Boundary Lyer Theory

138 VII. no~~ndnry-layer cqnntions for two-dirncnaionnl flow; houndary layer on a pink
This means that at the outcr edge there is a flow outward which is due to the fact
that the increasing boundary-layer thickness causes tho fluid to be displaced from
the wnll as it flows along it. There is no boundary-layer separation in tho present
case, as t,he pressure grndient is equal to zero.
J. St,cinhcuer [25] pr~hlisllctl a systctnatic rcvicw of t,hc solut.ions fro TJln.sius's
equation. 111 part,ic:ulnr, hc providcd a tli~n~~ssion of t~llr chnrnct.cr of the sol~rtions ill
the intcgrntion rnngc where r] < 0 in the presence of a varict,y of bountlnry conditions.
It turns out. that t,llcrc exist, three set* of solr~tions which differ from each other by
their nsyrnpt.otio ldmvior atf 7 + -m. Apart from t01r larninnr hountlnry layer on
a flat platme, the solutions which can I I givcn ~ a physically mcaningrul intcrprctnt,ion
include Inminar flow between t,wo parallel streams of which the two-dimensional
hnlf-jet. is a special ca.se (scc See. IXII), larninnr flow with suction or blowing nt right
angles (see Src. XIVb), as well ns tho laminar bonntla.ry Iaycr formed over a wn.ll
moving parallel to thc stream in the same or in the opposite direction.
Skin friction: Thcskin friction can be easily clctertninrtl from the precotling tlnta.
From qn. (7.19) we obtain for one side of the plate
wllcrr h is t h width and 1 is the Iengt,h of the plate. Now tho hlcal shearing stress
at the wall is given by
wiLh /" (0) -- a - 0.332 from Table 7.1. llence the ditncnsio~~lrss shearing st,rcss
1)ccornra :
Co~~sccjrmttly, from cqn. (7.30), t,lw ski11 friction of one sitlc I)ct:olnt.s
nntl for a. plnt,r wrttrd on I)of.h sitlrs:
1
It, is rrtn:l~kn.l)lr tI1:tt. fhn ski11 l'ric+iotl is 1rroport.ionnl t.o t 1 1 ~ powrr # of vclocit,y
whcrcns in rrcx?l)ing rnot.io~l t.hcrc was ~)roport,ionali(.y t,o the first, Imwrr of vclocit,y.
I~i~rt~l~rr, f,llr: t1m.g incrca.sos wil,h I.llo sclrtwc roof, of t,hc Icngt.l~ of the 1,la.t~. This
(.:I.II l)c i~~t.c~rprotc-tl :IS showing t,ha.t. t h (lownst.re:~rrl ~
1)ort.ions of the pI:~t.(: cont.ril)~~Le
11rol)ort.io11:1t.c-ly Irss to t.ho t,ot,nI tlr:lp t0wn t.hc portio~ls nra.r t.11~ Irntling rtlgr,
c. The bountlnry laycr along n flnt, plah 1 3!)
Tahle 7.1. The function /(v) for tho boundary layer along a flnt plate at zero incidence, after
L. Mowarth 1101