Unsteady flows in deformable pipes: the energy conservation law

We derive a quasi-one-dimensional energy equation that corresponds to the flow of a compressible viscous fluid in a deformable pipeline. To describe the flow of such a fluid in a pipeline, we couple this equation with the previously derived continuity and momentum equations as well as with the equations of state for the internal energies of the fluid, the pipe deformations, pressure, and the cross-sectional area of the pipe. The derivation of the equations is based on averaging over the pipeline cross section. The equations obtained are designed for numerical simulations of long-distance transportation of a compressible fluid.