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| We consider a heat flow through a thin (or long) pipe filled with incompressible viscous fluid. The process is described by a non-stationary convection-diffusion equation. The fluid in the pipe is cooled by the exterior medium and we describe the heat exchange simply by the Newton's cooling law . Assuming that the flow through a pipe has a given Poiseuille profile, we study the thermodynamic part of the system. Depending on the ratio between the pipe's thickness $\varepsilon$ and the Reynolds number $Re^\varepsilon$ , we obtain three different macroscopic models via rigorous asymptotic analysis. For small $Re^\varepsilon$ the fluid in the pipe is perfectly cooled, i.e. it assumes the temperature of the surrounding medium. For large $Re^\varepsilon$, the fluid is not cooled at all, i.e. it maintains the same temperature as it had when it entered the pipe. Between those two cases there is a critical value of $Re^\varepsilon$ when the macroscopic model is described by an ODE keeping the effects of the surrounding medium as well as the entering temperature. |
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