Interdisciplinary Applied Mathematics

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(6.4)



dp    d2u


dx ^ dy2


This equation is identical to the fully developed channel flow equation. However, in the slider problem, equation (6.4) is an approximation valid for R* ^ 1. Here, the flow is not fully developed but varies slowly in the streamwise direction due to gradual changes in the channel area. Since


du du ho dx / dy    L


the streamwise velocity is a function of y only to the leading-order approximation. The velocity distribution u(y) is obtained by integrating equation (6.4)    using    either    the    no-slip    or    slip    boundary conditions.    If    temperature


variations in the system are neglected, the density becomes a function of pressure, which varies mainly in the streamwise direction. Using the velocity distribution and the local density (p), the following equation for the mass flowrate (per channel width) is obtained for no-slip flows:


id



uoph



h3 p dp 12p dx



(6.5)


Since the flow is assumed isothermal, the density can be written as a function of the    pressure    using    the    equation    of    state    p = p/RT.    In    equation


(6.5), the left-hand side is constant, but the right-hand side is a function


of x. Furthermore, the mass flowrate is a function of the pressure gradient, which is    unknown.    Taking    the    gradient    of    (6.5),    we    obtain    the    Reynolds


equation for one-dimensional steady flow, i.e.,


d_ f з dp дх Г Pdx



6 d dx



(phU0).



(6.6)


Since the spatial variation of the channel height (h) is known, the pressure is the only unknown in this equation. The boundary conditions correspond to ambient pressure at the two ends of the slider bearing geometry, shown in Figure 6.3. The terms in parentheses on the left and right of equation (6.6) are proportional to the mass flowrate per channel width (divided by RT) in plane Poiseuille and Couette flows, respectively. Furthermore, in this form the Reynolds equation neglects the spatial variations across the slider bearing. Hence, a two-dimensional problem is reduced to a one-dimensional equation for pressure.

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