Boundary Lyer Theory

84
V. Exact. solul.ions of t.he Nnvirr-Stokrs equn1,ions
1. Parallel flow tltrough a straight channel nnd Col~eltc flow. A very simple
solut,ion of eqnat,ion (5.2) is obtained for the case of stm~tly flow ir~ ;I channel with
t,wo parallel flat walls, Vig. 5.1. 1,et. t,hc tlist.ancc bctwcen the :valls be denoted by 2 h,
so t.liat cqtl. (5.2) can Iw writ9tcn
Vig. 5. I . i'nrellnl flow with p:~rnholir
vclocit.y disl.ril~~~t.i~tl
Another simple solution of eqn (5.3) is obtained for tho so-mlled Couette
flow 1)ctween two pnrnllcl flat walls, one of which is at rest, the other moving in
its own plan? wit11 ;I velocity rJ, Fig. 6 2 With the boundary conditions
we obt.n.in the solution
y=O: u=O; y=h: u=U
\vhic-11 is shown in Fig. 5.2. Tn parLicular for a vanishing pressure gratlicrlt we have
'I'his p:l.rt.inllar case is known ns simple Couct,tc flow, or si~nplc? sllcar Ilow. The
gcrlrral casc of (:ouct,tc flow is i supcrposit,io~~ of this simple casc over the flow
between two fht, wn.IIs. 'I'l~c sllapc of t.I~c vclocit.y~profilc is tlctcm~incd by t,hc dirnoltsionlrss
pressure p~.:~tlicr~t, I
For 7' > 0, i. e., for n prrssurc tlccreasing in t01c tiircctior~ of mot,ion, the velocity
is posit.ivc over the whole witll,ll of tl~o channel. For ncget.ive values of P the velocity
over a porkion of the cl~:~nncl width can I,ccomc ncgntivo, t,l~at is, back-//OW may occur
near the wall wllich is at rest, and it is seen from Fig. 6.2 that t,his Irappens ~llcn
I' < - 1. Tn this case the dragging act.ion of the fast,cr layers exertetl on fluitl lmrt,iclcs
in the ncigl~bourl~ootl of the wall is insufficient to ovcrcomc thr, influence of
t,hc adverse pressure gradient. This type of Coucttc flow with a Iwcssurc gratlicnt
has some importance in the hgtlrodynamic theory of luhricatio~t. 'J'hc flow in tltc
narrow clcnrar~ce bctwcen journal and l~enrirlg is, by a.nd large, identical with Couct,tc
flow with a pressure gradient (c/. Scc. VTc).
2. The IIngen-Poiseuille theory or flow tl~rorrgh a pipe. Tl~e flow t.llrougl~ a
stmight, tl~bc of ciroular nross-scc:l.ion is the casc with rol.:~t~ionn.l sylnlnct.ry wllic*h
rorr(~+~)onds to tho prcc:otli~~g casc: of lawn-tli~~lcnsior~:~l flow t.llrol~gl~ :I oll:~.r~r~t!l. 1,~t.
tho z-axis be solcct.c:tl along thn axis oL' t,l~n pipc, Pig. 1.2, :md Ict, y t1onot.o 1,llo I.IL,I~:LI
eoortlinate mcasurcd from the axis outwards. The vclocit,y componcnt.~ in t.ltc
tnr~gt?nl,inI and radial directions arc zero; the velocity component pnrallcl to the
axis, denoted by 11, depcntls on y alone, a,nd the pressure is const#ant in every crosssection.
Of the thrcc Navicr-Stolres equations in cylint1ricn.l coordinates, cqns. (3.:16),
only the one for the axial tlircct,ion remains, nntl it, simplifies to
tlw boundary condition being u = 0 for y = R. The solution of cqn. (5.6) gives the
velocity distribution
1 dp
IL (y) = - - - - - (R2-- y2)
411 dz