Boundary Lyer Theory

94
V. Exnct sohltiono ol' tho Nnvicr-Slnltcq oqr~nt,ions
Tho velocity profile u (!y,t) thus has thc form of a damped harmonic oscillaLion, thc
amplitude of which is I/, c w i ? ; , in which a fluid layer at s distance y has a phase
lag y l/;t% with respect to the motion of the wall. Fig. 5.9 rcprcscnts - this -. motion
for scvcral instants of time. Two fluid layers, a clistanco 2 n/k = 2 n d2 v/n apart,
oscillate in pli~c. This distancc can be regarded as a kind of wave length of the
motion: it is somctimcs called the depth o/ penetration of tho viscous wave. The
layer which is carried by tho wall has a thidrncss of t11c order d - Jq and dccrcasos
for decreasing kinematic viscosity and increasing frequcncyt.
8. A rlnm of non-steady solutions. A general c1:rss of no11-stcntly sol111ions of the
Nnvinr-Stnltw ,scq~latio~ls which possran bor~ndnry-lcycr o11arnctr:r is ol~tainrd in the sr)ccinl mm
when tho velocity componcnta arc indopcndcnt of Lho longitudin:~l coordinnl,c, a. 'rhc systcn~ of
rrlr~nt.ions (8.02). writlnw for plnno flow. nasun)cs 1.11~ form
----. .-
t Tltc ROIIIL~OII in cqn. (5.2(in) roprcscr~t.s also t,l~c tcmporat~~rn c1intril)ution in Lhc rarth which is
muwd by t.hc pcricxlio Iluc*t.r~at.ion of I.ho k~npcraturc on tho surfncc, my, from clay 14) d:ly or over
t,hc scnfu~ns in a yc::w.
b. Other oxnct solulionn 95
If we now prescribe a cnnutRnt vclocity v, < 0 at thc wall (suction), wo notice that cqn. (5.27~)
is satisfied in~mediately hy a flow for whicl~ o = v, and that the prc.wuro p bcaorncs indcpnndcnt,
of uirnultrmco~~sly. Accordingly, we put - (l/e) (+/ax) = tI(J/cll,, whom 11(t) donotes bile frwstrrnm
vrlocity nt jr very largc dishnc:~ from t.1~ w:rll, nncl I~cncc obtain 1l1c followir~g clilTc.rcl~l.itrl
cqtmtion for u(y, 1):
au b 3t 1
dU azu
l,,, -. ..-- 1
ag dl ay2 ' (5.28)
According to .I. 'r. Stuart m2] thorecxista an oxnct soluI,ion ofccln. (5.28) for tllo arld,r:rry oxkrr~al
vclocity
'lll~is so1116ion is
whcro
Sllh~tituLing thc I.wt thrw cqu~tions ink cqn. (529, we am led ID n psrtinl diffcrrntial oq11st.ion
for the unknown function g(!/. 1) = g(7. 1); thin hnn 1110 forrn
Tllc following non-di~ncnsionnl varinhlcs hnvo been irltrodud in the prccocling:
I'ie. 5.9. Vrlocit,y rlistrihution in
the neighbourhood of an oscillating
wall (Stokes's second problem) Solutions of (5.32) hnve hccn ohtaincd by J. Wnhn (411 who crnploycri Lnplaoo transformations
and who restricted hirnuclf to severnl apecinl forms of the functior~ /(1). (:cncrally
speaking, the following cxternnl flows, U(1). hnve been incIudw1:
a) dnrnped nnrl undampcd oscillations,
h) stop-likc chnngc from one vnluo of vclocif.y to xnot.lwr,
c) linear incre.nuc from ono vnltlc to anoll~cr.
In the upncial c.wc whcn the cxlcrnnl flow is indcpcnclcnt, of time, /(t) - 0, cq~~ation (5.32)
I-~ds to the uirnple solution '(7, 7') = 0. This CDIIRP* v01oi~il.y prolilo from oqn. (5.30) to
I~oromc iclont.ic:rrl wiI.11 Llw nuyrnptoLic s11c1io11 prolilo givcw IILIAW ill WIII. (14.l;).
The preccding examples on one-tlimcnsional flows were very simplc, I)cca~~se tho
convective acceleration which renders thc equations non-linear vnnishcd idontically
everywhere. WG shall now proceed to examine sorno exact solutions in which thcsc
terms are retained, so that non-linear equations will havo to t)o considcrcd. We shall,
however, restrict oursclves to steady flows.
9. Stagnation in plane flow (Hiemenz flow). Tho first simple examplc of this
t,ype of flow, represented in Fig. 6.10, is that lending lo a shgnc~tion point in plane,