Let us consider a reference frame z = 0 from which the height of various points on the surface is measured. Because the surfaces are rough, the height of the points can be decomposed into two components, a mean height and a height variation due to roughness in the following way

where hmJ and hmB are the mean heights, while and are the height variations due to asperities. Then mean gap between the two surfaces is then

The regions defined by hl < 0 denote the contact regions. Similar to the film thickness, we can decompose the pressure p in the film into a part corresponding to the mean film thickness, pm, and another part corresponding to variation in the film thickness due to asperities, . If we use local surface averaging to homogenize the Reynolds equation in total pressure, the space averaged Reynolds equation takes the form

where K is the effective permeability tensor and C is the shear flow tensor. These are defined in terms of a set of variables b and c as

The operator 〈.〉 above refers to the local surface average. This average is taken over an area which is much smaller than the overall dimensions of the contacting surface but is large enough to include many asperities. The variables b and c are closure variables that can be obtained by the solution of the boundary value problems

Here, x is the position vector of a point on the surface and l is the length of the unit cell. The vector ei refers to the local directions on the averaging surface.

When turbulence effects are neglected, b and c are zero so that

where Φ and Φs are the pressure flow factor and shear flow factor, respectively. Moreover, Φf and Φfs are the shear stress factors for a Couette flow, and Φfp is the shear stress factor for a Poiseuille flow. The variables hm and pm are the average gap and the average pressure in the film obtained by the surface averaging procedure. Thus,

Now, before we use the averaged Reynolds equation, the main task is to first determine the factors defined above for the flow. One obvious way of determining these factors is to first solve for the closure variables b and c on a representative surface unit cell with the actual roughness pattern. Once the solution is obtained, perform the required averages over the surface to get the values of the different factors in the averaged equation. For a very general roughness pattern, the solution of the closure variables can only be obtained numerically. If we make certain assumptions on the roughness pattern, it is possible to solve the closure equations analytically. Then it is just a matter of computing the statistical averages of certain quantities based on the known pattern. Often, empirical relations are used to evaluate the flow factors. These relations are usually functions of the statistical properties of the surface. For example, one of the relations proposed by Patir and Cheng to compute the flow factors is

where H is the normalized gap height, hm/σ, with σ being the equivalent surface roughness of the contacting surfaces. The constants in the above equation depend on the orientation of the structure in the surface roughness relative to the flow direction. This orientation is defined by Pecklenik factor γ, which is the ratio of the correlation lengths along the flow and perpendicular to the flow

A value γ > 1 corresponds to the roughness pattern so that the ridges and valleys are oriented along the flow direction, while a value less than one corresponds to ridges and valleys perpendicular to the flow.

See the theory of the Reynolds Equation for details of the computation of journal and bearing forces.