Boundary Lyer Theory

56 11 I . 1)t.rivntion of t.lm cquntionrr of n~olinn of n rotnprcasibln viscous tlnitl
arltl r?n/8z 11avc posit,ive nonvanishing vnlnrs, the right, angle at A will distort owing
t,o t,l~e sl~pcrposit,ion of t.wo n~ot,ions, t,llc st.:~tc of affairs bcing illr~st,ral,etl in Fig. 3.5.
1L is clr:~r that, 1.hr right n.tiglr at. A now distorts at t,wiac the mtc
tlcscrilwtl by I wo of the orf-tliagon;~l . . t,rr~ns of matrix (3.15%). In general, t,hc thrcc
~ff-dii~gon~l t.rrms Ex!, - F,/,, F,, = d,,, :LII~ E,, = Fyr tlcsrxibc the rate of dist,ort,ion
of a right, nnglc locatrd in ;L plane nornmnl t,o the axis the index of which does not
nppt'ar ns n srll)script.. 'l'hr tlistort.ion is volume-preserving and affects only the
shape of t,hc rlcmcnt .
(lirrr~mstanrrs nro ilgain tli!fcrcnt in the pzrticulxr case when au/ay = - av/az
illrrwt.r;ct,ctl in Kg. 3.6. k'roni t.11~ preceding considerations and from the fact that.
t~ow 2,, - t) \\.e ran infsr n.t oncc tllat, the right angle at A remains undistorted.
'I'his is also rlrar from thr diagram which shows that the fluid element rotates with
rcsprrt, t,o t.llr rcfkrencr point A. Insla~rtnneo~~sly, this rotdon occurs without
dist.ortion ant1 call Iw dcsoril)rtl as a rigitl-l)otly rot,ntion. The instant,ancons nngulnr
vrlorit,y of this rot,at,ion in
It, is now rasy t.o see that the component. < of curl in from cqn. (3.15b), known as
t,hc vort.irit,y oft.11~ vclocit,y ficld, reproscnt,~ t.lw angular velocity of this instant.nneo~~s
rigitl-lmlj~ rrfi:.:t ion, and that,
c. The! rate ~t whir11 a Ihid elcnwnt is ~Lrainctl in flow 57
(a) A pure tmnslation t1escril)ctl by the vclocit.y components w, v, 1il of it,.
(b) A rigid-body rotation described by the con~poncnt,~ 5, 7, 5' of c~~rl icr.
(e) A volrrmctric dilatation tlcscrilmi by e -- tliv in, the iinmr dil:it,:~tions in
the tlircctior~ or the axes bcing described I)y d,, i, :tnd E,, rrs~)e(:t.ivoly.
((I) A tlist,ort.ion in shapc drscribctl by t.hr cornponcnt.~ i,, (,I,(: wit11 rnixt:tl
inclicrs.
Only tho last, two motions produce an intxinsic tleformation of n llr~itl olcme~lt,
surrour~tling tho rrfercncc point A, lhc first two causing a mere, general, tlisplacerncnt,
of its location.
T11c el~ment~s of matrix (3.15a) constit,rrte the componcnt.s of ;t symrnet~ric
tensor known :IS t,he rate-of-slmitt lensor; it,s mat.l~ematical propertics arc analogous
to those of the cq~rally symmct.ric st,ress t,cnsor. It is known from the theory of
elasticit$y 13, 71 or from general c:onsidcrations of hnsor algebra [I I ] tht wit,l~
every symmetric tensor it is possil)lc t.o associate three rnrlt.rrally orthogond pritt,cipnl
axes which tlctormine tlrrce mut,nnlly ortllogonal principnl plar~cs t,l~:~t is a privilcgctl
Cart,esian syst,crn of coordinat,cn. In t.llin syst,cm of coortlinatlcs, t,he stlrcss vert,or
or the inst~wt.t~ncor~s nrolion in tiny ono of the prinoip:~l planes is nornr:d lo it., LhaL
is, pnrallel t,o one of the axes. IVlrcn sr~cl~ a special system of c~ordinat~cs is used,
the n~at~rices (3.10) or (3.15a) retain their diagonal tmms onlv. DcnoLirra the valrrcs
of tlw respective romponer~ts by symbols with I);ir
matrircs
It slrould, finally, bc remembered that, such :L t,rnrrsli)r~i~:~liot~ of c~oortlin:~t.vs tlo~
not affect, the sum of the diagonal terms, so that
Iz'ig. 3.7. I'rincipial axes for
st.rrss ;ind ral.c: of sl.r:~in