Boundary Lyer Theory

124 VI. Very slow motion
of tho t,wo-tli~nc~~~sionnl potential flow past the givrn body. Tl111s ?I,, v, and p, satisfy
tho equations
J'irst wr nol,ioc all O~CO from 1.11~ soIut,ion (6.39) that 1.11~ cqr~at~ion of continuity and
the cquntion of motion in the z-tlirection are satsisfietf. The fact, that the equations of
motion in the z- and y-directions are also satisfied follows frorn thc potential character
of ?I,, and v,,. Tho functions ?I,, and v, satisfy the condition of irrotationality
so that the pot.cntin1 equations V2 71, = 0 and V2 v, = 0, where V2 = a2/i3z2 +
iI2/B?p2, are sal,isfied.
The first t\vo rqnnt.ions (6.3) reduce lo ap/az - /L a2u/az2 and ap/a?j = /L a2v/az2;
t.11r.y nrc, howcvcr, uat,isfied, as seen from C~IIS. (0.39). Thus eqns. (0.39) rcprcsent
a solt~t.ion of t.hn equations for creoping mol.ion. On tho othcr Irantl Ll~e flow rcprcsentctl
by rqns (6.39) has the same streamlines as potential flow al)out the botly, n.nd the
st.rcamlines for all parnllcl layers z = const arc congruent. The condition of no
slip at tho pla.1.c~ z = f h is seen t,o be satisfied by eqn. (6.39), but the condition
of no slip at the surfacc: of the body is not sat,isfied.
'rhr ml.io of incrt,i:~ t,o viscous forces in JTele-Shaw motion, just as in the casc
of i,l~c mot.ion of Irll)ricat,ing oil, is givc:~~ 1)y t.11~ reduced Itnynolds number
whc-rr 1, tlcnoks a n11amat.cxistit: lincar tlimc.nsion of l.hc botly in the R., ?/-plane.
If R* c:scwtls unit,y the inertia tmms Iwco~ne considrmlllc nntl the motion tlcvin1.r~
from 1.11~ sin~plc sol~~l~ion (6.39).
'1'11~ solt~t.ion given by oqn. (6.30) can bc improvotl in the same mannor as
Stolcc~s's solu1,ion for a sphcro or t.hc solution for very slow flow. The inertia t,crms arc
cnlcnl:~.l.otl from t11c first approsimni.ion and introd~lce;l into the cq~tat,ions ns
c!xt,rrn:rl forws, :~ntl an improvcd solution results. This was carricd out I)y F. Riegels
1181 for t-he casc of Tfclc-Shnw flow past, a circular rylindsr.
For R* > 1 fl~c st.rt:nmlinos in t.11~ various layers pnrallcl to the ~valls cease
to l)o congn~cnl.. Tho slow p;~rt,icles near t,ho t~vo plates are tlefleclcd more by
t,l~o 1)rcsmc:r of (.he hotly t11n.n i,hc fast,rr particlcfi near tho ccnt.rc. This causes t'hc
st,rcan~li~~rs t.o n.pprn.r somcwhnt blurrrtl a.rd i,hc phcnomcnon is more pronounced
at, tJtc rcnr of lhc botly than in fronl, of it,, Fig. G.6.
Solutiol~s in thr case of cwcping motion are inherently restricted to very small
Rrynoltls n1tni1)rrs 111 prineiplr it is possiblr to extend tho ficltl of applicat,ion
to I:~rger Reynolds numbers by successive approximnt,ion, as mentioned prcvionsly.
IIowevcr, in all cases the calculations become so complicated that it is not practicable
to carry out more than one step in the approximation. For this reason it is not
possilh to reach t11o region of motlcrntc Rcynolds numbors frorn this tlircctiol~.
, .lo , all intmts and purposes the region of moderate Re~nolds numher~ in which
1,110 in(:rh tincl viwo~~s forc~s nrn of ~~oIII~~I~~:LI~Ic m~~~~t~ilwlo
I~II~OII~IIOIII~ IJIC lia>l~l
of flow 11a.s not been cxtcnsivcly investigntcd by analytic means.
It is, therefore, the more useful to have the possibility of intograting thc
Navicr-Stokes cq~~ation for t,he othcr limiting casc of very large Rcynolds numbers. ,
'I'hns we arc lctl to the boundary-layer theory which will form the subjcc.1 of tho
lollowing chapters.
Fig. 6.6. IJcle-Shew flo~v
past circulnr cylinder nt
R* - 4, shr Iticpln [I:$]
Referencer
[I] Bauer, K.: Einfluss der endlichen Breite des Gleitlngcrs nuf Trngfiihigkeit uncl IIeibr~ng.
Forschg. 1ng.-Wes. 14, 48-02 (1943).
121 Constnntinescu, V.N.: Analynis of bearings oprmting in turl)ulrnt rcgin~e. 'l'rn~~s. i\SI\lE,
Serb D, J. Ilnsic Eng. 84, 130-151 (l!)(i2).
[3] Constnntinescu, V.N.: On the influence of inertin forces in tnrbulent and Inminnr selfacting
films. Trans. ASME, Series F, J. 1,llbricntion Technolo~y 92, 47:1--481 (1970).
[4] Constantinescu, V.N.: On gun lubrication in turbulent regin~e. Trans. ASMI':, Series 11,
J. Basic Eng. 86, 475-482 (1964).
[R] Frossel, W.: lteibl~ngs~viderntnl~d unrl Trngkrnft cin~s Gleitnch1111cs endlichrr Brrile. Po~.scIlg.
1ng.-Wcs. 13, 65--75 (1042).
[GI Giimbel, L., and Everling, 13.: Ileibung und Schn~ierung in1 Mnscl~incnbnu. 13crli11, 1025.
[7] Hole-Shnw, H.S.: Inve.st,igntio~~ of thc nnturc of surfncc renist.nncc of wntar nntl of st,ron~n
motion nndcr ccrtnin ox~icrin~cntnl conditions. T~OIIA. Innt. Nnv. Arch. XI, 25 (IA!)H); ~ co
nlso Nnture 58, 34 (1898) tint1 Proc. Roy. Innl. 16. 40 (1899).
(81 Knhlert, W.: Dcr Einllusx cler l'rlgheit.~ltrlfk bei der Irydrotly~~a~~~iscl~etl Schtniertnittolthcorie.
Uius. Brnunschweig 1947; 1ng.-Arrh. 16, 321 -342 (1948).
[9] Michell, A.G.M.: Z. Mnth. 11. Phys. 52, S. 123 (1905); mealso Ostwald's IClnssiker No. 218.
(101 Nnhtne, F.: Beitrrigc zur l~ydrodynnn~incl~e~~ l'hcorio der 1,agcrrcibung. 1ng.-Arch. 11,
191 -200 - (1040). \-
[I I] Oseen, C. W.: Uber die Stokcs'sche Forrnel und iiher einc verwnndte Aufgnbc in der Hydrodynamik.
Ark. f. Math. Astron. och Fys. 6, No. 29 (1!110).
[I21 Pmndtl, I,.: The mechnnics of viscous fluids. In W. F. Ihrand: Aerodynnn~ic Theory Ill,
34-208 (1035).
[I31 R,iegels, F.: Zur Kritik des Hele-Shnw-Vcrsucl~es. Diss. Gnttingen 1038; ZAMM 18, 05- 106
(1933).
1141 $nib&, E.A., and Mncken, N.A.: The fluid mechanics of Inbricntion. Annual Review of
Fluid Mech. (M. Van Dyke, ed.) 5, 185-212 (1973).
[IR] Snibel, E.A., nnd Mnoken. N.A.: Non-lnmirmr bchnvior in bcnrings. Critic-nl review of the
li(.rrnt,urr. l'mnx. ASMIC, Scric~ F, .I. I,~lliricntion 'I'ccl~nolo~y 96, 174---I81 (1974).