Boundary Lyer Theory

52
111. I)rr.ivnt.iot~ of tho rrlrtnt,ions of ~ndion of n cotnprc.ssiblc viscous fluid
'1'110 system of the Ihrcw cq~t:~t,ions (3.1 1 ) aotrt.air~s the six st,rrsscs a,, a,, a,,
T,,, t , s,,. Tho next tdc is t80 tlct,crlninc the relation between them and the
strair~s so as to enable 11s t,n introduc~ the vrlocity components u, v, w into eqn. (3.1 1).
Before giving this rclat,ion in See. 111 tl wo shall ir~vcstigat.~ t,l~c syst,om of st,mins
in great,cr detail.
c. The rntc nt which a fluid element i~ stmined in flow
Whrn a cont,inl~o~ls l)odq. of fluitl is rnatlc to flow, every rlcmcnt in it is, gcncrally
sptvdting, clis~,ln.cctl t.o a new posit,ion in t1he course of time. 1)uring t.his motion
rlcmpnts of flrlitl l)ccon~c st,minctl, ant1 since the mot,ion of t.hn flnicl is completely
tlet.rrminrt1 when the vclocit,y vect,or rrr is given as a funct,ion of t,ime and positpion,
tr, = ru(z,?/,z,t), them cxist Itincrnat,ic rrlnt,ions l)et,wcen the components of t,hc
r:~.tc of st.min and t,ltis function. 'Vhe rntc nt which an clement or fluid is strained
tlcprt~ls on t.11~ ~~1.lbz.t: n~otion of t,wo poin1.s wit.hin it. We, therefore, consitlor the
t,wo neigl~l~onring points A ant1 B whirh arc st~own in Fig. 3.2. Owing to t.he presence
of t,ho vrlority field, point A will be tlisplncctl to A' in t,ime dl by a distat~ce s = ro dt ;
sinrr, I~owevor, tho vclocity at B, imngincd at a dist,ancc dr from A, is different,
point H will move t,o B' displ:md from 1% by s -1- (1.9 = (ui -1-tlrri) dt. More explicitly,
if t,hr componcnt.~
l)oint, I 0. 'i'l~c rclat,ive velocity ol'
any point B with respect t,o A is now
antl tho field consists of planes x =- const which displace thcmsclvcs nniformly
wit,ll a velocit,y which is proportional Lo the clistancc tlx away from the plane x = 0.
An elementary parallelepiped with A anti R :~t its vertices placed in such a vclocity
field will be distorted in extension, its face BC receding from AD wil.11 nn inorcasing