Boundary Lyer Theory

76 1V. Genrrnl proprrtir~ of thr Nnvirr-Stokm rqustio~~n r. 'I'llr litr~iting cnw of wry small visoo~~s hrrr~ 7 7
Fig. 4.2n
Fig. 4.2. T'nt,tcrn of st,rcnmlinra 1)cIiiutl
a rect.nr~gulnr flat. plnto (lf/rt = 1.6) plnctrtl
nt. right, nnglr t.o th flow at. n Itry~~olds
nu~~iI)rr R = I' ![/I) = 6000, nfkr J. I m). 111 this case the [lroccss of m;~t,h(~lrl:~l,ic::~I
sin~plificntion of the tliffcrcntial cqn. (4.10) requircs a consitlcrablc amount of rart:.
It is not pcrlnissiblc simply t,o olnit (,lie viscous tcrrns, i. c., t,lle right,-h:~ntl side of
(:(in. (4.10). This woultl rntlnco the ortlcr of t,llc oqu:~t,iot~ from four to two, :LII(I t.110
solution of the simplifictl cquat.iou could not be made to salisfy the full bountl:lry
condit.iol~s of t.Iw originn.1 cquat.ion. The problem wl~icll was ontlincd in tllc prccctling
scr~t~cnccs belongs essentially to tl~e rcdrn of hou~cdnr?/-kr!/er lheory. We now proposc:
to tliscuss briefly the genrral st.at,r~nc~lt,s which can t)c made about the solutions
of the Nnvicr-St,oltes cqnnl,ions for t,ho special mse of small viscous forccs as cornpared
wit.ll t,hc incrt,in forccs, thnt is in t,11r limiting case of very Inrgr: 1tt:ynolds
n11m1)crs.
The following analogy rnny scrvc to illust,rate tl16 c:llsr:~cter of the solut.ions
of t,hc Navier-Stokcs cquat~ions for the litnit.ing c;~se or vt:ry small viscosil,y, i. c.,
of vrry small friction terms, as compared with t,llc inertia terms Tl~e tcrnprrat1lrcb
distribution O(r, y) aboul, a hot, I)otly in n fluid strrarn is clcscrihtl 1)y the following
tlilTrr'rrrntial rqrlation, Chap. XI 1.