Verified simulation of the stationary polymer fluid flows in the channel with elliptical cross-section

摘要

This work is aimed at the numerical analysis of the stationary non-isothermal flows of an incompressible viscoelastic polymer fluid in the channels with elliptical cross-sections. The description of such flows is done on the basis of the mesoscopic approach, and the resolving equations are derived. For solving them three algorithms, which use different techniques of constructing the approximate solutions, are designed: the least-squares collocation method based on the piecewise polynomial approximations, which lead to the overdetermined systems of linear algebraic equations; the finite element method, which uses weak formulations; and the non-local method without saturation, which operates with the global approximations in the elliptical coordinate system and with the matrix Sylvester equations. The proposed algorithms are verified by solving the test problem with the known analytical solution. Further we use them for the numerical analysis of the polymer fluid flows with its parameters varying in wide ranges. Comparison of the results obtained by the different algorithms shows their high performance and confirms that the solution of the considered non-linear problem exists and that it was computed accurately. The singularities of the obtained stationary solutions are analyzed. Taking them into account within the proposed algorithms enables us to increase the accuracy and the speed of simulations.