Experimental Physics EP1 MECHANICS – Fluid mechanics · PDF file Experimental Physics -...

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  • 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!