Boundary Lyer Theory

even in fluitls wit,lt vcry srnall viscosit,ics, unliltc in pcrfwt. fluiels, t.he rontlit.ion of
no slip near n, solill I~oundary prevails. 'l'l~is c:ot~dil~ion of no slip int,rotlures in many
(::~sos very hrgc tliscrcpar~cics in t,hc laws of moLiorl of perfect an(\ ronl fluids. In pnrt.icular,
t h vcry largc tliscrcpel~cy 1)ctwccn Llle vdr~o of' drag in a rral ant1 a pnrkct,
Iltti(1 I1:w its pl~ysical origin in the contlil,ion of no slip nwr :L wall.
'I'l~is 11oolc t1r:rls wil,l~ 1.11~ rnot,ior~ of llrlitls of'sm:~II visrosil,y, I)(-r:~llsr of t.l~c grc:~L
I~:~ct,ical itnporl.ance of' the problcln. Ihrirtg 1,llc course of lhc st~dy it will l~cconlc
clear how this p:trtJy consistent ant1 p:l,rl.ly tlivcrgcnt I)cl~aviour of pcrfrct and real
fluids can be cxpl:tinotl.
h. Viscosity
'I%(: II:L~,II~C of' vi~rosit~y can 11cst I)c vi~rdizcd with the :lid of t,ltc following cx-
~wrimnnt,: Consitlcr the ~not~ion of a fluid l)cl,\vccrt two very long pn.rallnl ~)latcs, one
of wl~inh is at rrst, the other moving wit,l~ n, constant velocity pnrallcl t,o itdf, as
sl~owu in Fig. 1 .l. 1,ct tJ1o clist.anco hctwcc~~ thc plates bc h,, the prrssnre Iwing const,nnt
t.l~rol~gl~ot~t tllc fluid. Exprrintcnt t.c~:rcltcs t.l~:rt t.11~ fluitl atll~rrcs l.o l)ot.l~ ~valls, so
I,II:II, it,s vclovity :rI, the lownr p1:~t.c is zero, :I,II(~ t,11:1t 3.t Lltc ltplwr ph1.c: is rt111al to
t.11~ vcloeit,y of the plate, IJ. Ir'rtrt~l~ermor~, I.llc vclocit.y tIist,ril)r~t,ion ill t,llc fluid
I)ct,wccn the pIat,cs is linear, so that, the fluid vclocit,y is proport,ion:ll tto t.ltc tlist,ancr ?/
from t 11c. lowvr platr, :~ntl we h:tvr
In ortlnr 1.0 s~lpport t,l~e motmion it is necessary to apply a I~n~~gc.nt,ial forcn t,o thn
tlpprr l)lnto, tho force 1)cing in cc~t~ilibriurn with tl~c f'rict~ional forces in t,l~c fluid.
It is Icnown from expcrimont,~ t,l~at tJtis forcc (ta.l~cn per unit awn of t,l~c plal,c)
is proprt.ion:~.I to t,hc velocity 1J of the 11l11)er plat.c, ant1 invcrsrly proport,ion:~l to
lhc tlist,:r.nrc~ h. 'l'llc 1'ricI.ion:ll force por nit, area, tlcnotctl by t (Srict.ional shearing
sl,rcw) is, t,licreCore, proport.ionn1 1.0 lJ/h, for which in general we may als? ssulist.itr~t,c
tlii/tl?/. 'l'ltc: 1)ro1)01.t~io11:rIil.y far:l.or I)ct,wcnn t ant1 d71 tly, wl~iclr we sl~all dc~~ot,c I)y ,u,
I
tlc11~1(1s or1 tho r~al~llrc of 1.110 ll~~i(l. 11, is ~rna.ll for. "lhiri" fluids, s11c11 nk wal.cr or
:~l(:ol~ol, I~ut I:qn in the case of vcry viscous liquids, srtclt as oil or glyccrinc. 'I'hl~s
wc 11;tve ol)t,:~inctl t,llc ftl~~tl:rrncnl,al rclnt,ion for fluid frict.ion in t,lte form
>
du
(1.2)
= fL ~ I Y .
Tl~r quantity p is n propertry of thc fluid and depcntls to n great cxl.cnt on it,s ternpcrnt,rlrc.
It is n rneasuro of tho i)i.~co,qit~y OF the fl~iid. '1'11~ I:LW of' friction givrtl by
cqn. (I .2) is 1znow11 :LS Nrwtotc's 1rr.v~ of friction. ICqn. (1.2) cnn bc rrg:~relvrl :I.R t,llc
c1rlinil.ion of visc:osit.jy. It. is, Ilowevcr, nccxssary to st.ross that the cxnrnplc cot~siclcrc:d
in IGg. 1.1 (:onstit~~t.rs :L p:~rt,ic~~larly simple case of fluit1 motion. A gnncr:~liz:~l,it,r~ of
this sitn111v e:rsc is cont,:~.inc(l in Stolccs's I:IW of fridion (cf. (!II:L~. I I I). '1'11~ ~limc~~~si~
of visrosi1,y c:all IIC tlotl~rc:c:tl wit,hol~t, diFlicull.y from cqn. (1.2)-I-. '1'110 sl~c:nritlg s1,rcw
is ~ncnsurcd in N/m2 =I J'n nrld tltc vcloc:it,y grntlicnt du/tl?y in ~ o I. c ~IVII(Y*
wllcre tho square 1~r;~(:Iccts arc IISC(~ to (Icr~ot~ 11ni1.s. '1'1~ :L~)OVC is not. 1hc o~~ly, or
even the most, witlcly, employctl unit of viscosit,y. l'riblc? 1 .I lists t,he various t~nits
togct.lrcr with thir conversion factors.
.15qn. (1.2) is rc1:rtctl t.o IIooltc's law for all c~l:ist,ic: solicl I)otly in w11ic:h rasc: tl~c
shearing sCrcss is proport,ional to the strain
Ilrrc (: denotes lhe n~oclnlus of shear, y the change in anglc bct.wc.cn tfwo linrs
wlliclt were originally nt right anglcs, nntl 6 tlcnotcs t.110 clisplr~ccmcnt, in t11c tlircc:t.ion
of a1)scissae. Wllcrcas in thc cnsc of an elastic solid th: sl~caring strcss is proporl.ional
t,o the nw~gniturle of the strain,, y, expcricnrc tcacl~cs tll:~t in tl~c case of fluitls it is
proport,ionnl t.o the vale of chnnrlc. of strain tly/tll. If' we put
we s1r;~ll obtain, as bcforr,
a11
t ' fl
?I!/
bccausc 5 = XI. Jlowcvcr, this analogy is not, complctc, I~cca~lsc t.llc: st,rc:ssas in :r
flt~itl tlepcntl on one const,atlt., t,l~c viscosit.y ti, wllcw-:is tllose irl :tn iso1,ropic vI:~sLic:
solicl tlnpcntl on two.