Boundary Lyer Theory

90
V. ICxnct sol~ttion~ of tho Nnvior-StOkcs cqi~:~liot~~ a. Parallcl flow 9 I
as derived by C. W. Osecn [21] and G. 1Ia1nel [I]]. This velocity distribution is
represented graphically in Pig. 5.5 Here 16 dcnotcs t,he circulation of the vortex
filamolt, at time 1 =1 0, i. c. at tho moment whcn viscosity is nssumed to bcgit~ it*
actiol~. An cxperimenLal investigation of this procoss was ~~ntlcrt~nlta~~ 11y A. Tirnmo
[40]. K. Kirdc 1171 mndc an nnnlytio study of the caso when the velocity distribution
in t,ho vortcx tlilTcr~ from I.hnt irnposctl hy pot,cnt,inl theory.
4. The suddenly necelernted plane wall; Stokes's first problem. We now procccd
to calcuhtc somo non-steady par;rllcl flows. Since the convcctivc acceleration terms
vanish itlcr~tic:aIly, the frict.ior~ forcos intrmct with tho local ncce1crnt.ion. Tho si~nplcst
flows of this clam occur when motion is stnrtcd irnpr~lsival~ from rest. We sl~all
begin with the c,wc of thc flow near a flat plntc which is s~tldcr~ly acce1cr:~tcd from
rest and n~ovcs in it,s own plane with a constant vclocitty [lo. This is onc of the pro-
I~lcms which wcro solvctl by (2. Stokes in his colcbr:rtccl memoir on per~tl~~lurr~s
[3ri]t. Selecting tho z-axis along the wall in the dirnction of U,, we obtain the
simplifiotl Navicr-SOnlccs oqmt.ion
'rho prrssuro in tho wh01o space is constant, and Lhe bol~nclirry conclit,iol~s are:
The cliIT(:rcnt.ial equation (5.17) is icIcnt.ical with the equation of hest conduction
which clcscribcs the propngatrinrl of Itoat, in tho space y > 0, whcn at time 1 = 0 the
wall y = 0 is sudtlcnly hcatcd to a t,cmpcr;~t,nre which oxccecls that in the surroundings.
'l'lle pnrl,i:~l tliffcrcntial oquation (5.17) can be retlucctl t.o an ortlinary diTcrt:ntial
cqu:~t.ior~ 11s tho sul)st.il,nt.ion
Y
(5.19) " --3' 2 1/
If wn, further, n.ssumc
x = Uoj(r]), (5.20)
wc o11I.air1 the followi~~g ordinary tliITorcntial equation for / (q):
t.hc complemenhry error /um%ion, erfc q, 11.w been tabulatedt. The velocity distribution
is rcpresontcd in Pig. 6.0, and it may bo notctl that tho vclocity profiles for
varying tinies arc 'similar', i. e., they can bc rctl~lccd to the same curve by changing t,hc
scalc ttlong the axis of ordinates. Thc cornplcmcntary error f~~nctior~ whicl~ appcnrs
iu eqn. (5.22) has a valuo of about 0.01 at 7 -- 2.0. %.king into accor~r~t tho tlcfir~ition
of t,l~c: t~hic:ltnoss of the: I~ot~ntl~try Inyor, 0, wc: ol)t.r~ir~
6=2qaJZx4 JZ. (5.23)
It is seen to be proportionnl to the sqnnrc root of tho protll~ot, of kir~ornnLio visc:osiOy
silt1 time.
This problem was gcncralized by E. J?ccker [3] to ir~clr~dc: more genrml rat.rs
of nccclnratio~~ as well as the cqses involving suctior~ or blowing or tho cfict of
compressil)ility.
Fig. 5.6. Vclocity dishibution
above a snddenly accelerated wall
5. Flow forn~ntion in Cmuette motion. The s~~bstiLuI.ion (5.10) which Imds to eqn. (5.21)
dm not, in general, lend to a sol~ttion of 1.hc so-cnllcd lwnt conduction cquntion (5.17) ir morc
complicntod boundnry contlilions aro itn~~osctl, Sincc cqn. (6.17) i~ linear, solution^ (i)r il, OILII
be obtained by the use or the 1,nplncc t,mnsfor~nation nnd by tnoro direct nlcl.hods dcvclopc:tl
in conncxion with tho study of the conduction of hcnt in solids. Mnny rc~~lkt obtni~~ctl, c. g.,
for the tcmperaturc vnriation in nn infinite or semi-infinite solid, cnn be tlircctly transposed
and uacd for the ~oIut,ion of problems in viscons flow. Thw the prcccding problem in which the
formation of tho boundary layer noar a suddenly accclcrakl wall has bwn invwtigntrcf can
also be nolvcd for tlw CDSC when the wall movur in a direction parallel to ar~otl~or flat w:dl at.
n ~ and t at a distantx, h from it. This is the problcm of flow forn~ation in Couettc motion, i. c.,
t Soe c. g. Shoppard. "The Probability Tnbgrnl", Rritish Atwoe. Adv. Sci.: Math. Tsblea
vol. vii (3039) and Works Project Administration "Tables of the Probability Function", New
York, 1041.