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In hydraulic engineering, it is common to find geometric transitions where a channel is not prismatic. Among these geometric\ud transitions, constrictions and obstructions are channel reaches in which a cross-section contraction is followed by an expansion. These nonprismatic\ud reaches are significant because they induce rapid variations of the flow conditions. In the literature, the characteristics of the\ud geometric transitions have been well studied for the case of the steady-state flow, but less attention has been dedicated to the unsteady\ud flow conditions. The present paper focuses on the exact solution of the dam-break problem in horizontal frictionless channels where constrictions\ud and obstructions are present. In order to find this solution, the geometric transition is assumed to be short with respect to the channel\ud length, and a stationary solution of the shallow water equations is used to describe the flow through the nonprismatic reach. The mathematical\ud analysis, carried out with the elementary theory of the nonlinear hyperbolic systems of partial differential equations, shows that the dam-break\ud solution always exists and that it is unique for the given initial conditions and geometric characteristics of the problem. The one-dimensional\ud mathematical model proves to be successful in capturing the main characteristics of the flow immediately outside the geometric transition, in\ud comparison with a two-dimensional numerical model. The exact solution is then used to reproduce a set of experimental dam-break results,\ud showing that the one-dimensional mathematical theory agrees with the laboratory data when the flow conditions through the constriction are\ud smooth. The exact solutions presented here allow the construction of a class of benchmarks for the one-dimensional numerical models that\ud simulate the flow propagation in channels with internal boundary conditions

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Background At the present time, in patients with liver cirrhosis and gastroesophageal varices, primary prophylaxis of variceal bleeding made with combination therapy with non-selective β-blockers (NSBBs) and endoscopic band ligation (EBL) is not recommended. The aim of this study was to evaluate if patients with worsening varices while on NSBBs regimen benefit, in terms of bleeding and survival, from adding treatment with EBL. Methods Patients with cirrhosis and endoscopic finding of gastroesophageal varices with high risk feature (increased variceal size and/or development of red signs) during primary prophylaxis with NSBBs, followed at the Unit of Internal Medicine and Hepatology, University and General Hospital of Padova, Italy, from 2012 to 2019, were retrospectively evaluated. When an increased bleeding risk of the varices was confirmed, patients maintained the pharmacological therapy alone or underwent also EBL. The primary endpoint of the study was the rate of variceal bleeding, the secondary endpoint was mortality at 30 months. Results Compared to patients treated only with NSBBs (n=56), in patients treated also with EBL (n=45), the 30‐month probability of variceal bleeding (29.1% vs 5.1%; P =0.036) was significantly reduced, while the probability of survival was similar (59.6% vs 65.7%; P=0.61). On multivariate analysis, treatment with EBL was found to be a weak protective factor for mortality (HR 0.47, P=0.044). Conclusion In patients with liver cirrhosis, when varices show endoscopic feature of increased haemorrhagic risk, adding EBL to NSBBs is effective in reducing the probability of first bleeding.