The Horizontal, Enclosed Shallow Cavity

We begin by considering the flow within a shallow, horizontal (a = 0) cavity as sketched in Fig. 6-7a. We assume that the ratio, d/L, is asymptotically small. We seek only the leading-order approximation within the shallow cavity. Hence the starting point for analysis is the thin-film equations, (6—1)—(6—3). In the present case of a 2D cavity, we can use a Cartesian coordinate system, and, for the present problem, we assume that the fluid is isothermal, so that the body-force term in (6-3) can be incorporated into the dynamic pressure, and hence plays no role in the fluid s motion. In this case, the governing equations become 

The characteristic scales that have been used to produce these nondimensionalized equations are 

The variables m(0), w(0), and p1 are the first terms in an asymptotic expansion [see, e.g., 6-68] for e - 0 (with e2Re P 1, where Re = UL/v). The shallow-cavity scaling that produces (6-119) (6 121) must, of course, break down in the neighborhood of the end walls. The ends of the cavity are impermeable, and thus the  

The difference in scaling between the central core of the thin cavity (6-122) and the vicinity of the end walls (6-123) means that the asymptotic solution for s c 1 is singular, and a different set of dimensionless equations and a different form for the asymptotic expansion for e c 1 must be obtained in the end regions. The distinct expansions in the core and end regions are then required to match in the region of overlapping validity. 

We have already noted that the governing equations are (6-119)—(6—121). The boundary conditions at the top and bottom walls are 

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