The Narrow-Gap Limit - Governing Equations and Solutions

the general problem of (5—5)—(5—10) is highly nonlinear and, for an arbitrary occentric cylinder geometry, it can only be solved numerically - i.e., for arbitrary e and X in the range 0 X 1.3 However, for Re = 0, an exact analytic solution can be obtained by a coordinate transformation. In addition, for Re / 0, there are two limiting cases for which we can use asymptotic methods to obtain approximate analytic solutions. These are slight eccentricity 

The geometry of the eccentric cylinder configuration in the narrow-gap limit is described by the limiting form as e - 0 of the general expressions (5-1) and (5-3). Thus, 

To see how the thin-gap approximation e C 1 simplifies this problem, it is again necessary to nondimensionalize the governing equations and boundary conditions. The characteristic velocity for the polar velocity component is clearly 

Furthermore, the length scale characteristic of velocity gradients in the thin gap is just the characteristic distance across the gap, 

The only dimensional quantity that remains to be nondimensionalized is the radial velocity component u r. To determine a characteristic scale for u r, we utilize the continuity equation (5-5) along with the dimensionless, forms, (5-11) and (5 12), for u 0 and y. For convenience, let us denote the characteristic magnitude of u r as V. Then, substituting u r = Vur plus (5-11) and (5 12), into (5-5), we obtain 

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