Boundary Lyer Theory

400 XIV. Boundary-layer control
Owing to the definilion of w, we must have Ze, Wc = 0, and for each component i we may
write 6hc law of maus conservation in the form
div (er wi) = div (e{ (W -t W{)] = 0.
(14.24)
Upon summing ovcr all components, we obtain the continuity equation
which has the faruilar form of eqn. (3.1).
tliv (e W) = 0
111 tho rrbsonco of nxbrnnl ficlds, tho clifiuuivo flow is drivcn, cssenLially, by conccntration
grndicnta as well as by tl~crmal difiusion which prorlucea n flow of musses in the presence of n
tenipcrntr~re gratliont. In tho case of a binary mixture, we may write the law of tliffusion in the form
c, W, - - Dl, (gmd c, -t kT grad In T) , (14.26)
whcrr I),, clonotes the cocfficient of biuary difiusion, kT is the thermal diffusion ratio, and c, = el/e
is tho masq conccntration of lhe first gas, assnmod to be the one which emerges from the wall.
'J'hc c:orflicinnt of binary tlifiusion dcpends only litllc on concentration and is affected by temperntnrc
in tho uamo way as the kincmntic viscosity. Thc thermal difiuaion ratio, kT, depends essentinily
on concont.rntion and is frequonlly npproxirnabd hy the rathcr crude relation
kT = me, (I -c,)
(14.27)
duc to Onsagcr, Furry and Jones. Here, thc cocfficient of thermal diffusion, a, is assurncd to be
a constnnt for wcry specific cornbinntion of gases.
Inserting cqn. (14.26) into thc law of mass conservalion, eqn. (14.24), written for the first
component, and taking inlo account cqn. (14.25), we obtain
Wr may now introdwe the normal honndary-layer simplificatiolls into the right-hand side of this
rqnnt.ion tl~ns t~rglrrtin~ krms in a/& with respect to those in a/&~. In this manner we obtain
1 he ro71rrnlmlion rpolzon
A corrcspontling oqunlion is valid for tl~c scxoncl component,; I~owcver, this second equation
brcon~cs txivinl wlmn (,hc niotlificd form of cqn. (14.28) is uscd because c, 4- c, = 1. For this
rrnson. t.11~ S~:COII~ (:q~l:ition is ropl:lc.crl by thc continuity equntior~ (14.25).
Tho ~non~rnt.rttn cq~~nlions
wril.tn~~
for :I, gas rnixluro :rre identicnl with those for a sirlgle gss antl are
a7' =0, (1 4.30)
a!/
wl~rro now Q nnd p clrpcntl 011 roncrntr:lt,ion in addition t.o their familiar dcpc~~dcnce on tetnpc.
rat.urr. . I
,
IIC rncrgy cqnat,ion for a gaseons mixt,nre must bk formulated with due rcgzrd being peid
tm Ihc r~ormal tl~orn~nl oo*~~lnction, to the transfor of h6at by diffusion, and to that by thermal
~~~I'IIS~OII. Itc~t~riclillfi our ~0114i~lcriltiolls 10 pcrfcrt plscs, w 1111rod11~o ~hc mixhrc ct~tl~alpy
h = cI ha -{- C, h, .
(14.31)
Sincv: tho tloriv:rl.icul is Irngl.l~y, wc rncmly qnot.c thc rrsult. in which the boundary-layer approxirnnt.ion8
hnvc nlrcntly I~ecn introtl~~cod:
Ilrrn R sI:intls for 1.l~: ~mivrrwl gns onnsln~~t~. If lll~rr~n~1I (~~~I'IIR~oII is ~~rglw:t.rd, ~.II(! nntl(-rIii~(.(l
trrn~s nrc: tlclotod. III the tlcrivaLion of this cqualiol~ IIRC 1111s lw(:n ~nnd(! of OIIHI~~(!~'H pri~wiplo
arrortling lo wl~ich tho corfficirnt of Wlo oonccntmLion grnrlicr~t in UIC brat-flux vrctor in tho
samr :IS thnt of thc trn~pcrat~ure gradient in tho mass flux.
and in view of oqn. (14.20). wc olhin tl~c conrlit.io11 t,l~:rl
(grad el -I- kT grad 111 7')
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