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Động lực học chất lưu (Fluid dynamics) doc

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Động lực học chất lưu (Nd: Fluid dynamics)
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#$!% !& '()*) +,#-.)/ '0"122
3+ 4' ,*
Cơ học về thể liên tục (Nd: Continuum mechanics)

Các định luật (Nd: Laws)
5"6 70!% '#-8 ,*4 9+,,2
5"6 / '0!% '#-8 ,*4 9++ +2
5"6  : '0!% '#-8 ,*4 9 *')2
; <  *)#-= *) >0)?@2
Cơ học chất rắn (Nd: Solid mechanics)
8<*
A ',<#-5*,,2B5"C D '#-E9*+ 2
; !F ';#-8+0)2
5G: 'HD #-I ,* 2B5G: '4J #-K 9 ,+0,* 2
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5"7 #-  '2B$ 0O(
(1)C!P '#-I0**)2
8F1G'Q)#-I**+ ,2
Cơ học chất lưu (Nd: Fluid mechanics)
8<0!#-I0,2
R 1<0!#-I0,,2BĐộng lực học chất lưu
5G: 'S+T#-5*9 , 2
8!F '*U -4*V5,
$/ &#-W,,)2
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Lưu biến học (Nd: Rheology; ?)
;  L &#-W,0,)?@2
8<0!Y '+ #-5+*90,2


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$[ 0!C #-=0**0'0?@2
8<0!\,#-I**90,?@2
5"0!C #-]+*)?@2B^!C C#-]+*?@2
Các nhà khoa học
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-4*B-X B5,
4bb

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+;+g8/Dh)#-4*h' **,2+0ifh 0J)#-'* 2jk)#-
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*90X2gs++#-@2o ' ': *nl ' '!%4p ',<n;,: '>
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* '4O0y1động lực học chất lưu0+/+Y 1 #-,,0 2.F1<0!+
'6>)Cdòng chất lưu (Nd: fluid flow)V+Y 1" c #- *0, 24S<0!
#<0P '#-0>,24<;2 ')d / '#- + 2g6 J  \o '\4
+Y 1 'L+;/ '0"1#," 'c G4SY ';4; ')d / '24
thủy động lực học (Nd: hydrodynamics)#," 'c G4S<0P ' ')d / '2g8F1

<0!\+/D+4G 'i '*/ ''L+4[;  0"4+Y+ #-++ ,? )
m0+4&_++ +`#,7;2)_++ `#,7 S2\ 'R0_/ '0!% '`2*c +))
#-**9? ),7;,7 S ! 2h 7/l '7#-+,,90X*?@2.
+P#-*0+? C0_0t`U'J) +0u 4&_*, `2>!v '7 'c  
+YU #-* ,2vCdC#- *,  '2 4J #- 02* 'Y '
' 'H,#- *,00*24DU #-+0 '2d! #-**0)2,";
z4o;J D#-9,, X   ?n4O)+& \ H ' 1#Y ' \
 ' e ' e++{{{20 H 'MQ+  <#4\1 <2. J 0D2gW* ',7
')c 0y. \O+;!%p '* ' ' |,!'Y '#-*99 ' * '?'Y '
Y ' @2n\,"'Y '!%ht0y !+/<0!0c i#-  ,902g
$/ '0"1<0!!*+/<*j\[7 '+0+ S 6 '+Y 1"m  )
+'L+ 0O  '[+4 t  '[+#-+*0 ,++*00X,2
+ 'L ,"Dl '6)#-90X+,*+ 24!%p 'd'6 "
m g}6+/ / '0"1<0!cd'L+4[;   S/; 
.<0! !04O 7,<w*1 '#- ,)24 [/ !+#-9  ,2
Y '' 4v' g
WS0,tthủy động lực học (Nd: hydrodynamics)\ 'R0'U\+4& \ ') )g*!&
C~G+!F.)/ '0"10L ' 'R4&/ '0"1<0!g$S )4u !%6 
 * 'c .4Sf.S#-,2/ '0"1<0! !0.)/ '0"1
#-+' )*) +,24; z  .)/ '0"#-)*) +,0)2•+6
S\6 : 'G 'i 'o ' ! '!%G 'i '4#-0,00 ,X00,
 '02<;g€•
Contents
• => ,990) +,
• g8+*,,04, +*,,090X
• gW,,4, 4,90X
• g‚5)4, ,)90X
• g^+ *4,*0 90X
• gƒ-X  4,  -X  90,
• g„5, 4,* , ,*,  )*, 90X,

• gZ' )*) +,
• g…†**h+ ,
• *+ 0') 90) +,
• g*+ 0')  +*,,090) +,
• g*+ 0') +*,,090) +,
• ‚50,
• ‚gI0,9,)
• ‚gZ+0> ,  ,
• ‚g‚),99090X
• ‚gI0**,
• ‚gƒI0 + 
• ‚g„‡0 ,
• ‚gZ,00 ,
• ]9* ,
• ƒ-,
• „=h* 00 ,
[edit] Các phương trình của động lực học chất lưu
8c S S 6 '#-9  0h+,2./ '0"1<0!0 0O6 +/
)c [0,"6 70!% ',"6 / '0!% ')C ; #-0 *++ +2
#+o '!%C ! 0OG U.-X 4S)d / '#--X ˆ,5 ^X9
Z 224,"6  : '0!% '#+o '!%C !$ 0OG <4S [/ '0"1
#-*+) +,22g-H ' )!%"*c F1zd #-0,,0+ ,24!%
C 6* 'F10!% 't#-> ++ ,24; !F '7 '#-' *0*04)2g
8j '!%dm #-h*,,2x 'p '$ 0y4O )d ]) 0,#-]) 0,
* ,**+2g
-' H 'Sn*c <0!!%'6 0J giả thuyết về thể liên tục (Nd: continuum
assumption)g8<0!'L+J t+4D+4& 44&4Od* g) c '6
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0x ',7;#-
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[edit] Dòng chảy nén được đối với dòng chảy không nén được (Nd: Compressible vs
incompressible flow)
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#-gg?,2

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[edit] Dòng chảy nhớt đối với dòng chảy không nhớt (Nd: Viscous vs inviscid flow)
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 2j ' c p '!F '*U l '6) &g-!!%+ 1n '0y#-
*h2ˆ‡0+*+/4Od* '+/<0!Y ' &,rY '*6>0"6  #-X00
h*  *'9*2g8!F '*U ck 4Sl '6)Y ' &0!F '*U 
=0*gZ/+YU !v '!%p 'T[* '/ '0"1<0!+); 0p '
!F '*U =0*h4Od#-X)9*+)24!F '*U 0&* '&+,
O/ &* '+/4p '+' 4&4Odg
8!F '*U =0*\d!%•+4&+/l '6)#-  '*0 '
,*+0 2d\!%#-'2!F '*U * 00gsl '6)#-90X20Y 'h)#-
** 024Y ' &n F!F '*U * 00\d!%p ',7>*!v 'l '
6)#-*'90X902g8l '6) !C!%'10l '6)C#- 0
90X,?C0C @2g
[edit] Dòng chảy ổn định đối với dòng chảy không ổn định (Nd: Steady vs unsteady
flow)


5"'60O.)/ '0"1.; <z  ])0'V)0*€•
s<6D+v' #-+*44,2.+/*!v 'l '6)SC +<l '
6)!%h+ !0+/dòng chảy ổn địnhgEl '6)*D 'z  wC S[ +n\
/; .<0!n+/d+* '[7 'Y ')z>v' g-CY '#-
†*X,2l '6)!%'10Y 'z  gW[+/l '6)*c '[0z  )Y 'z 
 #-**0*90X,,)* ,)2\dp)4 '+C#-9*+9
*9* 2!%1 gW;il '6)0&fC #-0+ *90X2*c f>#-4*2+/7
0z  * ' '+C+7 74&7g* '+/ '+C+7 
74&+/l '6) S f;,#-'* 90X2l '6) )0Y 'z  g
8l '6)Œ 0D f/ h)#-*0 20Y 'z  4S+T  'R#-)
9  2g) c +/l '6)/ h)\d0R D4S+T7 'c#-,,00)
, *)2gN€‚•
*!v ' 'u c #-* +902U#xt20R D4S+T7 'c C<6,"7 'c
#-,,,2S<C !&+/,")d 4Sv' #- *,9 +2g
$S )D\ 'R0<6/; 7 'c0Y 'zv' g!v 'U
*!v '* 'U #-+ 90207!% '!%> J+#-‘9 *,24 )
o 'Y 'z* '+/l '6)R D4S+T7 'cg
8l '6)z  !v 'mht0yfmŠ#-+**02F l '6)Y 'z 
 !F '"#- *X,,+0* ,)90X,2g8!F '*U Sw #-
'4*  '> ,2.+/ z  \;F !F '*U Sw .p ' 0
+/S#v' 2+Y '0%i '; z  .*!v 'l '6)#-9,)*0+
4 + , 0,,#+2 '4*  '> ,9,+*0+X '
4 '9, ,,990X902g
[edit] Dòng chảy lớp đối với dòng chảy cuộn xoáy (Nd: Laminar vs turbulent flow)
5"/ h)f," m/ '#-*0 20l '6)+!% cT; n,"  f,"
>)4l '#-**0 2)d / 'h)#-,24;  'u c #-* + ,,2
*Š* 'fdC gEl '6)+* '\," m/ 'Y '!%d0/U!%'100&fC 
#-0+ *2g) c  c jy*x '#-K,0 X4*2w\,"[ [ .
h)#-,2),"  UY ' < Cw*l '6) m/ '•[ 
!% ' )o '\d[ [ * 'l '6)0&gWS+T 1l '6) m/ '!v '

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0\d!%g8D C#-],* ,2Up)4Y ',<.+); +!%p '4; 
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#]“*[2g8 +))4O 6# !0*c +/C‡*,‡‚) '2\,7
]) 0,*[#"*c J) '. #-X '*22g$d'6 l '6)v
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,*,,,2p," m/ 'o '': ',"*)S  [470!% 'gZ/!F '1#-
+0')2G” 0,"'60Oh)0& #-0*'),+0 2#^=52T[* '0&
07f0&4P#-',2.,"'60Oh)*v#-),+0 2#E=52•+0+/,"
C%.,"" '+YU  m/ ']‡-54,"'60Oh)0& g
[edit] Các chất lưu theo Newton đối với các chất lưu không theo Newton (Nd:
Newtonian vs non-Newtonian fluids)
$G '#-5*2K,-X Q<)G ',<47/C D '*<' 'o+/)C ; 
4&<0!> / !C #-,XX,*,, *9,* *4*)0,
0 *0)*09*+ )9+0*90,?8J )>>{{{2 !0 !&4Y ';g8<0!

-X  )!%" '+YU n+/[,7#-99 2+!%'10/ &+p)/
4<0!)c [g
) c 4* ',74O0[ !0 o!F '#-+0, ,240D4Hh+: '
#-,0**,24+/,74O0[4 L4 &#-4,0,+*0,2#4gg++/,7<
*p '%#-0)+*,22\ H '," htC D 'VG ',<không theo NewtonGDF 
#-+*+0non-Newtonian,*,,,* 4*,2g84O0[ )'L+các chất lỏng
dính (Nd: sticky liquids) !00h+O '4<Y*F f v #-0* ,2+!%
'c G* '+Y 1 0!C 1g
[edit] Các dòng chảy hạ âm đối với transonic, siêu âm và siêu thanh (Nd: Subsonic
vs transonic, supersonic and hypersonic flows)
* ' Sl '6)*c *<fY '!v '#-**,*090X,2#4ggl '6). !&
>+/7 '2h6)*n,7+< Sl '6)!%> J+* '"m #-+ )90X,
9*0 *,2#4gg* ';/ '0"12h6)*n #-'9* ,2.,7Z
Z“)4!%> \#-* h,,92#l '6),cJ+2g8[ !% '+&h6)*n
C/,7Z )#-,Z +**'+,2#4gg,\ 'h ';#-,X4,2
l '6),cJ+; <z  * , #-* ,  ,0)2* '+/C/#-*'+2
l '6)4&Z' x '," ht\1Y 'J x '#-  >0*++0
4*2+," \* 'l '6),c 24,"ht0y+Œ* ',7C/l '
6) )+/*c '[0 Cg
[edit] Từ thủy động lực học (Nd: Magnetohydrodynamics)
Zi; .)/ '0"1
.)/ '0"10," 'c G+Y f ' #-+0,0 *),)24Sl '6).
<0!u [ #-0*00)  '2* '*!v '[ g84;i4S<0!
!C'L+)C!F '#-0,+,2+0D0P '#-0>+0,24 !&+7g8
!F '*U l '6)<0!!%'6L 'v4&!F '*U ZhX00.+Y [ 1
#-0*+' ,+2g
[edit] Các môn khoa học tương tự khác (Nd: Other approximations)
8\+/,70& +Y !F '"\d\74& / '0"1<0!gW* ',7
+Y !%p 'zC F !%0[cc !&g
• Tính tương đương Boussinesq (Nd: Boussinesq approximation)P> H ',"C c 4S

w*1 ' 'D*4[; 0" z#-4* ,  ,)h00) )
9*,2g-\!v '!%p '* ' 70!"#-9* 4 *0+,2+
n\)z4Sw*1 '0 Pg
• Học thuyết chất bôi trơn (Nd: Lubrication theory)4dòng chảy Hele-Shaww,7
;D #-,*20& .4p 'f+S f0R 4"#-+ 2d<)*x '
6  \#-* *+,2* '!F '*U 0 P44U4O)\d!%P>g
• Học thuyết thân mảnh khảnh (Nd: Slender-body theory)0+/!F '1#-
+0')2+!%p '* ' l '6)5,d!&0!% '0"*c )
*!v 'l '6)> +/4Od+6 6 * '+/<0! &g
• Các phương trình nước nông (Nd: Shallow-water equations)\d!%p 'd+Y6+/
0&<0!!F '7Y ' &4&+/S+T"+* '\) /#'* ,2S
+T0 Pg
• Các phương trình Boussinesq\dG 'i '!%,\ 'S+T#-,*9X4,2
*c 0&<0!)F 44&/7#-,0,2S+T7F #-,*2g
• Định luật Darcy!%p 'l '6)* '+Y*!v 'h7#-*,+240+
4[4&C ,7+!%; * 'U *c 4S*/ '0Œ#-X4*0,4*'
4*,4*0*X,2g
• * '[7 '>)#-* ',),+,2tính tương tự quasi-geostrophic (Nd: quasi-
geostrophic approximation)'6 +/,"J x ' ! 6#- 0+,*9
0 2'H) /,<#-*,,*'* ,240"8*0,g-\\;* ',"
'c G4S/ '0"1;>)d #-+,*) +,2g
[edit] Thuật ngữ học (Nd: Terminology) trong động lực học chất
lưu
s [+,<0* 'J+.," 'c G4S#-, *0,)926R 1<0!
4/ '0"1<0!gZ/,<\d!%L ' <\#- 92+1d+* '+/
7<0!#-)9902<<4[<0!\\)d / ')Y 'g•,<\d
!%x 'p '+/C#- *27 '* #-* 2/.) 'J #-
+**)0+ 2) S!F ' g
WO 'H#-5+9*+ 0')2+ C* ' 'c G/ '0"1<0!Y '
!%U+<)* '0R 4" 'c G!F '"g$T[4O 'H+!%p '* '

/ '0"1<0!UY '!%p '* 'R 1<0!g
[edit] Thuật ngữ học trong động lực học chất lưu không nén được
8 [+4S,< d#-0*,,*24,</ '#-) +*,,*2 6), 
!F '*U * 004 'd* '," 'c G4S<6l '6)<0!#(,< )
Y '60,<y 'RU !v '#-,0, ,2•j 'Y 'd!%x '
p '+/C7 '* )/.) 'J 2g$d* ,"+FLS+ 'wC ,<
* '/ '0"1<0! S'6p 'O 'H,<R #-,*,,*2dJ [
\4&,< d4,</ 'g•,<R '7 '[4&,<4\d!%L ' <\
+1d+* '+/*!v 'l '6)<0!g
* 'Aerodynamics (Nd: Khí động lực học)^g3g80 )4C€ƒ•Để phân biệt nó với các áp suất toàn
thể và động, áp suất thực của chất lưu, mà không liên quan đến sự chuyển động của nó mà với trạng thái
của nó (Nd: which is associated not with its motion but with its state), thường được chỉ đến như áp suất
tĩnh, nhưng ở nơi mà thuật ngữ áp suất được dùng một mình, nó chỉ đến áp suất tĩnh này.
Z/d+* '+/l '6)<0! F+l '6)QC d ' '0D#-‡  90
90XX*90X,+*,2#G07/0x ',7Y '+CO 4&4Od*  \+
!% j '* 'l '6)<0!2\+> *1 'T[#-,9,0,' 9 2g-\
> *1 'C  Œ+#-K,9,+* 2 \!%+/c T[•+/d+p
Q+fG1 'fU *[#-,'   2g•,<R nd+G1 '\y 'RT[4!%
c .;  \•,<G1 '#-,'  *,,*2g* 'l '6)Y ' e !%
,<G1 'n+/d+G1 '0x '4&,< d* ',7*!v 'l '6)g
[edit] Thuật ngữ học trong động lực học chất lưu nén được
* '+/<0! e !% !0Y ';,<4w*1 '0C)Ch *D '.
<0!g-' [+,< d#+o '!%C !,<G1 '2 [+4S
[/ d#)G1 '24w*1 ' d#)G1 '2o 'C)C* '<U 'c G 
4Sl '6)<0! e !%g$d* ,"+FLS+ 'wC  [/4w*1 ' S
'6p 'O 'H [/R #-,+**24w*1 'R #-, ,)2g-[
/R '7 '[4& [/4w*1 'R '7 '[4&w*1 '?46\d!%L ' <\
+1d+* '+/*!v 'l '6)<0!g
-[/4w*1 'n+/d+G1 '!%'10 [/G1 '4w*1 'G1 'g
Z/,"CO !F '"o '!%"[ 4&/;  [/ '0"1.<0! e 

!%g-S'6p 'O 'H 0) d#)G1 '24 *) d#)G1 '2g
8O 'H 0)R 4 *)R \4M !;Y 'i 'F  ! 'n F+j '!%
p 'j '\ 'RY ''U#-)+   '+* 2 0)4 *)+/
!F 'G '4C 'HfS 7#-*9h2–R #-,2– '!%p 'd* ,"+FL4&
!F 'G '#- **,2ˆ d#-02ˆ)ˆG1 '#-,'  2ˆ.j 'gE
S[ l '6)ˆ dˆ!%  'Rn,"!<0!4S*D ' 'wd— '
 *)#-), *00)* ' '90*,2 *) d#)G1 '2  'R0
0Y x '4& *)–R –g
[edit] Cũng hãy xem
[edit] Các lĩnh vực nghiên cứu
• (1)Cf0y)C4SJ+ #-‡,*)2
• s;/ '0"1
• ;  L.;#-‡*0,)2
• ( 'Y '1#-‡* ,2
• $/ '0"1<0!+); 
• 5"l '6)
• ()C/ '0"1#-(+) +,2
• .)0"1#-()*0,2
• .)1#-()*0')2
• .)R 1#-()*,,2
• $[ .)/ '0"1#-=0*)*) +,2
• .)/ '0"1#-Z' )*) +,2
• ^!C 1#-]0')2
• ^!% 't.)/ '0"1#-˜ +)*) +,2
[edit] Các phương trình và các khái niệm toán học (Nd: Mathematical equations and
concepts)
• (1)C,\ 'Y ';#-‡*)X4*)2
• N!F '*U * 00
• $ 0y4O )d ]) 0,
• N!F '*U  ‘+ V VZ )

• 5"!F '!F '#-*h+ 2,, ,>#/ z2
• 5"!F '!F ',, ,>#,\ ' !&2
• 8 0O6 
• 8!F '*U =0*#/ '0"1<0!2
• $ 0OE*)
• •,</ '#-E) +*,,*2
• R 1<0!#-I0,,2
• 8 0y(0+0‰
• 8!F '*U s*99
• N!F '*U Z  '
• N!F '*U 7 ”f6#-Z0,0> 2
• N!F '*U Z*, 
• 8!F '*U -4*V5,
• El '6)†, 
• $ 0ON,0
• $ 0ON,00
• El '6)C#-N 090X2
• •,<
• •,<R 
• $,<#-N*,,*2
• 8!F '*U =0*)C!F '7#-]04,2
• 5"J 0)#-+, 2]) 0,
• El '6)5,
• (+l '6)5,#-5,,*+9  2
• (+l '6)#-5*+9  2
• 8,*+0 ,*0 40 
[edit] Các loại dòng chảy chất lưu
• 5"D '#-84 ?U  !\0c > C  '!0Y540.-'2
• El '6) e !%
• El '6)8

• El '6)J t"#-I*+00*90X2
• El '6)Y ' e !%
• El '6)Y ' &#-K 4,90X2
• El '6)— ' [#-K,*+090X2
• El '6)0&fC #-^+ *90X2
• El '6)c n#-†  090X2
• El '6)G<#-5 *)90X2
• ([ !% ',c0P '#-5*90)2
• 5cJ+#-5*, 2
• El '6) <v#-* , 90X2
• * , 
• El '6) m/ '#-*0 90X2
• El '6)#-X,90X2
[edit] Các thuộc tính của chất lưu
• w*1 '#-E ,)2
• E ,; <z  .)/ '0"1
• 8<0!-X 
• 8<0!Y '-X 
• 5G: 'S+T
• $/ &f,[f, f )#-W,,)2
• •,<)F#-W**,,*2
• ([,7 e #-8+*,,0)2
[edit] Các hiện tượng của chất lưu
• ^&c #- *)0)*2
• ([G '8 #-8 99?\0c > &+/0D '+0D)x ' 
DJ .-'!%h+0p ',G '!v[>6F ,4&p 'J 4Y '!v 'g
8\x ', 'Cp '[G ' )#}'0N 5*22
• N 70!#-8 4 00?@2
• $//i#-8 4*' 2f*D '0!• 'a#-9* 2
• #^"26 #-E*'#9*22

• ; z  .)/ '0"1
• #^"2 J '#-^9#9*2? +))\ <C60Ln+T*c )  '+\
<C60Ln+T!&)Y '@2
• 8,\ 'S+TD!F '
• 5\ '],,)
• 5\ 'h ';#-5X42
• 50 
• 5,*9
• 5" m/ '#-*0 2
• ([G 'W *
• El 'h)# W*h2
• ; h)#-W*)2
• j !&#-*++*2
• ^"6 ,\ '#-4*'2
[edit] Các ứng dụng
• ™+1
• s;/ '0"1
• s1 >)d #-8*),*, ?@2
• -: '0!% '<0!#-I0X*2
• Z)+\.)0"#-()*0+ *)2
• s;!% '1#-Z*0')2
• sC *j '6#--40**?Dd0 ' \ '@2
• (6!F '1#-† '*)2
• WO0y10,+f)C!F '#-N0,+),,2
• s14S; e #-N +,2
• Z)F+fF+#-N+2
• 50,) +,
[edit] Linh tinh/ lặt vặt (Nd: Miscellaneous)
• 8< k+#-0 ,2> *1 '4S/ '0"1<0!
• $— '[ #-K,,*9?@2

• 57s0' V8* *
• L h)#-] ' 2
• ( '*J+ #-5 ***2
• ZT— '#-0 2
• N!F 'c  j '#-K++*, *)+2
• 5"*)h!&#-*',*?@2
[edit] References
• ‡, Eg3g#šš2gElementary Fluid Dynamicsg80*  N*,,gK5-š…ƒš„šg
• 0*}gsg#š„2gAn Introduction to Fluid Dynamicsg8+*'› 4*,)N*,,g
K5-ƒ„„‚š„g
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[edit] Notes
1. ^=*Z0#„2gThe Dawn of Fluid Dynamics: A Discipline Between Science and
Technologyg0)gghgK5-‚ƒƒ‚ƒg
2. ^5 '^(^–N*000‡Z]89*8+*,,0Z(E*(E=> ,–#^,
‡0+,- 0^**)2€•
3. ^5N#2'ƒg
4. ^5N#2'‚g
5. ^80 )^g3gAerodynamics'

[edit] External links
+8++ ,,+*0Fluid
dynamics
+8++ ,,+*0Fluid
mechanics
• I0,   ',4*0'00*,990+ 
• - 08++9*I0Z ,I0+,#-8IZI2   '90+, ,4*0,‘, 
90) +,# *0+9*+2
• ^,9I0E) +,,
• I0Z ,‡,**,9*),,,
4bb
Dimensionless numbers in fluid dynamics
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E ·E*·=*·=+ ·=•4•,·=0*·I*·}00·}*‰·}*,9·}•*0*·(' ·
s0' V8* *·s , ·^0·^X,·Z·Z* ' ·Z* ·-,,0·† ,*'·
Ne0·N* 0#+' ·*0 2·])0'·]) 0,#+' 2·]*, ·],·],,)·
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4bb
General subfields within physics
‡,,·‡'*),,#50),,2·‡,*),,·‡+,*),,·‡++00* 
0),,·),,#Z0),,b-*),,2·8+0),,·8  ,+*
),,·= ),,·=0*+' ,+·}),,·Z ,#80,,0+ ,b8  +
+ ,b˜ ++ ,bI0+ ,b†+ ,b*+) +,2·-0*),,·†,·
N*0),,·˜ +90*)·]04)#50*04)b} *0*04)2
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8'*,8  ++ ,Ÿ‡*) +,Ÿ8+0 ' * 'ŸI0) +,ŸI0
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