Stationary Branches and Oscillatory Regimes of the Debyaoui-Ersoy Model in a Variable Channel: Hydrostatic Limits, Phase Portraits, and Numerical Validation
Abstract
We investigate the stationary hydraulic regimes of the Debyaoui-Ersoy model in a rectangular channel with variable width and bottom topography, focusing on the interplay between hydrostatic and dispersive effects. Three configurations are considered: smooth subcritical flows, smooth transcritical flows, and transitions associated with an entropy-admissible hydraulic jump in the hydrostatic limit.
For the hydrostatic model, we establish an exact classification of stationary branches and construct a phase portrait in terms of the prescribed inflow discharge and downstream water depth. For the dispersive model, we derive the stationary equations and investigate their behaviour as the dispersion parameter tends to zero. Under explicit compactness and regularity assumptions, we establish convergence towards smooth hydrostatic profiles and obtain a quadratic error estimate for uniformly regular subcritical solutions. In contrast, we show that a dissipative hydrostatic jump cannot arise as a strong limit of uniformly positive and bounded stationary dispersive solutions. We also investigate the role of oscillations and total variation in this singular limit.
These analytical results are complemented by numerical experiments illustrating the hydrostatic-dispersive comparison and the structure of stationary branches. A direct finite-volume stationary solver is developed and validated against long-time simulations of the corresponding discrete evolution equations for the three hydraulic configurations. The resulting phase portraits and refinement studies reveal the influence of dispersion, oscillatory transitions, and numerical resolution on the computed equilibria.
Together, these results provide a theoretical and computational framework for understanding stationary hydraulic transitions and the hydrostatic limit of the Debyaoui-Ersoy model in variable channels.
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BibTeX
@misc{ersoy2026stationary,
title = {{Stationary Branches and Oscillatory Regimes of the Debyaoui-Ersoy Model in a Variable Channel: Hydrostatic Limits, Phase Portraits, and Numerical Validation}},
author = {Mehmet Ersoy},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05803261v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stationary Branches and Oscillatory Regimes of the Debyaoui-Ersoy Model in a Variable Channel: Hydrostatic Limits, Phase Portraits, and Numerical Validation
AU - Mehmet Ersoy
PY - 2026
DA - 2026/10/16
PB - HAL
UR - https://hal.science/hal-05803261v1
AB - We investigate the stationary hydraulic regimes of the Debyaoui-Ersoy model in a rectangular channel with variable width and bottom topography, focusing on the interplay between hydrostatic and dispersive effects. Three configurations are considered: smooth subcritical flows, smooth transcritical flows, and transitions associated with an entropy-admissible hydraulic jump in the hydrostatic limit. For the hydrostatic model, we establish an exact classification of stationary branches and construct a phase portrait in terms of the prescribed inflow discharge and downstream water depth. For the dispersive model, we derive the stationary equations and investigate their behaviour as the dispersion parameter tends to zero. Under explicit compactness and regularity assumptions, we establish convergence towards smooth hydrostatic profiles and obtain a quadratic error estimate for uniformly regular subcritical solutions. In contrast, we show that a dissipative hydrostatic jump cannot arise as a strong limit of uniformly positive and bounded stationary dispersive solutions. We also investigate the role of oscillations and total variation in this singular limit. These analytical results are complemented by numerical experiments illustrating the hydrostatic-dispersive comparison and the structure of stationary branches. A direct finite-volume stationary solver is developed and validated against long-time simulations of the corresponding discrete evolution equations for the three hydraulic configurations. The resulting phase portraits and refinement studies reveal the influence of dispersion, oscillatory transitions, and numerical resolution on the computed equilibria. Together, these results provide a theoretical and computational framework for understanding stationary hydraulic transitions and the hydrostatic limit of the Debyaoui-Ersoy model in variable channels.
ER -