!Fluid Mech 3

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!Fluid Mech 3

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  • 1

    3 ................................................ 2

    3.1 ........................................................................................................................ 2

    3.2 - ..................................... 2

    3.3 Darcy-Weisbach ...................... 5

    3.4 ............................................................................. 8

    3.5 9

    3.5.1 ...................................................... 9 3.5.2 .................................................................... 10 3.5.3 ......................................................................................... 11 3.5-1 ...................................................................................................... 11 3.5-2 ...................................................................................................... 14

    3.6 15

    3.6.1 ......................................................... 15 3.6.2 .................................................................... 15 3.6.3 ......................................................................................... 16 3.6-1 ...................................................................................................... 17 3.6-2 ...................................................................................................... 18 3.6.4 ..................................................... 19 3.6-3 ...................................................................................................... 21 3.6-4 ...................................................................................................... 22 3.6.5 Moody ...................................................................................... 23 3.6-5 ...................................................................................................... 25

    3.7 ...................................................................... 25 3.7-1 ...................................................................................................... 27

    3.8 - ................................. 27

    3.9 .................................................................................. 28 3.9.1 ........................................... 28 3.9.2 ....................................................... 29 3.9.3 ........................................................ 31 3.9.4 ...................................................... 33 3.9.5 ....................................................... 33 3.9.6 ........................................ 34 3.9.7 .......................................................................... 35 3.9.8 ..................................................................... 36

    3.10 ....................................................................... 36

  • 2

    3

    3.1 o

    .

    ,

    1. ,

    .

    2. () , ()

    , () f ()

    .

    3. Moody

    .

    3.2 - Reynolds (. . 1.4.2)

    (. 1.4.7) . 3.2-1, .

    ,

    .

    Le,

    .

    3.2-1.

  • 3

    Le .

    Le

    , , V D, .

    eF(L ,,,V,D) 0= (3.2-1)

    eL

    F( , Re) 0D

    = eL

    F(Re)D

    = (3.2-2)

    . Le Re, VD

    Re /

    =

    eL

    0.06ReD

    = Re

  • 4

    0

    50

    100

    150

    0 10000 20000 30000 40000 50000

    Re

    Le

    ()

    0

    10

    20

    30

    40

    50

    0.0E+00 2.0E+05 4.0E+05 6.0E+05 8.0E+05 1.0E+06

    Re

    Le

    ()

    3.2-2. Reynolds

    1. To

    , .

    2. To 20-40 D.

    1000 D,

    .

  • 5

    3.3 Darcy-Weisbach

    . 3.3-1,

    x 1 2.

    x .

    p=p1-p2 z=z1-z2

    1 2.

    3.3-1.

    hf x , 1 2.

    .

    hf w .

    w ,

    ,

    hf ,

    .

    1. ,

    . 3.3-1.

    1.

    1 2Q Q Q= = 1 1 2 2A V A V= 1 2V V V= = (3.3-1)

    2

    1 2

    DA A A

    4= = = (3.3-2)

    .

    .

  • 6

    2.

    2 2

    1 1 2 21 2 f 1 1 2 2 f

    p V p VH H h z z h

    2g 2g= + => + + = + + + (3.3-3)

    . (3.3-3) hf 1=2= (

    )

    2 2 2 2

    1 2 1 2f 1 2

    p p V V V Vp ph z z z ( ) z

    2g 2g 2g 2g

    = + + = + + = + +

    (3.3-4)

    .

    . . (3.3-4)

    . (3.3-1)

    1 2f 1 2p p p p

    h z z z+ (z+ )

    = + = = (3.3-5)

    .

    .

    3.

    x x x 1 1 2 2Fp F Fg (V Q V Q ) (VQ VQ) 0+ + = = = (3.3-6)

    3 .

    (i) xFp

    2 2 2 2 2

    x 1 2 1 1

    D D D D DFp p p p (p p) p

    4 4 4 4 4= = = (3.3-7)

    (ii) () xF

    x wF Dx= (3.3-8)

    (iii) xFg

    2

    x

    DFg mgsin Vgsin x sin

    4= = = (3.3-9)

    . (3.3-6) . (3.3-7), (3.3-8) (3.3-9)

    2 2

    w

    D Dp Dx x sin 0

    4 4 + = (3.3-10)

    z=x sin . (3.3-10)

  • 7

    w4 x p

    (z )D

    = + wD (p z)

    4 x

    += (3.3-11)

    . (3.3-5) (3.3-11) hf w

    fw

    p(z )

    hD D

    4 x 4 x

    += = wf

    4 xh

    D= (3.3-12)

    2. w (

    ), (V) (D ks)

    , .

    w s F(, ,V,D, k )= (3.3-13)

    ( .

    1.5.4)

    w s2

    kf VDF Re ,

    V 8 D

    = = =

    2

    w

    1 fV

    8= (3.3-14)

    f Darcy. O Henry Darcy (1803-1858)

    , 1857 ,

    .

    . (3.3-14) w . (3.3-12),

    2

    f

    x Vh f

    D 2g= (3.3-15)

    L, . (3.3-15)

    2

    f

    L Vh f

    D 2g= (3.3-16)

    . (3.3-16) Darcy-Weisbach,

    Julius Weisbach (1945), o

    .

    1. . (3.3-3) () , . . (1.2-

    10).

    2. . (3.3-12), (3.3-14) (3.3-16) 3

    (, ),

    ,

    . , . (3.3-12), (3.3-14) (3.3-16)

    .

  • 8

    3. . (3.3-14), f (Re)

    , ks/D,

    .

    4. . (3.3-16)

    , ,

    .

    f. . 3.5.

    3.4 . 3.4-1,

    .

    ,

    , .

    1.4.

    (. . 3.4-1).

    1.

    r 1 1 u

    (rv ) (v ) 0r r r x

    + + =

    (3.4-1)

    u, vr v ( x),

    r , .

    3.4-1. x

    2.

    xu p 1

    ru g (r)x x r r

    = + +

    (3.4-2)

    . (3.4-1) vr=0 v=0,

  • 9

    u

    0x

    =

    (3.4-3)

    . u r

    x, . .

    3. xg g sin = . (3.4-3) . (3.4-2),

    dp 1

    (p gsin ) (r)dx r r

    =

    1 dp(r) (p z)

    r r dx

    = +

    (3.4-4)

    . (3.4-4) r

    x. , .

    4. . (3.4-4) =0 r=0,

    r d r

    (p z) K2 dx 2

    = + = (3.4-5)

    , .

    d

    K (p z)dx

    = + (3.4-6)

    . 3.4-1

    (r=R=D/2),

    wR d D (p z)

    (p z)2 dx 4 x

    += + = (3.4-7)

    1. . (3.4-5) d

    (p z)dx

    + ,

    x, . .

    2. . (3.4-7) . (3.3-12). ,

    .

    f

    .

    3.5

    3.5.1

    . (1.2-14),

    ( )

  • 10

    du(r)

    dr

    = (3.5-1)

    . (3.4-5) . (3.5-1),

    du(r) r

    Kdr 2

    = K

    du(r) rdr2

    = (3.5-2)

    . (3.5-2)

    2

    1

    Ku(r) r C

    4= + (3.5-3)

    u=0 (r=R)

    C1

    2

    1

    KC - R

    4= (3.5-4)

    . (3.5-4) . (3.5-3),

    2 2 2 2 2 2K K K 1 du(r) r R (R r ) (p z)(R r )

    4 4 4 4 dx= = = + (3.5-5)

    . (3.5-5)

    (. . 3.4-1).

    3.5.2

    umax (r=0)

    . (3.5-5) r=0, .

    2 2

    max

    K 1 du R (p z)R

    4 4 dx= = + (3.5-6)

    . (3.5-5) . (3.5-6)

    2

    max 2

    ru(r) u (1 )

    R= (3.5-7)

    V . (3.3-1).

    Q

    VA

    = (3.3-1)

    Q . (3.5-7), .

    R 22max

    max 2

    0

    urQ udA u (1 )2rdr R

    R 2= = = (3.5-8)

  • 11

    max1

    V u2

    = (3.5-9)

    . .

    3.5.3

    . . (3.5-1),

    . (3.5-7) . (3.5-9)

    maxwr R

    2udu 2(2V) 8V

    dr R (D / 2) D== = = = (3.5-10)

    . (3.5-10) . (3.3-14)

    f

    w 2 2

    8V8( )

    8 64 64DfVDV V Re

    ( ) /

    = = = = (3.5-11)

    . (3.5-11) . (3.3-15),

    2

    f 2

    64 L V 32 Lh V

    VD D 2g g D( ) /

    = = (3.5-12)

    1. . (3.5-7)

    Hagen-Poiseuille G. Hagen (1839)

    J. L. Poiseuille (1841).

    2. , . (3.5-8),