516 A SIMPLE PROBLEM IN FORCED LUBRICATION [426
Of these (5) is satisfied by v in (1), and (6) is satisfied when w = 0 and p is independent of z. Also, with use of (2), (4) becomes
dp __ _ _ ,„ >
dr~~ r vrA3?1' ...........................
so that
QfiU, r-i /m
P-P* = ^jPlQ8f' ........................... (9)
where p: is the pressure at the outer radius ra. If the layer is open at r1} and we reckon only pressures above atmosphere, pl may be omitted.
The whole force sustained by the layer of fluid between the radii r0 and rl is independent of w, being given by
p_
(10)
If we suppose r0 = 0, so that the supply takes place on the axis itself, this becomes simply
/A8, .......................... ....(11)
but we have then to face an infinite pressure at the axis. In practice r0 would have to be finite though small, and would correspond to the radius of the perforation in the lower fixed plate, not much disturbing (11). In fact, if jp0 be the pressure of the feed corresponding to r0,
6/4*7, r, 2P . r,
-^TT log - = - ,log- ...................... (12)
TT/i3 ° rQ irr-i1 ° rfl v '
The moment of the forces due to viscosity, by which the rotation is resisted, has the expression
*=**/•/»•*©.- ^W-r/) ................ (13)
It may be worth remarking that if geometric similarity is preserved, so that r1, r0, h are .in constant ratios, a consideration of "dimensions" suffices to show that P is proportional to ^Ur^1, at least when we assume independence of the rotation (w) which does not influence u. A deficiency of viscosity may thus always be compensated by an increase of supply.
The work which must be done in unit time to maintain the rotation is Mm. In addition to this, there is the work required to introduce the feed of lubricant, represented by pJJ. Thus, altogether, for the work requiredter system.