Boundary Lyer Theory

and t,llc? clnsh now clet~otos tlilTrrcntia.tion with rcspcct, to (. Thc boundary contlitions
art?
( ~ 0 F=O; : t=oo: r=O (0.41)
whcro t.1~ consl;~rtl~ of inl.rgr:it,iotr was m:itle cqunl to I. 'l'llis li)llows if we p t
Ff(0) - 1, wlticlt is prrnlissihlo wil.ltottt losx ol'gc~lcrnlil,y Imnnsc of I,llo frcc cotlst.:inl.
a in t,l~c rrlat.icin Iictwoen f ~ n P. d 1Cq11atio11 (9.42) is n clill:rrnt.ial cqn:tt,ion of 1tic:t::~t.i'~
typc ancl can Iir int.rgrat.ctl in closctl t,rrms. \Ye oli1.ni11
I 11vr1 t ing this rqnnt ion wr obtain
Since, furt,llcr, tlP/tlE - I - 1:in11~
qn. (9.37) and is
F
1 - exp(-BE)
I. =t,anh E= - - -
1 4- cxp ( 3s) '
E, the vc1ocit.y (li~lril~~tl~ion (:all I I tloclucctl ~ from
1 - r (I t.an11~ 6) . (9.44)
.I
1.he vrlorily tlisl.rilwtiorl from cqn. (9.37) is soon plott.ctl in Pig. !).Is.
1L now rcn1:tins t,o dc(.crtninc. t81rc const:tn(. a, :LWI this ciln be (lone wit.11 the
aid of condition (!).:3R) wl~ich shtcs that t,l~c rnomcnl,um in 1.11~ x-tlirrcl ion is ronst,nnt..
(hnbining rqns. (9.44) :111(1 (0.36) we obhin
we shall assume that tho flux of momontum, J, for thc jet is given. It is proportional
to Lhr excess in pressure with which the jet leaves the slit. lrrtrodricing the kinematic
mo~nenlmt .I/@ = K, we have from eqn. (9.45)
Fig. 9.13. ,VrIcirit,y dist,ril~ulio~~ in x t,\\o-rJimrt~.
sion111 nn(J cire~ll~w frcc jcL fro111 cqns. (9.44)
:md (11.16) icspect~ivcly. For tho two-tlirner~.
xionnl jct [ = 0.275 KIP y/(v~)~/~, and for the
circnlar jct. C - 0.244 y/vz. I< and K'
t1twot.c: Ilir kincrnat.ic monwnt.um J/e
and, hencc, for tlio volocit,y distribution
K. Pnrnllcl &reams in hminar flow
r 7
Ihc transvcrsc: vclooil,y at thc bountlnry of Iht. jet is
-1 00
ant1 the volume-mtc of discltxrgc per unit height of slit bocorncs Q = e J v (I!/, or
- m
Q = 3.3010 (I< VX)"~.
(!).48)
Tlic volumc-rate of tlisclmrgo increases in the tlownstrcam direction, bccai~sc: flnid
particles are carried away with the jet owing Lo friction on its boundnrics. It also
increases with increasing momcnt~um.
The corre,sponcling rotationally symmct.rica1 casc in which the jet cmcrgcs from
n small circ~~lar orificc will be tliscussed in Chap. XI. The problem of t,hc twodirne~~sional
laminar compressible jet cmcrging from narrow slit was solvctl Iiy
S. 1. P.zi [4!)] nntl M. Z. JZrzywo1)locki [42].
Moasurcmrnts performed I)y TI:. N. Antlrntlo [I] for tho t,wo-tli~ncnsiot~:~I 1n.rnina.r
jct confirm t.he preceding thcorct~icd argurncnt vory well. 'l'llo jct rcn~ail~s laminar
np t,o R - 30 appro~irnat~rly, where the Ibynoltls number is rcfcrrctl to thc cfflrrx
vclority and to the widL11 ol' tho slit. Tho casc of a Lwo-tlinlensional ant1 t.llat of .z
circular trtrl~ulent jct is discusscd in Chap. XXIV. A comprchensivc review of all
probloms involving jets can be found in S. I. Pai's book [49].
g. Pnrnllel streoms in laminnr h w
Wo shall now 1)rirfly cxnminc the laycr 1)ctwccn two pnrallcl, Inminnr sl,rcnms
which move at tlifTercnt vclocitics, xntl so provitlc a htrtl~cr cxnrnplc of the npplicability
of the bountlnry-laycr equations. Thc forrn~~liition of thc problctn is scot1
il111sLraLctl in Fig. !).14: Two it~il~ially scp:ir:~Lc(l, ut~disO~~rI~~xl, prnllcl HL~~!ILIIIS whith
move with the vclocit.ics TJ1 nncl (I,, rcspcctivcly, l~cgin tm intcrc& thro11g11 frit:l.iorr.
It is possi1)lo Lo assurnc thnt the transition from the vclociLy U, to vclocity (I, talccs
in n narrow zone of mixing and that the transvcrsc vcl&ty component, v, is
everywhere smalc oomp,zrcd with the longitudinal velocity, 11. Consequently, the
boundnry-layer equation (9.1) can be usctl to describe the flow in thc zoncs I and 11,
and the pressure t1crm may be omitted.
In n manner analogous to that employed for thc boundary layer on a flnt platme
(Scc. VIIe), it is possible to obtain the ordinary tliffcrentinl equation