Past exam of the flows, fluctuations and complexity and symmetry and relativity paper of Part B of the physics course of the University of Oxford / 2011 / 2 / 4 / Solution

Put $A=2\mu a{F}_{0}k/(cf)$. In the specified near-wall limit the outer temperature is $T-T_0\simeq A/(bz)$, soThus the near-wall radiative heating layer has scaleThe comparison is an estimate of the breakdown of the outer solution, not a new inner temperature solution. It requires ${\delta }_C\ll b$ and $k\rho {\delta }_C\ll 1$; neglecting longitudinal relative to transverse thermal conduction additionally requires ${\delta }_C\ll a$ in the region to which the estimate is applied.