This dissertation studies a heat conduction problem in a pipe filled with incompressible viscous fluid whose length varies depending on the fluid temperature. We assume that the longitudinal dilatation of the pipe is described by a linear heat expansion law (length of the pipe varies proportionally to the integral of the mean temperature, with the coefficient of proportionality \(\varepsilon\), the heat expansion coefficient), i.e. the flow domain is not fixed, but varies depending on the fluid temperature. Fluid temperature is unknown and given as a solution of the standard heat conduction equation. Fluid velocity in the convection term is given and it corresponds to the Poiseuille flow. We also consider heat exchange with the surrounding medium and apply Newton’s law of cooling to describe it. When taking into account linear heat conduction equation and linear heat expansion law, we get an interesting nonlinear effect, where we try to find a solution to the problem posed on a domain that is varying due to that same unknown solution. Our first goal was to see under which conditions there is a solution to the proposed problem and when is the solution unique. For that purpose, we proved a priori estimates, and subsequently the existence and uniqueness result of the solution. Established conditions are realistic, since they impose dependence on the size of a parameter \(\varepsilon\), the heat expansion coefficient, which is small enough since it depends on the actual pipe material. Next goal was construction of an approximation of the solution by means of an asymptotic expansion in powers of the abovementioned small parameter \(\varepsilon\), followed by rigorous justification of the obtained model by proving the appropriate error estimate. Finally, we studied a special case of a thin pipe, where we constructed the approximation of the solution by means of an asymptotic expansion in powers of the pipe radius. We achieved the solution approximation for the first two terms and subsequently we justified the result by proving adequate error estimates.