Abstract
AbstractSteady two-dimensional nonlinear flexural–gravity hydraulic falls past a submerged obstruction on the bottom of a channel are considered. The fluid is assumed to be ideal and is covered above by a thin ice plate. Cosserat theory is used to model the sheet of ice as a thin elastic shell, and boundary integral equation techniques are then employed to find critical flow solutions. By utilising a second obstruction, solutions with a train of waves trapped between two obstructions are investigated.
Highlights
The classic hydroelastic problem concerning the interaction between a deformable body and a moving fluid is a widely studied area in mathematics and engineering, because of its vast range of physical applications (Korobkin, Parau & Vanden-Broeck 2011)
Transportation links over the ice plates are of importance (for example, railways have been placed over the river Ob and the lake Baikal in Siberia, Russia)
Fully nonlinear hydraulic fall solutions under an ice plate modelled by a thin elastic shell have been presented and discussed
Summary
The classic hydroelastic problem concerning the interaction between a deformable body and a moving fluid is a widely studied area in mathematics and engineering, because of its vast range of physical applications (Korobkin, Parau & Vanden-Broeck 2011) One such application involves the modelling of an ice plate floating on top of water as a thin elastic sheet (Squire et al 1996). Following Belward & Forbes (1993), Dias & Vanden-Broeck (2004) and Page et al (2014), the problem is reformulated as a system of nonlinear integrodifferential equations, which can be solved for the unknown fluid surface y = 1 + η(x) over arbitrarily shaped obstacles on the bottom of the channel. The two integral equations (2.12) and (2.13), together with the parametrised dynamic condition (2.10) and the parametric equation (2.9), form the system of integrodifferential equations to be solved, iteratively via Newton’s method, for the unknown variables Y (s), X (s) and φ (s) at the fluid–ice interface and u(x) on the channel bottom
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