Simulation of an unsteady incompressible fluid flow through a perforated pipeline

A mathematical model of the unsteady flow of an incompressible viscous fluid through a perforated pipeline is proposed, which is described by a system of nonlinear partial differential equations. In the framework of the model, the purpose is to determine the pressure and the flow rate of the fluid at the pipeline inlet, providing the flow rate and the pressure required at the pipeline outlet. By combining the system of the equations, the original problem is reduced to a boundary-value inverse problem for a nonlinear parabolic equation with respect to fluid flow rate. To solve the boundary inverse problem, the method of nonlocal perturbation of boundary conditions is proposed. A discrete analog of the inverse problem is obtained using the finite-difference approximation, and a special approach is suggested for solving the resulting system of difference equations. As a result, the difference problem for each discrete value of the time variable splits into two second-order difference problems and a linear equation with respect to an approximate value of the desired flow rate at the pipeline inlet. The absolutely stable Thomas method is used to numerically solve the obtained difference problems. After determining the flow rate distribution along the entire pipeline, the pressure at the pipeline inlet is also calculated using an explicit formula. Based on the proposed computational algorithm, the numerical experiments are performed for benchmark problems.