Viscous Fluid Flow in a Coaxial Pipe
In this paper, we propose a method to solve problems of stability of viscous incompressible fluid in pipes. The method is based on the R-functions with which the boundary function is obtained. Boundary function role is similar to the mesh in the finite element method, but it depends on shape of the region only and does not require any changes to increase accuracy. With the help of the boundary function and Chebyshev polynomials we design and elaborate structures for an approximate solution of the equations that satisfy the boundary conditions. This method is used for calculation of laminar steady flow in a pipe due to the constant pressure gradient and for solution of the linearized eigenvalue problem for perturbations. The pressure is handled by the Poisson equation. An example of flow stability in a rectangular duct is investigated. We obtain the eigenvalues and plots of perturbation velocity. The proposed algorithm does not depend on the shape of duct cross-section and can be used to investigate the stability of flows in arbitrary pipes including internal elements. This algorithm is simpler than the collocation method, tau method or spectral element method. Therefore, it should work faster and can be quickly adapted to new computer architectures, such as CUDA, OpenCL, Xeon Phi.