Cap 9
Cap 9
T Bassa velocita’
V Tw
4L8 u
E 4p.@gJX:Z)0
Vr),mw2 IU25?]1.,S>)6 25?:?A)lS9<94p.=<>)/?:8:mn8:<>2 ]58:IU25m@.=<98A]D) i|S>8L+ EU6 2D6JE 494>2n]>U2B?A)/S><94p.=<>)
1Z e
ρ
DV
+ ∇p = µ∇2 V + (λ + µ)∇(∇ · V) + ρg
gJX./3
Dt
ρCp
DT
−
Dp
= k∇2 T + µφ
gJX.-3
Il]D),IU6J8:HD8A),I 8Y6J82DCQE 8:?:8: 498A)8A6J4>);S9<p.=<98A]D) (V = 0) ?`_a2DCFEG.=HD8A),IU2[6J8-]D),IUS>25490;.=HD8A),IU2
Dt Dt
6 25?:?~.wCFEG.=IQ<98:<pn6J8Ymw),<>)wS98|498A6JEU]D2B.
∇p = ρ g
s ∞
gJX 0Q
Z1e;g
O),<9<94p.,25IU6 )@CQEU2DS9<1_ E ?:<98:m@.@.=?:?~.o gJX./3bS98|G.
ρ
DV
+ ∇p0 = µ∇2 V + (λ + µ)∇(∇ · V) + (ρ − ρ∞ )g
gJXhe3
D2 S>S>25IU6 ) p ?~.6J83|254>25I Hv.4p. ?~. + 4>2DS>S98A),IU22*?~.+ 4>2DSpS98A),IU2*8A6J4>);S9<p.=<98A]1. p X
Dt
ρ̃
D Ṽ
+ ∇p̃ =
1
∇2 Ṽ + 1 +
λ
∇ ∇ · Ṽ +
1
(ρ̃ − 1) j
gJX ;
D t̃ Re µ F r2
ρ̃
D T̃
=
1
∇2 T̃ + Ec
D p̃ Ec
+ ϕ̃
gJX.3
IU25?:?A2[CFEG.=?:8 j 4p.=+ + 4>2DS>25IQ<p.n8:?-03254>S>),4>26 25?:?~.6J8:4>25HD8A),IU2\6 25?:?~.B),49Hv. u 4p.10Q8:<p.=HD8A),IG.=?A2B2D6
D t̃ P rRe D t̃ Re
IFE mw254>)/6J8Yyz]>R32549< Ec =
V∞ 2
Cp ∆T0
?bIQE mw254>)6J872QIU),?A6 Sw4p.=+ + 4>2DS>25IF<p.8:?b4p.=+ +),49<>)q 4p.c?A2o),49H12o6|_ 8:IU2549HD8~.2o?A2
),49H12{0Q8ASp]D);S>2^2D6 8:?7IQE mw254>)6J8 4 4>),EU6 2tCQEU25?:?A)q4p.$?A2^ ),49H12{6|_ 8:IU2549HD8~.2{?~.$ ),49Hv.
Z1
Ec = 2
∆T a gJXhg3
yzS9+ 498:mn8~.=mw)),4p. ?A20,.=498~.=HD8A),I 8/6J86 25IUS>8:<p 8:I#E I HD8A),IU2 6 25?:?A20,.=498~.=HD8A),I 8w6J8
∆T 0
0325?A)]58:<p V X
∞
Vr),IQ0325HD8A),IU2IG.=<9E 4p.=?A2
y_7CQEU2DS9<>) 8:?B]1.,S>) 8:I ]5E 8 8:?sUE 8A6 ) IU),I k6 ),<p.=<>) 6J8BE IG. 0325?A)]58:<p + 4>),+ 498~.
2D6 8:?}mw),<>)k^),498 u 8:IG.=<>)6U.=?:?~.t ),49Hv.6J8 u .=?:?A2 u;u 8~.=mw25IF<>) ]D),IUS>2 u EU25IQ<>2
.=?:?A2\6J38 -254>25I H126J8Y6 25IUS>8:<pJX
(V = 0)
∞
Re = 1
ν
Fr = p 3 Ec =
ν2 gJX:Z /3
gl l2 Cp ∆T0
8A]D),4>6U.=IU6 )]>U2 +-254j?`_h.=498~. 2 +-254x?`_h.,]M
CFEG.Jik8:mnmw2D6J8~.=<>)0325ν498=G]11.5 .=4>2· ]>10U2 Mm /sec
−5 2
kmw),?:<>)/+ 8A]D]D),?A)/ν2\=?~.{10 gJX:Z;Zvmmw/sec
−6 2
);S9<94p./]>U2B?~.
6 25IUS98:<p/+ EU2DSpS>254>2 498:<>25IFE <p.8:IU6J8:+-25IU6 25IF<>2w6U.=?:?~./+ 4>2DS>S98A),IU2o 8:+-),<>2DS986J8L8:IU]D),mn+ 498M
∞
D T̃
=
1 2
∇ T̃
gJX:Z1e3
.D0325IU6 )w8:IQ<94>)Q6 ),<9<>)@8:?& (" D t̃ Pr
6
β∆T0 gl3
Gr =
[( .=?:?`_a2DSp.=mw2W6 25?S98AS><>25m@.}6J82DCFEG.=HD8A),I 8 S98Q032D6 2x]pU2x8:?J]1.=mn+-)7]58:IU25m@.=<98A]D)*6J8:+-25IU6 2
ν2
6J8 u .=?:?A2 u;u 8~.=mw25IQ<>) X(,L),8A]pUkw+-254 V = 100m/sec. S98&G. ∆T ' 5 K i-8:?&IFE mw254>)
6J8&yz]pR32549<*k\8:I u 25IU254p.=?A2S9E .@]58A25IQ<>25mw25IF<>2n+ 8A]D]D),?A)6U.@+),<>254T<94p.,S>]5E 4p.=4>2 IU25?:?`_a2DCFEG.M
∞ a
◦
D T̃
=
1
∇2 T̃
gJX:Z 3
Vr),mw2oS98x032D6 28:I CQEU2DS9<>)t]1.,S>)c?A22DCQEG.=HD8A),I 86J8b]D),IUS>25490;.=HD8A),IU26 25?:?~.cm@.,SpSp.c2
D t̃ P rRe
+ E 4p.=mw25IQ<>28:IU6J8A]1.=<98:03) X
I CQEU2DS9<>)q]1.,Sp)qIU),I k^+ 8: +);SpS98: 8:?A2^.,6 ),<9<p.=4>2{?`_ 8:+),<>2DS9876J8}8:IU]D),mn+ 498:mn8: 8:?:8:<p
m@.kw.=IU]D),4p.o?A2D]58:<>)o<94p.,S>]5E 4p.=4>2w?~.o6J8:+25IU6 25I Hv.^6 25?:?~.o6 25IUS98:<po6U.=?:?~.o<>25mn+254p.=<9E 4p.JX
IJ`.=<9<98+254 V = 300m/sec. S>8G. ∆T ' 50 K iY]pU2n4p.=+ + 4>2DS>25IF<p.?~.m@.,S>S98:m@.
6J38 |254>25I Hv./6J8<>25mn+-254p.=<9E 4p./.=?:?`_ 8:IQ<>2549IU)6 25?]1.=mn+) i2D68:?L<>2549mn8:IU2 β∆T (T̃ − T̃ )
∞ a
◦
D ρ̃
+ ρ̃∇ · Ṽ = 0
gJX:Z1g3
D t̃
ρ̃
D Ṽ
+ ∇p̃ =
1
∇2 Ṽ + 1 +
λ
∇ ∇ · Ṽ
gJX./3
D t̃ Re µ
ρ̃
D T̃
=
1
∇2 T̃ + Ec
D p̃ gJX./JZv
-}IU]pU28:IqCFEU2DS9<>)l]1.,S>)l8:?z]1.=mn+-)o]58:IU25m@.=<98A]D)oIU),Iq6J8:+-25IU6 2n6U.lCFEU25?:?A)<>2549mn8A]D)
D t̃ P rRe D t̃
]pU2[+ EUw2DS>Sp254>2*498AS>),?:<>)wS>25+G.=4p.=<p.=mw25IQ<>2;X
Z1-
4L?:EUSpS98.,6^.=?:<p. 0325?A)]58:<p
-}?:?`_h.=E mw25IQ<p.=4>2[6J8 8:?0;.=?A),4>2}6J8 ]54>2DS>]D27mw),?:<>)B4p.=+ 8A6U.=mw25IF<>2[27?A20;.=498~.M
HD8A),I 8Y6J8<>25mn+-254p.=<9E 4p.nM.=?:?`_ 8:IF<>2549IU)@6 25?-]1.=∆T
∞
mn+-)n6J8:0325IF<p.=IU)n]D);S u 4p.=IU6J8Y6U.BIU),Io+-),<>254
a
+ 8:n<94p.,S>]5E 4p.=4>2b?A2W0;.=498~.=HD8A),I 8G6J8U6 25IUS98:<p*]>U2IU2b]D),IUS>2 u EU),IU) X IU),?:<94>2.=IU]pU28 ]D)2.nM
]58A25IF<98-6J8G08AS>]D);S98:<p2}6J8]D),IU6JEU]58: 8:?:8:<pB<>2549mn8A]1.IU),I/+);SpS>),IU)+ 8:2DS>S>254>2}]D),IUS>8A6 254p.=<98
]D);S9<p.=IF<988:I@CFEG.=IQ<>)\S98U6 25032}<>25IU254j]D),IF<>)B6 25?:?~.[?A),4>)[0;.=498~.= 8:?:8:<p]D),I@?~.*<>25mn+254p.=<9E 4p.JX
?W]1.=mn+)c]58:IU25m@.=<98A]D)$6J8:+-25IU6 2/+-2549<p.=IF<>)$6U.{CQEU25?:?A){<>2549mn8A]D)t2D68:?bS98AS9<>25m@. gJXh3i
gJX ;i gJ.X 3iU]D),mn+ ?A25<p.=<>)@6U.=?:?`_a2DCQEG.=HD8A),IU2\6J8YS><p.=<>)n26U.=?:?A2*?A2 u;u 8 2 ) iU6 25032
2DS>S>254>2b498AS>),?:<>)\S98:m\E ?:<p.=IU21.=mw25IQ<>2;XxK*.=<9E 4p.=?:mw25IF<>27IU25?:?~.@ gJX ;+ EUBµ(T2DS>S>)254>2bk(T
<94p.,Sp]5E 4p.=<>)
8:?|<>2549mn8:IU2\6J8 u .=?:?A2 u;u 8~.=mw25IF<>) i-2DS>Sp25IU6 ) u 4p.=IU6 28:?|IFE mw254>)/6J8 4 4>),EU6 2;X
&# *
$,
, !
δ̃ 1 , δ̃ 1
gJ.X //3
Vr),IUS98A6 25498~.=mw)+-254/S>25mn+ ?:8A]58:<p?`_a2DCQEG.=HD8A),IU2 gJ.X 3@IU25?*]1.,S>) 8A6J8:mw25IUS98A),IG.=?A2$2
T
δ̃
!2
1
= 0(1)
)1003254>) δ̃ √
= 0( P r)
gJX./10Q
4>25?~.=HD8A),IU2o]pU2k0325498G]1.=<p.$CQEG.=IU6 ) P r 1 8:?b]>U2.,]D]1.,6 2l+-2548 " ' )
δ̃T Pr δ̃T
.,62DS>25mn+ 8A)t8:?bS>)6J8A){?:8ACFE 8A6 )cEUSp.=<>)t]D),mw2l4>2498 u 254p.=IF<>2oIU258W4>21.=<9<>),498bIFEU]5?A21.=498bG.
X
IQ032D]D2nIU25?z]1.,S>)l8:Iq]5E 8 δ̃/δ̃ 1 8:?<>2549mn8:IU2w]D),IQ0325<9<98:03)^6 ),mn8:IG.=IF<>2@knCFEU25?:?A)
P r = .005
8:I y 26 )104pw.103254>S98 T
δ̃
!2
1
=0
δ̃
!
1) 003254>) δ̃
= 0(P r)
gJX./;e3
7EU2DS9<p. 4>25?~.=HD8A),IU20;.=?A2[CFEG.=IU6 )8:?-sUE 8A6 )wG. P r 1 ]Dδ̃),mw2*IU25?L]1.,S>)6 258 )
δ̃T Pr δ̃T T
Pr =
ν ]D),I α = k
α ρC
)v0Q03254>)n]D),mw28:?|4p.=+ +),49<>) 4p.B?~.6J38 EUS98:08:<p6J8YCQEG.=IF<98:<pn6J8|mw),<>) 2*?~. 6J38 EUS>8:0Q8:<p
p
ρ u
∂u
+v
∂u
=
∂
µ
∂u
−
dp gJX./;
∂x ∂y ∂y ∂y dx
ρCp u
∂T
+v
∂T
=
∂
k
∂T
+u
dp
+µ
∂u
2
gJX./3
∂x ∂y ∂y ∂y dx ∂y
1Z ;e
6 )1032WS98QkW.=IU]>U2r<>25IQE <>)T]D),IQ<>)*6 25?:?~.0;.=498~.= 8:?:8:<pT6J8 µ 2 k X&OQ8J);S>S>254908Q]pU2r?~.b+ 4>2DS>S98A),IU2
IU),I koE IY_ 8:IU]D) u I 8:<p.$m@.qk6U.t498:<>25IU254>S98IU),<p.$8:I G.,S>2^.=?:?~.$Sp),?:E HD8A),IU26 25?}]1.=mn+-)
2DS9<>2549IU)q2]pU2^2DS>S>25IU6 ).=?:?`_ 8:IF<>2549IU)6 25?:?A)qS9<94p.=<>)$?:8:mn8:<>2 ∂p/∂y = 0 ?~.t+ 4>2DSpS98A),IU2
6J8:+25IU6 2[E I 8A]1.=mw25IQ<>2B6U. x X
Y22DCQEG.=HD8A),I 8x gJX./;3 gJX./37.,S>S98A25mw2.=?:?`_a2DCQEG.=HD8A),IU2\6J8S9<p.=<>)n2D6{.=?:?A2*?A2 u;u 8
µ = µ(T ) k = k(T )
gJ.X /;g3
]D);S9<98:<9E 8AS>]D),IU)wE IS>8AS9<>25m@.n6J8e2DCQEG.=HD8A),I 8IU25?:?A2[8:IU]D) u I 8:<>2 u, v, ρ, T, µ, k X
!*
τ =µ
∂u
gJX.-JZv
∂y y=0
q = −k
∂T
gJX.-/3
∂y
}<98:?:8:HDHv.=IU6 )@?A2w gJX.-3W2@ gJX.-JZvr8:?-sUEUS>Sp)n6J8]1.=?A),4>2[+ EUwS>]5498:03254>S>8
y=0
q=k
(Tw − T∞ ) τ gJX.--3
V∞ µ
Y)nSN ),49H1) τ + EUw.=IU]pU2[2DS>S>254>2[2DS9+ 4>2DS>Sp)B8:IlE I HD8A),IU2[6 25?Y]D)2.@]58A25IF<>26J8.=<9<9498:<>)
cf
τ=
1 2
ρ∞ V∞ cf
gJX.-10Q
2D6l8:?YsUEUS>S>)n6J8Y]1.=?A),4>28:IlE I HD8A),IU26 25?]D)2.@]58A25IF<>2B6J8.,6 6JE HD8A),IU2
2
h
q = h(Tw − T∞ )
gJX.-;e3
IQ<94>)Q6JEU]D25IU6 )^8:IJUIU2n8:?IFE mw254>){6J8zK}EUS>Sp25?:<[?A)Q]1.=?A2w2D6t8:?jIFE mw254>)^6J8 72QIU),?A6 S
?A)]1.=?A2
N ux =
hx
Rex =
ρ∞ V∞ x gJX.-;3
k µ
?~.o gJX.--3b498AS9E ?:<p.
N ux =
1
cf Rex
gJX.-;
]pU2kWIU),<p.*]D),mw2 )& * )/" & 2b]pU2b+-2549mw25<9<>26J8U),<9<>25IU254>28:?U]D)2.@]58A25IF<>2
2
.664
cf = √
Rex
2[CQE 8:IU6J8
p
N ux = .332 Rex
1Z
L.=?:03),?:<p.T498AS9E ?:<p.7+ 8:n]D),IF0325I 8A25IQ<>26 2UI 8:4>2W8:?QsUEUS>S>)*6J8]1.=?A),4>2;iQ.=I HD8A]>UkWmw2D6J8~.=IF<>2
8:?|IFE mw254>)@6J8LK}EUS>Sp25?:<1iJE <98:?:8:HDHv.=IU6 )w8:? & " & &
St =
h
=
Nu gJ.X -3
yzS>S>25IU6 ) P r = 1 iJ?`_h.=IG.=?A) u 8~.@6J8 }2IU),?A6 Sb+ EU@.=?:?A),4p.wS>]5498:03254>S98
ρ C V
∞ p ∞ ReP r
St = c
1
f
gJ.X -;g3
2
#*+
S98|G.
" ! !# " 2 #
∂ u2 ∂ u2 ∂2T ∂u ∂2u
ρCp u T+ +v T+ =k 2 +µ +u 2
∂x 2Cp ∂y 2Cp ∂y ∂y ∂y
]>U2;iU]D),I^Sp25mn+ ?:8A]58Y+G.,S>Sp. u;u 8L.=IG.=?:8:<98A]58`iS98-498A6JEU]D2B.
ρ u
∂T0
+v
∂T0
=
µ ∂ 2 T0
"
+ (P r − 1)
∂ 2 (u2 /2Cp )
#
gJX 0UZv
∂x ∂y P r ∂y 2 ∂y 2
K725?U]1.,Sp)*8:I]5E 8 i3CQEU2DS9<1_a2DCQEG.=HD8A),IU2b498AS9E ?:<p.T8A6 25IQ<98A]1..=?:?~.w gJX./;L8:I@.,S>S>25IJM
Hv./6J8 u 4p.,6J8A25IQ<>2n6J8P+ 4>r2DSp=S98A),1IU22\+-2549<p.=IF<>)oS>2 u k S>),?:E HD8A),IU2n6 25?:?~.t gJX./;i-?~.c gJX./3
Sp.=4pS>)6 6J8ASN .=<9<p.w6U.=?:?~.nS>),?:E HD8A),IU2
T0 = au + b
IU25?:?~.CFEG.=?A2*?A2[]D);S9<p.=IQ<98 a, b 6 2503),IU)w2DS>S>254>2*6 25<>2549mn8:IG.=<>2[8:Iomw)6 )n6U.S>)6 6J8ASN`.=4>27?A2
1Z
]D),IU6J8:HD8A),I 8&.=?Y]D),IF<>),49IU)
y=0 T0 = T = Tw , u=0
2
V∞
y=∞ T0 = T0∞ = T∞ + , u = V∞
2Cp
OQ8),<9<98A25IU2
T0∞ − Tw
a= b = Tw
V∞
2*?~.n<>25mn+254p.=<9E 4p.nk\CFE 8:IU6J86U.=<p.6U.
u u2
T = Tw + (T0∞ − Tw ) −
V∞ 2Cp
{2D6J8~.=IQ<>2?~.o gJX.-/3WS98|+ EUw+),8Y]1.=?A]D),?~.=4>2\8:?-sUEUS>S>)n6J8]1.=?A),4>2\.=?:?~.n+G.=4>25<>2
q|y=0 =
k
(Tw − T0∞ )
∂u
gJX 0 /3
,L),8A]pUk .=?:?~.+G.=4>25<>27k}+);S>8:<98:0,.JiF8:? sUEUS>S>)B6J8]1.=?A),4>2}k76J8:4>25<9<>)\6U.=?:?~.\+G.=4>25<>2
V∞ ∂y y=0
y T0 y T0 y T0
T V2 /2Cp T V 2 /2Cp T V 2 /2Cp
T T0 T T0 T T0
( )dT >0
dy ( )dT =0
d y y=0 ( dd Ty ) < 0
δT y=0 y=0
T T T
Tw Tw Tw
48 u E 4p.wgJX./ 0
1Z ;g
.=IU6U.=mw25IF<98LCQEG.=?:8:<p.=<98:08Y6 25?:?~. <>25mn+254p.=<9E 4p.2[6 25?:?~. <>25mn+-254p.=<9E 4p. <>),<p.=?A2\.=?:?`_ 8:IF<>2549IU)
6 25?:?A)TS><94p.=<>)7?:8:mn8:<>2rIU258J<94>2W]1.,S98J6J8 T T XVr),mw2rS>8Q032D6 2;i;S>),?A)TIU25? ]1.,S>)T6J8J+G.=4>25<>2
.,6J8~.=G.=<98A]1.o?~.l<>25mn+-254p.=<9E 4p.o<>),<p.=?A2wS>8m@.=IF<98A25IU2@]D);S><p.=IF<>2/.=?:?`_ 8:IQ<>2549IU)^6 25?:?A)S><94p.=<>)
<
w > 0∞
T T0
T T0
Taw T
4L8 u
E 4p.@gJX.- 0
(T8L]D),IUS>2 u EU25I Hv./?~.n<>25mn+-254p.=<9E 4p.l.,6J8~.=G.=<98A]1.@6J8+G.=4>25<>2 T 498AS>E ?:<p.wmn8:IU),4>2 6J8
X aw
T 0∞
L.=?A2B6J83-254>25I Hv.+ EU@2DS>S>254>2[CQEG.=IF<98G]1.=<p.n8:IQ<94>)Q6JEU]D25IU6 )/8:?-`.=<9<>),4>26J8|4>2D]5E +254>)
r=
T −T aw ∞ gJX 0 -3
]>U24p.=+ + 4>2DS>25IQ<p.l8:?&4p.=+ +),49<>)4p./?`_h.=E mw25IF<>)6J8&<>25mn+-254p.=<9E 4p.l6 )10E <>).=?:?`_h.=<9<9498:<>)2
T −T 0∞ ∞
Z
20 20
15 15
η η M=5
10 10
M=5
M=4 M=4
M=3
5 5 M=3
M=2
M=1 M=2
M=0 M=1
0 0
0 0.25 0.5 0.75 1 1 2 3 4 5 6
.3
u’
T’
4L8 u E 4p.@gJX 0 0
OQ8F);S>S>25490;.b]pU2;i=.=?F]54>2DSp]D254>2j6 25?3IQE mw254>)}6J8Ft.,]>Yi?~.b6J8AS9<9498: E HD8A),IU2r6 25?:?~.W0325?A)]58:<p
=. ?:?`_ 8:IF<>2549IU)6 25?:?A)oS9<94p.=<>)?:8:mn8:<>2<>25IU6 2w.6J8:0325IF<p.=4>2@S>25mn+ 4>2 + 8:c4>25<9<98:?:8:IU21.2 ]pU2?A)
S9+2DSpS>),4>2n6 25?:?A)S9<94p.=<>)o?:8:mn8:<>2.=E mw25IF<p. +254 M = 5 kw]58:4>]1. 003),?:<>2n+ 8:S9+2DSpS>)
]>U2tIU25?[]1.,S>)8:IU]D),mn+ 4>2DS>S98: 8:?A2vXVx8Akt6 )v0QE <>) .=?:?~.6J8:mn8:IFE HD8A),IU26 25?:?~.6 25IUS>8:<p
∞