Boundary Lyer Theory

'I'his c.quat,ion Lmnsforms into t1in.L for n flat plat^, cqn. (7.28), in (.he special case
whcn m = 0. The solut.ions of t,hc Falltncr-Slran eclun.tion (9.8) have been discussed
in tleta.il in 1611.
According t,o J. Sl~cinllcr~cr [631, nn interesting cxtensiot~ of 6hr solr~t.int~ of t,llr r 'nlltner-Sltnn
eq~lntion (9.8) which in vnlid for ret.nrded flows (P < 0) in cases when velocity dintrihtct.ions possest+
ing n velocity cxccns (I'(i1) > 1) with n~naxin~utn near the wall arc ndrnittcd. In RIIC~I cnscs, the
limit /'('I) = 1 for 11 -+ 00 is nttnincd nsympLot,icnlly "from abovr" rnthcr thnn "from hclow",
as was t,lle cnw RO far. SIICII uo1uI.ions can he interpreted pl~ysirnlly as corrrspo~lding to a laminar
wall-jet prod~~cctl in nn oxtnrnal strcnrn wit.11 n positive pressu~e grndient,. dplda: > 0. Ileferenro
[G3] drrnonnt,rntn~ t,hnt t.lw limiting cnne of ll~mo uolut,ions, oI)tnin~d W ~ I I,IIc I tnnxirnun~ velocity
cxccns tend^ 10 in fin it,^, trnnnforms illlo tllr wr-ll-known dl-sitnilnr nolttl.ion of n plro wnll-jet in
t,lm absence of nn cxtrrnnl vclociLy -- n cnnc trcr~tccl hy RI. 11. (:lartt:rl (ucc 1401 in (Illr~p. XI)-~when
we put, p = -2.
A pnrtirulnrly drt,niletl n~onogrnpll on exnrt., self-nirnilnr solt~t.ions for lnminnr Imlndary
lnyeru in two-din~cnsional nnd rot,ntionnlly symmetric nrrangemcnt,~, inrl~lsive of the nssocintrtl
thcrinnl boundnry lnycrn (am Chnp. XTl),wns prlhlinl~cd hyC. 1'. J>cwey nntl J. F. Grosn [141.
Their consitlcrnt.ionn inclntlc t,lle elTt:ct.s of con~presaibilit~ (nee Chnp. XIJI) wil.11 and mitl~out, hcn,t
tmnnfer, relate Lo vnryitlg vnlnes of t.he Prar~tlt,l number, and incJ~tde some rases of suction and
blowing.
K. 1(. Clien nnd P. A. Libby 191 cnrried out nn cxtx?nsivc invcst,ignlion of bo~~~~rlnry lnycrs
which are el~ornctorizcd by ~mnll clcpnrtnrcs from t.11~ nelf-ui~nilnr \vctlge-flow boutltlnry lnycrs
of tho I'nlknrr-Sltan type. Rvidcnt,ly, RII~II 1)ounrlnry Inyerrr nre no longer nolf-~in1iln.r.
b. Flow in n convergent channel
The case of potmt,ial flow given by thc eqlrnthn
U(s) = -2L
x
is related to flows pt~t a wedge, and also leads to 'similar' solutions. With > 0
it rcprcscnt,s two-dimengional mot,ion in n convergent ohnnncl with flat, walls (sink).
The volume of flow for a frill opening angle 2n and for a strnt,~~nl of ttnit
I~cight is ($ = 2 n ?I,, (Fig. 9.2). Int,rodncing t.he simi1nrit.y t,ransformat.ioti
Fig. 9.2. 1 % ~ in n ronvrrgrnt rhnnnrl
, I . ltr I)~IIIII~*L~.~ ~on~li1,ion~ rollow Prom c(ln. (V.3) nl~tl nrc?: /' : 0 nl. o, 0, I / I
1tt1(1 /" = 0 a(* 17 == w . 'I'lris is nlso :I j)nrl,icrrlar caso of I,llo clasa of 'similar' sol~lt~i~tt~
consitlcred in Chap. V111. ISquntiot~ (9.12) is obtnincd from 1.11~ more gcncral tlifli~rc~tltial
equation (8.15) for the case of 'similar' boundary layers, if we put a - 0, ntld .-.
4- 1. The example under consideration is one of the rare cases whcn the sol~ttior~ of'
tllc botrndary-layer equation can be 01)tdncd analytically in closcd form.
First,, upon ~nult~iplying cqn. (9.12) by 1" and integrating ol~ce, \vc? I1:1vc
where n is a ronstnnt of intrgmtiou. 1t.s value is zero, as /' .- 1 ant1 /" -- 0 lor
7
v;
f 00. '1'1111s
-- - -- --
T = (,I - 112 (I* + 2)
d 71
whrre the additive constmit of intrgration is seen to bc cq~lnl to zero in virw of
tile Im~ndary condition /' = I at 17 = oo . The int,egral ran be rxprcssctl ill closrtl
form as follows:
or, solving for 1' = w/11:
/' = = 3 t8anh2