Geodesic
Roll wave structure in long tubes with compliant walls
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems and methods in mechanics, Tome 300 (2018), pp. 205-215

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We consider a flow of a fluid in a long vertical tube with elastic walls and show that, for certain parameters of the flow, small perturbations of the flow at the inlet section of the tube give rise to roll waves. Depending on the properties of the closing relation, either regular or anomalous roll waves are formed. In the latter case, a roll wave is characterized by two strong discontinuities that connect regions of continuous flow. We present the results of numerical simulations of the development of a pulsatile flow mode for convex and nonconvex closing relations that demonstrate the formation of regular and anomalous roll waves. We also construct a two-parameter class of exact periodic solutions and obtain existence diagrams for roll waves.
Keywords: flow in elastic tube, hyperbolic equations, Whitham condition, roll waves. 
@article{TM_2018_300_a16,
     author = {A. A. Chesnokov and V. Yu. Liapidevskii},
     title = {Roll wave structure in long tubes with compliant walls},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {205--215},
     publisher = {mathdoc},
     volume = {300},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2018_300_a16/}
}
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A. A. Chesnokov; V. Yu. Liapidevskii. Roll wave structure in long tubes with compliant walls. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems and methods in mechanics, Tome 300 (2018), pp. 205-215. http://geodesic.mathdoc.fr/item/TM_2018_300_a16/