Boundary Lyer Theory

conclil.ion (4.14) plays n part wl~ich ia ein~ilnr tn 1.11~ no-slip condit.ion of n red fl~~id. 'Chc latter
cnn be saLislic4 by Ihc solutions of 1.11~ Nnvicr-Stokcs eq~~etions I~ut not by those of Euler'a
cquntio~~s. 'I'lw slowly-varying solnt,ion is trnnlngrr~~s In) tile frictionlcsn solution (ptrntinl flow)
whicl~ f:rils to satisfy the no-slip contlil.ion. 7'11~ f~rst-vnrying solntion rcprcscnts Lhe counlr?rpart
of tho bonndnry-lnycr ~oIuLion whicl~ ia delcrn~incd by t.ho prcscnm of viscosit.y; it clin'cn fron~
zero only in n narrow zone near tho wall (boundary Inyer). It is to bo nokd that the second
bonndnry condition (no slip at tho wall) can only be sal,inficd if this bountlnry-layer solution is
a.dclwl, t,l111s mnking tho whole sol~~bion phy~icnlly red.
This simple rxarnplc cxl~ibit,s thc sarnc matbcmstical Aaturcs M t.l~osc ch?usscd in 1,110
prcrcding cl~u.plcr. It is, nrrrnrly, not pcrn~iasil~lc si~n rly In onlit tl~c viscou~ tern18 ill tlw
Nnvicr-Stokm equation, wlmn performing the process or going over to t.he limit Tor very small
viscosit.y (vrry I;irge llcynolrln n~~rnbrr). This wn only bc: clonc: in tile intrgrnl solnl.ion itxlr.
We sha.ll tlcnionstratc latm in grcatcr c1cl:iil tht if, is not t1cc:c:ssary to rctain lhc
Cull Navirr-St.olrcs eqnnt,ions for the process of finding the limit for R -+m. For
lhc salte of n~athrmatical simplification il will provc possible to omit certain t.rrni~
in it, pnrticnlnrly certain small viscous tcrnls. It is, however, important to note that,
not all viscous trrma can bc ncglrctrtl. ns this woril~l depress tllc ordrr of tho Navier-
Stolrrs rqnntions
[l] Ackcrct, J.: Ubcr cxnkte J5sungen dcr Stokes-Navi~r- Glcicl~ungen inkomprrmihler 1Pliiimigkciton
bci vcriin~lcrbn (:rr~~r,l~c~li~~~rtngc?~~.
%1\;\11' 3, 259--271 (1952).
[In] Apeelt. C. ,i.: 'l'hc ~trnrly Ilrtiv of a viscous flnid pat n circulnr cylindcr at Reynolds numbers
40 and 44. Ikitisl~ AltC ItM 3175 (IWil).
(Lh] Allen, D.N. 1)c G., m~ci So~~t.hwcll, 1t.V.: ltclaxation methods npplicd to deternline the
mot,ion, in t,wo di~nm~iona, of R viscons flnid pnat n Bxetl cylinder. Q~mrt. J. Mecb. Appl.
MnLIl. 8, 12!)-145 (1!)55).
[lo] Coutnnccau, M., nnd Uo~lnrd, R.: 15xpcrirncntnl dckr~ninnt.ion of t.lw main fcnt,nrrs of the
vinrn~~n flow in ...~. tho wakn of R circular cvlinder in 11uifor111 tra~~slation. Par1 I. Stendy now.
~
JFM 78, 231 -256 (1977j.
[Id] (huta~~cenu. M., ~1r1 Ih~nrd, It.: RxprritnrnLal detcnnination of thc mnin fcnCuren of thc
visco~~s flr~w in tbe wake of R circulilr cylinder in uniform trnnslation. Part 2. Unsbndy flow.
.11W 79, 257- 272 (15377).
[2] Ihmnis, S.C.K.. and GRII-ZII Chang: Nn~ncrical soI~~t,it)ns for stcarly flo~ past x circ~~lnr
cylintlcr nt, I