Asymptotic analysis of the heat conduction problem in a dilated pipe

The goal of this paper is to propose new asymptotic models describing a viscous fluid flow through a pipe-like domain subjected to heating. The deformation of the pipe due to heat extension of its material is taken into account by considering a linear heat expansion law. The heat exchange between the fluid and the surrounding medium is prescribed by Newton's cooling law. An asymptotic analysis with respect to the small parameter $\varepsilon$ (the heat expansion coefficient of the material of the pipe) is performed for the governing coupled nonlinear problem. Motivated by applications, the special case of flow through a thin (or long) pipe is also addressed leading to an asymptotic approximation of the solution in the form of an explicit formula. In both settings, we rigorously justify the usage of the proposed effective models by proving the corresponding error estimates.