Boundary Lyer Theory

114 VI. Vcry slow motion
where r2 = z2 + y2 4- z2 has been introdneed for the sake of brevit,y. Tt, is easy to
verify that these expressions satisfy eqns. (6.3) and (6.4) and that t.ho velocity va-
ninhes at all pointa on the surfnco of t,ho sphere. The pressure on t,he surfnre becomes
'rho rnnximllm ant1 n~inimurn of prrssllrc occurs nt points P, and I'2, respectively,
thrir valnrs bring
3 11 uw
1)1.2- pcn - -1- --ji -
(G 7 1))
Tile prcssnrc distribr~t,ion along a 1ncridia.11 of t,hr sphere as well as alor~g the axis
of al)scissar, r, is shown in Fig. 6.1. '1'11~ shrnring-stress distribution over the sphere
can also be cnln~libtctl from the n.lmvo formulae. If, is found that the shearing st,ress
has it,s largoat value nt poirll /I whcro t = ij ,IL fJ,/I1 :m(I is r~(11al to the pressllre
riso nt PI or prrssurc tlccrrase nt /',. Tntrgmting tho pressure distribut.ion ant1 the
shrnring sl,rrss over the surfacr of tho sphrre we obt,nin t,ho t,ot,nl tlrng
'This is f,ltc vcry wcll known ,Wko.~ cr/udion for thc: tlrag of a spl~rrc. It, can I1v shown
t,l~at. ol~c t.llirtl of ihc t1r:t.g is tlrro t,o the prossure ,list,ril)~ition n ~ ~.II:L~, ~ d tho ron~ni~~ir~g
t.wo t,l~irtls nrr t111o t.o tho cxistctico of shcar. It is fr~rlhar rcrnark:~l)lc t,h:lt t,hc ctr:~g
is ~xo~)~rt,io~~nl to the first, powcr of vclocity. If a t1ra.g coefficient is fornled hy
rc.l;.rrillg f,llc tlr;la 1.0 t,hc tlyu:~mic hc:r.rl a Q 11,2,nntl t.he rrontd arca., :IS is dotlc:
in tllc c.:rsct of highrr Ib~~noI(Is r~n~nlwrs, or if we p~t.
b. Parallel flow pnst 8 spl~rrc 116
A coniparisot~ het,ween Stolzcs's equation nnd rxpcritnont was givan in Vig. 1.6
from which it is seen that is applies only to rases when R < 1. The pnt,tern of
strcamlinos in front of and behind the sphere must be the same, as by rcvvrsing
t,he direction of free flow, i. e., by changing the sign of vclocity con~ponents in cqns.
(6.3) and (6.4) t.he syston~ is transformed into it,self. The st,reamlincs in viscons
flow past. a sphcro are sl~own in Fig. 0.2. Thy were tlrnwn ns they woultl nl)pear
to an observer in front of whom the sphere is dragged with n constnnt, vclocity U,.
The sltrt,ch contains also velocit,y prolilcs at scvcral cross-s~ct~ions. It is scon f,l~nt
tho sphere drags with it a vrry witlr layer of flnitl wl~id~ rxtr~~tls over :iI~out, one
tliitrnclor on I)oth sitlns. At, vory high Itryrtoltls nurnl)ors tl~is I~o~ln~liiry li~y~r
I)ccornes very thin.
Ipig. 6.2. Sl.rcnnilincs nnd vrloci1.y di.st.ri-
brttinr~ in Stokm' snlut.ior1 for n spllcrr ill
pa r;dlrl flow
[Pig. 6.3. S1.rc;itnlirlr.s in llir flow
ORCVII'.~ improvcrnt~~~t: An ~ITI~)~~VI!ITI(-II~, of' St,ok~s's srv is', 11' nntl 711' nrr t,l~o pcrl,url):l (.ion f.cwns, : rr~tl :IS SIIV~I, stn;r ll wit 11 ~.c~sl~~~,,
t.0
(.he f'rcr st,rca~n vclocity (1,. It is to noted, I~owevcr, that, this is r~ot, tr~r ill 1,11c
irnnlrrliatr neighl~onrhootl of t,hc spherr. With tho ns~nnl~~i~iot~ (0.1 I ) 1.111- illrrt.ia
t.crms it) t,hr Na,vier-Stdokt,s rqns. (3.32) arc tlrcv~nl)osed in two ,pro~~l)s, r. g. :
allr av'
(loo , U, --
ax
, . . . and
a ~ ' , avr
"Iaz , IL ax
--, . . .