Least-squares finite element method for solving stokes equation with a point source magnetic field

Abstract

The formulation and numerical computation of the two-dimensional Stokes flow under the effect of a point source magnetic field are presented in this study. Stokes flow is also known as low Reynolds number, creeping flow, or non-inertial. Leastsquares finite element method (LSFEM) is successfully employed for the discretization of the Stokes equation. LSFEM has several advantages in terms of theory and computing, and it can create a symmetric and positive-definite algebraic system of equations that can be solved quickly and robustly using iterative approaches. However, LSFEM is having an issue where the low order nodal expansions tend to lock. Thus, the present study proposed the discretization of the problem domain using higher-order elements. The source codes for the Stokes equation with and without the point source magnetic field effect have been developed and verified against the existing benchmark solutions. The verification achieved an excellent agreement. The solution of the Stokes flow in a lid-driven cavity and a straight rectangular channel subjected to the point source magnetic field are conducted. The results concerning velocity contour and streamlines pattern are analysed. Firstly, the streamlines pattern in the lid-driven cavity problem shows the development of a vortex at the bottom-left corner cavity. The new vortex appeared as the secondary flow in cavity. As the magnetic number grows, the primary flow separates from the secondary flow. Secondly, when the straight rectangular channel problem was solved, a single vortex which is close to the point of the magnetic source. As the magnetic number increased, a new vortex appeared at the channel's upper wall. This shows that the point source magnetic field has a substantial impact on Stokes flow, as shown by the numerical simulation findings. Based on the current results, it can be concluded that the LSFEM can be used to solve Stokes flow problems with the effect of the point source magnetic field.