In the previous section, we considered the effect of asperities on the film flow. If the gap between the surfaces reaches the order of the height of the asperities, it is no longer possible to avoid the contact between these. This scenario is called a mixed lubrication condition. In this case, the total reaction force of the bearing is a combination of the reaction due to the film pressure and the pressure due to the contact between asperities.

Since, the force between the pair of asperities will be a function of the combined height z = z1 + z2 rather than the individual heights z1 and z2, we can combine the individual distributions to give the sum of the distributions as ϕ0(z). The expected total force can then be written as

Now introduce wp = z − hm and define the integration over r in the above expression as

If the asperities are assumed to be paraboloidal f(r) = r2/2r0, and 2f(r/2) = r2/4r0. Here r0 is the radius of curvature at the peak of the asperity. The Hertzian solution for a paraboloidal surface in contact is

Introduce z = sσ and a standardized height distribution

In order to evaluate the contact pressure and contact area, we need to evaluate the functions Fn. This requires the knowledge of the distribution ϕ0. Experimental investigations have shown that a Gaussian distribution is a close approximation. For a Gaussian distribution, Fn is defined as

When h is large and positive, it can be approximated as

and if h is large but negative as

The following recurrence relation can be used to obtain the higher values of Fn:

The following table summarizes the values of Fn: