We study a system of an elastic ball moving in the non-relativistic spacetime with a nontrivial causal structure produced by a wormhole-based time machine. For such a system it is possible to formulate a simple model of the so-called "grandfather paradox": for certain "paradoxical" initial conditions the standard straight trajectory of the ball would self-collide inconsistently. We analyze globally consistent solutions of local equations of motion, namely, we find all trajectories with one self-collision. It is demonstrated that all standard initial conditions have a consistent evolution, including those "paradoxical" ones, for which the inconsistent collision-free trajectory is superseded by a special consistent self-colliding trajectory. Moreover, it is shown that for a wide class of initial conditions more than one globally consistent evolution exist. The nontrivial causal structure thus breaks the uniqueness of the classical theory even for locally deterministic physical laws.

A three-dimensional numerical simulation of particle motion in a pipe with a rough bed is presented. The simulation based on the Lattice Boltzmann Method (LBM) employs the hybrid diffuse bounce-back approach to model moving boundaries. The bed of the pipe is formed by stationary spherical particles of the same size as the moving particles. Particle movements are induced by gravitational and hydrodynamic forces. To evaluate the hydrodynamic forces, the Momentum Exchange Algorithm is used. The LBM unified computational frame makes it possible to simulate both the particle motion and the fluid flow and to study mutual interactions of the carrier liquid flow and particles and the particle-bed and particle-particle collisions. The trajectories of simulated and experimental particles are compared. The Particle Tracking method is used to track particle motion. The correctness of the applied approach is assessed.

In numerical models of fluid flow with particles moving close to solid boundaries, the Basset force is usually calculated for the particle motion between particle-boundary collisions. The present study shows that the history force must also be taken into account regarding particle collisions with boundaries or with other particles. For saltation -the main mode of bed load transport -it is shown using calculations that two parts of the history force due to both particle motion in the fluid and to particle-bed collisions are comparable and substantially compensate one another. The calculations and comparison of the Basset force with other forces acting on a sand particle saltating in water flow are carried out for the different values of the transport stage. The conditions under which the Basset force can be neglected in numerical models of saltation are studied. V numerických modelech proudění tekutin s pevnými částicemi v blízkosti pevné stěny je Bassetova historická síla obvykle počítána pro pohyb částice mezi jejími jednotlivými kolisemi se dnem. Předložená studie ukazuje, že při výpočtu Bassetovy historické síly je nutné brát v úvahu kolisi částice s pevným dnem nebo s jinými částicemi. Pro saltaci, hlavní typ pohybu splavenin u dna koryta, je na základě použitých výpočtů ukázáno, že dvě části Bassetovy historické síly, tj. síly způsobené pohybem částice v tekutině a kolisí částice se dnem, jsou srovnatelné a mohou se vzájemně významně kompensovat. Výpočet Bassetovy historické síly a její srovnání s ostatními silami působícími na písčitou částici při jejím saltačním pohybu ve vodě je uskutečněn pro různé hodnoty tzv. transport stage (poměr aktuálního a kritického smykového napětí na dně). Zároveň byly studovány podmínky, za nichž může být Bassetova historická síla v numerických modelech zanedbána.KLÍČOVÁ SLOVA: Bassetova historická síla, pohyb splavenin, numerický model, kolise částice se dnem.

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Abstract. Motion of a number of spherical particles in a closed conduit is examined by means of both simulation and experiment. The bed of the conduit is covered by stationary spherical particles of the size of the moving particles. The flow is driven by experimentally measured velocity profiles which are inputs of the simulation. Altering input velocity profiles generates various trajectory patterns. The lattice Boltzmann method (LBM) based simulation is developed to study mutual interactions of the flow and the particles. The simulation enables to model both the particle motion and the fluid flow. The entropic LBM is employed to deal with the flow characterized by the high Reynolds number. The entropic modification of the LBM along with the enhanced refinement of the lattice grid yield an increase in demands on computational resources. Due to the inherently parallel nature of the LBM it can be handled by employing the Parallel Computing Toolbox (MATLAB) and other transformations enabling usage of the CUDA GPU computing technology. The trajectories of the particles determined within the LBM simulation are validated against data gained from the experiments. The compatibility of the simulation results with the outputs of experimental measurements is evaluated. The accuracy of the applied approach is assessed and stability and efficiency of the simulation is also considered.