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U = 0 = 0
ij
Ra
f(r, t)
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(s, n)
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dV= dxdydz
u2
uv
us
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us
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 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
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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
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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
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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
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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
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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
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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
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= (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
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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.
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= /
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
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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
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0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 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
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g
u/y
(n < 1)
(n = 1)
(n > 1)
P = F(S) P
S
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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
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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
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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
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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
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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
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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 ,
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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 .
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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
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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
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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
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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
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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)
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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
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= 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,
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
.
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U(r, t)
ui/xi
div U rot U
a
a
a
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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
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Vm
Am
V
A
Rn
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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)
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(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
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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
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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
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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
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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
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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
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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
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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
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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
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=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
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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
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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
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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
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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
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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
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+
+
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
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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.
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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.
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dVm dAm
dV
dA
Vm Am
V A
dVm V
V
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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
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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
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=
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.
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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
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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
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= 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
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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.
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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
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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
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=
+
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
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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
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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.
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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
.
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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
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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
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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 , . . .
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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
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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
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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
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()/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)
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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
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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.
,
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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()
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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
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(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