Method of regularized sources for axisymmetric Stokes flow problems

Abstract

Purpose – The purpose of this paper is to find solution of Stokes flow problems with Dirichlet and Neumann boundary conditions in axisymmetry using an efficient non-singular method of fundamental solutions that does not require an artificial boundary, i.e. source points of the fundamental solution coincide with the collocation points on the boundary. The fundamental solution of the Stokes pressure and velocity represents analytical solution of the flow due to a singular Dirac delta source in infinite space. Design/methodology/approach – Instead of the singular source, a non-singular source with a regularization parameter is employed. Regularized axisymmetric sources were derived from the regularized three-dimensional sources by integrating over the symmetry coordinate. The analytical expressions for related Stokes flow pressure and velocity around such regularized axisymmetric sources have been derived. The solution to the problem is sought as a linear combination of the fields due to the regularized sources that coincide with the boundary. The intensities of the sources are chosen in such a way that the solution complies with the boundary conditions. Findings – An axisymmetric driven cavity numerical example and the flow in a hollow tube and flow between two concentric tubes are chosen to assess the performance of the method. The results of the newly developed method of regularized sources in axisymmetry are compared with the results obtained by the fine-grid second-order classical finite difference method and analytical solution. The results converge with a finer discretization, however, as expected, they depend on the value of the regularization parameter. The method gives accurate results if the value of this parameter scales with the typical nodal distance on the boundary. Originality/value – Analytical expressions for the axisymmetric blobs are derived. The method of regularized sources is for the first time applied to axisymmetric Stokes flow problems.

A coupled domain–boundary type meshless method for phase-field modelling of dendritic solidification with the fluid flow

“…In the group of the domain-type meshless strong-form methods, also known as the meshless collocation methods, we find, for example, the diffuse approximate method (Sadat and Prax, 1996; Hatić et al , 2019; Talat et al , 2018) and the radial basis function generated finite difference (RBF-FD) method (Flyer et al , 2016; Bayona et al , 2017), also known as the local radial basis function collocation method (Šarler and Vertnik, 2006; Kosec and Šarler, 2011; Vertnik et al , 2019; Mramor et al , 2014; Hanoglu and Šarler, 2018; Mavrič and Šarler, 2015). Examples of boundary-type meshless methods are the local boundary integral equation method (Zhu et al , 1998), the boundary-point interpolation method (Gu and Liu, 2002), the boundary radial point interpolation method (Gu and Liu, 2003), the non-singular method of fundamental solutions (Liu and Šarler, 2018), method of regularised sources (MRS) (Wang et al , 2016), and modified MRS (MRSM) (Rek et al , 2021).…”