Boundary Lyer Theory

630 XX. 'I'r~rl~ulrmt. flow t.l~rouah pipc:s
j. Drag reduction by the nddition of polymers
Tn turb~~lcnt flow, t.11~ prrssnre drop in n pipe can be considerably reduced relative
to eqn. (20.00) by t,hr ntltlition of sn~nll qunntitics of polymer particles. In laminar
flow, similnr nrltlilivcs leave the pressure drop prncticnlly uncl~angrd. The extent of
drag rcd~tetion tlcpcnds on the tnolcculnr weight of t,he polymcr and on its concentration.
7~hc gmph of Fig. 20.29 dcucribcs 1.11~ rcsist,ancc cocflicicnt,, 1, as a function of
tho licynoltls nurnbcr R = S'd/vp - where vp is the kinematic viscosity of the solut,ion
- for dill'ercnt valucs of the conccntrntion c of the solution. 'rhe measurerncnts
were perfortnrtl by R.M1. I1nt,ct~son and P. 11. AbcrnntAy [50nJ. As the concent,ratio~~
incrcnscs, the rrt1ucc.d rrsislmce cocfficirnt tends t,o a curve of maximum reductiot~,
curvc (3), indicated by 1'. S. Virlc 1771. 'The diagram of Fig. 20.29 tlisplnys poitits
obtnincd at two measuring stations along the pipe. The diflkrence in t,he values
of thc resistnt~co coefficient nt hhosc two sections is cxplsinetl by the fact that the
polymer molecules ?re tmn npnrt in the turhulcnt st,rcarn resulting in an effective
tlecre~se in concentration in the downstrenm direct,ion. In spite of intensive
rcsearch into this phenomenon (c/. the pnpcr I)y M.T. IAnndahl [36]), no satisfactory
cxplanntion for its occurrence has yct been advanced. Nevertheless, experiment,^
demonst.rate unmiutnlrahly that thc retl~~ction in tlrng is linked to changes in thc
stxncture of tthc tnrlmlrncc. The proccss is bcst illustratetl with the aid of t.he experimcnt,n.lly
dct,erminctl ve1ocit.y-tlisttributsion Inws.
nn - kinrn~ntir visrvelty of t,hr rr~lutlon
0 - ~wnsl~ri~~~ alntlcrn at 214 cl frrrnl lolet
A - n~rnsrlrin~ st.nl.lon at 1641 d from inlrt
(1) lnn~irrnr flow with 1 = 64/R
(2) twt~elcnt now. Nnwtonlan nllla, elin. (20.6)
(3) nryn~ptots for rnaxin~nnl rlrn~rccl~~rlionnlter
1'. S. Virk [77),rqn. (20.46)
Fig. 20 2!). Ilroistn~~creorfficir~~t, 1,of ~111ootl1 pipes in t~~rbulctrt flow of polyn~ersolutiona ns nfunction
of II.~ynoIds nr~wbrr as ~nrns~~rrd by R. W. Pntcrson nntl F. H. Ahernathy r60]. Solntionn of
1mIyc*l I~ylcnr oxide of givrn roncentrntions r ; 1 pp111 dcsignatcq I g of polyn~rr per 106 g of watt-r
Arc:r)rrling l,o I'.S. Virlc [78, 793, it is necessary to distinguish three velocitydist.ril)~~l,ion
zones:
r
$
(1) 'I'hc Intninnr snl~lnyer (0 < 17 < 10); this corresponds t.o curve (I) in Fig. 20.4.
(2) 'IYic fnlly tlovrlopc.tl t.nrhnlent zone. TTcrc the distribr~t~ion follows the law nccortling
t.o rqn. (20.13) with A 1 = 2.5 rrgnrdlnss of th- physicnl propertics of the
solut,ion. 'l'hc c:onsl~:~.nl. 14 vnrics st,rongly wi1,h tlw conc~nt~r~tion.
(3) 'I'lic zonr tlcsrril)rrl as elastic^" 'I'his zone plnrc,s itsrlf I)ctnrrn 1 hr In n~innt.
sublayer and the fully tlrvcloprd turl)ulcnt region 1771. llerc fl~c vrlocity is
rrpresrntrtl by the log:withtnic Inw
. .
Illis law is vnlid For nll int,ent,s t~nd prposrs ILN fn,r :LS 1,111: c:c~tit.~~v of the* pil1c Sot,
sul'ficirnl.ly high concc~n tmtions. In iinalogy wit.l~ ccln. (20.30), we c:nn in tcg~.;~t.c it,
to drrive t.11~ rcsistn.nce formuln
( I] ;rkeret., J.: (:rcnzscl~ichtrn in gcrnclcn unrl gekriin~~nI.rti I)ill'~tsorrn. 111'1'~1bl-Sy1nposi11111
t(reiburg/Br. 1!)57 (H. Gvrtler, etl.). Ilcrlin, 1958, 22-37,
[In] Ackcwt, .I.: AspcoLq of inlarnnl Ilow. lfll~~iii nrrel~~ini(.~ of inf.c.rnnl flow ((:. Sovrirn. c.rl.).
lhcvit*r I'ublinhing Conri~nny, AIIIsI~~~~~IIIIII/I,~II~IoII/Nc~~
l'ork, l!)l;7, I - - 24.
[2] Atller, M.: Stroniung in gckriim~nten Jlohrcn. ZAMM Id, 257--275 (1!)34).
131 Bauer, B., and (Mnvics. F.: I':xprrin~cntdlc unci tlioorctiscl~o Untrrsncl~r~ngon iilwr din
Rol~rreibung von Heiza~nsserleitur~~~~. Mitt. d. J"rrnl~rizk,afL~wrkc:~l tl. IC'I'II Ziirich I!):l(i;
see nlso: F. Galnvics: Scl~\wizer Archiv 5, 12, 337 (1!)3!)).
[4] Becker, E.: Beitrag zur ~erechnung von Srk~~~~tl:irnIrii~~~r~r~ge~~.
ZAhlM Jfi. spccinl issue,
3--8 (1956); seo nlso: Mitt,. Max-l'lanck- Inst,. riir SI.rii~~~r~ng~forsrl~~t~~g 13 (1!)5li).
[4nj L