Tables 1 and 2 compare the slip length values obtained from the perturbation method and Refs. [6] and [18] to *β* obtained from the PDE toolbox. We note that, insofar as the perturbation analysis to $o(\epsilon \u0303)$, the mean temperatures along $y\u0303=0$ and $y\u0303\u2192\u221e$ are unchanged as per Eqs. (24) and (26). Thus, in Eq. (C8), the corresponding terms in the numerator, i.e., $T\u0303\xaf0(x\u0303,0)$ and $T\u03031\xaf(x\u0303,0)$, may be replaced by those as $y\u0303\u2192\u221e$. Thus, the slip may also be computed from Eq. (40) for *β*. However, once the magnitude of $\u03f5\u0303$ is large enough that terms of $o(\epsilon \u0303)$ become relevant, $T\u0303\xaf0(x\u0303,0)$ and $T\u0303\xaf1(x\u0303,0)$ may not be replaced by those as $y\u0303\u2192\u221e$, and thus, we do not expect $b\u0303t$ and *β* to agree as borne out by our results. The same argument applies for the case of an isothermal ridge. In summary, $\beta =b\u0303t+o(\epsilon \u0303)$.