Boundary Lyer Theory

88 V. Rxnct uolutions of tltc Nnvinr-Stokcs rqr~ntions n. Pnrallrl flow H!)
x 1-1.2
(1) A,-.--- (~nner rotntinp, outcr at ITS^). (5.lGn)
11, I -x2 x
TI is ~tol.t.worIhy IItnt, t,ltr vclocit.,y vnrirs strtmgl~r wil.l~ t,hc rnt,io x - rl/rz of t,hc I~WO
radii ill Cnsr I, whrrras for (hsr I1 it is almost intlcprnclc~~t of' it. MThcn x = rl/rz + 1 ,
Iwt.l~ c~tscvi tclttl t,o the linrar vrloeity tlistril)l~t.ion of (!oucstt.(, flo~, as it, ocrurrcd
1)cI.wc~vn l,wo flat plat,cs in thr rase rrprc~srntccl in Pig. 1.1 . The cc~nnt~ion of' Cnsr J
yicltls tho satrtr linlif fiw r1 -- 0, i. C. fnv x = O \vhcn 110 in~ier rylintlcl is prrsc~lt. In
(.Itis c,nsr, IJIt(~ Il~~itl 1~)(;11t>s insitlt: IIIC out,cr eylintlrr as n rigit1 I)otiy. Ilcncc il. is seen
lIln.1 ('nso I1 yicsltls n lineal vcloril~y tlislril)~~t,iotl POI. llir t,\\~ sy111pI.ot.i~ C~SCS x -- 0
n11t1 x -- 1. 'l'ltis I)rhnvio~ ~rtaltcs it rn.sy to ~intirrst,e.rltl why t,hc vclocit,y tlist.ribut8ions
for lhr. ot.lter, inl.crmrtliatc valurs of x tlilTc,r so liklc from n stmight line.
- x' - - r-r,
5 - 5
Fig. 5.4. Vclorily dist~ributinl~ it1 thr nnn~~lr~n l)c~t\r.c~r~~
I:tlrtl with tl~c i~itl of cqtm (5.l5n, b).
n) Cnsr I: irinrr cyli~ltler rot.nt.ing; orct,rr cylitlticr at, rcst, ro2 - 0
h) (he I I : inner cylitlclrr at rcst, tol = 0; o~tt,rr cylitlclrr rotati~l~
r, - r;uIius or i1111l.r 1.~1i11dcr. r, = r:uIim oI'o1111,r cyliwlpr
t,\\.o.c-o~lc.c.~ltri~.. ~.ot;tti~~~ ~~Jlill~l~~rs
>In ~~;lll~ll-
r
?J, = --.I. .
2nr
It, is sccn, therefore, t,ll,zt, t,he case of fric:l.iot~lrss flow ill t,hc r~eiglll)o~~rhootl of a
vort,cx line constitput.cs a. solut,iorl of t.hc Navirr-Stokes cquntior~s (c/. Scc. IVb).
In t,llis connexion it may be instn~ctive t,o n~cnt,iotl a11 cxnrnple of an cxnct nmslendy
solut.ion of the Navier-Stokcs cclunt,iotls, rlnmcly that which tlescribcs the
process of t1cca.y of n vortex t,hrough bhc acthl of viscosity. The distribr~t~ioll of t.l~e
t,angct~t~ial vrlority component 7~ wit,lr respect to t.l~e radial tlista~lce r and tirnc t
is give11 by
Pig. 5.5. Vclocit.y distribution at varying
times in tho ncigltbo~~rhood of n vortex
filament cnl~setl by tho action of viscosity
1; - circulnlinn or llw vortox nnrncttl nt 11 nio 1 =. 0
w1:c.n vircoslly Iwylnr lo ncl: ti.- I;/? n r.