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observation point

(b)

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Re = !ULµ

O Reynolds number "6,01 % 5*:%+ '#, 03$0,"10/8, #+ 9$%+ '1+ 3(,<µ"1+ 1>83%(+

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!u = !u x, y, z, t( ) = u x, y, z, t( ) x + v x, y, z, t( ) y+w x, y, z, t( ) z! = ! x, y, z, t( )T = T x, y, z, t( )

!R0 =

!R x0, y0, z0, t0( )

!R =!R x, y, z, t( )

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X

z

y

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x

z

y

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?&@*;&$%( (Eulerian) G*;&F;$C(

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L;#( -4+ Pascal

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Oh

(b)A D

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Water

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FA = ! fluidVg

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!F!"#"!$%&'( = FA "FW = 0

FA = %)*%$& 34 h

3g

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#

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B&$-(;"," ?;µ(6 - Navier-Stokes equation

B&$-(;"," M%7;F*&$6

!.(<: $3< ()1 T3<>9#) L6190+?: /E#+6 3+6 $1$KE-$µ9 '@), 63A-0+61 (-9": .?-"9: K6#".E: "@E9: 3+6 $3+(94+?1 () DA#) (): @61$µ".': (&1 -96#(81:

1

2

3

1

!! L;#( I&$-(;","6 -"6 µ89$6 F )(,%51/7 µ<20 m ",*+ $"()'%& 310'4$"6'01 )'0="$7. =)4. + -6>µ<: µ9($D+4': (): µAC$: (+6 -96#(+? 3-E39" 1$ 951$" µ)@E1.

H3+6 R(t) µ"$ (60$5$ 39-"+0' 3+6 39-"E09" #6%.9.-"µE1) µAC$ m (+6 -96#(+?. U 36.1<()($ $ 935#): µ3+-95 1$ 951$" 0-+1".A µ9($D$44<µ91).

L,#574'9' µ%K#* – !.@9;9' 90$-B1,#*

dmdt

=ddt

! dV = 0R(t )!

•! P"$ $#6µ359#() -+' ) 36.1<()($ $ >9&-95($" #($>9-', @)4.

.$" #61938: ) !75#&#) L61E09"$: %519($":

!! !t = 0!"!u = 0

!!!t+"# !

!u( ) = 0230,4," 5+%6#*&$7 (Continuity Equation)

! =""xx + "

"yy+ "

"zz

#

$%

&

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G(,%51/7 "91'<-(,)4 = "91@0,. 3(,<µ"1++ ">#'"$1/.+ 3(,<µ"1+ !

{H(=µ*+ µ"'0?%57+ '4+ %$µ7+} = {I=$%1)µ0 3(,<µ"#,} !

2

!! B*5-*;46 O1µ46 -4+ O*5-A%$ F )(,%51/7 3&,0µ4 F 9%( 0)/"6'01 )" .,0 $"()'* "6,01 6)4 µ" '% $(=µ* µ"'0?%57+ '4+ %$µ7+ '%(.

L,#574'9' /4µ7* – Navier-Stokes equation

d m!u( )dt

=!F

!DDt

!R t( )" !udV =

!FSS t( )" dA +

!FBR t( )" dV

G(,%51/7 "91'<-(,)4 = F96")4+ + F1>834+ + F?0$&'4'0+ !

2 L,#574'9' /4µ7* – Navier-Stokes equation

! !"!u"t+ !!u #$!u

R t( )% dV =!FSS t( )% dA +

!FBR t( )% dV

J10 K"('8,"1%, 0)(µ96")'% $"()'*:

! !"!u"t+!u #$( )

!u%

&'

(

)* = +$p + µ$2 !u + !g

! =""xx + "

"yy+ "

"zz

#

$%

&

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Navier-Stokes equation

!DDt

!R t( )" !udV =

!FSS t( )" dA +

!FBR t( )" dV

G(,%51/7 "91'<-(,)4 = F96")4+ + F1>834+ + F?0$&'4'0+ !

{H(=µ*+ µ"'0?%57+ '4+ %$µ7+} = {I=$%1)µ0 3(,<µ"#,} !

T% 05:"?$1/* <=$%1)µ0 *5#, '#, µ%$@8, ",.$:"10+ )" .,0 $"()'* 310'4$"6'01 )'0="$* µ" '4, 9<$%3% '%( -$*,%(. T3A-0+61 3+44+5 @"$K+-9(".+5 (?3+" 91E-%9"$: (3.0. 9#&(9-".', ."1)(".', @61$µ".<, 0)µ".'), 3+6 µ3+-+?1 1$ .$($#('#+61 ()1 9K$-µ+%' (+6: #(+ >98-)µ$ 3"+ 39-534+.). R#(<#+, #601A µ$: 91@"$KE-9" µ<1+ #9 E1$: (?3+: (): µ9($K+-A: 91E-%9"$: (3.0. µ9($K+-A >9-µ<()($:). <3+6 Q 951$" ) 9#&(9-".' 91E-%9"$ and W 951$" (+ E-%+ $3< (+ #&µ$(5@"+ -96#(+?.

3 L,#574'9' 1$-4=1,#*

d(E)

dt=

dQ

dt−

dW

dt

dEsys

dt=ddt

!ECV! dV + !"

!ur # n( )dSCS!

!.,9F91,* 6@$'9'* 41095F$

!.(<: $3< ()1 T3<>9#) L6190+?: /E#+6 3+6 $1$KE-$µ9 '@), 63A-0+61 (-9": .?-"9: K6#".E: "@E9: 3+6 $3+(94+?1 () DA#) (): @61$µ".': (&1 -96#(81:

!!!t+"# !

!u( ) = 0

!!!u!t+!u "#( )

!u$

%&

'

() = *#p + µ#2 !u + !g

1

2

3 dEsys

dt=ddt

!ECV! dV + !"

!ur # n( )dSCS!

!.,9F91,* 6@$'9'* 41095F$

H49: +" 97"#8#9": .51)#): µ3+-+?1 1$ 9.K-$#(+?1 µ$C5 #()1 $.<4+6>) µ+-K': <3+6:

∂

∂tU +

∂

∂xFx +

∂

∂yFy +

∂

∂zFz =

∂

∂xGx +

∂

∂yGy +

∂

∂zGz

U =

ρ

ρu

ρv

ρw

ρE

, Fx =

ρu

ρu2 + p

ρuv

ρuw

ρuH

, Fy =

ρv

ρvu

ρv2 + p

ρvw

ρvH

, Fz =

ρw

ρwu

ρwv

ρw2 + p

ρwH

Gx =

0

σxx

σxy

σxz

(σ · u)x + kTx

,Gy =

0

σyx

σyy

σyz

(σ · u)y + kTy

,Gz =

0

σzx

σzy

σzz

(σ · u)z + kTz

(σ · u)x = uσxx + vσxy + wσxz

(σ · u)y = uσyx + vσyy + wσyz

(σ · u)z = uσzx + vσzy + wσzz

σij = µ

[(

∂ui

∂xj+

∂uj

∂xi

)

− 23δij(∇ · u)

]

!.,9F91,* 6@$'9'* 41095F$

B" 97"#8#9": Navier-Stokes 951$" µ) %-$µµ".E: .$" 39-534+.9: %"$ 1$ 46>+?1. P"$ ;96(819"+ (@)4. µ #($>9-<) .$" $#6µ359#(+ (@)4. $ #($>9-<) -96#(<, 0&-5: 97&(9-".E: @61Aµ9":, +" Navier-Stokes µ3+-+?1 1$ 9.K-$#(+?1 &::

!!!u!t+ !!u "#( )

!u = $#p + µ#2 !u

!"!u = 0

P"$ &N.I" ;4( – Stokes Flow (Re<<1), +" <-+" 93"(A061#): µ3+-+?1 1$ $%1+)>+?1.

!"!u = 0

µ!2 !u "!p = 0

P"$ 8-;&J" ;4( (<3+6 +" @61Aµ9": (-"D': 951$" µ".-E:, @)4. - #61'>&: µ$.-"A $3< #(9-9E: 93"KA19"9:).

!!!u!t+ !!u "#( )

!u +#p = 0

!"!u = 0

µ! 0

!.@9;9' Bernoulli

F )(,%51/7 96")4, pT "6,01 )'0="$7 )" µ10 H%L/7 J$0µµ7 (streamlne) *'0, 4 $%7 "6,01: 1.! )'0="$7,

2.! 0)(µ96")'4,

3.! <'$1?4 (µ43",1/* 1>83"+),

4.! 3", 0)/"6'01 .$:% )'% $"()'* (9.-. 09* ">#'"$1/7 94:7),

5.! 3", (9<$-"1 µ"'0@%$< ="$µ*'4'0+.

pT = p+12!u2 + !gz

3#4%+1,=µ# – 8/7 #"> #64/HI9,/ +1.#µ1$7*

F 3">0µ",7 '%( )-7µ0'%+ "6,01 :"µ<'4 µ" ,"$*, /01 .-"1 "µ?03*, "91@<,"10+ B1 /01 .,0 µ1/$* 0/$%@&)1% "µ?03%& B2 )" ?<=%+ h 09* '4, "91@<,"1< '%(. A%1< "6,01 4 '0-&'4'0 '4+ $%7+ 09* '% 0/$%@&)1%;

21 ) 93"KA19"$ (): @97$µ91': 951$" #9 ?N+: z1 (<(9 ) #61+4".' 359#) #()1 93"KA19"$ 951$":

<3+6 pA ) $(µ+#K$"-".' 359#). L(+ $.-+K?#"+, ) 359#) 951$" 935#): pA, (+ ?N+: 951$" z2=z1-h .$" ) ($0?()($ u2:

26(A ($ 2 #)µ95$ µ3+-+?1 1$ #61@9>+?1 µ9 µ"$ -+I.' %-$µµ' .$" A-$

pT1 = p! +12!u1

2 + !gz1

pT 2 = p! +12!u2

2 + !gz2

pT1 = pT 2 ! u22 "u1

2 = 2g z1 " z2( ) = 2gh/9 DA#) () @"$('-)#) µAC$:, + -6>µ<: µ9($D+4': #()1 93"KA19"$ u1A1 951$" 5@""+: µ9 (+ -6>µ< µ9($D+4': #(+ $.-+K?#"+ u2A2.

u22 − u2

1 = u22

�

1 −A2

2

A21

�

= 2gh ⇒ u2 =

�

2gh

1 − A22/A

21

≈�

2gh,

T3+>E(+6µ9 <(" ) -+' 951$" #($>9-', (+ 19-< 951$" $#6µ359#(+ .$" $%1++?µ9 935@-$#) "78@+6: .$>8: .$" >9-µ".E: 93"@-A#9":. R: 9. (+?(+6, +" <-+" (): Bernouilli 951$" ".$1+3+")µE1+".

3#4%+1,=µ# – A;&7$#* Venturi

M,0+ )#57,0+ Venturi 0,0:/<2"1 '% $"()'* ,0 9"$<)"1 09* µ10 )'.,#)4 (-#$6+ /0µ60 0985"10 ",.$:"10+). C$"6'" µ10 ./@$0)4 :10 '4, 96")4 P2 ",'*+ '4+ )'.,#)4+, 8+ 9$%+ '4, 9(/,*'4'0 $"()'%& $, '0 "µ?03< B1 /01 B2 /01 '4, '0-&'4'0 $%7+ U1 /01 96")4+ P1 )'4, "6)%3%.

!K$-µ<C+1($: ()1 975#&#) Bernoulli ($%1+81($: () D$-?()($ .$" >9&-81($: $#6µ3"9#(<()($) E0+6µ:

P1 +12!U1

2 = P2 +12!U 2

2

/9 DA#) () @"$('-)#) µAC$:, + -6>µ<: µ9($D+4': #()1 93"KA19"$ u1A1 951$" 5@""+: µ9 (+ -6>µ< µ9($D+4': #(+ $.-+K?#"+ u2A2.

U1A1 =U2A2 !U2 =U1A1A2

P2 = P1 +12!U1

2 !12!U 21A

21

A22"

P2 = P1 +12!U1

2 1! A21

A22

#

$%

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!0B#4,95F! asimina.kazakidi@gmail.com