Mehanika Fluida B - Handout

7/27/2019 Mehanika Fluida B - Handout

1/200


2/200


3/200

U = 0 = 0

ij

Ra

f(r, t)


4/200

(s, n)


5/200

dV= dxdydz

u2

uv

us


6/200

us


7/200

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

M 1

M > 1

M 1


8/200

U(u, v, w)

p

T

r(x, y, z)

t

f = f(r, t) = f(x, y, z, t) f = U, , p, T

p

T


9/200

Newton

Euler

Bernoulli

DAlambert

Kirchhoff

Helmholtz

Reyleigh

Hagen

Poiseuille

Reynolds

Prandtl

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

= 0

= 0

U U

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T

U

U

T

T

Tw

U

T Tw T


10/200

d 1010 m

l = 107 m

V = 500m/s

5, 5 109

1 mm3

2, 7 1016

Kn = lL 0, 01 L

f

f = f(x,y ,z ,t)

km

mm


11/200

p

T

A

V

U

M

dV

dm

m

V m

m

V

A

U dV

dm

= (x, y, z, t)

= m/V

= lim

V0(VM)

m

V= (M).

dm = dV

dm = const

(x, y, z, t) = dm/dV


12/200

V T

Van-der-Waals

10d 109 m

r

r

F

V

d

+

F

r

r F

r

V

r F

V1 1

m = V = const

dm = dV = const

= const


13/200

t = t1 t = t2

m

Vm V

dV

dV

dm = dV = const

dV1 = dV2 = dV = const

1 = 2 = = const

= const m = const t1 t2

t = t1 t = t2

m V1m V2 = V1

1 dV1 2

= const

1 dV1 = 2 dV2

1 = 2dV2 = dV1 dV2 = dV1

= const m = const t1 t2

= (x, y, z, t)

p

T

s

s = (dV)p

1

dV dp > 0 d(dV) < 0,

dp < 0 d(dV) > 0.

dV = f(p, T)

d = F(p, T) dV = const

pdV +

(dV)

p= 0 = s = 1

p


14/200

= (p, T)

=1

s

voda 20 103 bar = 20 105 kPa,

vazduh 1 bar = 105

Pa

f(, p, T) = 0 = (p, T).

p = RT,

R

R = 287 J/kgK

p1 1 T1

1=

p

p1

T1T

.

= (p),

1

n=

p

p1 p

n= const,

n = 1

n =

n


15/200

y + l

y

y

x

y l u(y l)

u(y + l)

V

V

y

y

=V l

3

u

y.

viscosit

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

=F

A=

U0h

= u

y

u(y) = U0y

h u(x, y)

h

F

y

yy

u(y)

U0

u

= 0

= 0

0

0

Couette

u(x, y)

= u

y.


16/200

= /

106

Pa s

106

m2/s

vode = 1 mm2

/s

vazduha = 15mm2

/s

(T, p) (T)

T

T

T V

V T

= 13 V l

T

p


17/200

0=

T0 + TsT + Ts

T

T03/2

T

T0n

.

Sutherlend

p0 = 1bar T0 = 273, 16 K 0 = 17, 1 Pa s Ts = 122K Satherlend n

n = 1

n 0, 76

n

0.5 n 1, 56

0 < Tc < 100 C

0 = exp TA

T + TB TA

TB + T0 .

TA = 506K TB = 150K p0 = 1bar

T0 = 273, 16 K 0 = 1, 793 mPa s

uC = uA +u

y

A

dy

CC =u

ydy dt, AC = dy

tg(d) d = CC

AC

d

dt=

u

y

A

A

A

B

B

B

C

C

D

D

C

C

D

Dd

d

u(x, y)

x

y

y

dy

dy

uA

uC

yx

yx

u/y


18/200

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

F

d

A

F

A=

= G

=

G

R. Hooke (1635-1703)

= f()

f

f()

= k n

n < 1

n = 1

n > 1

< g > g


19/200

g

u/y

(n < 1)

(n = 1)

(n > 1)

P = F(S) P

S


20/200

V

A

M

M

M

M

V

A

m V

x

y

z

ij

k

dV

A

Rm

Rn

NT

F

Fn

n

t

r

n1

n2

A1

A2

pn

|n| = |t| = 1

pn1

pn2

npn

dA

pnn

pnt

m

p1 pn1(r; t; n1) p2 pn2(r; t; n2) A1 A2 n1 n2 pn pnn pnt dA

F(r, t)

F = limmM

Rmm

= limV0

Rm V

=1

lim

V0

RmV

=1

dRmdV

.

dRm

m

V

Rm


21/200

dRm = F dV = Rm =

V

F dV .

A

m

V

pn

pn = limA0MA

RnA

=

dRndA

= pn(r, t; n) .

dRn

dA n

Rn

dRn = pn dA = Rn =

A

pn dA .

pn

pn = pnxi +pnyj +pnzk = pnnn +pntt,

pnnn = limA0

N

A=

dN

dA

pntt = limA0

T

A=

dT

dA

pn dA =

pn dA =

pn =

pn .

M

n

pn

pn r

t

n

n

pn

Cauchy

V

V


22/200

m = V

a V

= F V

+ pnAn + pxAx + pyAy + pzAz

.

a

px py pz

x y

z

i j

k

px = px, py = py, pz = pz,

M

M

M

pn

px

px

py

py

pz

pz

n

aF

x

x

xx

y

y

y

y

zzz

z

x1

y1

z1

ij

k

ij

pyxipyyj

pyzk

k

Ax

Ax

Ay

Azxx

yy

zz

xz

xyyx

yzzxzy

r

px

pn p = x,y,z

n = nxi + nyj + nzk = cos(n, x)i + cos(n, y)j + cos(n, z)k

AxAn

= nx,AyAn

= ny,AzAn

= nz AAn

= n, = x, y, z

liml0

V

An= 0 V

An

l0 0, V l3 An l2.

An

pn = pxnx + pyny + pznz

n

px py pz

n

px


23/200

px

x

= pxxi

+ pxyj +pxzk

py pz

pnx = pxxnx +pyxny +pzxnz

pny = pxynx +pyyny +pzynz

pnz = pxznx +pyzny +pzznz

pnm = k pkmnk = pkmnk, m, k = x,y,z. k

pn = p1n1 + p2n2 + p3n3 = pini, i = 1, 2, 3.

pn n

pnj = pijni ,

pij = pij(r, t).

pij

P

pn = P n,

P = ||pij|| = P =

pxx pyx pzx

pxy pyy pzy

pxz pyz pzz

.

pij

i

j

pij

P

P = P(r, t)

pij kk

ij i = j


24/200

pxx = xx = 11 pxy = xy = 12

pyy = yy = 22 pxz = xz = 13

pzz = zz = 33 pyz = yz = 23

pnx

pny

pnz

=

xx yx zx

xy yy zy

xz

yz

zz

nx

ny

nz

.

I

ij = ji xy = yx, xz = zx, yz = zy

II

x

y

z

x

yz

xx + yy + zz = xx + yy + zz .

0xyz 0xyz

xx = xx

P

11 + 22 + 33 = 11 + 22 + 33

III


25/200

pnt = 0 ij = 0, i = j xy = xz = yz = 0 .

pn = pnnn nnn, px = xxi, . . . , pz = zzk.

nn(nxi + nyj + nzk) = xxnxi + yynyj + zznzk

nn = xx = yy = zz 11 = 11, . . . .

p

ij = 0

ij = 0 p = nn = xx = yy = zz .

ij = 0 pn = p n, px = pi, py = pj, pz = pk

ij = 0 ij U = 0

= 0

ij

= 0

xx = p + xx, yy = p + yy , zz = p + zz ,


26/200

ij i, j = 1, 2, 3

p = 13

(xx + yy + zz) .

U = 0

U = 0, = 0

pnt = 0

ij = 0

pnn = nn =

xx

= yy

=

zz = p

pn = p npx = pipy = pjpz = p k

pij = p ij P = p E

P = ||pij|| = p 0 00 p 0

0 0 p

E = 1 0 00 1 00 0 1

ij = 1, i = j0, i = j

U = 0, = 0

pn = pn + pnpij = p ij +pij

P = pE+ T

p = 13 (xx + yy + zz)

pn

T

pn = T n, pn = P n

T =

xx

yx

zx

xy yy

zy

xz yz

zz .


27/200

U = 0

U = 0, = 0

Rn =

A

pn dA = A

p n dA =

=

Vgrad dV P

A

nx(. . . )dA =

V

x(. . . )dV

U = 0, = 0

Rn =

A

pn dA =

A

(pxnx + pyny + pznz) dA

=

=

V

pxx

+pyy

+pzz

dV

pij =dRjdAi

i,j=x,y,z= pxz xz = dRz

dAx.

ij

x

y

z

xx

xy

xzdAi

pij i dAi j

j

Rj =

A

pijni dA =

V

pijxi

dV.

V

A

A

V


28/200

pn = pn, p = pi, = x,y,z.

p

p = p(r)

pz pz = p k

py py = pj

px

px =

(

pi ) = pi

x

y

z

pnn

M

pn = pnn n = p n, px = pi, py = pj, pz = p k

pnn = pxx = pyy = pzz =

p


29/200

SI

[Pa]

bar

1bar = 100000 Pa

105 Pa

p

pA

(pm)A

(pv)B

pB

pa

pa pb

pa = const

pa 1bar pa = 101325 Pa

p

pm

pv

pA = pa + (pm)A

pB = pa (pv)B

(pm)A

(pv)B


30/200

V

A

V

m dA

dV

f

n

dRn = dP = pndA

m V

Rn =

A

pn dA =

Apn dA =

V

p dV = P

P

A

pn dA = V

gradp dV.

V

A

i

Fi = 0 Rm + Rn = 0

V fdV +

V

gradp dV = 0,

V

f p

dV = 0.

V

gradp = f .

p = const

p = const

f(r)

p

f


31/200

p

f

p = const

f

f

f = f(r) f = const

p = const

f

p(r)

(r)

f(r)

= const ( )

= (p) (

)

= (p, T) (

)

f

f = const

f = f(r) (

)

p(r) = p(x,y ,z)

p(M)

p = const

P =

Ap n dA

p

x= fx,

p

y= fy

p

z= fz,

f = fxi + fyj + fzk, dr = dxi + dy j + dz k

p = f / dr dp =

f dr

dp = (fx dx + fy dy + fz dz)


32/200

f = g = {0, 0, g}

a

fin = a

zf = 2 r = 2xi + 2yj =

2x, 2y, 0

= const

f = g k f = g k z

z

f = g k

f

x,y,z

fx = fy = 0 fz = g

p = p(x,y ,z) =

(g dz ) + C

0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 01 1 1 1 1

0 0 0 0 0 01 1 1 1 1 1

z

x

0

H

p = pa

p = pa + gH

g


33/200

= const.

p = g z + C

C

x = z =

0 p = pa pa

p = pa + g z

z = const

g

g

p = pa

pm

C

C = pa + pm

pv C C = pa pv

a

H

g

ain

ain = a

f = g + ain = g a

f = gradp

f

f

a

fx = a, fy = 0 fz = g,


34/200

0011

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 01 1 1 1 10 0 0 0 01 1 1 1 1y

O

H

a

gf

p =con

stain

x

z

dp =

(a dx g dz) = p = (ax gz) + C

C

x = 0, z = 0 : p = pa = C = pa

p pa = (ax gz)

p = C, C = const. z = ag

x + C

= arctg ag ose

p = pa

C

z =a

gx.

D

H


35/200

0 00 00 00 00 00 00 00 01 11 11 11 11 11 11 11 1

0 00 00 00 00 00 00 00 01 11 11 11 11 11 11 11 1

0 00 00 00 00 00 00 00 01 11 11 11 11 11 11 11 1

0 00 00 00 00 00 00 00 01 11 11 11 11 11 11 11 1

0 00 00 00 00 00 00 00 01 11 11 11 11 11 11 11 10 00 00 00 00 00 00 00 01 11 11 11 11 11 11 11 1

D

Hr

x

y

z

z

r

rfc

fc

= const

= const

0xy

z

r

fc = fxi + fyj = 2

r = 2

(xi + y j)

z

r

fx = 2x; fy =

2y; fz = g

p = 12

2 (x2 + y2) r2

gz + C = 12

2r2 gz + C

r

C

x = 0, z = 0 : p = pa = C = pa

p pa =1

2 2r2 gz

r

z


36/200

z =2

2gr2 + C, C = const

p = pa

z =2

2gr2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

k

z

f = gk

dp = g dz p = RT

dp

p= g dz

R T(z)

z = z0 p = p0

p = p0 exp

zz0

g dz

RT(z)

.

T = T(z)

T = T0 = const

p = p0 exp

g

RT0(z z0)

T(z) = T0 zp = p0

1 z

T0

gR

p/ = p/0

z z0 = 1

p

g0

1

p

p0

1

11km


37/200

T(z) = T0 z, T 0 = 288 K, = 6, 5 K/km

p = RT

= 0

1 z

T0

gR1

t = 56, 5 C

zz

pp0T0 T

10km

dP

dAn

dP

dP = p n dA P =

Ap n dA = n

A

p dA.

dP = p dA P =

A

p dA.


38/200

dAn

dP

n

n = const

dP = p n dA P =

A

p n dA

P =

A

p

nxi + nyj + nzk

dA,

dPj = p nj dA = p dAj P =

Ap nj dA, j = x,y,z.

A

A

dA

n

V

dP

dP = p n dA P =

A

p n dA

A

PA

=

Vp dV =

V

fdV.

p(r) = p(x,y,z) gradp = f dp = Fidxi i =

1, 2, 3

p


39/200

P =

A

p dA =

A

(pa + gz)dA

Apa dA =

paA + g

A

zdA

paA

= (pa + gzC) pC

A paA = (pc pa) A gzCA

zC z

pc

pa

A

0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 01 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 10 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 01 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 10 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1

D

C

P

v

u

zzC

vC

vD

vC

dAd P

n

P = (pc pa) A.

180 < < 0

z

D


40/200

u

P vD =

A

vdP gzCA vD = gA

z v dA,

z = v sin

vD =1

vCA

A

v2dA =Iu

vCA,

Iu A u

Iu = IC + v2CA

vD =1

vCA v2CA + IC = vC+ vC vC vD vC =

IC

vCA

.

vC vC = zC/ sin zCA P/g

vC =IC sin

zCA=

gIC sin

P

vC =IC

vc A gIC sin

P

Ic P

u

uc =I

uc A


41/200

pmpm pvpv

h1h1h1h1

h2 h2

h2h2

h3

1 1

11

2

22

2 3P Pvcvc

C

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

00

0

0

0

11

1

1

1

000

0

0

0

111

1

1

1

000

0

0

0

111

1

1

1

000

0

0

0

111

1

1

1

000

0

0

0

111

1

1

1

00

0

0

0

11

1

1

1

000

0

0

0

111

1

1

1

pmpm pvpv

h1h1h1h1

h2 h2

h2h2

h3

1 1

11

2

22

2 3P P

PP1

P2

vc vcvc

C C C2

pc = pa pv + 1gh1 + 2gh2

P = (pc pa)A = (pv + 1gh1 + 2gh2)A

zc =pc pa

2g= pv

2g+

12

h1 + h2

P

= 90

vc zc = IczcA

pv 1gh1 + 2gh2

P


42/200

vc

D

vc

pc = pa +pm + 1gh1 + 2gh2

P = (pc pa)A = (pm + 1gh1 + 2gh2)A

zc =pc pa

2g

=pm

2g

+1

2

h1 + h2

P

= 90

vc zc = IczcA

P2

P1

pc = pa pv + 1gh1 + 2gh2 + 3gh3

P = (pc pa)A = (pv + 1gh1 + 2gh2 + 3gh3)A2

zc =pc pa

2g= pv

3g+

13

h1 +23

h2 + h3

P = 90

vc zc = IczcA

pc = pa +pm + 1gh1 + 2gh2

P = (pc pa)A = (pm + 1gh1 + 2gh2)A2


43/200

zc =pc pa

2g=

pm2g

+12

h1 + h2

pm pv

P1 = pmA P2 = pvA

P2

P2

P2 = pvA

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 01 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1

0000000000000000011111111111111111

0000000000000000011111111111111111

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 01 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 10 0 0 0 0 0 0 01 1 1 1 1 1 1 1 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 pm

pmpv

pv

paP1 P2

x z


44/200

d P

dA

dP

0 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 1

0

dP

dP

dP

dA

z

zs

zs

ys

xs

xs

A

dAx

dAz

n

n

dPx = dPxi

dPz

dA

A

z

n

n = i cos(n,i) + k cos(n, k) = i cos + k cos

n

d P

dA

x

z

cos > 0 cos > 0

d P = d Px + d Pz dPxi + dPz k = g z (i cos + k cos )dA

dPx = g z (dA cos ) dAx

Px = gAx

z dAx Px = g zcx Ax

Ax A x y

z

zcx Ax


45/200

dPz

dPz = g z (dA cos )

dAz Pz = g

A

zdAz Pz = gV

V

Pz V

Px Pz

Ax

x

Px = (pcx pa)Ax g zcx Ax

Px = g (H + R)R2

R

x

zH

zcx

Ax

x

x Cx

DxPx

x

Pz1

Pz2

Pz2 Pz1


46/200

Pz1 Pz2 Pz1

Pz1 = gVz1 = gVz1 = g

1

2R2(H + R) 1

4 4

3R3

= gR2

H

2+

R

6

Vz1 R H + R

R

Pz2

Pz2 = gVz2 = gVz2 = g

1

2R2(H+ R) +

1

4 4

3R3

= gR2

H

2+

5R

6

R

x

zH

zcx

Pz1

Pz2

Vz1 Vz2

Pz2 Pz1 Pz

Pz = Pz2 Pz1 = gR2

H

2+

5R

6

gR2

H

2+

R

6

=

2

3gR3

x

z

PxPx

PzPz

P


47/200

P =

P2x + P

2z =

[ g (H + R)R2]2 +

2

3gR3

2

= gR2

(H + R)2 +

4R2

9

Pz1

Pz2

Px

Px1 Px2

R

x

z

H

x

Cx

Px1 Px2

4R

3Ax1 = Ax2

Ax1 Ax2

Vz

x

Px1 Px2

Px1 = Px2 = g

H +

4R

3

R2

2

x

Px = Px1 Px2 = 0

dA1 dA

1


48/200

dPx

A

x

Pz

Pz = gVz = g

R2H +

1

2 4

3R3

= gR2

H +

2R

3

Px = 0 P P = Pz

Pv

P

N

PP

A1A1

D

V

A2

P

A1

A2

Pv =

Ap n dA =

A1

p n dA1 P

+

A2

p n dA2 PN

= P + PN

P

P

N = PN


49/200

P = Pv PN = Pv + PN = gV + PN

V

V

V

n

P =ni=1

PNi + Pv =ni=1

PNi gV

PVPVPV

PNPN

PN

P

P

R L H

PN1 Pv1

PN1 = (pc1 pa)A1 = g

H R2

R

2 L

Pv1 = gV1 = g

R2

4 1

2R2

L = gR2

4 1

2

L


50/200

x

y

R

HPv1 Pv2

P =?

V1

V2PN1

PN2

PN3

(b)

P

x y

Px = PN1 cos45 = PN1

2

2= g

H R

2

RL

Py = PN1 sin 45 Pv1 = g

H R

2

RL gR2

4 1

2

L = g

H R

4

RL

P

P =

P2x + P2y = gRL

H R

2

2+

H R

4

2

(b)

PN2 = g

H R

2

RL , P N3 = gHRL

Pv2 = gV2 = g 14

R2L

P

Px = PN2 = g

H R

2

RL

Py = PN3

Pv2 = gHRL

g

1

4R2L = gH

R

4RL


51/200

PG

P

PN

R

i

Fi = 0 PN + R

+ PG

= 0

PN PG = g V R

PG V PN

P = R

P = PN + PG = PN + g V

n

P =ni=1

PN + PG =ni=1

PN + g V

R

L

H

R

L

V


52/200

x

y

R

H

PG1 PG2V1

V2

PN1PN2

PN3

(b)

PN1 = g

H R

2

R

2 L

PG1 = gV1 = g

3

4R2 +

1

2R2

L = g

3

4 +

1

2

R2L

x y

Px = PN1 cos45 = g

H R

2

RL

Py = PG1 + PN1 sin45

= g34 + 12R2L + gH R2 RL = g3R4 + HRL

P =

P2x + P2y = gRL

H R

2

2+

3R

4 + H

2

PN2 PN3 PG2

PN2 = g

H R

2

RL; PN3 = gHRL

PG2 = gV2 = 34

gR2L

P

Px = PN2 = g

H R

2

RL

Py = PN3 + PG2 = g

3R

4 + H

RL


53/200

D

pm

0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 00 00 00 0

0 0

0 0

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 11 11 11 1

1 1

1 1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 00 00 00 0

0 0

0 0

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 11 11 11 1

1 1

1 1

0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1

D

D

l

l

dP

n

u

p

z

pm

pm

Pz =Az

pm dAz = pm Az = pmD2

4 = 0,

pmD2

4= pD p = pmD

4< doz.

p

l

P = pm

Dl

pm Dl = u 2l u = pmD2

< doz

u > p

pmD2 doz

.


54/200

U(r, t)

ui/xi

div U rot U

a

a

a


55/200

U = 0

= 0

"The principle is most important, not the detail"

Theodore von Karman, 1954

U = 0 = 0

Rn

U

ui/xj

a

a = DU /Dt

D()

Dt=

()

t+ ( U )()

Ra


56/200

Vm

Am

V

A

Rn


57/200

J. DAlambert

pij = 0 ij = 0 ii = 0 i, j = x,y ,z T = 0 pij = pij

U(u,v,w) p

T

Lagrange

U = U[ x(t), y(t), z(t), t ] = U[ r(r0, t), t ],

r[ xi(t)] i = 1, 2, 3 t

M(r)

t

N(r1)

t1 M

U = limt1t

r1 rt1 t = limt0

r(t + t) rt

= r.

Euler

U = U(x,y ,z ,t) = U(r, t),

x,y,z

r(x,y ,z)


58/200

(r0, t)

(x0, y0, z0, t)

(r, t)

(x,y ,z ,t)

dr

dt

= U

dx

dt

= u(x,y ,z ,t),dy

dt

= v(x,y ,z ,t),dz

dt

= w(x,y ,z ,t).

dxk = uk(xi, t) dt, i, k = 1, 2, 3

x1,2,3 = x,y ,z u1,2,3 = u,v,w x2 = y

u3 = w

M0

M

t0 tt

U(r0, t0)

U(r0, t)

M0 M r0 t0 t

r = r(r0, t)

S

U dl = 0 dxu(x,y ,z ,t)

=dy

v(x,y ,z ,t)=

dz

w(x,y ,z ,t),

x,y ,z t


59/200

f1(x,y ,z ,t) = C1, f2(x,y ,z ,t) = C2,

C1 C2

M

r

dl

U(r, ti)

U(r, tj)

U(r, tk)

U(u,v,w)dl(dx, dy, dz)

S

ti

U(r, tl) l = 1, 2, . . . , i , . . . , j , . . . , k , . . .

. . . , ti, . . . , tj , . . . , tk, . . . S

ti dl

U(M1, t) = U(M1, t)

lm Am Vm

m

Vm

Am

V A

A

V


60/200

A

Vm

t

Vm Am

V

Vm(t) = V

t

V

m

A

m

V

A

Vm

Am

V A

V = Vm(t)

A = Am(t)

tt

Vm = Vm(t)

Am = Am(t)

Vm Am

V

A

A

Vm

Vm Am A

t = t t

dVm

U dV

dVm V U dV

Vm V Am A

dVm dV


61/200

A

V

n

nU

U

AA

VdA

dA

dV

|U|

t

A

dA

n

dV

dV =

Ut

n

dA = U n dA t

V =A

U n dA t = t A

U n dA

Vdef= lim

t0

V

t=

A

U n dA

U

A

A

m

mdef= lim

t0

m

t=

A

U n dA

V =

A

uini dA , m =

A

uini dA

A

div U


62/200

A

V

A

U n dA =

Vdiv UdV

A

uini dA =

V

uixi

dV.

div U

H. Helmholtz, 1821-1894

0

r

r +dr

dr

t

U(r, t)

U +dU

dU{dui} i = 1, 2, 3 dr{dxj} j = 1, 2, 3

U +dU = U + (

dr ) U ui + dui = ui + ui

xjdxj

U

ui

(ui/xj) dxj ui

xj i, j = 1, 2, 3

ui/xj

3 3


63/200

uixj

=1

2 uixj

+ujxi +

1

2 uixj

ujxi

= Sij + ij

Sij

ij

Sij =1

2 ui

xj

+uj

xi

ui/xj i = j

ABCD dx1 dx2

dt

ABCD

A AB

C D

B

C D

x1

x2

dx1

dx2

dx1 + u1dt

dx2 + u2dt

ui/xi i Sij t t + dt

A

B

C

D

A(0, 0) A(0, 0), D(dx1, dx2) D

dx1 +u1x1

dx1dt, dx2 +u2x2

dx2dt

B(dx1, 0) B

dx1 +u1x1

dx1dt, 0

, C(0, dx2) C

0, dx2 +

u2x2

dx2dt

x1 = x, x2 = y, x3 = z

u1 = u, u2 = v, u3 = w

x

u

y

v

z

w


64/200

0x1 0x2 AB

AB = dx1 + [u1(dx1) u1(0)] dt dx1 + u1x1

dx1dt.

AB

dt

d(AB)

AB=

u1x1

dt d(dx1)dx1

=u1x1

dt

ABCD

dA

A

=d(AB)

AB

+d(AC)

AC

= u1x1 +u2

x2 dt = div Udt.

x3

V

dV

V=

u1x1

+u2x2

+u3x3

dt =

uixi

dt = (div U)dt.

div U

Vm = dVm

d(dVm)dVm

= div Udt div U = d(dVm)dt

1dVm

.

div U

dVV = 0 d(dVm)dVm = 0 div U

ui/xj Sij i = j

A(0, 0) A(0, 0)B(dx1, 0) B

dx1,

u2x1

dx1dt

C(0, dx2) C

u1x2

dx2dt, dx2

D(dx1, dx2) D

dx1 +u1x2

dx2dt, dx2 +u2x1

dx1dt

d

AB

dt

d tg(d) = [u2(dx1) u2(0)] dtdx1

u2x1

dt


65/200

d

dt=

u2x1

d/dt =

u1/x2 d/dt

AB

AC

d

dt= d d

dt=

u2x1

+u1x2

= 2S12

A AB

CD

B

C

D

x1

x2

dx1

dx2

d

d

ui/xj i = j Sij t

t + dt

ui/xj i = j Sij

S

Sij

Sij i = j

u1x1

+u1x1

+u1x1

uixi

= div U

Sij i = j

ij

ij = 0 i = j ii = 0 i

ij = ji

Sij = 0 ui/xj = uj/xi

d u2x1 dt

d u1x2 dt = d

d = (d d) = 0

AB

AC


66/200

A

B

CD

A

B

C

D

x1x1

x2x2

dx1dx1

dx2dx2

d

d

u2(dx1)dt

u1(dx2)dt

ui/xj i = j ui/xj = uj/xi

t + dt

d =u2x1

dt =1

2

u2x1

u1x2

dt = 21dt

21 d/dt

AB

AC

S12

ij = 0, i = j ij = ji 12 = 21, 23 = 32, 31 = 13

||ij || =

0 12 1321 0 23

31 32 0

1

2

3

=

12

u3x2

u2x3

12

u1x3

u3x1

12

u3x2

u1x2

||ij||

||pij|| ||Sij|| ||pij || ||Sij|| ||ij ||

(1, 2, 3)

d

dt=

1

2

u2x1

u1x2

= 21 = 3, 13 = 2, 32 = 1

= xi + yj + z k


67/200

=1

2

w

y v

z

i +

u

z w

x

j +

v

x u

y

k

ui/xj

rot U

rot U = 2 (rot U U)

rot U

||

= rot U = 2

U = 0

= 0

U = 0

rotgrad = 0 0

(x,y ,z ,t)

= 0

rotU = 0

U = grad

div U = 0

= 0 U = grad

divgrad = 0 2 = 0

2

x2 +

2

y 2 +

2

z 2 = 0


68/200

Pierre Simon de Laplace, 1749-1827

x0y

U

U

x

y

r

r

r

u

v

R0

= 0

U =k

r

U = r

(r,,z)

rotU =1

r

r(ru) k

ur = uz = 0, u = u(r)

U = u

z = 12r

r(rur)

=

,

U = R, r < R

0,

U = R2/r, r > R

Oz


69/200

U = r ur = uz = 0, u = r = U

u = r sin = y, v = r cos = x.

v

x u

y= 2 rot U = 2 k ( = z)

U = k/r u = k sin r

= ky

x2 + y2, v =

cos

r= k

x

x2 + y2

k = const

v

x u

y= 0

z = 0.

u =

x, v =

y

divU=0,

2

x2+

2

y 2= 0 = 0.

rotUn

U

dl

l

dA

rotU

A

=

l

U dl =A

rot U n dA A

2 n dA.

= 0


70/200

l = 2R2

l

r = 0

M1(r1) M2(r2) t t

U(r1, t) U(r2, t)

t = t + t

M2(r2) r2 = r1 + U(r1, t)t + O (t)2 U(r2, t)

U

t = t t

U(r, t)

U(r1, t) U(r1, t)

U(r2, t) U(r1, t)

U = U(r2, t) U(r1, t) =

U

tt +

U

xx +

U

yy +

U

zz,

x y

z

r2 r1

M1(r1)

M2(r2)

U(r1, t)

U(r2, t)

U(r1, t)

U(r1, t)

r2 = r1 + U(r1, t)t + O

(t)2

(r2 r1) {x, y, z}

a

limt0

Ut

= limt0

Ut

+ Ux

xt

+ Uy

yt

+ Uz

zt

= Ut

+ u Ux

+ v Uy

+ w Uz

DUDt


71/200

DU

Dt=

U

t+ ( U )U

DU

Dt

Ra

U

t

Ra

(U )U

Ra

D()

Dt=

()

t+ ( U )() = ()

t+ ui

()

xi

T(r, t)

DT

Dt=

T

t+ ( U

)T =

T

t+ U

gradT

DT /Dt

T/t

T

U gradT

Df

Dt=

f

t+ u

f

x+ v

f

y+ w

f

z


72/200

f(r, t)

U(r, t)

r

U(r, t)

a(r, t) = U

t (r ) +U

xi

dxi

dt

=U

t

+ uiU

xi

r

D = dt

t+ (dr ), DU = dt

U

t+ (dr )U , D

U

Dt= a(r, t)

D

Ra

x

DU

Dt a Ra

u

t+ u

u

x+

1

2v

v

x+

u

y

+

1

2w

u

z+

w

x

+

1

2w

u

z w

x

1

2v

v

x u

y

ut

+

u

x+ v

u

y+ w

u

z

2Sxy 2Sxz 2y 2z

Ra,x = ax


73/200

+

+

y

z

0x

Ra,x

ut + x U2

2 + 2(wy vz)

Ra,y Ra,z

Ra

U

t+ grad

U2

2

U rot U

D

U

Dt a

+

U

U |U| =

u2 + v2 + w2.

(U

)U

1

2

gradU2

U

rot U


74/200

Ra

U

t+ grad

U2

2+ 2 U

rot U = 2

(U )U

as

U

U

s

Ut + U

Us (L) t

t

U(s, t)

dt

s + ds = s + Udt

(T)

(L)

t

(S)

ds

(L)

U(t + dt, s + ds)

dt dU = U(t+dt, s+ds)U(s, t) = Us Udt+ Ut dt

as =dU

dt DU

Dt=

U

t+ U

U

s.


75/200

a

as an

0

an = U2

/R

dm = dV

(U2/R)dV

0

s

s U

U

R

nds

=Udt

ds=

Udt

U(t, s)

U(t + dt, s + ds)

(S)

(S)

(T)

t : U(s)

t + dt :U(s + ds)

U

t= 0

U

t= 0

t : (S) (L)

a = DU /Dt

(s, n)

(S)

s

U

s

n

(S)

R

(T)

(L)

t

as

U

t

s

t + dt

s + ds = s + Udt

U = |U|

dt

dU =U(s + Udt)

U(s) = U

sUdt

as =dU

dt DU

Dt= UU

s

dm = dV

dV UUs

= s

U2

2

dV

= const

s

U

2

2

dV

D()

Dt=

()

t+ (U)(); U = |U|s = Us; U = U(s, t); (U)() = (|U|s )() = U

s()

D()

Dt=

()

t+ U

()

s. (U = |U|)

a =D

Dt(Us) =

U

t+ U

U

s

s +

U2

Rn = ass + ann

Us = U

Ut + U

Us =

DUDt

a =DU

Dt=

DU

Dts +

U2

Rn =

U

t+ U

U

s

s +

U2

Rn.


76/200

dVm dAm

dV

dA

Vm Am

V A

dVm V

V


77/200

n

n

n

U

U

m

t

t :

V

V

uini < 0

uini < 0

uini > 0

uini > 0

A

U n dA

V

tdV

A

AAm

V = Vm

dA

dA

Vm V t Am A A V


78/200

m

Vm

D

Dt

Vm

dV = 0

D

Dt

Vm

dm =

(M)

D

Dt(dm) = 0

D

Dt(dm) D

Dt(dV) = 0

D

Dt

(Vm)

dV =

(V)

D

DtdV+

(V)

D(dV)

Dt=

=

V

D

Dt+ div U

dV = 0

div U =D(dV)

Dt

1

dV

V

D

Dt+ div U = 0

t+ div(U) = 0

DDt

= t

+ U grad

div(U) = div U + U

D

Dt+

uixi

= 0

t+

xi(ui) = 0

D

Dt(dV) =

D

DtdV +

D(dV)

Dt=

D

Dt+ div U = 0


79/200

=

V

tdV =

A

U n dA ddt

V

dV =

Auini dA

m

V

A

d

dt

V

dV =

AU n dA

d

dt

V

dV =

t

V

dV =

V

tdV

A

U n dA =

V (U)dV

V

V

t+ divU

dV = 0.

V

t+ div(U) = 0 D

Dt+ U = 0.


80/200

divU =1

dV

D(dV)

Dt= 1

D

Dt

D/Dt

t+ U grad

DDt

+div U = 0.

D

Dt= 0 = div U = 0 ui

xi= 0

1

D

Dt+

1

dV

D(dV)

Dt= 0.

div U = 0

D

Dt= 0

= const

i = cont i = 1, 2, . . . , k , . . . i = k

t+ U grad = 0

/t

U

grad


81/200

2 = rotU = 0

(x,y ,z ,t)

U = grad

= 0

w = 0 /z = 0 /t = 0 u(x, y) v(x, y)

div U = 0 = ux

+v

y= 0

(x, y)

u =

y, v =

x

u =

y=

x, v =

x=

y

2

x2+

2

y 2= 0 = 0 .

Cauchy-Riemann

w(z) =

(x, y) + i(x, y)

z = x + iy

i =1

B

A

x

y

A

B

= const

U

U

nn n

dl

dldl

u

v

dx

dy

a

b

n(a, b)dl(dx, dy)

dl n = adx + bdy|n| = a2 + b2 = 1

n =

dy

dl, dx

dl

= const = const


82/200

= const

d =

xdx +

ydy = vdx + udy = 0 =

dy

dx

=const

=v

u.

= const

U grad = u x

+ v

y= 0

= const

d =

xdx +

ydy = udx + vdy = 0 =

dy

dx

=const

= uv

.

= const

= const

grad

grad =

x

x

+

y

y

=

vu + uv = 0

= const

V

0z

VAB =

BA

(U n)

dVdl =

BA

ydy +

x

(dx)

=

BA

d = B A

A

B

f(r, t)

Vm = V(t) V t


83/200

Vm

D

Dt

Vm=V(t)

fdV

f = f(r, t)

Vm

dV = dm

D

Dt

Vm=V(t)

fdV =D

Dt

M

Vm

fdm =

M

Vm

D

Dt(fdm)

=M

Df

Dt dm =

Vm

Df

Dt dV.

dm

f

Df /Dt

Vm = Vm(t)

V

t

Vm

D

Dt

Vm

fdV =

V

Df

DtdV

D

Dt

Vm

fdV =D

Dt

M

f vdm( )

=

V

D

Dt(f v)dm

V

DDt

(f v)dm =

V

DfDt

dV +

V

fD(dV)Dt

=

V

DfDt

+ fdiv U

dV.


84/200

D

Dt

Vm

fdV =

V

Df

Dt+ fdiv U dV

Vm V

D

Dt

Vm

fdV =

V

f

t+ U f + f U

(fU)

dV

=

V

f

t+ div(fU)

dV =

V

f

t+

xi(f ui)

dV.

f

V

V

fdV =

A

nfdA

V

f

xi=

A

nifdA.

A

V

n

D

Dt

Vm

fdV =

V

f

tdV +

A

f U n dA

f

f

f

f =

f = 1


85/200

t + dt

U n dA dt

U n dA dtt t : V = Vm, A = Am

f

f

tA

Am

n

n

n

n

U

U

dAdA

M

uini > 0

uini < 0

V

f

tdV d

dt

V

fdV

f

f

V

A

fU n dA = A

f uini dA f f U

A

f(r, t) Vm V t A Am

V

A

f 1

V A

A

U n dA =

Vdiv UdV

A

V

div U


86/200

=

+

D

Dt

Vm

UdV =

Vm

fdV +

Am

p n dA

DK

Dt Ra Rm Rn

Vm

DU

Dt f + gradp

dV = 0

V

DU

Dt= f gradp


87/200

DuiDt

= fi pxi

t

(ui) +

xj(uiuj) = fi p

xi

t+

(uj)

xj= 0

U = 0

f

f = U

V

(U)

tdV +

A

U(U n) dA =

V

fdV

A

p n dA

V

A

t

V

UdV =

AU(U n)dA +

V

fdV

Apn dA

I II III IV


88/200

I

V

II

A

A

III

Rm V

IV

Rn A

V

f

f = grad

( f) = 0

V

fdV =

V

graddV =

A

n dA

f = grad

A

U(U n)dA =

A( p)n dA

Ri

Ri =

AU(U n) dA,

U

Ri + Rm + Rn = 0.


89/200

t

V

ui dV =

Auiujnj dA +

V

fi dV

Apni dA,

V

(ui)

t=

V

xj(uiuj)dV +

V

fi p

xi

dV.

U = ui + vj + wk

x

i = 1

u1 = u

V

(u)

t=

A

u(U n)dA +

VfxdV

A

pnxdA.

x

V

(u)

t+ (uU) fx + p

x

dV = 0

i = 1

y

z

f = 0

= 0

(uU) =

p

x

,

(v U) =

p

y

,

(w U) =

p

z

.


90/200

V

(r U)t

dV +

A

(r U)(U n)dA = Mm,n

r

DDt

Vm

e + U22dV =

V f UdV

Ap U n dA +

V qdV,

e

U2/2

q

q > 0

q < 0

=

+ +

+

t

V

e +U2

2 dV =

A

e +U2

2 U

n dA +

V

f

UdV

A

p U

n +

V

qdV


91/200

t

e +

U2

2

+

e +

U2

2

U

= f U (pU) + q.

q = 0

f = 0

D

Dt

e +

U2

2

= f U gradp Updiv U .

e

= (p)

T(r, t)

f(p, , T) = 0


92/200

U p

= const

U

p

= (p)p/ = constp/ = const

p/

= const

U p T

f(p, , T)

p = RT

e = cvT

Clapeyron

p = RT

dp

p d

dT

T= 0

(Un)U uj ui/xj

t = t0

t = t0 : f(r, t0) = f0(r), f =U , p , , T , . . .


93/200

U n = UA n !

U UA A n

A

A

U n = 0

!

Un = U n

A

F(r, t) = 0 n = gradF|gradF|

F(r, t)

n

A

U gradF = UA gradF F(r, t) = 0

xi(t) F[x(t), y(t), z(t), t] = 0

F


94/200

dF/dt = 0

dxi/dt = uA,i UA(uA, vA, wA)

F

t+ uA

F

x+ vA

F

x+ wA

F

z= 0 UA gradF = F

t.

DF

Dt=

F

x+ U gradF = 0

F(r, t) = 0.

F(r) =

0

U gradF = 0

F(r) = 0,

a priori

U

= 0

A

A

F(r, t) = 0

F(r) = 0


95/200

d

dt

V

dV +

A

U n dA = 0

t+ div(U) = 0

t

V

UdV =

AU(U n)dA +

V

fdV

Apn dA

Du

Dt= Fx p

x,

Dv

Dt= Fy p

y,

Dw

Dt= Fz p

z

t

V

e +U2

2 dV =

A

e +U2

2 Un dA +

V

f

UdV

A

p U

n +

V

qdV

t

e +

U2

2

+

e +

U2

2

U

= f U (pU) + q

e = cvT

p = RT


96/200

()/t = 0

q = 0

f = 0

h0 = h +U2

2= const

h = e + p

h0

e = cvT h = cpT cv =R

1 cp =R

1 =cpcv

f

f = f(r)

rot f = 0

f = gradf f = f(r)

G =

f

rot f = 0 f = gradf = gradG

= (p)

= (p) 1

gradp = grad

dp

pG

pG = p f ( = const)


97/200

B

B U2

2 f +

dp

(p) U

2

2+ G +

dp

(p)

U2/2 U U /2 f = G G f

(dp/)

1gradp

grad[(dp/)] (dp/)

U

t +1

2 gradU2

2U = f1

gradp

/t = 0

= (p)

f = gradf

grad

U2

2 f +

dp

= 2 U .

s

gradB s = 2s (U ) gradB s = 2 (U s) gradB s = 0,

U

s = U /|U|

gradB s


98/200

B

gradB s = 0 Bx

sx +B

ysy +

B

zsz = 0 dsB = 0

B U2

2+ G +

dp

(p)= CB() (const = const)

CB() = const

= 0

z

g =

gk

f =

gz

G = gz

U2

2+ gz +

p

= CB()

U2

2g+ z +

p

g= const.

,


99/200

gradB

B =

dB

dl= 0,

d()/dl

dl = 0

B = const

U

B

p + U2

2= const

z = const

U = 0

B = 0

(

) = 0

(

) U

U

2

2+ G +

dp

= CB = const

U

U2

2+

dp

= CB()


100/200

T = const

p/ = const

p/p1 = /1

U2

2+

p11

lnp

p1=

U2

2+

p11

ln

1= const

p/p1 = (/1)

/1 = (p/p1)1/

U2

2

1p11

1

p

p1

1

= const;

U2

2

1p11

1

1

1

= const;

U2 U212

+

1 p

p

1 = 0.

(s, n)

U = Us

DU

Dt= f 1

gradp DU

Dts +

U2

Rn = grad

F

dp

U = Us

UDU

Dt= U grad

F

dp

.

D

Dt

U U2

=

D

Dt

F

dp

t

F

dp

D

Dt

U2

2 F +

dp

=

t

F

dp

()

t= 0 D

Dt

U2

2 F +

dp

= 0

U2

2 F +

dp

= const


101/200

(s, n)

s

n

U

t+ U

U

s= Fs 1

p

s

U2

R= fn 1

p

n

fs = gz/s fn = gz/n f = g z

n

f = gz

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Mehanika Fluida B - Handout