Geodesic
Unsteady flows in deformable pipes: the energy conservation law
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems and methods in mechanics, Tome 300 (2018), pp. 76-85

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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. 
@article{TM_2018_300_a4,
     author = {A. T. Il'ichev and S. I. Sumskoi and V. A. Shargatov},
     title = {Unsteady flows in deformable pipes: the energy conservation law},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {76--85},
     publisher = {mathdoc},
     volume = {300},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2018_300_a4/}
}
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A. T. Il'ichev; S. I. Sumskoi; V. A. Shargatov. Unsteady flows in deformable pipes: the energy conservation law. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems and methods in mechanics, Tome 300 (2018), pp. 76-85. http://geodesic.mathdoc.fr/item/TM_2018_300_a4/