Mecánica de Fluidos

5/19/2018 Mecnica de Fluidos

1/95

058:0160 Chapters 3 & 4

Professor Fred Stern Fall 2006 1

Chapters 3 & 4: Integral Relations for a Control ol!"e

and #ifferential Relations for Fl!id Flo$


2/95

058:0160 Chapters 3 & 4



3/95

058:0160 Chapters 3 & 4


Re%nolds ransport heore" 'R(

)eed relationship *et$een ( )sysBdtd

and +hanges in

==CVCV

ddmcvB ,

1 - ti"e rate of +hange of . in C - =CV

ddt

d

dt

cvdB

2 - net o!tfl!/ of . fro" C a+ross CS -

R

CS

V n dA

dAnVddt

d

dt

dBR

CSCV

SYS +=

eneral for" R for "oing defor"ing +ontrol ol!"e

Spe+ial Cases:


4/95

058:0160 Chapters 3 & 4


1( )ondefor"ing C "oing at +onstant elo+it%

( ) dAnVdtdt

dBR

CSCV

SYS +=

2( Fi/ed C

( ) dAnVdtdt

dB

CSCV

SYS +=

reens heore":CV CS

b d b n dA =

( ) ( )

+

= dVtdt

dB

CV

SYS

Sin+e C fi/ed and ar*itrar% 0li"d gies differential e,

3( Stead% Flo$: 0=

t

4( nifor" flo$ a+ross dis+rete CS 'stead% or !nstead%(

=CSCS

dAnVdAnV (- inlet, + outlet)


5/95

058:0160 Chapters 3 & 4


Contin!it% !ation:

. - - "ass of s%ste"

7 - 1

0=dt

dM*% definition s%ste" - fi/ed a"o!nt of "ass

Integral For":

dAnVddt

d

dt

dM

CS

R

CV +== 0

dAnVddt

d

CS

R

CV

=

Rate of decrease of mass in CV net rate of mass outflo! across CS

)ote si"plifi+ations for nondefor"ing C fi/ed Cstead% flo$ and !nifor" flo$ a+ross dis+rete CS

In+o"pressi*le Fl!id: 9 - +onstant

dAnVddt

d

CS

R

CV

=

+onseration of ol!"e;


6/95

058:0160 Chapters 3 & 4


Spe+ial +ase C for" +ontin!it% e!ation:

Fi/ed C

0CV CS

d V n dAt

+ = and !nifor" flo$ oer dis+rete inlet


7/95

058:0160 Chapters 3 & 4

Professor Fred Stern Fall 2006 =

0

1=

+

+

+

unit$erc%an&eofrate

'

!

y

v

(

u

V

unit$er

c%an&eofrate

#t

#

Called the +ontin!it% e!ation sin+e the i"pli+ation is that

9 and Vare +ontin!o!s f!n+tions of /,

In+o"pressi*le Fl!id: 9 - +onstant

0

0

=

+

+

=

'

!

y

v

(

u

V

P3,15 >ater ass!"ed in+o"pressi*le flo$s steadil%thro!gh the ro!nd pipe in Fig, P3,15, he entran+e

elo+it% is +onstant 0u )= and the e/it elo+it%

appro/i"ates t!r*!lent flo$ ( )1 =

"a/ 1u u r R= , #eter"ine

the ratio "a/) u for this flo$,


8/95

058:0160 Chapters 3 & 4


Stead% flo$ nondefor"ing or "oing C one inlet

!nifor" flo$ and one o!tlet non!nifor" flo$( )

1 =2

0 "a/0

0 1 2R

) R u r R rdr = + 2 2

0 "a/

4?0

60) R u R

= +

0

"a/

4?

60

)

u=

( ) ( )15 =

"a/ "a/0

2 2

1 12 2 1 1

1 12 1

= =

R

u rdr u r R r R

R R

=

+ +

2

"a/

= =2 0

15 8u R

= 2

"a/4?260

u R=


9/95

058:0160 Chapters 3 & 4

Professor Fred Stern Fall 2006 ?

P3,12 he pipe flo$ in Fig, P3,12 fills a +%lindri+al tan@

as sho$n, At ti"e t-0 the $ater depth in the tan@ is

30+", sti"ate the ti"e re!ired to fill the re"ainder of

the tan@,

nstead% flo$ defor"ing C one inlet one o!tlet

!nifor" flo$

1 20CV

dd " "

dt = +

2 2

1 20 4 4CV

d d d

d V Vdt

= +( ) ( )

2

4

#t % t

=


10/95

058:0160 Chapters 3 & 4


( )2 2

2 104 4

# d% dV V

dt

= +

( )

2

2 1 0,0153d% d V Vdt # = =

0,=46

0,0153 0,0153

d%dt s= = =

Stead% flo$ one inlet and t$o e/its $ith !nifor" flo$

1 2 30 " " "= + +

( )2

3 1 2 1 24

d" " " V V

= =

( )

2

2

1 2

4

4

#d%

dtd"

V V

= =

( )

2

1 2

#d%d

V V

=

3

A

*" V n dA

s=

ol!"e of fl!id per !nit ti"e thro!gh A


11/95

058:0160 Chapters 3 & 4


P4,1= A reasona*le appro/i"ation for the t$o

di"ensional in+o"pressi*le la"inar *o!ndar% la%er on

the flat s!rfa+e in Fig,P4,1= is

2

2

2y yu )

= for

y

$here 1 2C(= C const='a( Ass!"ing a noslip +ondition at the $all find an

e/pression for the elo+it% +o"ponent ( )v ( y for y ,'*( he find the "a/i"!" al!e of v at the station 1( m= for the parti+!lar +ase of flo$ $hen 3) m s= and

1,1cm= ,

0u v( y

+ =

( )2 2 32 2v u

) y yy ( (

= = +

( )2 2 30

2y

(v ) y y dy =


12/95

058:0160 Chapters 3 & 4


'a(2 3

2 32

2 3(

y yv )

=

1 2

C(= 1 22 2

(

C(

(

= =

'*( Sin+e 0yv = at y =( )"a/

2 1 1

2 2 3

)v v y

(

= = =

3 1,10,0055

6 6

)m s

(

= = =

o"ent!" !ation:

. - V- "o"ent!" 7 - V

Integral For":

{

' (

31 2

R

CV CS

d MV d V d V V n dA +

dt dt

= + = 1 4 2 4 3 1 44 2 4 43

+ - e+tor s!" of all for+es a+ting on C

- F.B Fs

F.- .od% for+es $hi+h a+t on entire C of fl!id d!e

to e/ternal for+e field s!+h as grait% or

ele+trostati+ or "agneti+ for+es, For+e per !nit

ol!"e,

Fs- S!rfa+e for+es $hi+h a+t on entire CS d!e to

nor"al 'press!re and is+o!s stress( and


13/95

058:0160 Chapters 3 & 4


tangential 'is+o!s stresses( stresses, For+e per

!nit area,

>hen CS +!ts thro!gh solids Fs"a% also in+l!de FR-

rea+tion for+es e,g, rea+tion for+e re!ired to hold nole

or *end $hen CS +!ts thro!gh *olts holding nolehere ( ) V

V Vt t t

= +

andDD D

V V VV ui V v,V !- V = = + + is a tensor' ( ' ( ' ( ' ( ' (V V VV uV vV !V

( y '

= = + +

VVVV += ('

- 0 +ontin!it%


14/95

058:0160 Chapters 3 & 4


( )CV

VV V V V d +

t t

+ + + =

Sin+eV #V

V Vt #t

+ =

= +d#t

V#

CV

= f#t

V# $er elemental fluid volume

sbffa +=

bf - *od% for+e per !nit ol!"e

sf - s!rfa+e for+e per !nit ol!"e

.od% for+es are d!e to e/ternal fields s!+h as grait% or

"agneti+ fields, Eere $e onl% +onsider a graitational

field that isd(dyd'&+d+

&ravbody = =

and D& &-= for

i,e, Dbody

f &-=

S!rfa+e For+es are d!e to the stresses that a+t on the sides

of the +ontrol s!rfa+es

i,i,i, $ +=

++

+=

'''y'(

y'yyy(

('(y((

$

$

$

is+o!s stre)or"al press!re

g

S%""etri+ i, ,i =

2ndorder tensor


15/95

058:0160 Chapters 3 & 4


As sho$n *efore for p alone it is not the stresses

the"seles that +a!se a net for+e *!t their gradients,

S%""etr% +ondition fro" re!ire"ent that for ele"ental

fl!id ol!"e stresses the"seles +a!se no rotation,

fff

$s+=

Re+all $f$ = *ased on 1storder S, f is "ore +o"ple/

sin+e i, is a 2ndorder tensor *!t si"ilarl% as for p the

for+e is d!e to stress gradients and are deried *ased on

1storder S,

GGG

GGG

GGG

-,i

-,i

-,i

'''y'('

y'yyy(y

('(y(((

++=

++=

++=

' ( ' ( ' (s ( y '+ d(dyd'( y '

= + +

Res!ltant stress

on ea+h fa+e


16/95

058:0160 Chapters 3 & 4


' ( ' ( ' (( y 'sf

( y '

= + +

and si"ilarl% for

G' ( ' ( ' (

G' ( ' ( ' (

G' ( ' ( ' (

(( y( '(s

(y yy 'y

(' y' ''

f i( y '

,( y '

-( y '

= + +

+ + +

+ + +

i, i,s,

f(

= =

/

%

dyd'd((

((

((

+

y(

y( dy d(d'y

+

y(d(d'

((dyd'


17/95

058:0160 Chapters 3 & 4

Professor Fred Stern Fall 2006 1=

' ( ' ( ' (

' ( ' ( ' (

' ( ' ( ' (

s( (( (y ('

sy y( yy y'

s' '( 'y ''

f( y '

f ( y '

f( y '

= + +

= + +

= + +

P!tting together the a*oe res!lts

Di,

#Va &-

#t

= = +

)e/t $e need to relate the stresses Hito the fl!id "otion

i,e, the elo+it% field, o this end $e e/a"ine the

relatie "otion *et$een t$o neigh*oring fl!id parti+les,

J .: VdrVdVV +=+ 1storder a%lor Series

,i,

'y(

'y(

'y(

d(e

d'

dy

d(

!!!

vvv

uuu

VdrdV =

==

*od% for+e d!e

to grait%"otion

s!rfa+e for+e - p B is+o!s ter"s

'd!e to stress gradients(

.

A '!$( - V

relatie "otion defor"ation rate

tensor - i,e

dr


18/95

058:0160 Chapters 3 & 4


1 1

2 2

, ,i i ii, i, i,

, , i , i

u uu u ue

( ( ( ( (

symmetric $art anit symmetric $arti, ,i i, ,i

= = + + = +

= =

1 442 4 43 1 442 4 43

1 10 ' ( ' (

2 2

1 1' ( 0 ' (

2 2

1 1' ( ' ( 0

2 2

y ( ' (

i, ( y ' y

( ' y '

u v u !

v u v ! ri&id body rotation

of fluid element

! u ! v

= =

6 47 48

142 43

1 42 43

!%ere rotation about ais rotation about y ais. rotation about ' ais

)ote that the +o"ponents of iare related to the orti+it%e+tor define *%:

DD D' ( ' ( ' (

2 22

y ' ' ( ( yV ! v i u ! , v u -

= = + + 14 2 4314 2 43 14 2 43

- 2 ang!lar elo+it% of fl!id ele"ent


19/95

058:0160 Chapters 3 & 4

Professor Fred Stern Fall 2006 1?

1 1' ( ' (

2 2

1 1' ( ' (

2 2

1 1' ( ' (

2 2

i,

( y ( ' (

( y y ' y

( ' y ' '

rate of strain tensor

u u v u !

v u v v !

! u ! v !

=

+ +

= + + + +

( y 'u v ! V + + = - elon&ation (or volumetric dilatation)

of fluid element

1 #

#t

=

('2

1(y

vu + - distortion $rt '/%( plane

('2

1('

!u + - distortion $rt '/( plane

('2

1y'

!v + - distortion $rt '%( plane

h!s general "otion +onsists of:

1( p!re translation des+ri*ed *% V

2( rigid*od% rotation des+ri*ed *% K /

3( ol!"etri+ dilatation des+ri*ed *% V

4( distortion in shape des+ri*ed *% i i

It is no$ ne+essar% to "a@e +ertain post!lates +on+erning

the relationship *et$een the fl!id stress tensor 'Hi( andrateofdefor"ation tensor 'ei(, hese post!lates are

*ased on ph%si+al reasoning and e/peri"ental

o*serations and hae *een erified e/peri"entall% een

for e/tre"e +onditions, For a )e$tonian fl!id:


20/95

058:0160 Chapters 3 & 4


1( >hen the fl!id is at rest the stress is h%drostati+ and

the press!re is the ther"od%na"i+ press!re

2( Hiis linearl% related to eiand onl% depends on ei,

3( Sin+e there is no shearing a+tion in rigid *od%

rotation it +a!ses no shear stress,

4( here is no preferred dire+tion in the fl!id so thatthe fl!id properties are point f!n+tions '+ondition of

isotrop%(,


21/95

058:0160 Chapters 3 & 4


sing state"ents 13

i,i,mni,i, -$ +=

imn- 4thorder tensor $ith 81 +o"ponents s!+h that ea+h

stress is linearl% related to all nine +o"ponents of Li,

Eo$eer state"ent '4( re!ires that the fl!id has no

dire+tional preferen+e i,e, Hiis independent of rotation of

+oordinate s%ste" $hi+h "eans imnis an isotropi+ tensor- een order tensor "ade !p of prod!+ts of 0i,

i,mn i, mn im ,n in ,m- = + +

scalars=('

Mastl% the s%""etr% +ondition Hi - Hire!ires:

@i"n- @i"n N - O - is+osit%

{2i, i, i, mm i,$

V

= + +

and O +an *e f!rther related if one +onsiders "ean

nor"al stress s, ther"od%na"i+ p,

3 '2 3 (ii

$ V = + +


22/95

058:0160 Chapters 3 & 4


1 2

3 3ii$ V

$ mean

normal stress

= + +

=123

2

3$ $ V

= +

In+o"pressi*le flo$: $$= and a*sol!te press!re is

indeter"inant sin+e there is no e!ation of state for p,

!ations of "otion deter"ine $ ,

Co"pressi*le flo$: $$ and - *!l@ is+osit% "!st *e

deter"ined ho$eer it is a er% diffi+!lt "eas!re"ent

re!iring large1 1# #

V#t #t

= =

e,g, $ithin sho+@

$aes,

Sto@es E%pothesis also s!pported @ineti+ theor%

"onotoni+ gas,

$$=

= 3

2

2 23

i, i, i,$ V = + + eneraliation dy

du= for 3# flo$,

+

=i

,

,

i

i,

(

u

(

u ,i relates s%ear stress to strain rate


23/95

058:0160 Chapters 3 & 4


2 12 2

3 3

i iii

i i

u u$ V $ V

( (

normal viscous stress

= + = + + 1 4 442 4 4 43

>here the nor"al is+o!s stress is the differen+e *et$een

the e/tension rate in the /i dire+tion and aerage

e/pansion at a point, Qnl% differen+es fro" the aerage -

+

+

'

!

y

v

(

u

3

1 generate nor"al is+o!s stresses, For

in+o"pressi*le fl!ids aerage - 0 i,e, 0V = ,

)on)e$tonian fl!ids:

i,i, for s"all strain rates

$hi+h $or@s $ell for

air $ater et+, )e$tonian fl!ids

{ {

ni, i, i,

tnon linear %istory effect

+

)on)e$tonian

is+oeslasti+ "aterials


24/95

058:0160 Chapters 3 & 4


)on)e$tonian fl!ids in+l!de:

'1( Pol%"ers "ole+!les $ith large "ole+!lar

$eights and for" long +hains +oiled together

in spong% *all shapes that defor" !nder shear,

'2( "!lsions and sl!rries +ontaining s!spended

parti+les s!+h as *lood and $ater


25/95

058:0160 Chapters 3 & 4


22 22

,i i

i, i

, , , i , ,

uu uu V

( ( ( ( ( (

= + = = =

For in+o"pressi*le flo$ 0V =

2D

D D

#V&- $ V

#t$ !%ere $ $ '

$ie'ometric $ressure

= +

= +1 42 43

For O - 0

D#V & - $#t

= !ler !ation

)S e!ations for 9 O +onstant

2D#V $ V#t

= +

2D#V

V V $ V #t

+ = +

1on-linear 2ndorder 3#4, as is t%e case for 5, 6 not constant

Co"*ine $ith V for 4 e!ations for 4 !n@no$ns V pand +an *e al*eit diffi+!lt soled s!*e+t to initial and

*o!ndar% +onditions for V p at t - t0and on all

*o!ndaries i,e, $ell posed; I.P,


26/95

058:0160 Chapters 3 & 4


Appli+ation of C o"ent!" !ation:

RCV CS

d

+ V d V V n dAdt = +

SB +++ += 'in+l!des rea+tion for+es(

)ote:

1, e+tor e!ation

2, n - o!t$ard !nit nor"al: RV n 0 inlet 0 o!tlet

3, 1# o"ent!" fl!/

( ) ( )i ii iout inCS

V V n dA m V m V =

& &

>here iV i are ass!"ed !nifor" oer dis+rete inlets

and o!tlets

i i ni im V A=&

{ ( ) ( )i ii i inout

CV

d+ V d m V m V

dtnet force inlet momentumoutlet momentum

time rate of c%an&eonCV flu(flu(of momentum in CV

= + & &1 42 431 4 2 4 31 4 2 4 3


27/95

058:0160 Chapters 3 & 4


4, o"ent!" fl!/ +orrelation fa+tors

2 2

avu V n dA u dA AV

a(ial flo! !it%non uniformvelocity $rofile

= =

14 2 43

>here

2

1

avCS

udA

A V

=

1av

CS

"V u dA

AA= =

Ma"inar pipe flo$:

12

2

0 021 1r r

u ) )R R =

0,53avV )= 34=

!r*!lent pipe flo$:

m

R

r)u = 10 1 1? 5

m

( )0

2

1 '2 (avV )

m m=

+ + =1=m Vav 7829


28/95

058:0160 Chapters 3 & 4


( ) ( )

(22('21'2

21 22

mm

mm

++++= m:;< = :792

5, Constant p +a!ses no for+e herefore

se pgage- pat"pa*sol!te

0$CS CV

+ $n dA $ d= = = for p - +onstant

6, At et e/it to at"osphere pgage- 0,

=, Choose C +aref!ll% $ith CS nor"al to flo$ and

indi+ating +oordinate s%ste" and +on C si"ilar

as free *od% diagra" !sed in d%na"i+s,

8, an% appli+ations !s!all% $ith +ontin!it% and

energ% e!ations, Caref!l pra+ti+e is needed for"aster%,

a, Stead% and !nstead% deeloping and f!ll%

deeloped pipe flo$

*, "pt%ing or filling tan@s

+, For+es on transitions

d, For+es on fi/ed and "oing anes

e, E%dra!li+ !"p

f, .o!ndar% Ma%er and *l!ff *od% drag

g, Ro+@et or et prop!lsion

h, )ole


29/95

058:0160 Chapters 3 & 4


i, Propeller

, >aterha""er

First relate !"a/to 0!sing +ontin!it% e!ation

( ) += R m

drrRruR)

0"a/

2

0 210

( )0 "a/20

11 2

Rm

avr) u r dr V

RR

= =

T


30/95

058:0160 Chapters 3 & 4


"a/

2

'1 ('2 (av

V um m

=+ + " - 1


31/95

058:0160 Chapters 3 & 4


22

0

2

21 02,(' R)R$$+

turb =

Re+onsider the pro*le" for f!ll% deeloped flo$:

Contin!it%:

0in out

in out

m m

m m m

+ == =

& &

& & & or U - +onstant

o"ent!":

2

1 2

1 1 2 2

2 1

' (

' ( ' (

' (

0

(+ $ $ R + uV A

u u A u u A

" u u

= =

= +=

=

( ) 21 2 2 0!$

$ $ R R (

=14 2 43

=

d(

d$R!

2 or for s"aller C r R

=

d(

d$r

2

(valid for laminar or turbulent flo!, but assume laminar)

===

d(

d$r

dr

du

dy

du

2 % - Rr '!all coord7)

=

d(

d$r

dr

du

2

Co"plete anal%sis

!sing CF#V


32/95

058:0160 Chapters 3 & 4


cd(

d$ru +

=

4

2

0(' ==Rru

= d(d$R

c 4

2

=

d

d$rRru

4('

22

=

d(

d$Ru

4

2

"a/

=

2

2

"a/ 1('

R

ruru

==

d(

d$Rdrrru"

R

82('

4

0

2

"a/

28ave

" R d$ uV

A d(

= = =

2

8 4

2 2

ave ave!

V VR d$ R

d( R R

= = =

2

8 32 64 64

Re

!

ave ave ave

fV RV V #

= = = =

Re aveV #

=


33/95

058:0160 Chapters 3 & 4


/a+t sol!tion of )S for la"inar f!ll% deeloped pipe

flo$


34/95

058:0160 Chapters 3 & 4


a@ing "oing C at speed s- WR X en+losing et and

*!+@et:

Contin!it%: 0in out m m + =& &

Rin out

CS

m m m V n dA= = = & & &

Inlet DR ,V V R n i= =

Q!tlet G' (R ,V V R n i= =

o"ent!":

> buc-et out in+ + mu mu= = & &

2

' ( ' (

2 ' (

2 ' (

buc-et , ,

,

, ,

+ m V R V R

m V R

A V R

=

=

=

&

' (, ,m A V R

=

22 ' (buc-et , ,

3 R+ A R V R= =


35/95

058:0160 Chapters 3 & 4


0=d

d3 for

3

,VR = 3"a/

8

2= , ,

3 A V=

If infinite n!"*er of *!+@ets: , ,m A V=&

3

"a/

2 ' (

2 ' (

10

2 2

buc-et , , ,

, , ,

,

, ,

+ A V V R

3 A V R V R

Vd3for R 3 A V

d

=

=

= = =

C +ontin!it% e!ation for stead% in+o"pressi*le flo$

one inlet and o!tlet A - +onstant

in out

V ndA V ndA m " = = = &

in out " "=

( ) ( )ave avein out V A V A=

For A - +onstant ( ) ( )ave avein out V V=

( ) ( )in out

+ V V n dA V V n dA = + Pipe:

( ) ( )(in out

+ u V n dA u V n dA = + ( ) ( )2 2ave ave

in out AV AV = +

( )ave out in"V = +hange in shape !

all et "ass flo$

res!lt in $or@,


36/95

058:0160 Chapters 3 & 4


ane:

( )out in out in+ m V V V V= = &

( ) ( )2( out in in+ m u u m u= = & &

+hange in dire+tion !

Appli+ation of differential "o"ent!" e!ation:

1, )S alid *oth la"inar and t!r*!lent flo$ ho$eer

"an% order of "agnit!de differen+e in te"poral and

spatial resol!tion i,e, t!r*!lent flo$ re!ires er%s"all ti"e and spatial s+ales

2, Ma"inar flo$ Re+rit-)

1000

Re Re+rit insta*ilit%

3, !r*!lent flo$ Retransition 10 or 20 Re+rit

Rando" "otion s!peri"posed on "ean +oherent

str!+t!res,

Cas+ade: energ% fro" large s+ale dissipates at

s"allest s+ales d!e to is+osit%,Yol"ogoro h%pothesis for s"allest s+ales

4, )o e/a+t sol!tions for t!r*!lent flo$: RA)S #S

MS #)S 'all CF#(


37/95

058:0160 Chapters 3 & 4


5, 80 e/a+t sol!tions for si"ple la"inar flo$s are

"ostl% linear 0V V =

a, Co!ette flo$ - shear drien

*, Stead% d!+t flo$ - Poise!ille flo$

+, nstead% d!+t flo$

d, nstead% "oing $alls

e, As%"ptoti+ s!+tion

f, >inddrien flo$sg, Si"ilarit% sol!tions, et+,

6, Also "an% e/a+t sol!tions for lo$ Re Sto@es and

high Re .M appro/i"ations

=, Can also !se CF# for non si"ple la"inar flo$s

8, AF# or CF# re!ires $ell posed I.P therefore

e/a+t sol!tions are !sef!l for set!p of I.P ph%si+s

and erifi+ation CF# sin+e "odeling errors %ield

S- 0 and onl% errors are n!"eri+al errors S)

i,e, ass!"e anal%ti+al sol!tion - tr!th +alled

anal%ti+al *en+h"ar@

M'(-M'(-MA'A(-0 si"ilarl% for IC&.C

ZS- S [ - ZS)B ZS

S2- S)

2B S2


38/95

058:0160 Chapters 3 & 4


nerg% !ation:

. - - energ%

7 - e - d


39/95

058:0160 Chapters 3 & 4


vd? dA V = & - viscous force velocity

v

CS? V dA=

&

( )' (


40/95

058:0160 Chapters 3 & 4


#efine @ineti+ energ% +orre+tion fa+tor

32

21' (2 2

ave

aveA A

VVdA V V n dA mA V

= =

Ma"inar flo$:

=

2

0 1

R

r)u

Vave97A = ; D2

!r*!lent flo$:m

R

r)u

= 1

0

( ) ( )3 3

1 2

4'1 3 ('2 3 (

m m

m m

+ +=

+ +

m:;< D:79A8 as !it% =, DE: for

turbulent flo!

2 2

D D' < ( ' < (2 2

s ave aveout in

? V V"u $ &' u $ &'

m m = + + + + + +

& &

& &

Met in - 1 o!t - 2 V- Vae and diide *% g

2 21 1 2 21 1 2 2

2 2$ t *

$ $V ' % V ' % %

& & & &

+ + + = + + + +


41/95

058:0160 Chapters 3 & 4


$s tt $

?? ?% %

&m &m &m= =

&& &

& & &

2 1

1' (

*

"% u u

& m&=

&

&

%* t%ermal ener&y (ot%er terms re$resent mec%anical ener&y

1 1 2 2m AV A V = =&

Ass!"ing no heat transfer "e+hani+al energ% +onerted

to ther"al energ% thro!gh is+osit% and +an not *e

re+oered therefore it is referred to as head loss 0

$hi+h +an *e sho$n fro" 2ndla$ of ther"od%na"i+s,

1# energ% e!ation +an *e +onsidered as "odified.erno!lli e!ation for hpht and hM,

Appli+ation of 1# nerg% e!ation f!ll% deeloped pipe

flo$ $itho!t hpor ht,


42/95

058:0160 Chapters 3 & 4


1 21 2

2

2

' ( ' (

2

1

8

2

*

! !

ave

ave*

$ $ * d% ' ' $ ' * d(

& & d(

*f& R

V

V*% f

# &

= + = + =

= =

=

For la"inar flo$

2

8 32!

ave ave

fV RV

= =

2

32 ave*

*V%

#

= Vae e/a+t sol!tionV

For t!r*!lent flo$ Re+rit ] 2000 Retrans] 3000

f-f 'Re @


43/95

058:0160 Chapters 3 & 4


%* %f+ F%m $here

2

2m

V% @

&

@ loss coefficient

=

=

h"- so +alled; "inor losses e,g, entran+e


44/95

058:0160 Chapters 3 & 4


.erno!lli negle+t g p2-pa2 2

1 1 2 2

1 1

2 2$ V $ V + = + %*9, 'constant

( )2 21 2 2 11

2$ $ V V= + 2 21

,=? ??8101 000 '5,22 2,0? (2

$ = +

1 110020$ 3a=

)ote: 2 2 2

2 2 3 3 4 42 2 2

$ V $ V $ V

+ = + = +

2 3 4 2 3 4a$ $ $ $ V V V= = = = =

2 2 3 3 4 40CS

V A A V A V A V = = +432

AAA +=

3 3 3 4 4 40 ' (yCS

+ VV A V V A V V A = = = + 2 2

3 3 4 4V A V A = 43 AA =

'*( For the ro!nd et gien in the pro*le" state"ent

2 2

2 2

2

,=? ?8? ,02 4254

(+ + mV V 1

A

= = = & 142 43

12 3

2 41,4 < 10,3


45/95

058:0160 Chapters 3 & 4


Contin!it% e!ation *et$een points 1 and 22

21 1 2 2 1 2

1

#V A V A V V

#

= =

2

1

241,4

5V

=

1 6,63


46/95

058:0160 Chapters 3 & 4


'a(2

21 2

2

V' '

&= + 01101 212 ==== ''%*

2 1 22 ' (V & ' ' = 11_81,?_2= sm


47/95

058:0160 Chapters 3 & 4


'+(

2 2

2 21 2 2

2 2

V V*' ' f

& # &= + + D2:

( )2

21 2 1 here ( )i,#V

V V & V #t

=

D' ( ii,

,

ii,

,

u#u- J

#t (

u$ V

(

= +

1 2 3

D' < ( ' ( 0ii,,

u# #$u $ - J dissi$ation function

#t #t (

+ = + + =

123

++=#t#$J-

#t#% ('

S!""ar% # for +o"pressi*le non+onstant propert%

fl!id flo$

Cont, ' ( 0Vt

+ =

o", i,#V

& $#t

= +

D

i, i, i,V

& &-

= +

=

' < ($ # #

$#t #t

= +

"o"ent!" e!ation

+ontin!it% e!ation


50/95

058:0160 Chapters 3 & 4


nerg% ++= (' J-#t#$

#t

#%

Pri"ar% aria*les: p V

A!/iliar% relations: 9 - 9 'p( O - O 'p(

h - h 'p( @ - @ 'p(

Restri+tie Ass!"ptions:

1( Contin!!"2( )e$tonian fl!ids

3( her"od%na"i+ e!ili*ri!"

4( f - 9g D-

5( heat +ond!+tion follo$s Fo!rier`s la$

6( no internal heat so!r+es

For in+o"pressi*le +onstant propert% fl!id flo$

Dvdu c dJ = cv, 6, , 5 E constant

+= J-#t

#Jc

v

2

For stati+ fl!id or Vs"all

J-t

Jc

$

2=

%eat conduction eIuation (also valid for solids)


51/95

058:0160 Chapters 3 & 4


S!""ar% # for in+o"pressi*le +onstant propert% fl!id

flo$ '+] +p(

0V =

2D#V &- $ V#t

= + ellipti+;

+= J-#t

#Jc

$

2 $here,

i

i,

(

u

=

Contin!it% and "o"ent!" !n+o!pled fro" energ%

therefore sole separatel% and !se sol!tion post fa+to to

get ,

For +o"pressi*le flo$ 9 soled fro" +ontin!it% e!ation

fro" energ% e!ation and p - '9( fro" e!ation of

state 'eg ideal gas la$(, For in+o"pressi*le flo$ 9 -

+onstant and !n+o!pled fro" +ontin!it% and "o"ent!"

e!ations the latter of $hi+h +ontains $ s!+h that

referen+e p is ar*itrar% and spe+ified post fa+to 'i,e, for

in+o"pressi*le flo$ there is no +onne+tion *et$een p and

9(, he +onne+tion is *et$een$

and0V

= i,e, asol!tion for p re!ires 0V = ,

)S 21 D

#V$ V

#t

= + D$ $ '= +


52/95

058:0160 Chapters 3 & 4


('1S 'See deriation details on p,81(

2 21 ,i

, i

uu#V $#t ( (

= +

For 0V = :

i

,

,

i

(

u

(

u$

= 2

Poisson e!ation deter"ines press!re !p to additie

+onstant,

Appro/i"ate odels:

1( Sto@es Flo$

For lo$ Re 1 ] 0)*

V V

=


53/95

058:0160 Chapters 3 & 4


0V =21V

$ Vt

= +

0(' 2 = $1S

2( .o!ndar% Ma%er !ations

For high Re 1 and atta+hed *o!ndar% la%ers or

f!ll% deeloped free shear flo$s '$a@es ets "i/ing

la%ers( v )


54/95

058:0160 Chapters 3 & 4


3( Inis+id Flo$

( ) 0

b

' ( '

Vt#V

& $ 4uler 4Iuation nonlinear %y$erboli#t

#% #$- J $ V J un-no!ns and % - f $

#t #t

+ =

=

= + =

4( Inis+id In+o"pressi*le Irrotational

2

0

0 0 b b

V V

V linear elli$tic

= =

= =

!ler !ation .erno!lli !ation:

2

2$ V &' const

+ + =

an% elegant sol!tions: Mapla+e e!ation !sing

s!perposition ele"entar% sol!tions separation of

aria*les +o"ple/ aria*les for 2# and .o!ndar%

le"ent "ethods,


55/95

058:0160 Chapters 3 & 4


Co!ette Shear Flo$s: 1# shear flo$ *et$een s!rfa+es of

li@e geo"etr% 'parallel plates or rotating +%linders(,

Stead% Flo$ .et$een Parallel Plates: Combined Couette

and 3oiseuille lo!7

0

0

0

( y '

(

V

u v !

u

=+ + =

=2D

#V$ V

#t = + 0=+++

'y( !uvuuu

t

u

D0 ( yy$ u= ++= J-

#t

#Jc

$

2 0( y '

JuJ vJ !J

t

+ + + =

2

0yyy

u-J +=

(note inertia terms vanis% in%erently and 5 is absent

from eIuations)

2 2 2

2 2

2

2 2 2

' ( ' ( ' (

' (

( y '

( y y ' ' (

( y '

y

u v !

v u ! v u !

u v !

u

= + +

+ + + + + +

+ + +

=


56/95

058:0160 Chapters 3 & 4


1on-dimensionali'e eIuations, but dro$ N

)uu


57/95

058:0160 Chapters 3 & 4


Sol!tion depends on2

D(

%B $

)= :

. 0 D($ is opposite to

. 0,5 *a+@flo$ o++!rs near lo$er $all

c.c 1 flo$ approa+hes para*oli+ profilePress!re gradient effe+t

22 3 4Pr Pr Pr 1 '1 ( '1 ( ' ( '1 (

2 8 6 12

c c c4 4 B 4 BJ y y y y y= + + +

G55555555555555555555555H

1ote usually 3r4cis Iuite small

Substance 3r4 c dissi$ation

Air 0,001 er% s"all

>ater 0,02d

Pr

Brin-man

4Brc

=

=

Cr!de oil 20 large

P!re

+ond!+tion

rises d!e to

is+o!s dissipation

#o"inant ter"

for .


58/95

058:0160 Chapters 3 & 4


Shear Stress

1( D 0($ = i,e, p!re Co!ette Flo$

2 21 1

2 2

y

f

uC

= =_

_ 1< 2y

u =

%)% Re

1==

2


59/95

058:0160 Chapters 3 & 4


_ _21 '1 (2

u B y= __ _y

u By= y%

B)u

y 2= u)=

>here"a/2D

(

u%B $

) )= =

#i"ensional for" ( )2 2

"a/

1D 1

2

% yu $

%

u

=

14 2 43 "a/34

%udyu"%

%

==

"a/

3

2

2u

%

"u ==

%u

%

u

%

B)

lo!er%

B)

u$$er%

B)u

!

%yy!

32

"a/

===

+=

===

6Re6

2

1 02 ==== %f!

f C3or%u

)

C

/+ept for n!"eri+al +onstants sa"e as for +ir+!lar pipe,

2

,

,

u lam

u turb


60/95

058:0160 Chapters 3 & 4


Rate of heat transfer at the $alls:

%

)JJ

%

-J-I

%yy!

4

('

2

2

01 ==

B - !pper - lo$er

Eeat transfer +oeffi+ient:

( )1 0!I

J J=

212

Br-

%1u ==

For .r 2 *oth !pper & lo$er $alls "!st *e +ooled to

"aintain 1and 0


61/95

058:0160 Chapters 3 & 4


Conseration of Ang!lar o"ent!": "o"ent for" of

"o"ent!" e!ation 'not ne$ +onseration la$V(

0

sys

B O r V dm= = = an&ular momentum of system aboutinertial coordinate system 97

r V=

( ) ( )

0

CV CS

d O d

r V d r V V n dAdt dt = + == 0M vector sum all eternal moments a$$lied CV

due to bot% B and S7

For !nifor" flo$ a+ross dis+rete inlet


62/95

058:0160 Chapters 3 & 4


a@e inertial fra"e 0 as fi/ed to earth s!+h that CS

"oing at Vs- R i

2 0D DV V i R i= 2 Dr R,=

1 0

DV V -= 1 D0r ,=

0$i$e

"V

A=

2 12 10 inD ' ( ' (P outM J - r V m r V m= = & &

out inm m "= =& &

0D D' (' (oJ - R V R - " =

0 0

2

V J

R "R

= interestin&ly, even for J99, /maV9;R

Retarding tor!e d!e to*earing fri+tion


63/95

058:0160 Chapters 3 & 4


#ifferential !ation of Conseration of Ang!lar

o"ent!":

Appl% C for" for fi/ed C:

'&- ang!lar a++elerationQ - "o"ent of inertia

2 2 2 2'

d( d( dy dyQ a dy b dy c d( d d( = + &

( )' y yQ ddy = &

Sin+e3 3 2 2

12 12Q d(dy dyd( d(dy d( dy

= + = +

2 2

12 ' (y y(d( dy

+ =

&

0 0li"

d( dy y((y = si"ilarl% '((' = 'yy' =

i,e ,ii, = stress tensor is symmetric (stressest%emselves cause no rotation)


64/95

058:0160 Chapters 3 & 4


.o!ndar% Conditions for is+o!sFlo$ Pro*le"

he # to *e dis+!ssed ne/t +onstit!te an I.P for

a s%ste" of 2ndorder nonlinear P# $hi+h re!ire

.C for their sol!tion depending on ph%si+al pro*le"

and appropriate appro/i"ations,

%pes of .o!ndaries:

1, Solid S!rfa+e2, Interfa+e

3, Inlet


65/95

058:0160 Chapters 3 & 4


1, Solid S!rfa+e

a, Mi!id

- "ean free path fl!id "otion therefore

"aros+opi+ ie$ is no slip; +ondition i,e, no

relatie "otion or te"perat!re differen+e *et$een

li!id and solid,

liIuid solid V V= solidliIuid JJ

=

/+eption is for +onta+t line for $hi+h anal%sis is

si"ilar to that for gas,

*, as

S"ooth $all

Ro!gh $all

!

!

dy

dulu =

Spe+!lar refle+tionConseration of

tangential "o"ent!"

!$-0-fl!id elo+it% at$all

#iff!se refle+tion,

Ma+@ of refle+tedtangential "o"ent!"

*alan+ed *% !$


66/95

058:0160 Chapters 3 & 4


!

!

dy

du =

al

32

= lo! density limit

!

!

au

2

3=

a)Ma = 21

2

!f

C

)

=

f! CMa)u =5,< =

Eigh Re: Cf E 9799A

Say Ma E 29

Mo$ Re: Cf E 7HRe-:;2 Re;

2

1

Re

4,

(

! Ma

)

u=

Signifi+ant slip possi*le at lo$ Re high a:

E%personi+ M Pro*le";

Si"ilar for :

Eigh Re: J&as J!

Mo$ Re( )

,8=&as !

f

r !

J JMaC

J J

=

air- driing

01,0hi+h leads to:

Eel"holt heore" 3: )o fl!id parti+le +an hae

rotation if it did not originall% rotate, Qr e!ialentl% in

the a*sen+e of rotational for+es a fl!id that is initiall%

irrotational re"ains irrotational, In general $e +an

+on+l!de that orti+es are presered as ti"e passes, Qnl%

thro!gh the a+tion of is+osit% +an the% de+a% ordisappear,

Yelins Cir+!lation heore" and Eel"holt

heore" 3 are er% i"portant in the st!d% of inis+id

flo$, he i"portant +on+l!sion is rea+hed that a fl!id that

he +ir+!lation of a

"aterial loop neer

+hanges


78/95

058:0160 Chapters 3 & 4

Professor Fred Stern Fall 2006 =8

is initiall% irrotational re"ains irrotational $hi+h is the

!stifi+ation for idealflo$ theor%,

In a real is+o!s fl!id orti+it% is generated *%

is+o!s for+es, is+o!s for+es are large near solid

s!rfa+es as a res!lt of the noslip +ondition, Qn the

s!rfa+e there is a dire+t relationship *et$een the is+o!s

shear stress and the orti+it%,

Consider a 1# flo$ near a $all:

12 12

22 22

32 32

2 0

0

u v u

y ( y

v

y

! v

y '

= = + =

= = =

= = + =

>hi+h sho$s that

dy

u(

= 0== 'y

Eo$eer fro" the definition orti+it% $e also see that

he is+o!s stresses are gien *

i, ,n $here i, i, =

11 1 12 2 13 3

21 1 22 2 23 3

31 1 32 2 33 3

(

y

'

n n n

n n n

n n n

+ + =

+ + =

+ + =)Q: the onl% +o"ponent of

is , A+t!all% this is a

general res!lt in that it +an *e

sho$n that s!rfa+eis

perpendi+!lar to the li"iting

strea"line,


79/95

058:0160 Chapters 3 & 4

Professor Fred Stern Fall 2006 =?

( '

u

y

= =

i,e, the $all orti+it% is dire+tl% proportional to the $all

shear stress, his anal%sis +an *e easil% e/tended for

general 3d flo$,

i, , i, ,n n = at a fi/ed solid $all

r!e sin+e at a $all $ith +oordinate /21 3

0( (

= =

and

fro" +ontin!it%2

0v

(

=

Qn+e orti+it% is generated its s!*se!ent *ehaior is

goerned *% the orti+it% e!ation,

)S ( ) 2here0

V

V

=

= V =

If 0V V V = + =

hen 2 0 = J%e 7d7e7 foris t%e *a$lace 4I7

And V A = Since ( ) 0= A

2 ' (

V A

A A

= =

= +

he irrotational part of

the elo+it% field +an *e

e/pressed as the gradient

of a s+alar

Again *% e+tor identit%


84/95

058:0160 Chapters 3 & 4


i,e = A2

he sol!tion of this e!ation is

= dR

A

4

1

h!s 31

4

RV d

R

=

>hi+h is @no$n as the .iotSaart la$,

he .iotSaart la$ +an *e !sed to +o"p!te the elo+it%

field ind!+ed *% a @no$n orti+it% field, It has "an%

!sef!l appli+ations in+l!ding in ideal flo$ theor% 'e,g,

$hen applied to line orti+es and orte/ sheets it for"s

the *asis of +o"p!ting the elo+it% field in orte/latti+eand orte/sheet liftings!rfa+e "ethods(,

he i"portant +on+l!sion fro" the Eel"holt

de+o"position is that an% in+o"pressi*le flo$ +an *e

tho!ght of as the e+tor s!" of rotational and irrotational

+o"ponents, h!s a sol!tion for irrotational part V

represents at least part of an e/a+t sol!tion, nder +ertain+onditions high Re flo$ a*o!t slender *odies $ith

atta+hed thin *o!ndar% la%er and $a@e Vis s"all oer

"!+h of the flo$ field s!+h that V is a good

appro/i"ation to v , his is pro*a*l% the strongest


85/95

058:0160 Chapters 3 & 4


!stifi+ation for idealflo$ theor%, 'incom$ressible,

inviscid, and irrotational flo!),

)oninertial Referen+e Fra"e

h!s far $e hae ass!"ed !se of an inertial referen+e

fra"e 'i,e, fi/ed $ith respe+t to the distant stars in

deriing the C and differential for" of the "o"ent!"

e!ation(, Eo$eer in "an% +ases noninertial referen+efra"es are !sef!l 'e,g, rotational "a+hiner% ehi+le

d%na"i+s geoph%si+al appli+ations et+(,

i rel

dVa a

dt= +

i rel

dV+ ma m adt

= = +

rel

dV+ ma m

dt = 1 2 3

rRSi +=

i,e )e$ton`s la$

applies to non

inertial fra"e $ith

addition of @no$n

inertial for+e ter"s


86/95

058:0160 Chapters 3 & 4


i

d RV V r

dt= + +

2

22 ' (i

rel

dV d R d a r V r

dt dt dt

dVa

dt

= + + + +

= +

2

2

dt

Rd acceleration (,y,')

rdt

d - an&ular acceleration (,y,')

2 V - Coriolis acceleration

(' r - centri$etal acceleration (-W2*, !%ere * normal

distance from r to ais of rotation W)7

Sin+e R and W ass!"ed @no$n altho!gh "ore

+o"pli+ated $e are si"pl% adding @no$n

inho"ogeneities to the "o"ent!" e!ation,

C for" of o"ent!" e!ation for noninertial

+oordinates:

rel R

CV CV CS

d+ a d V d v V n dAdt

= +

#ifferential for" of o"ent!" e!ation for noninertial

+oordinates:

rdterm from fact t%at

(,y,') rotatin& at W(t)7


87/95

058:0160 Chapters 3 & 4


2D 23

rel ,i i,

,

#Va & $ V

#t

+ = +

( ) vrrRa rel +++=

2

All ter"s in rela seldo" a+t in !nison 'e,g, geoph%si+al

flo$s(:

R ] 0 earth not a++elerating relatie to distant stars

] 0 for earth

( )r ] 0 g nearl% +onstant $ith latit!de

v2 "ost i"portantV

1

0 '2 (idVa R Vdt

= + 00

tVVV tV *

= =

2

0 00

0

V * V

R RossbyV *

= = =

if M is large i,e, +o"para*le to the

order of "agnit!de of the earth

radi!s R01 then Coriolis ter" is

larger than the inertia ter"s and is

i"portant,


88/95

058:0160 Chapters 3 & 4


/a"ple of )oninertial Coordinates:

eoph%si+al fl!ids d%na"i+s

At"osphere and o+eans are nat!rall% st!died !sing non

inertial +oordinate s%ste" rotating $ith the earth, $o

pri"ar% for+es are Coriolis for+e and *!o%an+% for+e d!e

to densit% stratifi+ation 9 - 9'(, .oth are st!died !sing

.o!ssines appro/i"ations '9 - +onstant e/+ept

( ) DJ &- ter" and O @ Cp- +onstant( and thin la%er on

rotating s!rfa+e ass!"ption *O)? ] ,

#ifferen+es *et$een at"ospheri+al o+eans: lateral

*o!ndaries '+ontinents( in o+eans +!rrents in o+ean 'g!lf

and Y!roshio strea"( along $estern *o!ndaries +lo!ds

and latent heat release in at"osphere d!e to "oist!re

+ondensation o+ean- 0,1]1 or 2 "


89/95

058:0160 Chapters 3 & 4


Coriolis for+e - 2 V

-!vu

-,i

'y(

GGG

- ( )

+ +ossinsin+os2

GGG

u-u,v!i

-GGG

+os2 -u,fuifv + sin2=f

f 0 northern he"isphere

f 0 so!thern he"isphere

f - at poles

0 sin+e $

- planetar%orti+it%

- 2 _ erti+al

+o"ponent WPerson

spins at

W


90/95

058:0160 Chapters 3 & 4

Professor Fred Stern Fall 2006 ?0

f - 0 at e!ator

!ations of otion

0V =

!

&

'

$

#t

#!

vy

$fu

#t

#v

u(

$fv

#t

#u

2

00

2

0

2

0

1

1

1

+

=

+=+

+

=

('100

JJ= p 9 - pert!r*ation fro" h%drostati+

+ondition

eostrophi+ Flo$: !asistead% larges+ale "otions in

at"osphere or o+ean far fro" *o!ndaries

(

$fv

=

0

1

y

$fu

=0

1

2

] 0#V )

#t *

] 0 ' (f V f) M - horiontal s+ales

Ross*% n!"*er - f*)

At"osphere: ] 10 "


91/95

058:0160 Chapters 3 & 4


herefore negle+t#V

#tand sin+e there are no *o!ndaries

negle+t2V ,

"o"ent!" $

&'

=

*aro+lini+ 'i,e, p - p'((

and +an *e !sed to eli"inate p in a*oe e!ations

$here*% '!( - f''(( $hi+h is +alled ther"al $ind *!t

not +onsidered here,

If $e negle+t 9-9'( effe+ts '!( - f'p( and +an *e

deter"ined fro" "eas!red p'/%(, )ot alid near thee!ator 'B 3o( $here f is s"all,

( )0

1D D D D D D$ $ $ $u i v , $ i , i ,f y ( ( y

+ = + +

- 0

i,e Vis perpendi+!lar to $ horiontal elo+it% is

along 'and not a+ross( lines of +onstant horiontal

press!re $hi+h is reason iso*ars and strea" lines

+oin+ide on a $eather "apV


92/95

058:0160 Chapters 3 & 4


@"an Ma%er on Free S!rfa+e: effe+ts of fri+tion near

*o!ndaries

is+o!s la%ers:

S!dden a++eleration flat plate: t yyu u=

3,64 t =

Qs+illating flat plate: yyt uu =

6,5


93/95

058:0160 Chapters 3 & 4


='

u at - 0

0='

v at - 0

0(' =vu at -

!ltipl% e!ation *% 1=i and add to !e!ation:

2

2

d V i f V

dt

= V u i v

com$le( velocity

= +

='1 ( < '1 (


94/95

058:0160 Chapters 3 & 4


F, )ansen '1?02( o*sered drifting ar+ti+ i+e drifted 20

400to the right of the $ind $hi+h he attri*!ted to

Coriolis a++eleration, Eis st!dent @"an '1?05( deried

the sol!tion,

Re+all f 0 in so!thern he"isphere so the drift is to the

left of \,


95/95

058:0160 Chapters 3 & 4


Si"ilar sol!tion for i"p!lsie $ind:

0(0'00 ====== 'u'u'uuu'''t

tu

20=

la"inar sol!tion0

0' 6 < 20 (

!indu V m s J C = = = 97H m;s after one min7, 27 m;s

after one %our

t!r*!lent sol!tion(more realistic)u9972 m;s after : %r ( Y v!ind)

For @"an la%er si"ilar +onditions m - 400)

Ma"inar sol!tion !0- 2,= "

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