In figure 3 (Vynnycky & Ipek 2009), a comparison was made of the analytical results derived in §4, the results of the numerical computations of the equations derived in §3 and those presented in Ipek *et al.* (2007). There was considerable discrepancy between the analytical and the numerical results; this was attributed to short-circuiting effects which, it was believed, the analytical model was unable to capture. Here, with a modicum of analysis, it is demonstrated that the real reason is that the equations in §4 do not capture the role of convection, and that appropriate analysis of the equations derived in §3 gives the desired result.

For a slender geometry, *ϕ*^{bulk} satisfies
1.1subject to boundary conditions (3.72) and (3.73). The solution to equation (1.1) is given by
where *A*_{L},*A*_{U},*B*_{L} and *B*_{U} are constants to be determined; although *A*_{L},*A*_{U},*B*_{L} and *B*_{U} should, in general, be functions of *Y*, the computations in Ipek *et al.* (2007) suggest that, for this problem, they will be constant to a good approximation. Applying equations (3.72) and (3.73) gives
1.2
1.3
and
1.4
1.5in equations (1.4) and (1.5), and denote the potential at the cathodic and anodic portions of the strip, respectively. Since these are unknown, two more equations are needed. First of all, the requirement that all of the current leaving the anode enters the cathode gives
1.6Furthermore, this should be equal to the current that passes through the strip; hence,
1.7

There are now six equations (1.2)–(1.7) in six unknowns: and Omitting the details, these equations can be re-arranged to give one equation in one unknown:
1.8where
with
and
with
and
Now, since *μ*≫1, we find that
and the dimensional current density, [*i*], can be recovered as
1.9

The result is given in figure 1 and indicates that the new analytical estimate gives excellent agreement with the results of the earlier computations; the old result (equation (4.10)) is also given.

## Acknowledgements

The author acknowledges the support of the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland Mathematics Initiative grant 06/MI/005.

- Received March 9, 2011.
- Accepted April 5, 2011.

- This journal is © 2011 The Royal Society