Page 63 - Tribology in Machine Design

50 Tribology in machine design

where F is the total tangential force resisting relative motion and A is the
area of the surface CD wetted by the lubricant. This is the Petroff law and
gives a good approximation to friction losses at high speeds and light loads,
under conditions of lubrication, that is when interacting surfaces are
completely separated by the fluid film. It does not apply when the
lubrication is with an imperfect film, that is when boundary lubrication
conditions apply.

2.13.3. Viscous flow between very close parallel surfaces
Figure 2.20 represents a viscous fluid, flowing between two stationary
parallel plane boundaries of infinite extent, so that edge effects can be
neglected. The axes Ox and Oy are parallel and perpendicular, respectively,
to the direction of flow, and Ox represents a plane midway between the
boundaries. Let us consider the forces acting on a flat rectangular element
of width 6x, thickness <5y, and unit length in a direction perpendicular to the
plane of the paper. Let
tangential drag per unit area at y = q,
tangential drag per unit area at y + 6y = q + dq,
I net tangential drag on the element = 6qdx,
normal pressure per unit area at x=p,
normal pressure per unit area at x + dx=p + dp,
Figure 2.20 net normal load on the ends of the element = dpdy.

Hence the surrounding fluid exerts a net forward drag on the element of
amount 6q6x, which must be equivalent to the net resisting load dpdy acting
on the ends of the element, so that

Combining this result with the viscosity equation q =// dv/dy, we obtain the
fundamental equation for pressure

Rewriting this equation, and integrating twice with respect to y and keeping
x constant