Experimental Physics EP1 MECHANICS – Fluid mechanics€¦ · Experimental Physics - Mechanics -...

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

Fl lD

Tensile stress úûù

êëé= 2mN

AF

StrainllD

=degree of deformation

Young’s or elastic modulus úû

ùêëé

D==U 2/

/strainstress

mN

llAF

stre

ss

strain

Hook’s law

Elastic limit

Plastic flow

Fracture point

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

Aluminum 70 (90)

Glass 50-90 (50)

Silicon 185Steel 200 (520)

Diamond 1220


Shear modulus

Fh

xD

Shearmodulus úû

ùêëé

D= 2/

/mN

hxAF

G

Shear stress úûù

êëé= 2mN

AF

Shear strain qtan=D

=hx

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


Bulk modulus

Bulk modulus úû

ùêëé

D-= 2/

/mN

VVAFB

Volume stress

(pressure) úûù

êëé= 2mN

AF

Volume strainVVD

=

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?


FluidsØ Elastic stressØ Shear stressØ Bulk stress

PressureAFP =

Vacu

um

FA

Gas

Vacu

um

topF

Vacu

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:

dPdV

VB11

-=ºk


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


Hydraulic pressure

1F 2F

1A 2A1

11 AFP =

2

22 A

FP ==

atmAPF = 0=P

h

ghP r=atm


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.

aa2

a2

a

tF

tF

bF

bF

( )220 22 aghaPFt ×+×-= r

h( ) 22

0 22 aahgaPFb ×++×= rVgagF rr == 3

net 2buoyant force

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

Equilibrium (floating):f

wwf V

VgVgVrrrr =Þ=

''

T

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

331

kg/m1025.01 ´»÷÷ø

öççè

æ+=

-

mgT

wrr


Surface tension

cohesiveF

dAdE

=s - specific surface energy

[ ]mN

mJ== 2s - surface tension

mg

y

l

0yyD

ymgUg D-=D

( ) 2×D×-=--=D ylAAU ifs ss lmg2

=s

4

topF

dRR +

R

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

dRdUF s-=

APF =

RP s2cohesive =


Capillary effects

cqlgs

sls sgs

sgsllg cos ssqs =+c

The Young equation:

Contact angle

Degree ofwetting

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

θ = 0 Perfect wetting strong weak

0 < θ < 90° high wettabilitystrong strongweak weak

90°≤θ<180° low wettability weak strong

θ = 180° perfectlynon-wetting weak strong

cq

cF

ghRR lc rppqs 22cos =×h

gRh

l

c

rqs cos2

= capillary action


Viscous flow

A

zD

vD dzdvA

zvAfv hh -=DD

-=

vf viscosity or dynamic viscosity

P P

v vP vf PP D+

P PP D+

R r

x xx D+

0pnet, =+ vfF2

pnet, arPF p×D-=

aav dr

dvxrf D×= ph 2

òò DD

=rvr

Raa dv

Px

drr0

2h

( )xPrR

vr DD-

-=h4

22laminar flow

4

8R

xPIv DD

-=hp

Hagen-Poiseuille law:


-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

-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

30h

/ mm

sqrt (t) / s-1/2

y=9.2x


10-6 10-50

20

40

60

slop

e

tube diameter

B


Continuity1A

1v2A

2v

tvAV D=D 11 tvA D= 22

vAIV = - volume flow rate = constant

- continuity equation

?-v

m12=l

m3=L

cm2=D

cm1=dn

t

ttnn AvAv =

mglmvn =2

21

glDdvt 2

2

÷øö

çèæ=

m/s3=tv

What pressure should the pump provide?


Bernoulli’s equation

1h2h

m

21 mghmghU -=D

( )22212vvmEk -=D

1

2

2v

1v

Pipes are full

2F

( )22222 xAPdxFW D--=-= ò

2xD

1F

( )11111 xAPdxFW D-== òVPVPVPVPW 121122 -=-=D

kEUW D+D=D

222

1212

12112 VvVvVghVghVPVP rrrr -+-=-

212

111

222

122 vghPvghP rrrr ++=++ const2

21 =++ vghP rr

Bernoulli’s equation

1xD


Applications of Bernoulli’s equation

h

222

100 vPghP rr +»+

1

2

ghv 22 =

Torichelli’s law

1P 1v2P 2v

2A1A

222

12

212

11 vPvP rr +=+ const2

21 =+ vP r

Magnus effect


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!

Experimental Physics EP1 MECHANICS – Fluid mechanics€¦ · Experimental Physics - Mechanics -...

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