Generalised solutions to the Benjamin problem

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We generalise the classical Benjamin solution (Benjamin, J. Fluid Mech., vol. 31, 1968, pp. 209-248) modelling the flow in a horizontal duct of finite depth in situations where the flow contains a region spanning the depth of the duct, and a region in which the surface detaches from the ceiling of the duct as a free surface. It is shown that the Benjamin solution belongs to a one-parameter family of similar solutions, which are divided into two types: solutions that describe potential flows where the free surface of the fluid is deflected from the duct ceiling at a zero angle; and solutions that admit the formation of a vortex flow region in the vicinity of the point of fluid separation from the duct ceiling. It is shown that this one-parameter family of solutions is the limit of a two-parameter family of solutions in which part of the uniform flow energy is converted into energy of the small-scale fluid motion. Based on the local hydrostatic approximation, the applicability of the constructed solutions is discussed.

Original languageEnglish
Article numberR1
Number of pages12
JournalJournal of Fluid Mechanics
Early online date15 Apr 2020
Publication statusPublished - 25 Jun 2020


  • channel flow
  • vortex dynamics


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