dy

dx p p p

gp p y y

p

p p e

dy

Atmospheric PressureAtmospheric Pressure

00 0

h p p gh

A.P. – an Improved VersionA.P. – an Improved Version

0 0 0

0 0

00

0 0 0

( )

( )

pV nRT

p T

p T

T T y y

Tdp pg g

dy p T y y

Temperature decreases 6oC for each 1,000 meters of elevation

A.P. – an Improved VersionA.P. – an Improved Version

0 0

0

0 0

0 0

0 0 0

0 0

0 0 0

00

( )

' '

' ( ' )

1

p y

p y

gT p

g T dydp

p p T y y

g Tdp dy

p p T y y

hp p T

0

000

gh

pp e

Temperature CorrectionsTemperature Corrections

0 50 kmT

Pascal’s PrinciplePascal’s Principle

Pressure applied to an enclosed fluid is transmitted undimished to every portion of the fluid and to the walls of the containing vessel.

F

p

d

ext extp p gd p p

pext

Hydraulic LeverHydraulic Lever

/ / ,i i o o i i o oF A F A A d A d

Fi

mgAi Ao

Fo

di

do

Hydraulic BrakeHydraulic Brake

Fred Duesenberg originated hydraulic brakes on his 1914 racing cars and Duesenberg was the first automotive marque to use the technology on a passenger car in 1921. In 1918 Malcolm Lougheed (who later changed the spelling of his name to Lockheed) developed a hydraulic brake system.

Archimedes’ PrincipleArchimedes’ Principle

A body wholly or partially immersed in a fluid is buoyed up by a force equal in magnitude to the weight of the fluid displaced by the body.

FB

mg

B fluid fluid

object

F Vg m g

mg Vg

Two CommentsTwo Comments

Buoyant force volume/density

Center of buoyancy (not fixed)Center of gravity

B fluid fluid

object

F Vg m g

mg Vg

Question: Galileo’s ScaleQuestion: Galileo’s Scale

Can you invent a scale that one can use to read the percentage of silver in the gold crown without explicit calculations?

Fluid DynamicsFluid Dynamics

Aerodynamics (gases in motion) Hydrodynamics (liquids in motion)

– Blaise Pascal

– Daniel Bernoulli, Hydrodynamica (1738)

– Leonhard Euler

– Lagrange, d’Alembert, Laplace, von Helmholtz

Airplane, petroleum, weather, traffic

The Microscopic ApproachThe Microscopic Approach

N particles ri(t), vi(t); interaction V(ri-rj)

The Macroscopic ApproachThe Macroscopic Approach

A fluid is regarded as a continuous medium. Any small volume element in the fluid is always

supposed to be so large that it still contains a very large number of molecules.

When we speak of the displacement of fluid at a point, we meant not the displacement of an individual molecule, but that of a volume element containing many molecules, though still regarded as a point.

Euler’s SolutionEuler’s Solution

For fluid at a point at a time:

State of the fluid: described by parameters p, T.

However, laws of mechanics applied to particles, not to points in space.

Field

ρ(x , y , z ,t ), v⃗ (x , y , z ,t )

Ideal FluidsIdeal Fluids

Steady: velocity, density and pressure not change in time; no turbulence

Incompressible: constant density Nonviscous: no internal friction

between adjacent layers Irrotational: no particle rotation

about the center of mass

StreamlinesStreamlines

Paths of particles

P → Q → R v tangent to the streamline No crossing of streamlines

Pv Rv

Qv

PQ

R

Mass FluxMass Flux

Tube of flow: bundle of streamlines

2v

A1 A2

1vP

Q

11 1 1 1 1 1 1

1

mass flux m

m A v t A vt

Conservation of MassConservation of Mass

IF: no sources and no sinks/drains

– Narrower tube == larger speed, fast

– Wider tube == smaller speed, slow

Example of equation of continuity. Also conservation of charge in E&M

1 1 1 2 2 2

1 1 2 2

constant

co fornstant incompressible fluid,

A v A v

A v A v

What Accelerates the Fluid?What Accelerates the Fluid?

Acceleration due to pressure difference.Bernoulli’s Principle = Work-Energy Theorem

Work-Energy TheoremWork-Energy Theorem

Steady, incompressible, nonviscous, irrotational

Bernoulli’s EquationBernoulli’s Equation

1 1 2 2

2 21 1 1 2 2 2 2 2 1 1

2 21 1 1 2 2 2

2

1 1

2 21 1

2 21

constant2

m A x A x

p A x p A x mv mgy mv mgy

p v gy p v gy

p v gy

kinetic E, potential E, external work

CommentsComments

Restrictions of the Bernoulli’s equation:– Points 1 and 2 lie on the same streamline.

– The fluid has constant density.

– The flow is steady.

– There is no friction.

When streamlines are parallel the pressure is constant across them (if we ignore gravity and assume velocity is constant over the cross-section).

The Venturi MeterThe Venturi Meter

Speed changes as diameter changes. Can be used to measure the speed of the fluid flow.

2 21 1 2 2 1 1 2 2

1 1,

2 2p v p v v A v A

Bend it like BeckhamBend it like Beckham

Dynamic lift

http://www.tudou.com/programs/view/qLaZ-A0Pk_g/

Banana Free KickBanana Free Kick

~ 5mDistance 25 mInitial v = 25 m/sFlight time 1sSpin at 10 rev/sLift force ~ 4 NBall mass ~ 400 g a = 10 m/s2 A swing of 5 m!