# Internal Flow Presentation

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!"#\$%"&' )'*+, )-./0 123456/37 !!

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General Concept of Flows in Pipe

!  As a uniform flow enters a pipe, the velocity at the pipe walls must decrease to zero (no-slip boundary condition). Continuity indicates that the velocity at the center must increase.

!  Thus, the velocity profile is changing continuously from the pipe entrance until it reaches a fully developed condition. This distance, L, is called the

entrance length.

Dr. M. Khosravy

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!  For fully developed flows ( x>>L), flows become parallel, , the mean pressure remains constant over the pipe cross-section

v = (u(y),0,0)

General Concept of Flows in Pipe

Dr. M. Khosravy

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General Concept of Flows in Pipe

"  )-897 /6 5 -86: ;/;2 595? =>8@ ;/;2 26A>5632 560 2B/A

>2:/86C  !""#D 5>2 A42 -/@/A >27.-A7 8= E8.605>? -5?2> F897G

#42>2 5>2 A98 A?;27 8= ;/;2 F897H -5@/65> 560 A.>E.-26A

Dr. M. Khosravy

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"   I>/J35- %2?68-07 6.@E2> D=8> F89 /6 5 >8.60 ;/;2

%2 K LMNN H -5@/65>

LMNN O %2 O PNNN H A>567/J865-

%2 Q PNNN H A.>E.-26A

"  "8A2 A45A A4272 R5-.27 5>2 5;;>8B/@5A2G

"  )8> 5 :/R26 5;;-/35J86C %23> 02;2607 .;86

"   S/;2 >8.:46277 "  T/E>5J867

"  U;7A>25@ F.3A.5J867C 0/7A.>E56327 E A42 F89D

General Concept of Flows in Pipe

5

Dr. M. Khosravy

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General Concept of Flows in Pipe

"  +42A42> A42 F89 /7 -5@/65> 8> A.>E.-26A 02;2607 86 A42

%2?68-07 6.@E2>C 942>2 \$% /7 A42 3>877V723J865- @256

R2-83/A? 02W620 E?

"  #>567/J86 =>8@ -5@/65> A8 A.>E.-26A =8> F897 /6 3/>3.-5> ;/;2 8=

0/5@2A2> & 833.> 5A '(XLMNN

U  m   =

1

A u dA

A

!!

Dr. M. Khosravy

U  m

U  m

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General Concept of Flows in Pipe

"  )8> ;/;27 9/A4 R5>/5E-2 0/5@2A2>C % /7 7J-- A42 75@2 0.2 A8 38672>R5J86 8= @577C E.A ) * , ) -

D2

V2

2

1

V1

D1

m m

7

Dr. M. Khosravy

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General Concept of Flows in Pipe

"  +426 ;/;2 F89 /7 A.>E.-26AG #42 R2-83/A? /7 .67A250/-? >5608@

5608@-? 9/A4 J@2DC A42 F89 /7 345>53A2>/Y20 E? A42

@256 5:20D R2-83/A? 02W620 57H

"  Z.2 A8 A.>E.-26A @/B/6:C A42 R2-83/A? ;>8W-2 8= A.>E.-26A ;/;2

F89 /7 @8>2 .6/=8>@ A426 A45A 8= -5@/65> F89G

v ( y) = lim T !"

1 2T

v( y, t )dt  #T

Dr. M. Khosravy

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Dr. M. Khosravy

"  [2632C A42 @256 R2-83/A? :>50/26A 5A A42 95-- =8> A.>E.-26A

F89 /7 -5>:2> A456 -5@/65> F89G

"  #42 95-- 7425> 7A>277C C/7 5 =.63J86 8= A42 R2-83/A? :>50/26AG

#42 :>25A2> A42 3456:2 /6 9/A4 >27;23A A8 ? 5A A42 95--C A42

4/:42> /7 A42 95-- 7425> 7A>277G #42>2=8>2C A42 95-- 7425> 7A>277

560 A42 =>/3J865- -87727 5>2 4/:42> /6 A.>E.-26A F89G

w !

u

General Concept of Flows in Pipe

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Poiseuille Flow

Dr. M. Khosravy



"   I867/02> A42 7A250?C =.--? 02R2-8;20 -5@/65> F89 /6 5 7A>5/:4A

;/;2 8= 3/>3.-5> 3>877 723J86 9/A4 3867A56A 0/5@2A2>C &G

"  #42 388>0/65A2 /7 348726 7.34 A45A !  /7 5-86: A42 ;/;2 560 .  /7 /6

A42 >50/.7 0/>23J86 9/A4 A42 8>/:/6 5A A42 326A2> 8= A42 ;/;2G

!

D b

y

x1 p 2 p

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Dr. M. Khosravy



"  )8> 5 386A>8- R8-.@2 8= 5 3?-/602> 625> A42 ;/;2 326A2>C A42

E5-5632 8= @8@26A.@ /6 /6A2:>5- =8>@ /6  ! V0/>23J86 >2\./>27

A45A A42 ;>277.>2 =8>32C

53J6: 86 A42 =5327 8= A42 3?-/602> E2 2\.5- A8 A42 7425> 7A>277 53J6: 86 A42 3/>3.@=2>26J5- 5>25C 42632

"  !6 5338>05632 9/A4 A42 -59 8= =>/3J86

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"  #42>2=8>2H

9426 /7 3867A56A

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Poiseuille Flow

Dr. M. Khosravy



"  #42 R2-83/A? 0/7A>/E.J86 /7 ;5>5E8-/3 8R2> A42 >50/.7C 560 A42 @5B/@.@

R2-83/A? 86 A42 ;/;2 5B/7 E238@27H

"  #42>2=8>2C

"  #42 R8-.@2 F89 >5A2 /7H

um   = !   1

4 µ

dp

dx  R

2

u

u m

=1!   y

2

R 2

Q =   u( y)2!  ydy"  =   # 14µ "    dp

dx  R2 # y

2( )2!  ydy

Q = #   !

8µ

dp

dx  R4 =

1

2 um!  R

2

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Poiseuille Flow

Dr. M. Khosravy



"  #42 F89 >5A2 /7 ;>8;8>J865- A8 A42 W>7A ;892> 8= A42 ;>277.>2 :>50/26A 560 A8 A42 =8.>A4 ;892> 8= A42 >50/.7 8= A42 ;/;2G

Z2W62 @256 R2-83/A? 57

"  #42>2=8>2C

"  #4/7 78-.J86 833.>7 /6 ;>53J32 57 -86: 57C

U m   =

Q

!  R2

U m   = 1

2 u m   =

1

8 µ  !  dp

dx

"  # \$

%  & '   R

2

Re   =U  m

D

v

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Poiseuille Flow

Dr. M. Khosravy



"  #42 >2-5J86 E2A9226 A42 62:5JR2 ;>277.>2 :>50/26A 560 A42 @256 R2-83/A? 8= A42 F89 /7 >2;>2726A20 /6 26:/622>/6: 5;;-/35J86 E? /6A>80.3/6: 5 >27/7A5632 382]3/26A 8= ;/;2 F89C 5 G

"  #4/7 382]3/26A /7 5 686V0/@267/865- 62:5JR2 ;>277.>2 :>50/26A .7/6: A42 0?65@/3 4250 57 ;>277.>2 735-2 560 A42 ;/;2 0/5@2A2> 57 -26:A4 735-2C /G2GC

"  !6A>80.3/6: A42 5E8R2 2B;>277/86 =8>

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Poiseuille Flow

Dr. M. Khosravy



"  &A A42 95--C

"  ,8C

"   &7 5 >27.-AC A42 95-- =>/3J86 382]3/26A /7H

!   w   = !

µ  du

dy  y= R

= ! µ  dp

dx

2 y

4 µ   y= R

= !

R

2

dp

dx

C  f    = ! w

1

2  " U m

2

=

f

4 =

16

Re

2! w

R = !

dp

dx =

f

D

1

2  " U m

2

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Dr. M. Khosravy



"

)8> F897 /6 ;/;27C A42 A8A5- 262>:? ;2> .6/A 8= @577 /7

:/R26 E? 942>2 A42 38>>23J86 =53A8> /7

02W620 57C

9/A4 E2/6: A42 @577 F89 >5A2 560 & /7 A42 3>877 723J865-

5>25G

p

!  + " U

m

2

2 + gy

!  " #

\$  % &

!   = ! u 2

" udA A"  !mU

m

2

!m =  ! Q

!

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Dr. M. Khosravy



"

,8 A42 A8A5- 4250 -877 E2A9226 723J86 ^ 560 L 8= ;/;27 /7H

"  4-X4250 -877 0.2 A8 =>/3J865- 2_23A7 /6 =.--? 02R2-8;20 F89 /6 3867A56A

5>25 3860./A7

"  4-@X@/68> -87727 0.2 A8 26A>56327C W`6:7C 5>25 3456:27C 2A37G

h lt    =

p 1

!  +"

1

U  m1 2

2 + gy

1

% %

(  (   )

p 2

!  +"

2

U  m2 2

2 + gy

2

% %

(  (

h lt    = h

l   + h

lm

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Dr. M. Khosravy



"

,8C =8> 5 =.--? 02R2-8;20 F89 A4>8.:4 5 3867A56AV5>25 ;/;2C

"  &60 /= . *1. -C

hl   =  p

1  !  p

2

!

+ g( y1 ! y2 )

hl   =  p 1  ! p2

!  = " p

!

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Dr. M. Khosravy



"  )8> -5@/65> F89C

"   [2632

! dp

dx =

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