Boundary Lyer Theory

the of how the velocity profilc varion with tirne tcnding nsyn~ptotically to the linear
diutribtrtion nlrown in Fig. 1.1. The diITcrcntinl cqriation is the same en before, cqn. (5.17),
lmt with modified I)o~~r~clary conditions which now are:
'rllr solutio~~ of eqn. (5.17) which sntinficn tho bor~ndary :d initial ro~~ditior~s ran Iw
oht.:~inrd in t.l~c form of a ucrio~ of con~plcn~c~~tnry error fun~l~ions
7x1
11 'y'
- = x erfc r2 n 4- 711 - x crfc [2 (71 -1 I) ?I# - ?I]
('0 ,,-I, , -.I3
rrfc - rrfc (2 q1 - tl) -1- crfc (2 -1- 71) - rrfc (4 11, - - 11) 1- rrfc (4 71, .1- 71) - . . . 4- . . .
wllerc 71, := h,/2 1/ F i (lot~~t.cn
(5.24)
the cli~nenniol~lcsn tlistancc between t,l~c two wnlllr. 'Tho solut,ion
is represellted in Iiig. 6.7. 'rlw corly profiles nre &ill aplwoxi~nntely similar and rc~nain so, an
long nn t,llr bolllldary layer l~ns not sprcad to the stationary wall. The s~lcceeding vcloc:ity r)rofilcn
:).re no lo~~grr "similer" a~~tl tc~~cl nsynptotirnlly to t,lre linrar distrihrrt~ion of tile skndy st&?.
Exact solrtt,ions for non-stratly Coric.l.t,c flow werc rlcrivcd I)g .I. St.rinl~c~irr (331
for 1I1r (xsr WIII~II OIW ol' 1,110 wdls is ILI, ITS^, in ;I, shn(ly flow II,II(~ is 111v11 SII(I(I~II~.S
wc:r~lv~.at,c:cl to R givc.11, c:onstnut, vcloc:it,,y. 'l'o (lo t,his, il, is Iicbvc:ssal,y 1.0 solve! ['(lit.
(5.17), whirli is itlcnt,ical with tho one-dirnrnsior~nl Iicat conrluci.ion cqtlat,ion, l)y
lncnrls of n l~otiric~r. srrirs. A spccic'll CR,SC it, t.llis class of soluf.iotls is t.hd whrn t'l1~
moving wnll is sutltlrnly st.oppctl so t,l~al, it rcprrscnt,s tho decay of (h~ot,t,c flo~.
a. I';L~IIVI flow n:!
lnyor ncnr tho wall. 'The influonce of vi~cosity rcnrhcs the pipe ccnf.rr only in the 1:rt.c.r st,:~grn
of motion, antl tho velocity profile tonds asy~npLoLically to tho pt~roldic tlistribt~l.io~~ for ste:rtly
flow. The corresponding solut,ion for an nnn~tlar circul~rr cross-section was given 113. W. Murller 1201.
,, 1 IIC nccrlcr~~th of 11 I111id ovrr ~,IIc wl~nlc ICII~I,II of pipe di~c~~sst%(l 11~re IIIIIH~,
din~ir~g~~inl~ctl from t.lic acrcIornt.io~~ of n fluid in tl~c illlet j~ortio~~a or
,.41rcsrlllly
a pipe in ~IJ*:L~I,~ IIOW. 'I&
rechngnlar ve1ocit.y profile whicl~ exists in the entrance ucct.iol~ is grncl~lnlly transfor~~~ed as
t.he fluid progresses through the pipe with x increasing, antl tends, ~~ndcr the influence of viscosity,
to nssnine the Hngen-Poiscnillc parabolic diaI.ribntion. Since I~c?rc a@z :t 0 tho flo\rs is not
onc-rli~nensiond, nncl the vdocity depends on x, nu \vrll ns on t.ho rndi~rs. Thin proh~n wak
rlisrusricd by 11. Srl~lichl.ing [DO), who gave t,l~o solrlliolr for L\vo-tlin~c~~sio~~nl Ilo~ tl1ro11~11
n st.r:~igl~t. rhannel, antl by I,. Srhiller 1291, ;~nd B. 1'1111nin 1241 for nxinlly symrr~rt.rir,nl Ilow (rirc~~lar
pipr): srr nlno Sew. IX i nnd X 111.
Fig. 5.8. Vclocit,y profilc in n rircrrlnr pipe d~~ri~~g
ncc~rlrration,
art given by 1'. Szgtnnnski [87]; T .- v //I12