trabajo potencial final!!!!! 2

8/18/2019 Trabajo Potencial Final!!!!! 2

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FLUJO POTENCIAL: MODELAMIENTO NUMERICO

UNIVERSIDAD NACIONAL MAYOR DE SAN MARCOS(Universidad del Perú, DECANA DE AMÉRICA)

“MODELAMIENO NUMÉRICO DE !LU"O #IDIMENSIONAL EN UN CANALCON CUARO PILARES$

(“TWO-DIMENSIONAL NUMERICAL MODELING FLOW IN A CHANNEL WITH FOURPILLARS”)

RESUMENEl comportm!"#to $" lo% &'!$o% '# co# lo% $!"r%o% "%t'$!o%r"l!*$o% "# l m"c+#!c $" &'!$o% %!,'" %!"#$o '# !#t"rro,#t" "#m'co% "%c"#r!o% o %!t'c!o#"% r"l"%. Trtr"mo% "# "%t" pr"%"#t"tr/0o $" co#oc"r '# poco c"rc $" "%t" comportm!"#to$"t"rm!##$o l $!r"cc!1# 2 "l %"#t!$o 3'" "0"rc"# l% l4#"% $" corr!"#t" lo lr,o $" '# c#l co# c'"rpo% pro#'#c!$o%. Lo clc'lr"mo%c!"#$o '%o $" '# %o5t6r" $" pro,rmc!1# 3'" #o% p"rm!t!r+o/%"rr l ,r+7c $" "%t% l4#"%. Tm/!8# clc'lr"mo% l $!"r,"#c!%4 como lo% "ctor"% $" "loc!$$ 2 cmpo% $" pr"%!o#"% $" "%t"5"#1m"#o. E%to %"r!r+ $" l,'# m#"r pr po$"r "r "lcomportm!"#to ,"#"rl $" to$o% "%to% pr+m"tro% "# "%t" c%oprt!c'lr 2 $"%p"0r l% $'$% c"rc $" l %ol'c!1#.

Summary T" /"!or o5 &'!$% ""# 6!t t" r!o'% %t'$!"% !# &'!$ m"c#!c%r"m!#% 3'"%t!o# !# m#2 %c"#r!o% or r"l %!t't!o#%. T!% pp"r$!%c'%%"$ !# t!% to l"r# l!ttl" /o't t!% /"!or $"t"rm!#!#, t"$!r"ct!o# #$ %"#%" "9"rc!%!#, t" po6"r l!#"% lo#, c##"l 6!t

pro#o'#c"$ /o$!"%. W" clc'lt" !t '%!#, pro,rmm!#, %o5t6r" ttllo6 '% to o/%"r" t" ,rp o5 t"%" l!#"%. Al%o 6" clc'lt" t"$!"r,"#c" #$ "loc!t2 "ctor% #$ pr"%%'r" 7"l$% o5 t!% p"#om"#o#. T!% 6!ll !# %om" 62 to %"" t" ,"#"rl /"!or o5 ll t"%" prm"t"r%!# t!% prt!c'lr c%" #$ t" $o'/t% /o't t" %ol't!o#.

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%& INRODUCCION

L %ol'c!1# $" pro/l"m% $" $!%";o 2 mo$"lm!"#to "# "l +r" $" lm"c+#!c $" &'!$o%< como "l &'0o %o/r" %'p"r7c!"% c'r% o r"ct% lolr,o $" c#l"% ' otr% ,"om"tr4% $" p%o $" '# &'!$o !mpl!c "lco#oc!m!"#to $"c'$o $"l p"r7l $" "loc!$$ 2 $" pr"%!1#. Noo/%t#t" l% %ol'c!o#"% $!%po#!/l"% "%t+# l!m!t$% %1lo pr l,'#o%%!%t"m% (,"om"tr4% %!mpl"%) 2 &'0o% '#!$!r"cc!o#l"%.U# /'"# lt"r#t! %!mpl" pr "%t% l!m!t#t"% l pr"%"#t l t"or4$"l &'0o pot"#c!l 3'" #o%otro% "mo% pr"#$!$o "# cl%" co# lo%$!5"r"#t"% m8to$o% $" %ol'c!1# 2 "%to pro"" '# /'"# pro9!mc!1# lr"%'lt$o $" "%to% pro/l"m% $" $!%";o 2 p"rm!t" o/t"#"r '# !%!1#

mpl! $" l% %!t'c!o#"% r"l"% $" &'0o.

E# "l pr"%"#t" tr/0o 7#l /or$r"mo% l %!m'lc!1# $" '# &'0o tr8% $" '# c#l co# $!5"r"#t"% ,"om"tr4% p'"%t% "# "ll. Lo 3'"#o%otro% /'%cmo% "% o/%"rr como "% "l comportm!"#to $"l &'!$ocomo %" comport# l% l4#"% $" corr!"#t" lo% "ctor"% $" "loc!$$ "#$!r"cc!o#"% $" "%t%< tm/!8# co#oc"r"mo% l% $!5"r"#c!% $" pr"%!1# l$!"r,"#c! 3'" p'"$" oc'rr!r co# l% ,"om"tr4% $$%< tm/!8#mo$"lr"mo% "l mo!m!"#to $" prt4c'l% $" %"$!m"#to l !#,r"%r lc#l. =C'+l "% %' mo!m!"#to po%!c!1# !#!c!l 2 po%!c!1# 7#l> %! "% 3'"

"%t% %" 3'"$# %"$!m"#t$% "# l% ,"om"tr4% c"rr$% to$o "%to lor"mo% co# l 2'$ $" '# %o5t6r" 3'" #o% p"rm!t" $!%";r 2mo$"lr pr o/%"rr "l comportm!"#to c%! r"l $"l &'0o $" '# &'!$o.

'& O#"EIVOS

O*e+ivs -eneralesN'"%tro o/0"t!o "% "%t'$!r "l comportm!"#to $!#+m!co $"l &'0o"%t'$!$o.

C'+l "% "l comportm!"#to $" '# &'0o tr8% $" '# ,"om"tr4 $$"# #'"%tro c%o "% l $" '# c#l m'2 lr,o co# '# #co m'2co#%!$"r/l" 2 "# "ll% p'"%t '#o% p!lr"% %!m8tr!c% "#tr" %! 2 m+%3'" to$o "% o/%"rr l% l4#"% $" corr!"#t" c'2o comportm!"#to "% tr8% $" "%t% p!l"t% c'l "% l $!r"cc!1# $" %'% "ctor"% $""loc!$$ tm/!8# %" "r+# l% $!5"r"#c!% $" pr"%!o#"% "#tr" l% l4#"%$" corr!"#t" 2 l% $!"r,"#c!% 3'" %" pro$'c"# "# '# co3'" co# l%

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p!l"t% tm/!8# "r"mo% "l r"corr!$o $" '# c#t!$$ $" prt4c'l "# "l&'!$o to$o "%to lo r"mo% co# "l %o5t6r" “MATLA@” "# "l c'l!#tro$'c!r"mo% l% $!m"#%!o#"% $" l ,"om"tr4 l% "c'c!o#"% pr "lc+lc'lo $" "loc!$$ pr"%!1# "tc. L% co#$!c!o#"% pr 3'" "l pro,rmm'"%tr" '# r"%'lt$o %"m"0#t" '# &'0o r"l.

O*e+ivs Es.e/01/sL% "c'c!o#"% 3'" '%r"mo% %" /%# pr!#c!plm"#t" "# l "c'c!1# $"Lplc" pl!c$o% l mo$"lm!"#to #'m8r!co c'2 5'#c!1# "% clc'lrc!"rt c#t!$$ $" coor$"#$% l% 3'" #o%otro% llmr"mo% #o$o% "#"l c'l co# l 2'$ $"l “Mtl/” po#$r"mo% '# mll "# c'l "#c0rco# lo% #o$o% pr '# m2or $" 5c!l!$$ co# lo c'l clc'lr"mo% l%

"loc!$$"% (pr"!o% c+lc'lo% m#o% 3'" r"mo%) l c'l %"r!r+#pr r"pr"%"#tr "l &'!$o ,rc!% "%t" %o5t6r" r"%ol"r"mo% lo%$!5"r"#t"% pro/l"m% 3'" %" #o% pr"%"#t"#.

2& MEODOLO-IA

¿CÓMO SE REALIZÓ EL TRABAJO?R"l!*mo% "l $!%";o 2 mo$"lm!"#to "# %o5t6r" MATLA@ "# "l c'l't!l!*mo% l% %!,'!"#t"% "c'c!o#"%B

α =( ∆ x∆ y )2

nx= L

∆ x+1

ny= H

∆ y +1

A=−2(1+α )

Ψ i , j(k +1)=

−(∝Ψ i , j−1(k ) +Ψ i−1, j

(k ) +Ψ i+1, j(k ) +∝Ψ i , j+1

(k ) ) A

!UNDAMENO EORICO

L0neas de /rrien+e

E% '# c'r !m,!#r! 3'" co#"ct '# %"r!" $" p'#to% "# "l "%pc!o"# '# !#%t#t" $$o $" tl 5orm 3'" to$% l% prt4c'l% 3'" "%t+#

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%o/r" l c'r "# "%" !#%t#t" t!"#"# "loc!$$"% c'2o% "ctor"% %o#t#,"#t"% l m!%m.

Ve/+r vel/idad

E9pr"% "l $"%pl*m!"#to $" '# o/0"to por '#!$$ $" t!"mpo< por t#topr $"7#!r "l "ctor "loc!$$ $"/"# co#%!$"rr%" l $!r"cc!1# $"l$"%pl*m!"#to 2 "l m1$'lo "l c'l %" $"#om!# c"l"r!$$ o rp!$"*.

Ca3. de .resines

R"pr"%"#t l $!%tr!/'c!1# "%pc!l $" l pr"%!1# mo%tr#$o c!"rtr!c!1# "# '# r",!1# $"l "%pc!o.

Diver4en/ia

E% l $!5"r"#c! "#tr" "l &'0o %l!"#t" 2 "l &'0o "#tr#t" $" '# cmpo"ctor!l %o/r" l %'p"r7c!" 3'" ro$" '# ol'm"# $" co#trol.

!5n/i6n Crrien+e

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D"%cr!/" l tr2"ctor! 3'" t!"#" '# prt4c'l $" '# &'!$o "%t 5'#c!1#Ψ "%t+ r"pr"%"#t$ "# t8rm!#o% $" l% coor$"#$% (92)

u=−d Ψ

dy , v=d Ψ

dx

MÉODO DE SOLUCI7N “DI!ERENCIAS !INIAS$

El m8to$o $" $!5"r"#c!% 7#!t% %" 't!l!*$o tr$!c!o#lm"#t" prr"%ol"r "c'c!o#"% %!mpl!c!$$ co#c"pt'l 2 %' 5c!l!$$ $"pro,rmc!1# "# '# comp't$or 2 3'" "# "%t" m8to$o l% "c'c!o#"%$!5"r"#c!l"% %" tr#%5orm# $!r"ctm"#t" "# "c'c!o#"% pro9!m$%"# $!5"r"#c!% 7#!t%. E# t8rm!#o% ,"#"rl"% lo% p%o% %",'!r pro/t"#"r l %ol'c!1# #'m8r!c %o#B

(:) S"l"cc!o#r '# m8to$o $" $!5"r"#c!% 7#!t% $"c'$o pr"%t" t!po $" "c'c!o#"%.

(?) D!%cr"t!*r l% "c'c!o#"% $!5"r"#c!l"%.() R"%ol"r "l %!%t"m $" "c'c!o#"% l,"/r!c% %4 o/t"#!$o.

El o/0"t!o $" "%t" m8to$o "% ll"r '# $"r!$ $" l 5ormδ

δ

l 5orm ∆∆

Por "0"mploB

lim ¿∆ x →0∆ f ( x)

∆ xδφ

δx =¿

Prt!"#$o 3'"B ∆ x →0

∆ f ( x )∆ x

= f

1−f

0

x1− x0=

f 2−f

1

x2− x1=

f 2−f

0

x2− x0

atrasado adelantado centrado

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d2

f ( x)

δ x2 =

d

dx (df ( x)

dx ) x1=( df ( x)dx ) x2−(

df ( x )dx ) x 1

∆ x

d2

f ( x)δ x

2 = f 2−2 f 1+ f 0∆ x

2

:"r $"r!$ →2 puntos

?$ $"r!$ →3 puntos

"r $"r!$ →5 puntos

S! ,"#"rl!*mo%

df

dx=

f i+1−f i∆ x

=f i+1, j−f i , j

∆ x =

f i+1, j−f i−1, j2∆ x

d2

f

d x2 =

f i+1−2 f i+f i−1∆ x

2

∇2Ψ =0→d2

Ψ

d x2 +

d2

Ψ

d y2 =0

u=dφ

dx=−d Ψ

dy

v=dφ

dy =

−d Ψ dx

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C5and el .rle3a es idi3ensinalf → f ( x , y ) → La función f se expresa de la siguiente anera

f i , j →i :variaci!nen x

j :variación en y

NODOS8E9!%t"# #o$o% 3'" #o pr"%"#t co#tcto co# #o$o% "c!#o%.

ECUACI7N DE LAPLACE∇

2Ψ =0 ,s i Ψ : funcióncorriente

DEMOSTRACION DE LA ECUACIN DE LAPLACE DISCRETIADA

df dx

= f i , j−f i−1, j∆ x

= f i+1, j−f i , j∆ x

= f i+1, j−f i−1, j2∆ x

df

dy=

f i , j−f i , j−1∆ y

=f i , j +1−f i , j

∆ y =

f i , j +1−f i , j−12∆ y

d2

f

d x2 =

f i+1, j−2 f i , j+ f i−1, j

∆ x2

d2

f

d y2=

f i , j+1−2 f i , j+ f i , j−1∆ y

2

De la e/5a/i6n de La.la/e∇

2Ψ =0

d2

Ψ

d x2 +

d2

Ψ

d y2 =0

Dis/re+i9and ls +:r3ins8

Ψ i+1, j−2Ψ i , j+Ψ i−1, j∆ x

2 +

Ψ i, j+1−2Ψ i , j+Ψ i , j−1∆ y

2 =0

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∝=( ∆ x∆ y )2

Ψ

(¿¿ i , j+1−2Ψ i , j+Ψ i , j−1)=0Ψ i+1, j−2Ψ i , j+Ψ i−1, j+

∆ x2

∆ y2 ¿

Ψ

(¿¿ i , j+1−2Ψ i , j+Ψ i , j−1)=0Ψ i+1, j−2Ψ i, j+Ψ i , j+∝¿

Ψ i , j−1+Ψ i , j−1−2 (1+∝ ) Ψ i , j+Ψ i+1, j+∝Ψ i , j+1=0

A=−2(1+∝)

Ψ i , j−1+Ψ i , j−1+ A Ψ i , j+Ψ i+1, j+∝Ψ i , j+1=0

ENONCES8

Ψ i , j=Ψ i , j−1+Ψ i , j−1+Ψ i+1, j+∝Ψ i , j+1

A

COMO SOLUCIONAMOS Y CALCULAMOS

∆ x=2 , ∝=( ∆ x∆ y )2

→∝=1 ,u "=4 /s

∆ y=2

nx= L

∆ x+1→ nx=

680

2+1=341

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ny= H

∆ y +1→ ny=

100

2+1=101

A=−2 (1+∝ )=−2(1+1)=−4

-ENERANDO LA MALLAS" $"/" '/!cr l% coor$"#$% $" #9: #9? #9 #9 #2: #2? #2 2#2.E%t/l"c!"#$o l% co#$!c!o#"% $" 5ro#t"r pr "l "0" “9”

• nx1=

220

2+1=111

• nx2=

240

2+1=121

• nx3=

420

2+1=211

• nx 4=

440

2+1=221

E%t/l"c!"#$o l% co#$!c!o#"% $" 5ro#t"r pr "l "0" “2”

• ny 1=

50

2+1=26

• ny 2=

50

2+1=26

ES;UEMAI


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El /lo3'" %'p"r!or (:) t!"#" coor$"#$% (92)

(92)K :< ?< ?< :

El /lo3'" %'p"r!or (?) t!"#" coor$"#$% (92)

(92)K ::< :?< :?< ::

CONDICIONES DE !RONERA

A-UAS ARRI#A Y A#A"O

Ψ 1, j=# " ( j−1) ∆ y, j :1,2,3,$$%,51

Ψ nx , j=# " ( j−1 ) ∆ y , j :1,2,3,$ $.,51

!RONERA IN!ERIOR

Ψ i ,1=0i :1,2,3,4,5, $ $ $ $ $ ,341

!RONERA SUPERIOR

Ψ i ,ny

=400 i :1,2,3,4,5, $ $ $ $ $ ,:

EN LOS PILARES8

#LO;UE IN!ERIOR (%)

Ψ (111, j )=0 , j=1,2,3,.......26

Ψ (i , 26)=0 , i=111.112.113$ $ .121

#LO;UE IN!ERIOR (')

Ψ (211 , j )=0 % j=1,2,3, $ $ $ .. ,26

Ψ (i , 26)=0 , i=211%212,213$ $%221

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#LO;UE SUPERIOR (%)

Ψ (56 , j )=400 , j=26,27,28,$ $ $ $ ..51

Ψ (i , 26)

=400 ,i=56,57,58, $ $ $66

#LO;UE SUPERIOR (')

Ψ (150 , j )=400 , j=26,27,28, $ $ $ $ ..51

Ψ (i , 26)=400 ,i=150,151,152, $ $ $ $ .160

SOLUCIONANDO LA ECUACI7N DE LAPLACE DISCREILCULO DE RANSPORE DESEDIMENOS

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CONSIDERACION8• El %"$!m"#to %" m'"" %o/r" '# &'0o pot"#c!l• El p"%o $"l %"$!m"#to "% c%! $"%pr"c!/l" (#o %" pr"%"#t

%"$!m"#tc!1#).El %"$!m"#to "% co#%!$"r$o como '# prt4c'l.• S" $"/" co#oc"r l% compo#"#t"% $" "loc!$$ $" &'0o (').• E%t% compo#"#t"% %" o/t!"#"# $" l %ol'c!1# $" l "c'c!1# $"

Lplc".

∇2=Ψ =0

E#to#c"%

u=d Ψ

dy , v=

−d Ψ dx

Pr clc'lr l tr2"ctor! $" l% prt4c'l% %" "mpl"# l"c'c!1# /+%!c $" c!#"m+t!c.

v=d&

dt entonces &= xi+ yj+ 'k

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4. ANALISIS Y RESULTADOS

PRIMER CASOB CUATRO PILARES CON UN FLUO UNIFORME

CUADRO DE ELOCIDADES

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CUADRO DE TRANSPORTE DE SEDIMENTOS

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CUADRO DE PRESIONES

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CUADRO DE DIERGENCIA

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SEGUNDO CASOB REDUCCION DE LA SALIDA

LINEAS DE CORRIENTES

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CUADRO DE TRANSPORTE DE SEDIMENTOS CON REDUCCION DE SALIDA

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CUADRO DE PRESIONES

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CUADRO DIERGENCIA

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TERCER CASO INGRESO DE FLUO SUPERIORB

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CUADRO DE TRANSPORTE DE SEDIMENTOS

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CUADRO DE PRESIONES

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CUARTO CASOB FLUO UNIFORME CON ENTRADA DE FLUO INFERIOR REDUCCION DE FLUO EN LA SALIDA

TRANSPORTE DE SEDIMENTOS

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TRANSPORTE DE SEDIMENTOS EN LA ULTIMA PARTE DEL TRAMO DELCANAL

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TOMA DE LA SALIDA DEL FLUO EN LA PARTE DE REDUCCION DE SALIDA

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CUADRO DE PRESIONES

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EL PROGRAMA

clo%" llcl"r llroK:<$9K?<

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P(:BN9:BN2-:)K<5or !K?BN9 5or 0K?BN2 P(!0)K(("l(!0))?-("l(!-:0))?)(ro?)< "#$"#$X7,'r"Xco#to'r5(92P)Xcolor/rXt!tl"(ZCmpo $" Pr"%!o#"%Z)

COMENARIOS

• Cr""mo% 3'" "%t" tr/0o #o% %!r!1 "# c'#to t"#"r co#oc!m!"#to%$" mo$"lm!"#to p'"% #t"% #o t"#4mo% l %'7c!"#t" cpc!$$

pr po$"r r"l!*rlo.• Lo% port"% $$o% $"l pro5"%or "# c$ #c" "# or% $" cl%"

5'"ro# $" ,r# 2'$ 2 3'" #o po$4mo% %/"r %! "%t+/mo%#*#$o "# lo corr"cto.

• E%t+/mo% poco pr"oc'p$o p'"% p"#%+/mo% 3'" "l t!"mpo #o% 0',r4 "# co#tr p"ro lo %'p!mo% m#"0r co# "l po2o $" to$o%.

DISCUSI7N DE LOS RESULADOS

El "co 3'" "l c#l 2 l% p!l"t% %"# l!%o #o 3'!"r" $"c!r 3'" %!"mpr""l &'0o comportr $" "% m#"r 2 3'" %olo "%tmo% %'po#!"#$or',o%!$$ !,'l c"ro "# to$o "l c#l pr prop1%!to% $" #'"%tro%"%t'$!o% 2 mo$"l$o% $" lo% ,r+7co%. Por3'" %/"mo% 3'" "# r"l!$$#o %'c"$"# co%% %4 p'"% %!"mpr" "#co#trr"mo% r',o%!$$"% co#prt4c'l% &ot#t"% %'m+#$o%" "llo l,'#% m#c% 3'" po$r4#$"t"ctr%" $"#tro $"l c#l 3'" p'"$"# lt"rr l &'0o #o o/%t#t"

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tm/!8# po$r4mo% "#co#trr &'0o% #o l!mp!o% 3'" "#tr# c#l"% co#r',o%!$$"% 2 "%o proocr4 '# ,r# t'r/'l"#c! $"#tro $"l c#l $"lc'l tmpoco #o t"#"mo% lo% "%t'$!o% pr po$"r r"l!*r %' mo$"l$o.

• C'#$o rmmo% "l c'"rpo $"l pro,rm 2 %' $"/!$ comp!lc!1# "# "lt"m $" $!5"r"#c! $" pr"%!o#"% 3'" "% l "l'c!1# "#tr" l4#"% $"corr!"#t" po$"mo% o/%"rr 3'" #o 2 '# clr $!%tr!/'c!1# $"pr"%!o#"% 3'" $"/"r4 %"r %!m8tr!c %! "% 3'" tommo% "l &'0o'#!5orm" 2 3'" "%t %" r"prt" t#to ,'% rr!/ 2 ,'% /0o 2 ll!m!tc!1# $" l% 5ro#t"r% co% 3'" #o% 5lt1 '# m2or pr"c!%!1# "#l,'# prt" $"l c'"rpo $"l pro,rm (5lt $" pr"c!%!1# "# "l '%o $"mll%). Tm/!8# por l 5lt $" co#oc!m!"#to% "# "l c'r%o $"pro,rmc!1#.

• AL 7#l $" lo% r"%'lt$o% po$"mo% co#cl'!r 3'" l mo$"lc!1# $" '#&'0o '#!5orm" tr8% $" '#o% p!lr"% %!m8tr!co% #o% 2'$# $r '#5orm m+% clr c1mo %" comport l% l4#"% $" corr!"#t" 3'" %o# l$!r"cc!1# 3'" "l &'0o tom %4 tm/!8# po$"mo% %!m'lr l tr2"ctor!$" l% l4#"% $" corr!"#t" co# c'l3'!"r ,"om"tr4 3'" %" pr"%"#t%t l% m+% %!mpl" %t l m+% compl"0.

?& CONCLUSIONES

U# /'"# "9ct!t'$ $"l mo$"lo !$ro$!#+m!co #o 5'" po%!/l" por l5lt $" !mpl"m"#tc!1# $" mll% m+% p"3'";% 2 %/"r c1mo't!l!*rl%< %!# "m/r,o tl !#5ormc!1# $$ por "l pro,rm %!r" pr"#t"#$"r 2 compro/r 3'" "%to% r"%'lt$o% %" "#c'"#tr# $"#tro $" lo%l4m!t"% r"l"% 3'" #o%otro% "%p"r+/mo%.

Pr #'"%tr% co#$!c!o#"% $" 5ro#t"r% "9!%t"#t" "# #'"%tro $!%";o%!r"# $" m'c 2'$ 2 3'" "# "l pr!#c!p!o #o /4mo% %'m!$o%

"%t% 2 "% por "llo 3'" "l &'0o #o %" o/%"r$ clrm"#t" "%t%l!m!tc!1# $# '# m2or $!%tr!/'c!1# $" l% "loc!$$"% "# c$ #o$oclc'l$o co# r"%p"cto ,'% rr!/ 2 ,'% /0o "l &'0o "#pr!#c!p!o "% !,'l 3'" l 7#l.

Lo% m8to$o% pr l $"t"rm!#c!1# $" l% r"$"% $" &'0o %o#B

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) M8to$o% A#l4t!co%B r"%'lt#t"% $" l !#t",rc!1# $" l "c'c!1#$!5"r"#c!l $"l &'0o. Solm"#t" pl!c/l" "# l,'#o% c%o% %!mpl"%<$"/!$o l compl"0!$$ $"l trtm!"#to mt"m+t!co.

/) Sol'c!1# N'm8r!cB pl!cc!1# $" lo% m8to$o% #'m8r!co% pr l%ol'c!1# $" l "c'c!1# $" Lplc" tr8% $" pro,rm% $" or$"#$or.Por "0"mplo "l mo$"lo $" lo% El"m"#to% F!#!to%B cr"$ '# r"$ $""l"m"#to% 7#!to% %" p'"$" clc'lr co# r*o#/l" pr"c!%!1# l cr,totl "# c$ p'#to.

@& #I#LIO-RA!IA

- Fo9 6. : INTRODUCCION A LA MEC^NICA DE FLUIDOS t"$!c!1#pp:-J

- M'#%o# R. FUNDAMENTOS DE MEC^NICA DE FLUIDOS "r "$!c!1#

pp:-?- Sm"% H. MEC^NICA DE FLUIDOS "r "$!c!1# pp-:

- Pott"r C. MEC^NICA DE FLUIDOS ERA "$!c!1# pp-?:- U#!"r%!$$ M!coc# $" S# N!col+% $" H!$l,o (C!'$$ $"

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- U#!. Notr" Dm" ?ttpB666.#$."$'`m%"#M"cFl.p$5M"c#!c MEC^NICA DEFLUIDOS co#%'lt$o :??:

- U#!. T8c#!c $" Am/to ? U#!"r%!$$ Nc!o#l A't1#om $"M89!co?JttpBmmc?.,"o7%!c.'#m.m9c'r%o%pp$$mM"to$oD"El"m"#to%F!#!to%.p$ m8to$o $" "l"m"#to% 7#!to% co#%'lt$o:??:

- D4* . Fl'0o "# t'/"r4%.? ttp%B666.2o't'/".com6tc>KS-22c/U6 co#%'lt$o.?:??:.

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trabajo potencial final!!!!! 2

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