Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 1

Experimental Physics EP1 MECHANICS

– Fluid mechanics –

Rustem Valiullin

https://bloch.physgeo.uni-leipzig.de/amr/

Experimental Physics - Mechanics - Elasticity 2

Elastic modulus

A

F

F l lD

Tensile stress úû ù

êë é= 2m N

A F

Strain l lD

= degree of deformation

Young’s or elastic modulus úû

ù êë é

D ==U 2/

/ strain stress

m N

ll AF

st re

ss

strain

Hook’s law

Elastic limit

Plastic flow

Fracture point

Material Y, GPa (TS, MPa) Rubber 0.01-0.1 Wood ~10 (50)

Aluminum 70 (90)

Glass 50-90 (50)

Silicon 185 Steel 200 (520)

Diamond 1220

Experimental Physics - Mechanics - Elasticity 3

Shear modulus

F h

xD

Shear modulus úû

ù êë é

D = 2/

/ m N

hx AF

G

Shear stress úû ù

êë é= 2m N

A F

Shear strain qtan=D= h x

Material G, GPa Rubber 0.0006 Aluminum 25.5 Glass 26 Steel 80 Diamond 478

Experimental Physics - Mechanics - Elasticity 4

Bulk modulus

Bulk modulus úû

ù êë é

D -= 2/

/ m N

VV AFB

Volume stress

(pressure) úû ù

êë é= 2m N

A F

Volume strain V VD

=

Material G, GPa

Air ~10-4

Water 2.2

Glass 35-50

Steel 160

Diamond 442

Water expands by about 9%

upon freezing. What pressure

would it then create in a bottle?

~ 200 MPa

1 km down a see the pressure of

water rises by 10 MPa. What is the

water density there?

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 5

Fluids Ø Elastic stress Ø Shear stress Ø Bulk stress

Pressure A FP =

Va cu

um

F A

G as

Va cu

um

topF

Va cu

um

botF

h

AhgFmgFF r+=+= toptopbot hgPP r=- topbot

Pascal’s principle: Pressure applied to an enclosed fluid is transmitted to every point of the fluid and to the walls of the container.

Compressibility:

dP dV

VB 11

-=ºk

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 6

Basic equation of hydrostatics

!(#) !(# + )

&'()*, , = ! # &. − ! # + &.

! # + − ! # = &! = 0!0#

&'()*, , = − 0! 0# &. = −

0! 0# &1

&'()*, 2 = − 0! 03 &3&. = −

0! 03 &1

&'()*, 4 = − 0! 05 &5&. = −

0! 05 &1

678& ! = 9: = :⃗

Under equilibrium: 678& ! ≡ 0!0# ̂> +

0! 03 ̂? +

0! 05

@A

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 7

Hydraulic pressure

1F 2F

1A 2A 1

1 1 A FP =

2

2 2 A

FP ==

atmAPF = 0=P

h

ghP r=atm

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 8

Buoyancy

Archimed’s principle: A body submerged in a fluid is acted (buoyed up) by a force equal to the weight of the displaced fluid.

a a2

a2

a

tF

tF

bF

bF

( )220 22 aghaPFt ×+×-= r

h ( ) 220 22 aahgaPFb ×++×= r

VgagF rr == 3net 2 buoyant force

Equilibrium (under water): wfwf gVgV rrrr =Þ=

Equilibrium (floating): f

w wf V

VgVgV r rrr =Þ= ''

T

N285.0=mg N855.0=T ?-r

33 1

kg/m1025.01 ´»÷÷ ø

ö çç è

æ +=

-

mg T

wrr

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 9

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 10

Surface tension

cohesiveF

dA dE

=s - specific surface energy

[ ] m N

m J == 2s - surface tension

mg

y

l

0y yD

ymgUg D-=D

( ) 2×D×-=--=D ylAAU ifs ss l mg 2

=s

4

topF

dRR +

R

( )22 4)(4 RdRRdUs pps -+=

dR dUF s-=

A PF =

R P s2cohesive =

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 11

Capillary effects

cq lgs

sls sgs

sgsllg cos ssqs =+c

The Young equation:

Contact angle

Degree of wetting

Strength of interaction: Sol.-Liq. Liq.-Liq.

θ = 0 Perfect wetting strong weak

0 < θ < 90° high wettability strong strong weak weak

90°≤θ

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 12

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 13

Viscous flow

A

zD

vD dz dvA

z vAfv hh -=D

D -=

vf viscosity or dynamic viscosity

P P

v v P vf PP D+

P PP D+

R r

x xx D+

0pnet, =+ vfF 2

pnet, arPF p×D-=

a av dr

dvxrf D×= ph 2

òò D D

= rvr

R aa dvP

x drr

0

2h

( ) x PrR

vr D D-

-= h4

22laminar flow

4

8 R

x PIv D D

-= h p

Hagen-Poiseuille law:

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 14

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 15

-0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6

0

5

10

15

20

25

30

h / m

m

sqrt (t) / s-1/2

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 16

-0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6

0

5

10

15

20

25

30 h

/ m m

sqrt (t) / s-1/2

y=9.2x

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 17

10-6 10-5 0

20

40

60

sl op

e

tube diameter

B

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 18

Continuity 1A

1v 2A

2v

tvAV D=D 11 tvA D= 22

vAIV = - volume flow rate = constant - continuity equation

?-v

m12=l

m3=L

cm2=D

cm1=d n

t

ttnn AvAv =

mglmvn = 2

2 1

gl D dvt 2

2

÷ ø ö

ç è æ=

m/s3=tv

What pressure should the pump provide?

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 19

Bernoulli’s equation

1h 2h

m

21 mghmghU -=D

( )22212 vv mEk -=D

1

2

2v

1v

Pipes are full

2F

( )22222 xAPdxFW D--=-= ò

2xD

1F

( )11111 xAPdxFW D-== ò VPVPVPVPW 121122 -=-=DkEUW D+D=D

2 22

12 12

1 2112 VvVvVghVghVPVP rrrr -+-=-

2 12

1 11

2 22

1 22 vghPvghP rrrr ++=++ const

2 2 1 =++ vghP rr

Bernoulli’s equation

1xD

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 20

Applications of Bernoulli’s equation

h

2 22

1 00 vPghP rr +»+

1

2

ghv 22 =

Torichelli’s law

1P 1v 2P 2v

2A 1A

2 22

1 2

2 12

1 1 vPvP rr +=+ const

2 2 1 =+ vP r

Magnus effect

Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 21

Ø Pascal’s principle: pressure applied to an enclosed liquid is

transmitted to any point.

Ø Archimed’s principle: a body submerged in fluid is buoyed up

with a force equal to the weight of the displaced fluid.

ØIntermolecular forces: surface tension and capillarity.

Ø For steady-state flow of an incompressible fluid

the volume flow rate is constant (continuity).

Ø Bernoulli’s equation - conservation of total

mechanical energy applied to fluids.

Ø Pouseuille’s law: fluid velocity is inversely

proportional to the distance from the wall.

To remember!