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Full text of "Scientific Papers - Vi"

1918]                            ON  THE THEORY  OF  LUBRICATION                                 525
When y = 0, we get from (3) and (5)
du        , 4/i
which represents the tangential traction exercised by the liquid upon the moving plane.
It may be remarked that in the case of a simple shearing motion Q=-^hU, making H — h, and accordingly
dp/ das = 0,    du/dy = I7fh. Our equations allow for a different value of Q and a pressure variable with as.
So far we have regarded h as absolutely constant. But it is evident that Reynolds' equation (5) remains approximately applicable to the lubrication problem in two dimensions even when h is variable, though always very small, provided that the changes are not too sudden, so being measured circumferentially and y normally to the opposed surfaces. If the whole changes of direction are large, as in the journal bearing with a large arc of contact, complication arises in the reckoning of the resultant forces operative upon the solid parts concerned ; but this does not interfere with the applicability of (5) when h is suitably expressed as a function of ay. In the present paper we confine ourselves to the case where one surface (at y = 0) may be treated as absolutely plane. The second surface is supposed to be limited at x = a and at ai = b, where h is equal to hv and h,2 respectively, and the pressure at both these places is taken to be zero.
For the total pressure, or load, (P) we have
on integration by parts with regard to the evanescence of p at both limits. Hence by (5)
P       [bccdx    ,
Again, by direct integration of (5),
rb fjr           rb rfr
0—1   — -_ 77
h*            !»"
'<'                    J      n
by which H is determined. It is the thickness of the layer at the place, or places, where p is a maximum or a minimum. A change in the sign of U reverses also that of P.
Again, if x be the value of cc which gives the point of application of the resultant force,
ro                           rb      (fifl
x . P =     pxdoo = — -H   x~ ;/  dx,
J a                            J a      tf'®at, we are to make Q - 0 and (4) gives the rise of pressure as as decreases algebraically. But for the present purpose Q is to be taken finite. Denoting 2Q/Z7by H, we write (4)