formulas navier stokes
2
Transcript of formulas navier stokes
8/7/2019 formulas navier stokes
http://slidepdf.com/reader/full/formulas-navier-stokes 1/2
T h e E q u a t i o n o f C o n t i n u i t y a n d t h e E q u a t i o n o f M o t i o n i n
C a r t e s i a n , c y l i n d r i c a l , a n d s p h e r i c a l c o o r d i n a t e s
C M 3 1 0 F a l l 1 9 9 8 F a i t h A . M o r r i s o n
C o n t i n u i t y E q u a t i o n , C a r t e s i a n c o o r d i n a t e s
@
@ t
+
v
x
@
@ x
+ v
y
@
@ y
+ v
z
@
@ z
+
@ v
x
@ x
+
@ v
y
@ y
+
@ v
z
@ z
= 0
C o n t i n u i t y E q u a t i o n , c y l i n d r i c a l c o o r d i n a t e s
@
@ t
+
1
r
@ ( r v
r
)
@ r
+
1
r
@ ( v
)
@
+
@ ( v
z
)
@ z
= 0
C o n t i n u i t y E q u a t i o n , s p h e r i c a l c o o r d i n a t e s
@
@ t
+
1
r
2
@ ( r
2
v
r
)
@ r
+
1
r s i n
@ ( v
s i n )
@
+
1
r s i n
@ ( v
)
@
= 0
E q u a t i o n o f M o t i o n f o r a n i n c o m p r e s s i b l e u i d , 3 c o m p o n e n t s i n C a r t e s i a n c o o r d i n a t e s
@ v
x
@ t
+ v
x
@ v
x
@ x
+ v
y
@ v
x
@ y
+ v
z
@ v
x
@ z
= ;
@ P
@ x
;
@
x x
@ x
+
@
y x
@ y
+
@
z x
@ z
+ g
x
@ v
y
@ t
+ v
x
@ v
y
@ x
+ v
y
@ v
y
@ y
+ v
z
@ v
y
@ z
= ;
@ P
@ y
;
@
x y
@ x
+
@
y y
@ y
+
@
z y
@ z
+ g
y
@ v
z
@ t
+ v
x
@ v
z
@ x
+ v
y
@ v
z
@ y
+ v
z
@ v
z
@ z
= ;
@ P
@ z
;
@
x z
@ x
+
@
y z
@ y
+
@
z z
@ z
+ g
z
E q u a t i o n o f M o t i o n f o r a n i n c o m p r e s s i b l e u i d , 3 c o m p o n e n t s i n c y l i n d r i c a l c o o r d i n a t e s
@ v
r
@ t
+ v
r
@ v
r
@ r
+
v
r
@ v
r
@
;
v
2
r
+ v
z
@ v
r
@ z
= ;
@ P
@ r
;
1
r
@ ( r
r r
)
@ r
+
1
r
@
r
@
;
r
+
@
r z
@ z
+ g
r
@ v
@ t
+ v
r
@ v
@ r
+
v
r
@ v
@
+
v
v
r
r
+ v
z
@ v
@ z
= ;
1
r
@ P
@
;
1
r
2
@ ( r
2
r
)
@ r
+
1
r
@
@
+
@
z
@ z
+ g
@ v
z
@ t
+ v
r
@ v
z
@ r
+
v
r
@ v
z
@
+ v
z
@ v
z
@ z
= ;
@ P
@ z
;
1
r
@ ( r
r z
)
@ r
+
1
r
@
z
@
+
@
z z
@ z
+ g
z
E q u a t i o n o f M o t i o n f o r a n i n c o m p r e s s i b l e u i d , 3 c o m p o n e n t s i n s p h e r i c a l c o o r d i n a t e s
@ v
r
@ t
+ v
r
@ v
r
@ r
+
v
r
@ v
r
@
+
v
r s i n
@ v
r
@
;
v
2
+ v
2
r
= ;
@ P
@ r
;
1
r
2
@ ( r
2
r r
)
@ r
+
1
r s i n
@ (
r
s i n )
@
+
1
r s i n
@
r
@
;
+
r
+ g
r
@ v
@ t
+ v
r
@ v
@ r
+
v
r
@ v
@
+
v
r s i n
@ v
@
+
v
r
v
r
;
v
2
c o t
r
= ;
1
r
@ P
@
;
1
r
2
@ ( r
2
r
)
@ r
+
1
r s i n
@ (
s i n )
@
+
1
r s i n
@
@
+
r
r
;
( c o t )
r
+ g
@ v
@ t
+ v
r
@ v
@ r
+
v
r
@ v
@
+
v
r s i n
@ v
@
+
v
r
v
r
+
v
v
c o t
r
= ;
1
r s i n
@ P
@
;
1
r
2
@ ( r
2
r
)
@ r
+
1
r
@
@
+
1
r s i n
@
@
+
r
r
+
( 2 c o t )
r
+ g
1
8/7/2019 formulas navier stokes
http://slidepdf.com/reader/full/formulas-navier-stokes 2/2
E q u a t i o n o f M o t i o n f o r i n c o m p r e s s i b l e , N e w t o n i a n u i d ( N a v i e r - S t o k e s e q u a t i o n ) 3 c o m p o n e n t s i n
C a r t e s i a n c o o r d i n a t e s
@ v
x
@ t
+ v
x
@ v
x
@ x
+ v
y
@ v
x
@ y
+ v
z
@ v
x
@ z
= ;
@ P
@ x
+
@
2
v
x
@ x
2
+
@
2
v
x
@ y
2
+
@
2
v
x
@ z
2
+ g
x
@ v
y
@ t
+ v
x
@ v
y
@ x
+ v
y
@ v
y
@ y
+ v
z
@ v
y
@ z
= ;
@ P
@ y
+
@
2
v
y
@ x
2
+
@
2
v
y
@ y
2
+
@
2
v
y
@ z
2
+ g
y
@ v
z
@ t
+ v
x
@ v
z
@ x
+ v
y
@ v
z
@ y
+ v
z
@ v
z
@ z
= ;
@ P
@ z
+
@
2
v
z
@ x
2
+
@
2
v
z
@ y
2
+
@
2
v
z
@ z
2
+ g
z
E q u a t i o n o f M o t i o n f o r i n c o m p r e s s i b l e , N e w t o n i a n u i d ( N a v i e r - S t o k e s e q u a t i o n ) , 3 c o m p o n e n t s i n
c y l i n d r i c a l c o o r d i n a t e s
@ v
r
@ t
+ v
r
@ v
r
@ r
+
v
r
@ v
r
@
;
v
2
r
+ v
z
@ v
r
@ z
= ;
@ P
@ r
+
@
@ r
1
r
@ ( r v
r
)
@ r
+
1
r
2
@
2
v
r
@
2
;
2
r
2
@ v
@
+
@
2
v
r
@ z
2
+ g
r
@ v
@ t
+ v
r
@ v
@ r
+
v
r
@ v
@
+
v
r
v
r
+ v
z
@ v
@ z
= ;
1
r
@ P
@
+
@
@ r
1
r
@ ( r v
)
@ r
+
1
r
2
@
2
v
@
2
+
2
r
2
@ v
r
@
+
@
2
v
@ z
2
+ g
@ v
z
@ t
+ v
r
@ v
z
@ r
+
v
r
@ v
z
@
+ v
z
@ v
z
@ z
= ;
@ P
@ z
+
1
r
@
@ r
r
@ v
z
@ r
+
1
r
2
@
2
v
z
@
2
+
@
2
v
z
@ z
2
+ g
z
E q u a t i o n o f M o t i o n f o r i n c o m p r e s s i b l e , N e w t o n i a n u i d ( N a v i e r - S t o k e s e q u a t i o n ) , 3 c o m p o n e n t s i n
s p h e r i c a l c o o r d i n a t e s
@ v
r
@ t
+ v
r
@ v
r
@ r
+
v
r
@ v
r
@
+
v
r s i n
@ v
r
@
;
v
2
+ v
2
r
= ;
@ P
@ r
+
r
2
v
r
;
2 v
r
r
2
;
2
r
2
v
@
;
2 v
c o t
r
2
;
2
r
2
s i n
@ v
@
+ g
r
@ v
@ t
+ v
r
@ v
@ r
+
v
r
@ v
@
+
v
r s i n
@ v
@
+
v
r
v
r
;
v
2
c o t
r
= ;
1
r
@ P
@
+
r
2
v
+
2
r
2
@ v
r
@
;
v
r
2
s i n
2
;
2 c o s
r
2
s i n
2
@ v
@
+ g
@ v
@ t
+ v
r
@ v
@ r
+
v
r
@ v
@
+
v
r s i n
@ v
@
+
v
r
v
r
+
v
v
c o t
r
= ;
1
r s i n
@ P
@
+
r
2
v
;
v
r
2
s i n
+
2
r
2
s i n
@ v
r
@
+
2 c o s
r
2
s i n
2
@ v
@
+ g
w h e r e , i n t h e s e e q u a t i o n s , r
1
r
2
@
@ r
;
r
2
@
@ r
+
1
r
2
s i n
@
@
;
s i n
@
@
+
1
r
2
s i n
2
@
2
@
2
.
R e f e r e n c e : R . B . B i r d , W . E . S t e w a r t , a n d E . N . L i g h t f o o t , T r a n s p o r t P h e n o m e n a , W i l e y : N Y , 1 9 6 0 .
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