Boundary Lyer Theory

206 XIT. l'l~rr~nnl bo1111r11iry lnycra in Inminnr flow
'1'111- t.vr111 tl lC7./tlt rc:prrsrnls n st~l)st.ar~t id tlcrivativc which consist,^ of a local ant1
:I cwr~vc.c.l.ivr c.o~~l.ril)l~t.io~~.
Ilrrrs lr I.l/n~ sc~ t l q l tlcr~ot,cs the 1h0r1n:~l (:o~~(l~~t:I.ivit,y of 1.11~ (Il~id. 'l'he rtcg:~l.ivc
. .
slgn s~g~~ifics ~,II:L(, thc hrnt Ilux is rccltol~cvl :IS positivr in the tliroc1,ion of the t,cmpc-
r 3
I Ilr rl~nt~gc ill t.l~r f.ot.nl rl~c~fiy. tllCT, (:on!iist,s of a chnngo tlE : p,,l l'tlt. in t.hc intcrn:d
c~~crgy :tntl n c.l~:wgc in Iti~~rl.ic: oi~crgy by at1 ntnount, (1 { 6 0 LI I ' ( ~ L ~ 1- v2 4 w2)}, if
I IIV ~ I : I I I ~ ill V tl~r l)oi~v~~tial cwrgy (Iuc to a displ:~ccrn~~~t it1 Ihc gr:~vit:~t,im:~l Iirhl
is ~~rglrcl(~l. I lvncc
Tho nrgst,ivc sign is a.tltletl in order to follow t,hc sign convcnt,ion of cqn. (1 2.1) accortling
to wl~ich work adtlcd to tho fluid from t,l~r! oui,sitlc is rlcgativc. Thc tol.:d worlc
j)erforlnrcl by t.hc normal :rnd shearing shrssrs prr unit! time rsrl now l)c writ.t,c:n :IS
JIrrc a,, n,), . . . , s,, rlcuok tho rlorlnn.1 ant1 sl~rari~~g si.rcmt:s itit.rot111rrtl r:~rlivr ill
eqns. (3.20) and (3.26). Substituting eqtls. (12.3), (1 2.4) ant1 (I 2.0) into cqn. (1 8.1 ),
nntl pcrfnrming s numl)cr of obvious simplificst.ions, i~~cl~ltling those inI.rotl~lcrtl by
cqn. (3.1 I), wc ol~t,ain, after some calculnt,ion, the: followi~~g oncrg.y o(j~~:~I.iot~s 01' 1.l1c
flow:
Jlcrr @ rrpresrnts the tlissipatiorl firnct,ion give11 I)y
Rquat.ion (1 2.7) enjoys gcnrral valitlit,y, I)ut in most pr:1,c4ir:d rnsos it is possil~lc to sirnplify
it still furt01cr. Jrl doing so, it, is ncccssnry carcfrtlly l.o(list,il~g~~isl~ 1)ctwcc11 l.hc rnsr
of a pcrfcc:t, gas n.nd t.l~st of :LII incom~)rrssil~lc Iluitl. 'I'll(- f.l~c~r~notlyt~n~~~ic? proprrt,ic,s
of t,l~c Int,t,rr do wol ronsl.itnt.e n 1irnit.ing m.sc of 1.11~ prnl)c.~.l irs of the f'orl~lrr.
tJic va.riat,ion in iJ~o int,crn:d energy of :L prrfrc:t gas is clc I-- c,, ti'/', wllcrrns l,l1:11~ 01'
it,s ontl~allly is tlh. c,,dT. Tlin corrcspo~~tling v:~r.iat.iot~s for nn inc-olr~prrssil)l(~ fl~litl
:1ro (if? = c (17' a,ll(I dl& .r c (17' -1- (l/@)dp .
M'itlr the nit1 of' Illis rqu:~t,io~t :III~~ of
c, (IT - c,, d 7' 1 (1
111 li~~t.,