Determination of the flow field in the phenomenon of falling of sediment particles in near-stagnant water using the two-dimensional Burger equation solution with the fully implicit finite difference method

Physical processes are dependent on various parameters that are modeled in their mathematical language with their own equation. Since some nonlinear partial differential equations are so difficult to solve, hence obtain their analytical solution except in certain conditions is not possible and such equations can be solved numerically. The equation that is considered in this research is the Burger equation in time-dependent nonlinear two-dimensional mode, which models the phenomenon of velocity of falling of particle in the stagnant or near-stagnant fluid, such as sedimentary water behind a dam. In this research, first, this equation discretized using the fully implicit finite difference method to solve the two-dimensional Burger equation, which is an unconditional stable method, then it programmed. Also, the accuracy of the equation solution results with another numerical method (finite element method) is compared which implies the consistency of the simpler method of finite difference to the more complex method of finite element. Numerical results have been obtained for different viscosities and times, and their role in the particle velocity has been examined parametrically. Generally, the results showed that increasing viscosity and time, lead to decreasing in fall of the particle velocity. Within the increasing the time, locus of the maximum velocities approach to the bottom of the bed and the end of the length in the depth and longitudinal directions, respectively. Also, the negative depth velocity (upward flow) was observed, especially at the edges near the bed, indicating that the particles are suspended in some positions and times.