Asymmetric and Moving-Frame Approaches to MHD Equations

TL;DR

This paper develops twelve solution families for the incompressible viscous MHD equations using asymmetric and moving frame methods, revealing vortex characteristics and aiding numerical algorithm development.

Contribution

Introduces twelve novel solution families for MHD equations employing asymmetric and moving frame techniques, including singular and analytic solutions.

Findings

A family of singular solutions reflecting vortex features

Globally analytic solutions with respect to spatial variables

Potential to improve numerical algorithms for practical models

Abstract

The magnetohydrodynamic (MHD) equations of incompressible viscous fluids with finite electrical conductivity describe the motion of viscous electrically conducting fluids in a magnetic field. In this paper, we find twelve families of solutions of these equations by Xu's asymmetric and moving frame methods. A family of singular solutions may reflect basic characteristics of vortices. The other solutions are globally analytic with respect to the spacial variables. In particular, Bernoulli equation and Wronskian determinants play important roles in our approaches. Our solutions may also help engineers to develop more effective algorithms to find physical numeric solutions to practical models.