Analytic geometry asymptotes

Now, 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 fully developed concentration (or temperature) profile discussed in Section 8.2.2 is characterized by a uniform value of Sherwood (or Nusselt) number, that is, independent of axial distance. Young and Finlayson [108] solved the heat and mass transfer problem numerically using the orthogonal collocation technique and calculated the asymptotic (for long distances) Sherwood and Nusselt numbers for several channel geometries. In terms of the analytical development that we have been presenting, it can be calculated for laminar flows from... 

The solution given by Eq. (3.6.13) still requires numerical evaluation. An analytical solution has been obtained [34] that is exact for a infinite slab and is approximately correct for a particle of a general geometry. From this analytical solution, useful asymptotic and approximate solutions have been obtained for the maximum temperature rise. 

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