Boundary Lyer Theory

120
VI. Vcry ulow motion
and for the pressure distribution
2 (I-z)
p(x) = po + 6pU-----. h2(2 a-1)
(6.29)
The relations hecome somewhat simpler if the shape of the channel is described
by t,he gap widths hl and h, at inlet and exit, respect,ively, see Fig. 6.4. 'The c1lnr:lc-
t,erist,ic witlt,h now becomes equal to the harmonic mean
and the condition for positive pressure excess, eqn. (6.28), now requires that t,he
channel should be convergent. In this notation, the pressure tlistribution is given by
and the result.ant of the pressure forces can be con~putecl by int,egration, when we
obtain
with k .= h,/h,. The resr~lt~ant of the shezring stresses can be calc~~latctl in a similar
manner:
1
It is interesting to note [el that the resultant pressure force possesses a maximum
for k = 2.2 approximately, when its value is
and whcn
Tl~c corfficicnt of frict,ion F/P is propor16onal to hz/Z and can be made very small.
The coordi~~ates of the centre of pressure, x,, can be shown t,o be equal to
For small angles of inclinat.ion between block and . h c (k w I), tile pressure distri-
hution from cqn. (6.29) is nearly parabolic, the charact,erist.ic thicltness and cent,re
of prcsssnre being very nearly at z = 1 t. Pni,t,ing hm = h(4 1) we cnn find that the
pressure tli[l:ronce 1)ccomcs
If we compare t,his rcs~llt with t.hat for crccping nlotion past, n sphcro in cqn. (6.71)),
we not,ice that in the case of t.he slipper the pressure tlifTcrrnec is grcalrr I I n ~ f:rct,or
(lll~,,,)~. Since Ilh,,, is of the order of 500 t.o 1000 (1 = 4, A,, =x 0.004 to 0.00s ill).
t,he prevailing prcssurcs are seen to assume vcry large val~~es-1. 'l'hc occll~~c?t~c~: of
sucl~ high pressures in slow viscous motion is a pocwlinr proprrt,y of 1.11~ (,,yp(: of flow
(~nrot~t~t~~~rotl
ill I~~lriwdon. At, l h tmmo tirn~: itf is tw!op~iz~!(l I II:I~, I,IIc :III~CI~~ li~~.ttt