Boundary region determination constraints

The boundary of R is determined by t i = 0. Individual segments in the boundary of R are determined by/m = 0, m G M. Values of the uncertain variables 6 lying inside feasible region R allow the control variables z to be adjusted so that all the feasibility constraints can be satisfied. For values of 8 lying outside the feasible region, the control variables cannot be adjusted to satisfy all the feasibility constraints. 

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. 

Hence the velocity and pressure distributions at 0(1) can be completely determined within the core region (away from the end walls), to within an arbitrary constant for p 0). In fact, the flow is a simple unidirectional flow, as is appropriate for traction-driven flow between two plane surfaces. The turning flow that must occur near the ends of the cavity influences the core flow only in the sense that the presence of impermeable end walls requires a pressure gradient in the opposite direction to the boundary motion in order to satisfy the zero-mass-flux constraint. But now, a remarkable feature of the domain perturbation procedure is that we can use our knowledge of the unidirectional flow that is appropriate for an undeformed interface at 0(1) to directly determine the 0(5) contribution to the interface shape function in (6-159a) without having to determine any other feature of the solution at 0(5). 

Since the Fluid Film cannot sustain a negative pressure oF any practical signiFicance, the oil must become cavitated. The problem oF determining the Free boundary separating the lubricated and the cavitated region has been studied by others. He have chosen the approach oF the variational inequality C71. The so-called Reynolds boundary condition is realized by the non-negativity constraint ... 

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