The paper deals with a non-linear problem of long water waves approaching a sloping beach. In order to describe the phenomenon we apply the Lagrange’s system of material variables. With these variables it is much easier to solve boundary conditions, especially conditions on a shoreline. The formulation is based on the fundamental assumption for long waves propagating in shallow water of constant depth that vertical material lines of fluid particles remain vertical during entire motion of the fluid. The analysis is confined to one – dimensional case of unsteady water motion within a ’triangular’ body of fluid. The partial differential equations of fluid motion, obtained by means of a variational procedure, are then substituted by a system of equations resulting from a perturbation scheme with the second order expansion with respect to a small parameter. In this way the original problem has been reduced to a system of linear partial differential equations with variable coefficients. The latter equations are, in turn, substituted by a system of difference equations, which are then integrated in a discrete time space by means of the Wilson-µ method. The procedure developed in this paper may be a convenient tool in analysing non-breaking waves propagating in coastal zones of seas. Moreover, the model can also deliver useful results for cases when breaking of waves near a shoreline may be expected.
The paper describes the behavior of the liquid in a container that moves with a constant speed along a track consisting of three arcs. Such a complicated track shape generates complex form of inertia forces acting on the liquid and generates the sloshing effect. The behavior of the tank container vehicle is affected by the time-dependent inertia forces associated with the transient sloshing motion of the liquid in the non-inertial frame. These internal excitations, acting on a tank construction, can cause a loss of stability of the vehicle. For that reason, the authors analyze the dynamic loads acting on the walls of the tank truck container. The variation of the position of the liquid cargo gravity center, that depends on the filling level of the container, is also analyzed. The simulations were performed according to the varying fill level, which was 20%, 50% and 80% of a liquid in the whole tank volume. The simulations were carried out for a one-compartment container. Another aim of this study was the investigation of the influence of container division (tank with one, two and three compartments) on behavior of the liquid. These simulations considered only the half-filled container which was treated as a dangerous configuration prohibited by the law regulations for one-compartment tank. The results of simulation are presented in the form of visualization of temporary liquid free surface shape, variation of forces and moments, as well as frequency analysis. The results of simulation were analyzed, and some general conclusion were derived, providing the material for future investigation and modifications of the law regulations.