Solution by means of the classical thin-film analysis

applying the normal-stress condition (6-140), we obtain 

Because it is clear from this result that dp(())/dx is independent of z, we can integrate (6-119) to obtain 

At this point, we could apply the continuity equation, (6-120), to obtain a version of the Reynolds equation for the pressure. However, as in the preceding subsection, we can 

Equation (6-148), plus the boundary conditions (6 142) and the integral constraint (6 143), is sufficient to determine h(x). We should note that we do not necessarily expect Eq. (6-148) to hold all the way to the end walls atx = 0 andx = 1, for it was derived by means of the governing equation, (6-119), (6-120) and (6-137), and these are valid only for the core region of the shallow cavity. Nevertheless, we will at least temporarily ignore this fact and integrate (6-148) over the whole domain, with the promise to return to this issue later. Qualitatively, we can see that the interface deformation is determined by a balance between the nonuniform pressure associated with the flow in the cavity, e g., Eq. (6 145), which tends to deform the interface, and the effects of capillary and gravitational forces, both of which tend to maintain the interface in its flat, undeformed state, i.e., h = 1. 

The governing Eq., (6 148), is highly nonlinear. Although it can be solved numerically, we restrict our considerations to deformations that are a small perturbation from the undeformed state, namely, 

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