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
Experimental Physics - Mechanics - Elasticity 3
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
Experimental Physics - Mechanics - Elasticity 4
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?
Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 5
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
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 2A1
11 AFP =
2
22 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.
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
Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 9
Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 10
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 =
Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 11
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
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 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:
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
30h
/ mm
sqrt (t) / s-1/2
y=9.2x
Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 17
10-6 10-50
20
40
60
slop
e
tube diameter
B
Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 18
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?
Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 19
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
Experimental Physics - Mechanics - Fluid Mechanics – Static and Dynamic Behaviors 20
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
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!