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2016. 09. 26부산대학교 화공생명공학부(PNU)
현 규 (Kyu Hyun)
ž How to obtain the velocity profiles for laminar flow of fluids in simple flow systems.
ž Apply only to steady flow = the pressure, density, and velocity components do not change with time
ž Only with “Laminar flow”
§ 2.1 Shell momentum balances and boundary conditions§ 2.2 Flow of a falling film§ 2.3 Flow through a circular tube§ 2.4 Flow through an annulus§ 2.5 Flow of two adjacent immiscible fluids§ 2.6 Creeping flow around a sphere
ž The combined (convective + molecular) momentum flux
τδvvπvvΦ ++=+= prr
úúú
û
ù
êêê
ë
é
++
++
úúú
û
ù
êêê
ë
é
zzzyzx
yzyyyx
xzxyxx
zzyzxz
zyyyxy
zxyxxx
pp
p
vvvvvvvvvvvvvvvvvv
ttttttttt
rrrrrrrrr
steady flow
ž Procedure for solving viscous flow problems
Step 1. Identify velocity component and spatial variables
Step 2. Write shell momentum balance
Step 3. Obtain differential equation for momentum flux
Step 4. Get momentum flux distribution
Step 5. Obtain differential equation for velocity
Step 6. Get velocity distribution
Step 7. Get additional quantities
integration
integration
Maximum velocity, average velocity
ž The most commonly used boundary conditions
• Solid-Fluid Interface
• liquid-liquid Interface
• liquid-gas Interface
Assumptions and conditions
• No end effect
• Velocity
• Pressure , Gravity Force
• Newton’s law of viscosity
BLW >>&
0=== wvyuu ),(
)(xpp =
Lower Plate
Upper Plate
x
y
B-
B+W
L
Lower Plate
Upper PlateW
L
x
y
xDyD
• In volume element, there is no velocity change in z-direction but with length W. => 2-D problems
• We consider momentum balance in volume element.
유체역학및 열전달 9
x
y
xuuuu xxxxxx ¶¶
-=+=F mrtr 22
÷÷ø
öççè
涶
+¶¶
-=+=Fxv
yuvuuu yxxyyx mrtr
gΦgτVV
gτVVV
rrr
rrr
+Ñ-×-Ñ=+Ñ-+×-Ñ=
+Ñ-×Ñ-×-Ñ=¶
¶
pp
pt
)(
][)()(τVVΦ += r
Total stress consist of Convective momentum and Molecular stress
Rate of x-momentum inacross surface at x
xxxyW FD )(
Rate of x-momentum out across surface at x+Δx
xxxxyWD+
FD )(
Rate of x-momentum inacross surface at y
yyxxW FD )(
Rate of x-momentum out across surface at y+Δy
yyyxxWD+
FD )(
Pressure in across surface at x
xpyW )( D
Pressure out across surface at x+Δx xx
pyWD+
D )(
[ ] [ ] [ ] 0=D-D+FD-FD+FD-FDD+D+D+ xxxyyyxyyxxxxxxxx pyWpyWxWxWyWyW )()()()()()(
• There is no accumulation (steady state)• In red volume element, the force is considered as follow; (only x-direction velocity thus only x-direction force)
0=D
-+
D
F-F+
D
F-FD+D+D+
xpp
yxxxxyyyxyyxxxxxxxx
0=¶¶
+¶
F¶+
¶F¶
xp
yxyxxxτVVΦ += r
xuuuu xxxxxx ¶¶
-=+=F mrtr 22
÷÷ø
öççè
涶
+¶¶
-=+=Fxv
yuvuuu yxxyyx mrtr
02
=¶¶
+÷÷ø
öççè
涶
-¶¶
+¶¶
xp
yu
yxu mr
xp
yu
¶¶
=¶¶
\ 2
2
m
xp
yu
¶¶
=¶¶
\ 2
2
m
02
2
Cconstxp
yu
==¶¶
=¶¶
\ .m
LppLxppxat
====
,, 00
0
00
==
=¶¶
=
uByxuyat
,
,
0Cxp=
¶¶
10 CxCp +=0
0 pxLppp L +
-=
02
2
Cyu=
¶¶m 2
0 CyCyu
+=¶¶
m 320
2Cy
Lppu L +
-=
m
úúû
ù
êêë
é÷øö
çèæ-
-=
220 12 B
yLBppu L
m)(
úúû
ù
êêë
é÷øö
çèæ-=
úúû
ù
êêë
é÷øö
çèæ-
D=
úúû
ù
êêë
é÷øö
çèæ-
-=
222220 11
21
2 Byu
By
LpB
By
LBppu L
max
)(mm
Average velocity
ò= udSS
V 1
max
max
max
u
yyBBu
WdyBy
BWu
uWdyBW
V
B
B
B
B
B
B
32
31
2
12
21
322
2
=
úûù
êëé -=
úúû
ù
êêë
é÷øö
çèæ-=
=
+
--
-
ò
ò
ò
Lower Plate
Upper PlateW
WdydS =
dy
BWS 2=
32
=maxuV 주황색 면으로 흘러
들어가는 유체에 대해서
B-
B+
Assumptions and conditions
• No end effect
• Velocity
• Pressure
• Newton’s law of viscosity
d>>LW &
0),( === yxzz vvxvv
)(xpp =
)/( dxdvzzxxz mtt -==
z
x
yz x
0
Cartesian coordinate
Assumptions:
When the fluid has constant density
z
x
Assumptions and conditions
• No end effect
• Velocity
• Pressure
• Newton’s law of viscosity
RL >>
0),( === qvvrvv rzz
)(zpp =
)/( drdvzzrrz mtt -==
rz
zzzzzzz rrrrD+
FD-FD )()( pp 22
rrrzrrz zrrzrD+
FDD+-FD ))(()( pp 22
zzzprrprr
D+D-D )()( pp 22
Area= zrDp2
Area= rrDp2
shear stress on r-plane
gΦgτVV
gτVVV
rrr
rrr
+Ñ-×-Ñ=+Ñ-+×-Ñ=
+Ñ-×Ñ-×-Ñ=¶
¶
pp
pt
)(
][)()( τVVΦ += rConvective momentum과Molecular stress항을 묶어서Total stress라고 가정하자.
Rate of z-momentum in across annular surface at z zzzrr FD )( p2
Rate of z-momentum out across annular surface at z+Δz
Rate of z-momentum in across cylindrical surface at r
Rate of z-momentum in across cylindrical surface at r+Δr
Pressure across surface at z zprr )( Dp2
Pressure across surface at z+Δz
rz
zD
rD
zzzzrrD+
FD )( p2
rrzzr FD )( p2
rrrzzrrD+
FDD+ ))(( p2
zzp)rrπ(
Δ+Δ2
[ ][ ] [ ] 02222
22
=D-D+FDD+-FD+
FD-FD
D+D+
D+
zzzrrrzrrz
zzzzzzz
prrprrzrrzr
rrrr
)()())(()(
)()(
pppp
pp
( ) ( )0=
D
-+
D
F-F+
D
F-FD+D+D+
zpp
rrrr
zr zzzrzrzrrzzzzzzzz
( )zpr
xrr
rzz
rz ¶¶
-¶F¶
-=F¶¶
τVVΦ += r
zuuuu zzzzzz ¶¶
-=+=F mrtr 22
÷øö
çèæ
¶¶
+¶¶
-=+=Fru
zuuuuu r
rrzzrrz mrtr
zpr
rur
r ¶¶
=÷øö
çèæ
¶¶
¶¶
\m
zrDDp2 로 나누어 주면
zpr
zur
rur
r ¶¶
-¶
¶-=÷
øö
çèæ
¶¶
-¶¶ )( 2rm
)(ruu =Q
유체역학및열전달 Seminar 28 (Kyu Hyun, PNU)
Assumptions:
When the fluid has constant density
úúû
ù
êêë
é÷øö
çèæ-=
úúû
ù
êêë
é÷øö
çèæ-
D=
úúû
ù
êêë
é÷øö
çèæ-
-=
222220 11
41
4 Rru
Rr
LpR
Rr
LRppu L
max
)(mm
Average velocity
ò= udSS
V 1
( )
242
2
21
21
4
4
0
324
0
2
2
2
maxmax
max
max
uRRu
drrrRRu
rdrRr
Ru
rdruR
V
R
R
=×=
-=
úúû
ù
êêë
é÷øö
çèæ-=
=
ò
ò
ò
pp
pp
21
=maxuV
2RS p=
rdrdS p2=
유체역학및열전달 Seminar 33 (Kyu Hyun, PNU)
ž Assumptions for Hagen –Poiseuille equation
Assumptions and conditions
• No end effect
• Velocity
• Pressure
• Newton’s law of viscosity
RL >>
0),( === qvvrvv rzz
)(zpp =
)/( drdvzzrrz mtt -==
Assumptions and conditions are same like normal circular tube
Assumptions and conditions• No end effect
• Velocity
• Pressure
• Newton’s law of viscosity
bLW >>&0),( === yxzz vvxvv
Lppp L /)( 0 -=
)/( dxdvzzxxz mtt -==Assumptions and conditions are same like falling film
0,0 ¹¹ qvvr
“Creeping flow” = very slow flow=“stoke flow”
1.0Re <= ¥
mrDv
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