524 ON THE THEORY OF LUBRICATION [428
so. As an alternative to an inclined plane surface, consideration is given to a broken surface consisting of two parts, each of which is parallel to the first plane surface but at a different distance from it. It appears that this is the form which must be approached if we wish the total pressure supported to be a maximum, when the length of the bearing and the closest approacli are prescribed. In these questions we may anticipate that our calculations correspond pretty closely with what actually happens, — more than can be said of some branches of hydrodynamics.
In forming the necessary equation it is best, following Sommerfeld, to begin with the simplest possible case. The layer of fluid is contained between two parallel planes at y = 0 and at ?/ = />. The motion is everywhere parallel to x, so that the velocity-component u alone occurs, v and w being everywhere zero. Moreover u is a function of y only. The tangential traction acting across an element of area represented by doc is /j, (dujdy) dx, where p is the viscosity, so that the element of volume (dxdy) is subject to the force IJL(dzujdyz}dxdy. Since there is no acceleration, this force is balanced by that due to the pressure, viz. - (dp / 'dx) dxdy, and thus
In this equation p is independent of y, since there are in this direction neither motion nor components of traction, and (1), which may also be derived directly from the general hydrodynamical equations, is immediately integrable. We have
where A and B are constants of integration. We now suppose that when y — 0, u •= - U, and that when y = h, u = 0. Thus
das ^ '
The whole flow of liquid, regarded as incompressible, between 0 and h is
fh , _ h* dp hU _ n Jo y~~12^~~2~-~y'
where Q is a constant, so that
dx h3 \! U
If we suppose the passage to be absolutely blocked at a place where as is negatively great, 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)
dponsider the phases represented by the factor eik (x~^'] in P. For the point 0, x = 0, r = p, and the exponential factor is e~ikt>. As in the ordinary theory of specular reflection, the same is true for every point in the plane AOA and therefore for the element of surface at A A whose volume is 27rRdRd£. For points in a plane