(2)

For the boundary conditions, putting f(x, z) as limiting the lateral area of the lubricant, the conditions at the surfaces may be expressed thus:—

when y = 0,   u = U0,   w = 0,   v = 0

when y = h,   u = U1,   w = 0,   v1, = U1 dh+ V1
dx

when ƒ(x, z) = 0,   p = p0

(3)

Then, integrating the equations (2) over y, and determining the constants by equations (3), we have, since by the second of equations (2) p is independent of y,

u = 1 dp(y − h) y + U0 h − y+ U1 y
2μ dxh h
w = 1 dp(y − h) y
2μ dz

(4)