An analytical asymptotic solution to a conjugate heat transfer problem

In this paper, an asymptotic solution to the conjugate heat transfer problem with a flush-mounted heat source on the fluid–solid interface, in the case that the bottom of the solid is perfectly insulated and the velocity profile in the fluid is linear, is presented. The lowest order terms of the asymptotic solution can be naturally classified into contributions from pure convection, from the interaction of convection and the conduction in the solid and from the interaction of convection and the conduction in the fluid. It was found that downstream of the heat source the two leading order terms of the asymptotic expansion stem from pure convection, and that the leading term decays as O(x−2/3), which confirms the result from the analysis by Liu et al. [Int. J. Heat Mass Transfer 37 (17) (1994) 2809] in the case of an adiabatic wall. The third term, however, is a contribution from the interaction of conduction in the solid and convection. If we furthermore neglect the conduction in the fluid we have been able to find the asymptotic solution upstream of the heat source as well, and in this case we find that the temperature decays exponentially with the distance from the heat source. Our results show good agreement with numerical solutions to the problem.