Boundary Lyer Theory

of flltitl is strcsscd in thrcc nlutually pcrpcndicirlar tlire~t~ions, and its faccs arc
displaactl instantaneortsl~ also in tlrrec niut.ually perpendicular directions, as suggested
by Pigs. 3.7a antl I). This tlocs not., of course, moan tlmt bllcre exist no shearing
strrssrs in ot.Ilrr pl:~nrs or t,llat. t,l~o sltapc of t.lw clcmrnt rcmr~ins ~lntlistortrd.
11. Rclntion between stress and rote of deformation
It sl~nuld, 1)crhaps, 1~ st.rcsscd once more that the cql~at~ions which relate tho
surface forces to the flow ficld must, be ol)tainetl by a pcrc~pt~ivc interpretation of'
experimental resulB and that our intcrcst is restricted to isotropic and Newtonian
fluids. l'hc consitlcmt,ions of the precctling section provided 11s with a useful mathcmatical
franlcwork which allows us now to statc thc rcquircmcnt~s suggcstcd by
experimcnt.~ in a somewhat, morc prerisc form.
When the fluit1 is at rcst, it dcvclops a uniform ficld of hydroslatic st,rcss
(nrgat.ivc prrssurc - p) which is idcnt.ica1 witli the thermodynamic pressure.
When the fluid is in mot.ion, t,l~c equation of statc still tlcternlines a pressure at,
ovcry point ("principlc of local st,atc" 141), and it is rnnvcnicnt to consider t,he
tlcviat,oric normal strcssrs
togrtl~cr wit,lt the nnc:lrangotl shearing strosscs. The six q~~ant.itirs so obtjainctl
cor~st,itrct,c a symnlct,rit: strcss tensor thc cxistcnrc of which is tluc to the tnotiorl
hccnusc at rcst. :ill it.s componcnls vanislt irlcnt.irally. l'rom what Ilns bcen snit1
brfnrc it follows that tllc components or this tlcviat,oric tcnsor arc creatctl solcly
by t,hc componcnls of the ra1.c-or-shin tensor, t.11at is to thc exclusion of the cornponcnt.~
v, 17, m of vc~1ocit.y as wc:ll as of the componcnts (, ?I, 5 of vort,icit,y. This
is rclnivnlent. to s:r\,ing I,l~:rt the inst.nnt.:lncolts t.r:tnslat.ion [component motion (a)]
as wrll as tho inst,:~.rtt,ancou rigitl-hotly rot,:~tion Icompor~c~~t motion (I))] of' an
t,lrtnrnt of llrtitl protlucc no surface fnrccs on it in atltlit.ion to the exi~t~ing cotn-
~wmrnt.s of l~ytlrost.:tl~ic prrsstlm. 'l'hc procctling st.atcmcnL, cvidcnl.ly, rnorrly rcprcscnt,~
x prcciso 1oc::d formnlnt.ion of wltat we cxpoct Lo observe whcn :I Gnitc I)otly
of Ilt~id performs n gcncral mot,ion wl~icll is ir~tlist~ing~rishablc from tltnt of an
~:q,~~ivalcnt rigill hn(ly. We thus cmncl~dr th:it the erprrssions for tho components
a, , 0,'. . . ., T ~, of blrc tlcviatoric st.rcss t.cwsor can ront,airr in lhcm only the velocit,y
gratlicnt,~ aupx, . . ., alll/az in approprintc combinations which we now procccd to
clet,crmine. 'I'ltcsr: rclntions are postnlnt.ct1 t,o Oc lincar; thcy must rcmain unchangccl
by a rotation of the syst.crn of coortlirlalcs or by an intcrct~ange of nxcs 1.0 ensure
isot,rolry. Isotaopy also rctluirc-s that at every point, in t,hc continuum, tfhc principal
a.xcs of t,llo strrss t,cnsor must roinridc with the principal axes of the rate-of-strain
t,crisor, for, ot.I~(~rwisr, :I. prrfcrrcd diroct,ion wollld 1)c introduced. rllltc simplest
way to arllicvc otlr aim is to select an arlrilmrf point, in the cont,inunrn antl t,o
itn:~girin Llr:~.t, the locnl syst.crn of roordir~at~cs ?, ?7,'2 has been provisionally so clloscn
as t.o coincitlc with the t,llrcc common princ:ipal axes of tltc two trcnsor~. 'l'hc corn-
I)ot~i.ttt.s of t,l~v vo1ocit.v lic*ltl in t,his syst,c:m of coordinntcs are dcnototl by ?(., ii, 111.
of which conicides with it and on the sum of tho three, each with a different factor
of ~moportionalit,y. Thus we record, dircctJy in terms of tho spacc-clcrivat,ives, tJ~at,
'l'!.,? rlmlt,iti?s ?I,, 0, :~nd 4, r], C do not appcnr in ti~csc cxprcssiorts for t.11~ ~.r:~.sotls
jtlst explained. In each expression, the lnst term represents thc appropriate rate
of lincar dilatat,ion. that is, in essence, n change in allape, and tho first, terrn rcllrescnb
the vollrmetric clilatfation, that is the rate of change in volume, in csscnct.,
a change in density. Thc factors 2 kl the last terms are not essential, beirrg mcrely
convenient to facilit,ate thc interpretation, as we shall see 1at.er. Tlrc fact