Equacoes de Navier-Stokes Navier-Stokes
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7/23/2019 Equacoes de Navier-Stokes Navier-Stokes
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, ,
E
2012
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7/23/2019 Equacoes de Navier-Stokes Navier-Stokes
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A .
D , ,
. I . E (
).
C C, :
D
Dt = +
vf T%
, v , f ,
, T% . D Dt ,
: ( )D
Dt t
= +
v
A :
( )1
2 22
ji kij ij
j i k
uu up p
x x x
= + + = + + +
T I v I% % %% , p
, I% ( ,
), , %
, . I% % :
1 0 0
0 1 0
0 0 1
ij
= =
I% 0
1ij
se i j
se i j
= =
D K.
( )1 1
2 2
xx xy xzTji
yx yy yz
j i
zx zy zz
uu
x x
= = + = +
v v%
, ,
, :
( ) ( )( )2D pDt t
= + = + +
v v v v f I v I% %% ,
( ) + = + A B A B% %% % , H 2
03
+ = ,
, :
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( ) ( ) ( )2
3
TDp
Dt t
= + = + +
v vv v f v v v I%
E .
(
), ,
. E :
2
3
ji i i k j i ij
j i j j i k
uu u u upu f
t x x x x x x
+ = + +
( ) : 0 =v
0t
=
v , :
( ) ( )( )Tp = + + v v f v v . :
ji ij i
j i j j i
uu upu f
x x x x x
= + +
. .
(
, .. ; ),
, , :
( ) ( )21
3
Dp
Dt t
= + = + +
v vv v f v v
. :
2
2
1
3
i i i k j i
j i i j k
u u u upu f
t x x x x x
+ = + +
,
:
( ) 2D
pDt t
= + = +
v vv v f v , :
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2
2
i i ij i
j i i
u u upu f
t x x x
+ = +
. E ,
.
D ( ), , E E
B.
, .
.
, :
( )D
pDt t
= + =
v vv v f
E ( ) ( )
21 1
2 2 = = + v v v v v v
curl= = v v ,
:
( )21
2p
t
+ + = +
vv f v
:
( )212
i qii kji j i pq
i i p
u uu pf u
t x x x
+ + = +
C
1 123, 231,312
1 132,213,321
0 paraalgum indice repetido
ijk
se ijk
se ijk
=
= =
( LC)
,
. E E
E ( ).
D ,
, p .
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A, I, ,
, 0= = v ,
:
21
2p
t
+ + =
vv f
(
..), .. U= f , = v ,
( ),t = r , e ,tr
, E :
( ) 212
U pt
+ =
v
A
:
210
2
pU
t
+ + =
v
E , ( )21
2
pU f t
t
+ + =
v , ( )f t
. :
21
2
pU cte
+ =v
C B. U mgz= ,
z 0z = .
C d
( )m t : dM d= ,
dM d= v v . A :( )m t
d
v .
L , :
( )
Fora resultante atuando sobre Foras de corpo o volume
o volume de controle material + Foras de superficiem t
Dd
Dt
= =
v
A ( ),
(, ).
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C , :
( ) ( )( )
( )
Variao temporal da Foras de corpo Fora de superficiequantidade
do movimento
m m m
n
t t A t
Dd d dA
Dt
= + v g t1442443 14243 14243
( )n
t n
:
( )
( )
( )
Foras por und. rea atuando sobre uma superf. com normal // a
x
Foras por und. rea atuando sobre uma superf. com normal // a
y
Foras por und. r
xx xy xzi
yx yy yzj
zx zy zzk
iT jT kT
iT jT kT
iT jT kT
= + +
= + +
= + +
t
t
t ea atuando sobre uma superf. com normal // a z
E, T% :( ) ( ) ( )
=i j k
i j k + + T t t t%
A,( )n
= = t T n n T% % . L
:
=
xx xy xz
yx yy yz
zx zy zz
T T T
T T T p
T T T
= +
T I % % %
A, :
( ) ( ) ( )m m mt t A t
Dd d dA
Dt
= + v g n%
E ( )2p = + + T I v I% % %% , :
( ) ( ) ( ) ( )
( )( )
22
3m m m m m
t t A t A t A t
Dd d p dA dA dA
Dt
= + + v g I n n v I n% %%
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C ,
,
:
( )1
2 22
ji kij ij
j i k
uu up p
x x x
= + + = + + +
T I v I% % %%
A :
( )1 ( )
3 3xx yy zz
Tzp T T T= + + =
T%
( )
2
3
p p
= +
v, ,
. ,
20
3
+ =
. , (K , 1953)
,
. E
, (1954),
, .
A ( )
v , ,
. E ,
, , ( ) 0ou = =v
.
, : GH, M, ,
, J F E 117, . 1, . 35, 1995
