Numerical method for solving a non-local problem on pipeline transportation of viscous liquid

The paper deals with a process of unsteady axisymmetric flow of incompressible viscous liquid in the cylindrical pipeline, described by the nonlinear system of Navier-Stokes differential equations. The set of equations is transformed into one linear parabolic equation with an initial and natural boundary condition on the pipeline axis. We face a problem for determining velocity distribution in a cross-section of the pipeline based on the desired law of time variation of the pressure drop along the pipeline. As in case of liquid flow in pipes it’s practically impossible to define interaction models of fluid with a solid boundary, the boundary condition on the pipe wall is considered as unknown. For the problem accuracy an additional condition in the form of integral flow characteristic is specified. In other words, the law of time variation of volumetric flow rate in the pipeline is specified. This problem is related to nonlocal problems with an integral condition for partial differential equations. The specified integral condition is differentiated in time and the obtained ratio with the help of the differential equation is transformed into a local boundary condition. As a result, the set task is altered to a direct problem with local conditions. The finite difference method is applied for numerical solution of the boundary value problem. For this purpose, we create a discrete analog of the problem in the form of an implicit difference scheme using the integral method. A computational algorithm of solving the obtained difference equation system is suggested. Numerical experiments for test problems have been conducted to check the efficiency of practical application of the suggested computational algorithm. The computational algorithm has also been tried on the data of steady flow of viscous incompressible liquid in the pipeline.