Then, differentiating equations (4) with respect to x and z respectively, and substituting in the 4th of equations (2), and integrating from y = 0 to y = h, so that only the values of v at the surfaces may be required, we have for the differential equation of normal pressure at any point x, z, between the boundaries:—
| d | ( h³ | dp | ) + | d | ( h³ | dp | ) = 6μ { (U0 + U1) | dh | + 2V1 } |
| dx | dz | dz | dz | dx |
(5)
Again differentiating equations (4), with respect to x and z respectively, and substituting in the 5th and 6th of equations (2), and putting fx and fz for the intensities of the tangential stresses at the lower and upper surfaces:—
| ƒx = μ (U1 − U0) | 1 | ± | h | dp | |
| h | 2 | dx |
| ƒx = ± | h | dp | |
| 2 | dx |
(6)
Equations (5) and (6) are the general equations for the stresses at the boundaries at x, z, when h is a continuous function of x and z, μ and ρ being constant.
For the integration of equations (6) to get the resultant stresses and moments on the solid boundaries, so as to obtain the conditions of their equilibrium, it is necessary to know how x and z at any point on the boundary enter into h, as well as the equation ƒ(x, z) = 0, which determines the limits of the lubricating film. If y, the normal to one of the surfaces, has not the same direction for all points of this surface, in other words, if the surface is not plane, x and z become curvilinear co-ordinates, at all points perpendicular to y. Since, for lubrication, one of the surfaces must be plane, cylindrical, or a surface of revolution, we may put x = Rθ, y = r − R, and z perpendicular to the plane of motion. Then, if the data are sufficient, the resultant stresses and moments between the surfaces are obtained by integrating the intensity of the stress and moments of intensity of stress over the surface.