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