ÿþJ . S z a n t y r W y k Ba d n r 1 4 Z a m k n i t y u k Ba d r ó w n a D
m e c h a n i k i p By n ó w
P r z e d s t a w i o n e p o w y |e j r ó w n a n i a t w o r z z a m k n i t y u k Ba d r ó w n a D
m e c h a n i k i p By n ó w , k t ó r y m o |e b y z a s t o s o w a n y d o o p i s u k o n k r e t n y c h
p r z e p By w ó w i u z y s k a n i a , w d r o d z e r o z w i z a n i a t e g o u k Ba d u , i n f o r m a c j i
o w a r t o [c i a c h i n t e r e s u j c y c h n a s p a r a m e t r ó w t e g o p r z e p By w u .
K o n k r e t n a p o s t a u k Ba d u r ó w n a D z a l e |y o d p r z y j t e g o m o d e l u p By n u .
P r z y p a d e k 1 : p By n n i e [c i [l i w y o s t a Be j l e p k o [c i
Z a m k n i t y u k Ba d r ó w n a D t w o r z :
- r ó w n a n i e z a c h o w a n i a m a s y
d i v u = 0
D u
- r ó w n a n i e z a c h o w a n i a p d u
Á = Áf - g r a d p + µ "u
D t
R a z e m s t o c z t e r y r ó w n a n i a s k a l a r n e z c z t e r e m a n i e w i a d o m y m i :
- c i [n i e n i e p
- s k Ba d o w e p r d k o [c i u x , u y , u z
W t y m p r z y p a d k u p o l e t e m p e r a t u r y n i e w p By w a n a p r z e p By w , a l e
s a m o j e s t u z a l e |n i o n e o d p o l a p r d k o [c i p r z e p By w u p o p r z e z r ó w n a n i e
b i l a n s u e n t r o p i i w p o s t a c i :
ëø öø
"T "T "T "T
z
ìø
Ác ìø + u x + u y + u z ÷ø = T s M + »"T
÷ø
"t "x "y "z
íø øø
T p o s t a r ó w n a n i a m o |n a u z y s k a p o d s t a w i a j c d o o r y g i n a l n e g o
r ó w n a n i a z a l e |n o [ d l a e n e r g i i w e w n t r z n e j :
e = c T + e 0
W p r z y p a d k u g d y l e p k o [ p By n u z a l e |y o d t e m p e r a t u r y , r ó w n a n i e
b i l a n s u e n t r o p i i j e s t s p r z |o n e z r ó w n a n i a m i z a c h o w a n i a m a s y i
z a c h o w a n i a p d u p o p r z e z z a l e |n o [:
µ = µ ( T )
M a m y w t e d y u k Ba d s z e [c i u r ó w n a D z s z e [c i o m a n i e w i a d o m y m i :
- c i [n i e n i e p - s k Ba d o w e p r d k o [c i
u x , u y , u z
- t e m p e r a t u r a T - w s p ó Bc z y n n i k l e p k o [c i µ
P r z y p a d e k 2 : p By n [c i [l i w y
W t y m p r z y p a d k u z a m k n i t y u k Ba d r ó w n a D t w o r z :
"Á
+ d i v ( Áu ) = 0
- r ó w n a n i e z a c h o w a n i a m a s y
"t
D u 2
öø
- r ó w n a n i e z a c h o w a n i a p d u Á = Áf - g r a d p - g r a d ëø µ d i v u + d i v ( 2 µ [ D ] )
ìø ÷ø
D t 3
íø øø
D e p D p
z
z
- r ó w n a n i e b i l a n s u e n t r o p i i Á = T s M + + »"T
- r ó w n a n i e b i l a n s u e n t r o p i i Á = T s M + + »"T
D t Á D t
Á
T
e =
- r ó w n a n i e e n e r g i i w e w n t r z n e j
v
+"c ( T ) d T
T 0
p
Z f u n k c j a [c i [l i w o [c i
- r ó w n a n i e s t a n u
= Z ( p , T ) R T
R s t a Ba g a z o w a
Á
c V = c V ( T )
µ = µ ( T )
- d o d a t k o w e z a l e |n o [c i
W t y m p r z y p a d k u m a m y u k Ba d d z i e w i c i u r ó w n a D z d z i e w i c i o m a
n i e w i a d o m y m i :
- g s t o [ Á
- c i [n i e n i e p - e n e r g i a w e w n t r z n a e
- t e m p e r a t u r a T - w s p ó Bc z y n n i k l e p k o [c i µ
c V
- c i e p Bo w Ba [c i w e
u x , u y , u z
- s k Ba d o w e p r d k o [c i
Z a Bo |o n o , |e w s p ó Bc z y n n i k p r z e w o d n i c t w a c i e p l n e g o » m a w a r t o [
s t a B.
s t a B.
W a r u n k i b r z e g o w e i p o c z t k o w e
D l a u m o |l i w i e n i a r o z w i z a n i a p o w y |s z y c h u k Ba d ó w r ó w n a D
k o n i e c z n e j e s t o k r e [l e n i e o d p o w i e d n i c h w a r u n k ó w b r z e g o w y c h
o r a z ( d l a p r z e p By w ó w n i e s t a c j o n a r n y c h ) w a r u n k ó w p o c z t k o w y c h .
W a r u n k i t e s p o t r z e b n e d o w y z n a c z e n i a d o w o l n y c h s t a By c h i
d o w o l n y c h f u n k c j i w p r o w a d z o n y c h p o d c z a s c a Bk o w a n i a r ó w n a D.
W a r u n k i b r z e g o w e
W a r u n k i b r z e g o w e n a p o w i e r z c h n i c i a Ba s t a Be g o n i e p r z e n i k l i w e g o
- p By n n i e l e p k i p r d k o [ n o r m a l n a r ó w n a z e r o u n = 0
- p By n l e p k i p r d k o [ r ó w n a z e r o
u = 0
- d a n a t e m p e r a t u r a T l u b s t r u m i e D c i e p Ba j
W a r u n k i b r z e g o w e n a p o w i e r z c h n i c i a Ba s t a Be g o p o r o w a t e g o
- p r d k o [ s t y c z n a r ó w n a z e r o u = 0
- p r d k o [ s t y c z n a r ó w n a z e r o u t = 0
- p r d k o [ n o r m a l n a z a d a n a
u n = f ( x , y , z )
- d a n a t e m p e r a t u r a T l u b s t r u m i e D c i e p Ba j
W a r u n k i b r z e g o w e n a p o w i e r z c h n i r o z d z i a Bu d w ó c h p By n ó w
2
u 1 = u n
- p By n n i e l e p k i
n
- p By n l e p k i
u 1 = u 2
J e |e l i r ó w n a n i e p o w i e r z c h n i r o z d z i a Bu p By n ó w m a p o s t a :
F ( x , y , z , t ) = 0
t o k i n e m a t y c z n y w a r u n e k b r z e g o w y m a p o s t a :
"F "F "F "F
+ u x + u y + u z = 0
"t "x "y "z
W a r u n k i w n i e s k o Dc z o n o [c i z a d a j e s i w p r z y p a d k u o p By w u o b i e k t u
s t r u g , w k t ó r e j p o l e p r d k o [c i z d a l a o d o b i e k t u j e s t j e d n o r o d n e .
s t r u g , w k t ó r e j p o l e p r d k o [c i z d a l a o d o b i e k t u j e s t j e d n o r o d n e .
p = p ( ")
T = T ( ")
u = u ( ")
W a r u n k i p o c z t k o w e
W a r u n k i p o c z t k o w e d o t y c z z j a w i s k n i e s t a c j o n a r n y c h i p o w i n n y
b y p o d a n e d l a w s z y s t k i c h p u n k t ó w p r z e s t r z e n i w y p e Bn i o n e j p By n e m
d l a c h w i l i
t = t 0
D l a p r z e p By w u p By n u n i e [c i [l i w e g o :
p = p ( x , y , z , t 0 )
u = u ( x , y , z , t 0 )
D l a p r z e p By w u p By n u n i e [c i [l i w e g o o l e p k o [c i z a l e |n e j o d t e m p e r a t u r y
d o d a t k o w o :
µ = µ ( x , y , z , t 0 )
T = T ( x , y , z , t 0 )
D l a p r z e p By w u p By n u [c i [l i w e g o d o d a t k o w o :
D l a p r z e p By w u p By n u [c i [l i w e g o d o d a t k o w o :
c V = c V ( x , y , z , t 0 )
Á = Á( x , y , z , t 0 )
e = e ( x , y , z , t 0 )
W a r u n k i p o c z t k o w e p o w i n n y b y n i e s p r z e c z n e z w a r u n k a m i
b r z e g o w y m i .
S c h e m a t w a r u n k ó w b r z e g o w y c h d l a p r z e p By w u z e w n t r z n e g o
S c h e m a t w a r u n k ó w b r z e g o w y c h d l a p r z e p By w u w e w n t r z n e g o