Criticality in a nearly circular cylinder

The problem of determining the critical conditions in a non-circular cylinder of infinite length is examined, in which the reaction is of zero order and where the Frank-Kamenetskii approximation to the reaction rate has been made. The generators are assumed parallel and the boundary of a cross section is given by $r = 1 + \epsilon \cos \theta (\epsilon \simeq 1)$, where $(r, \theta)$ are plane polar coordinates. The surface conditions assumed are (i) uniform temperature and (ii) newtonian cooling. By suitable expansions of the interior temperature and the Frank-Kamenetskii parameter $\delta$, it is shown that the critical value of $\delta$ for (ii) is \begin{equation*}\begin{split} \delta_{\operatorname{crit}} (\epsilon, B) = \frac{8\sigma_c}{(1+\sigma_c)^2} \exp \Big\lbrack &- \frac{4\sigma_c}{B(1+\sigma_c)}\Big\rbrack \\ &x\Big\lbrack 1 - \epsilon^2\{\frac{\sigma^2_c}{1+\sigma_c} + \frac{1}{B} \sigma_c \frac{(1+3\sigma_c)}{(1+\sigma_c)^2}\Big\} + O(\epsilon^4)\Big\rbrack,\end{split}\end{equation*} where $\sigma_c = -2/B + (1 + 4/B^2)^\frac{1}{2}$, and where $B$ is a Biot number proportional to the surface-heat transfer coefficient. For (i), the simple result $\delta_{\operatorname{crit}}(\epsilon, \infty) = 2 - \epsilon^2 + O(\epsilon^4)$, is obtained.