Boundary Lyer Theory

80 111. l)rrivnt.ior~ of the cquationu of lnot,ion of a con~presaiblc viscouu fluid
where div rrt has been used for hrrvity. 'J'hc rcatlrr may notice the regularity with
whir11 the indices x, y, z, the componrnfs n, v, in, antl tl~r coortlinntrs x, y, z arc
permutcdt.
Applying t,l~ese equations t,o the si~nplc casc rcprescnt.ctl in Fig. I .I, we rccovcr
eqn. (1.2) and so confirm that t,hc precctling more gcnernl rrlat,ion rccluces to
Newt,on's law of friction in t01r casc of simple shear ant1 docs, t,l~orcfore, const.itxtc
it,s proper gcnoralimtion. At the samr timr, we identify tshe factor 11. with the viscosity
of t,he fluid, amply disc~~ssetl in Scc 1 h, antl, incid~nt~ally, justify the factor 2 previously
inserted int,o eqns. (3.21). The physical significance of the second factor, 1, requires
furt.Iicr tliscilsaion, t~ut we 11ot.c that, it, plays no part in an incompressible fluid when
div 119 = 0; it then disappears from the equat,ions a.lt.oget,hcr, ant1 so is seen to be
in~port.nnt for r~ompressible Ruitls only.
e. Stokes's hypothesis
Althougl~ the problem l,l~at we arc about to discuss has arise11 more than a
ccntury and a half ago, the physical intcrpretatiort of the second fact,or, 1, in
eqns. (3.21) or (8.22a, b) and for flows in which tliv rcJ does not vanish ident,ically,
is still being disputed, even though the vabe which should be given to it in the
ioorkirtg eq~u~fio?~~ is not. l'his numerical VRIIIC is determined with the aid of a. hypothsis
:~tlvancod by G. G. St,oltcs in 1845 11.71. Without, for Lhe nlomcnt,, concerning
o~~rsclvcs with the physical reasons which just.ify Stokes's h?yjvath~s~:s, we first st.ate
that according t.o it,, it is neerssary to assume
This rclatrs the value of the fartor 1 to the visrosity, 14, of thr romprrssible fluid
and redures thr number of propertics whic41 rhamct~erize the field of stresscs in
a flowing romprcssiblc fluid from t,wo to onr, that is to thr same num1)cr as is
rrq~~irctl for rcn incomprrssihlr f111itl
Subst,it.nting t,l~is v:duc ir~t,o eqr~s. (3.22a), we ol~tairl the normal corni)oncnt,s
of tlevin.t.orio stmss :
aw
a,' = - /L div IJ 2 ,u az ,
3
t 'hc aboyc ncL of six cqnnl.iona can be oontrac:tcd to a single one in Cartesian-hnsor notation
(wit.l~ Einflkin's .sn~nrnnt,ion convention):
u'IIc~(. tlw Kronrrkrr tlrlta dl, - 0 for i + j nntl dij - I for i -- J .
the ~l~raring stresse~ remairhg unrhangrtl. Malting usr of eqrls. (3.20), wr obtain
tl~r so-railed conrtitutioe eqlantion for an isotropir, Newtonian fluid
in it,s final form, 11ot,ing that p reprcser~t.~ the local t,l~errnotlynarnic: prrssurrl-
Regartled as a pure hypothesis, or ever1 guess, eqn. (3.23) can certainly be
:~cceptctl on tho ground that the working eqr~at.ions which result from the substitut.ion
of cqns. (3.26a,b) into (3.11) have been si~bject~cd to an unusually 1;trge number
of cxpcritnentnl verifications, even ~~ntlcr quite cxt,remc conditions, as t,he reader
will cor~crtlc after having studied this book. Thus, even if it should not rrprescnt,
thr state of affairs exact.ly, it certainly constitut.cs an rxcellent approximation.
Since the deviatoric components are the only ones which arise in motion,
t,l~cy rcprrscr~t those components of stmss which produce dissipation in all isothertnnl
flow, t,l~crc bcit~g further dissipatior~ in a t,cmperature field ~ IIC t,o thermal cor~tIuct,ion,
(%:L~I. XI I . Fl~rt.hcrmore, since t,hc S:~cbor 1 occurs only in tho normal cornpo~~rnt,~
cr,', a,', a,' wl~ich also cont~i~in the thcrnmdynaniic pressure, cqrls. (3.20), it I)ccomcs
vlvar t,l~at. t,hc p11ysic:d significnncc of 1 is connectctl with t.he nicchanism of tlissip:\t,ion
\\.IIPI~ t.he volume of t,hc fluid clcrncnt is changcyl at a finite rate as well :as
\\.it,h t.11~ r.rl:~t.ion I)rt,wrrn the tohl st.rcss tensor :wtl t.l~c:rnmotlynnmic: II~OHSII~O.
f. Bulk viscosity nrd tl~errnodynamic pressure
\Ye now ~rvcrt to the genrral tlisc~~ssion, wiL11orrt ncccssarily arcaptir~g th(~
\aldiI y of Stokrs's hgpothcsis, but, confine it to the casc wl~erl no shearing str~ssrs
arr irivolvctl, 11cmusc their physical signifiranre arid origin is rlcar Conseq~lrntly,
6
In the compact tcnaorial notation wr would write