Numerical solution of the three-dimensional Burger’s equation by using the DQ-FD combined method in the determination of the 3D velocity of the flow

Abstract

Abstract In this paper, the differential quadrature and the finite difference combined method (DQ-FDM) was applied to solve the three-dimensional Burger’s equation in the determination of the 3D velocity of the flow; so that spatial terms were discretized by the differential quadrature method, and the temporal term was discretized by the finite difference method, and the resulting nonlinear equations were solved using the Newton–Raphson method. All variables were considered as dimensionless in this equation. The solution results were compared with solution results of the two-dimensional equation in the two other numerical methods available in the literature which provided an acceptable accuracy. Also, the results of the mentioned numerical method were compared with those of the fully implicit finite difference method that was solved for larger than or equal viscosities of 0.1. The results showed that by increasing time and viscosity, the longitudinal, depth and transverse velocities were decreased. The occurrence of the upward flow was observed especially in the $$\upsilon$$ υ = 0.05 in the close of the bed, end of the length and width that in the presence of very fine particles of the clay and silt shows suspension of these particles in some spaces. The position of the longitudinal, depth and transverse velocities in the plan for the passing plates through the section depth for different viscosities and times showed that by increasing viscosity and time, the position of the maximum velocities became closer to the middle of the section width. Also, stream lines were plotted in all of sections and then analyzed.

Keywords

• Partial differential equation
• Three-dimensional Burger’s equation
• Numerical solution methods
• DQ-FD method
• Newton–Raphson method