navier's equations
TRANSCRIPT
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Applied combustion engineering(Prof. S.noda)
Student name: Subhan UllahStudent ID: M135117
Department: Me!an"a#
$o%o!a&!" 'n"er&"t% of $e!no#og%
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)))%)*
%)%%%*
*)*%**
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Navier-Stokes Equations
+e ,ou#d #"-e to der"e t!e differential form of the momentum conservation lawfor
"nompre&&"#e f#o,&.+e &!a## f"r&t !oo&e an "nf"n"te&"ma##% &ma## ontro#%*/ = around t!e po"nt (* % ) and t!en &!r"n- t!"& o#ume to a po"nt %
a##o,"ng 0. ea## from t!e Newtons second law of motion:
,!ere t!e left hand side represents the sum of the eternal forces and the right hand
side represents the inertia forces. S"ne ,e !ae a#read% der"ed t!e e*pre&&"on for
Dt
/Da =
a& t!e ae#erat"on at a po"nt,e## u&e t!"& re&u#t !ere.
et u& re"e, &ome fat& aout &tre&&e&. $!e &tre&& ten&or t!at %ou ,ere taug!t "n t!e
&trengt! of mater"a#& and repre&ented % t!e matr"* e#o, !a& omponent&.
ote t!at repre&ent& a normalstressand repre&ent& a shearstress. S"ne &tre&&
"& def"ned a& a fore oer an area each stress component has two subscripts. 8or
e*amp#e*
***
9
8= repre&ent& a fore "n t!e *d"ret"on t!at "& a#&o at"ng oer an area
t!at "& "n t!e *d"ret"on. o, an area ma! be considered positive or negative"depending on whether the outward normal to the area points in the positive or
negative coordinate direction#8or e*amp#e "n t!e d"agram e#o, ,e !ae !o&en apo&"t"e and a negat"e %area.
$!u& t!e &tre&& due to t!e fore 8%&!o,n on t!e po&"t"e %fae ,"## repre&ent a po&"t"e
&tre&&. S"m"#ar#% a po&"t"e &tre&& on t!e negat"e %fae mu&t e due to a fore po"nt"ng
am88 S; =+
$
%!
&!
!
'
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...)2
%()t%*(
%2
1
2
%)t%*(
%)t%*()t
2
%%*(
2
2
%%2
%%
%%%%
+
=
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"n t!e negat"e %d"ret"on P%. +e ma% e a#e to repre&ent &!ear &tre&&e& t!e &ame ,a%.
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0erivation of Navier1s Equations
Now let us focus on onl! the forces that act in the !-direction("n&tead of t!o&e t!at aton %fae&). =ne t!"& re&u#t "& der"ed ,e ma% e a#e to ,r"te &"m"#ar e*pre&&"on& for
t!e * and d"ret"on&. 8or t!"& purpo&e u&e t!e fo##o,"ng &-et!:
ote t!at t!e aoe &!o,n are t!e onl! possible surface forces in the !-direction on thecontrol volume. $!erefore:
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2od! %orcea#ong )%*%;;%8% =
o, ,e are read% to ,r"te t!e !-component of the momentum equation:
S"ne t!e ma&& of t!e &ma## ontro# o#ume !o&en "&%*/ =
and
,
%
*
u
t
Dt
Da%
+
+
+
== t!u&
+
+
+
=+
+
+
z
vw
y
vv
x
vu
t
vzyxzyxBzyx
zyx y
zyyyxy
?ane##"ng from ot! &"de& ,e get:
+
+
+
=+
+
+
)
(,
%
((
*
(u
t
(;
)%* %
)%%%*%
$!e first three terms on the left hand side represent surface forces per unit volume and t!e fourth one is the bod! force per unit volume#S"m"#"ar#% t!e r"g!t !and &"de
term& o##et"e#% repre&ent the inertia force per unit volume.
et u& e*am"ne t!e term*
*%
one aga"n. $!"& term "& t!e &urfae fore on t!e ontro#
o#ume t!at ar"&e& due to t!e net shear force between two . surfaces but acting
!
'
)%2
*
*
*%*%
+
%*2
)
))%)%
)%2
*
*
*%*%
%*2
)
)
)%)%
+
)*2
%
%
%%%%
)*2
%
%
%%%%
+
%
*
)*2
%
%
%%%%
)*2
%
%
%%%%S%
8
+=
)%2*
**%
*%)%
2*
**%
*%
++
%*2
)
)
)%)%
%*2
)
)
)%)%
++
)*2
%
)2)%
2
*
*2)*
2
%
%2
)%*%%%
+
+
=
)%*)%*
)%%%*%
+
+
=
%;%S% ma88 =+
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along the !-direction. "-e,"&e t!e &eond and t!"rd term& are due to t!e norma# fore
and &!ear fore et,een t,o %&urfae& and t,o &urfae& re&pet"e#%.
ote there is a pattern which emerges from the summation of all the !-forces on thecontrol volume. $!e &tre&& term&@ &eond &u&r"pt& are a## % repre&ent"ng t!e d"ret"on
of t!e fore ut t!e f"r&t &u&r"pt& t!at repre&ent ,!"! area& t!e fore& at on !angefrom * to for ea! &ue&&"e term. S"m"#ar#% t!e der"at"e& !ange a#&o from * to for
ea! term. $!e etor omponent "& repre&ented % % and t!e e#o"t% omponent "&
aord"ng#% . "-e,"&e t!e *and t!e eAuat"on& ma% a#&o e ,r"tten rat!er t!ander"ed. 8"na##% a## t!ree eAuat"on& are repre&ented %:
$!e aoe t!ree eAuat"on& are o##et"e#% a##ed the Navier1s equations named after
t!e"r or"g"nator. ote t!at t!e&e eAuat"on& !ae 4 "ndependent ar"a#e& (* % and t)ut
34 dependent variables +u" v" w" and the stress components, . +e &!a## a&&ume t!at t!eod% fore omponent& (,!"! "& u&ua##% due to gra"t% "n me!an"a# eng"neer"ng
pro#em&) are -no,n. $!erefore t!e Navier1s equationsare notsolvable &"ne t!ere are
more un-no,n& "n t!e&e eAuat"on& t!an t!e numer of eAuat"on&.
$o ma-e t!e&e eAuat"on& &o#a#e Stokespropo&ed a &et of constitutive relationsg"en
e#o,. $!e&e re#at"on& together with the continuit! equationder"ed efore ma-e t!e
a"er& eAuat"on& &o#a#e. $!e Sto-e& re#at"on& "n(artesian (oordinatesare:
+
+
+
=+
+
+
)
u,
%
u(
*
uu
t
u;
)%* *
)*%***
+
+
+
=+
+
+
)(,
%((
*(u
t(;
)%* %
)%%%*%
+
+
+
=+
+
+
)
,,
%
,(
*
,u
t
,;
)%* )
))%)*)
+
==)
(
%
,)%%)
+
==
%
u
*
(%**%
+
== *,
)
u*))*
%
(2/
3
2p%%
+=
),2/
32p))
+=
*
u2/
3
2p**
+=
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In t!e aoe re#at"on& "& t!e d!namicviscosit!and p"& t!e thermod!namic
pressure. 9#&o note t!at &"ne t!e &tre&& ten&or "& &%mmetr" (".e. *%B%* et) t!e f"r&t
t!ree re#at"on& are ,r"tten "n a more ompat form rat!er t!an u&"ng 6 #"ne&.
S"ne our fou& "n t!"& #a&& "& "nompre&&"#e f#o,& ,e ma% e a#e to &et 0/ = "n
t!e aoe re#at"on& % t!e u&e of t!e ont"nu"t% eAuat"on for "nompre&&"#e f#o,&. If ,e&u&t"tute t!e aoe eAuat"on "nto t!e a"er& eAuat"on& and repeated#% u&e t!e
ont"nu"t% eAuat"on for "nompre&&"#e f#o,& (t!"& ,or- "& #eft out for %ou to tr% a& an
e*er"&e) ,e ota"n t!e &"mp#"f"ed form of t!e a"er& eAuat"on& a&:
In t!e aoe eAuat"on& popu#ar#% a##ed the Navier-Stokes equations ,e !ae on#% 4dependent ar"a#e& (u , and p) one aga"n a&&um"ng ;* ;% and ;to e -no,n.
$!u& add"ng t!e ont"nu"t% eAuat"on to t!"& &et:
+e no, !ae a &et of 4 eAuat"on& "n 4 dependent ar"a#e& or un-no,n&. $!erefore t!e
f#u"d d%nam" pro#em& for "nompre&&"#e f#o,& are no, #ear#% &een to e &o#a#e.
+
+
+
=
+
+
+
2
2
2
2
2
2
*)
u
%
u
*
u
*
p;
)
u,
%
u(
*
uu
t
u
+
+
+
=
+
+
+
2
2
2
2
2
2
%)
(
%
(
*
(
%
p;
)
(,
%
((
*
(u
t
(
+
+
+
=
+
+
+
2
2
2
2
2
2
))
,
%
,
*
,
)
p;
)
,,
%
,(
*
,u
t
,
0
,
%
*
u/ =
+
+
=
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0/= +A,
/p;Dt
/D 2 += +2,
+
+
+=
+
+
+
2
2
2
2
2
2
)
/
%
/
*
/p;
)
/,
%
/(
*
/u
t
/
0/=
Page C 1/26/2014
In #ater !apter& our attempt ,"## e to &o#e t!"& &et of 4 eAuat"on& "n 4 un-no,n&
ana#%t"a##% for ar"ou& "nterna# and e*terna# f#o, onf"gurat"on&. 5emember that these
equations ma! not be applied to compressible flow problems# %or those" we need to
derive the Navier-Stokes equations without the eplicit use of the incompressible
continuit! equation#
=ften t!e ont"nu"t% eAuat"on and t!e "nompre&&"#e a"erSto-e& eAuat"on& are ,r"tten
"n etor form a&:
=r &"mp#%:
ote t!at t!e f"r&t eAuat"on +A, "& a scalarequation(S"ne
,
%
*
u/
+
+
= "& a
&a#ar). >o,eer t!e &eond eAuat"on +2) !"de& 3 eAuat"on& (one ea! for t!e * % and omponent&) "nto a &"ng#e eAuat"on. $!u& ,e &t"## !ae 4 eAuat"on& "n 4 un-no,n&.
+e !ae a#read% d"&u&&ed t!e p!%&"a# &"gn"f"ane of t!e eAuat"on +A,. Auat"on +2,"&t!e etor form of t!e e,ton& &eond #a, of mot"on (e*ept ea! term !a& een
d""ded % dur"ng t!e der"at"on). $!u& t!e #eft !and &"de repre&ent& "nert"a fore per
un"t o#ume ,!"#e ea! of t!e t!ree term& "n t!e r"g!t !and &"de repre&ent& a t%pe of
e*terna# fore per un"t o#ume (od% fore pre&&ure fore and "&ou& fore
re&pet""#%).
Eample +use of Navier1s equation 6 Stokes1 laws,:
&roblem:
/he velocit! profile for full! developed laminar flow between two parallel platesseparated b!
distance 4b is given b!
=
2
2
ma*.
%1uu where umais the centerline velocit! +at ! 7
8,# 0etermine the shear force per unit volume on a fluid element in the -direction#
%ind the maimum value of this quantit! for this flow" when b =1 m, umax=2 m/s and
=10-1N sec/m2(SAE-10W).
uma*
*
%
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%
u
%
u
*
(%*
=
+
=
0)
u
*
,)* =
+
=
0
00
Page 1/26/2014
Solution:
9iven:
e#o"t% Prof"#e u(%)
9t % B 0 u B uma*
9t % B u B 0
%ind:
("))%)*%*
+
("") $!e ma*"mum a#ue of t!e t!e re&u#t "n (")
+0iscussion:$!"& pro#em& #anguage "nd"ate& u&e of t!e a"erSto-e& eAuat"on.
doe& not !o#d ot!er,"&e owever if the Navier-Stokes form +2, is used" we
must test for incompressibilit! since +2,.
9&&ume: B 0 B , &"ne t!e e#o"t% prof"#e g"en "& on#% "n t!e *
d"ret"on.
?!e- "nompre&&""#"t%: 0
,0
%
0
*
u=
=
=
0/ =
$!u& t!e pro#em ma% e &o#ed % &"mp#% a#u#at"ng u2 . +e &!a## a## t!"&
part Met!od II)
;ethod