Finite element based direct and iterative approach to investigate a magneto-micropolar flow through a rectangular channel

Abstract

This article aims to numerically compute the magnetic induced flow in the framework of higher order continuum through its implementation using FreeFem++. The flow through a rectangular channel is assumed laminar, incompressible and under constant applied pressure. External magnetic field is applied through the boundary of the channel. To compute the physical response theory of micropolar continuum is taken into consideration and governing dynamics is described in the form of coupled partial differential equations in velocity, induction and microrotation field variables along with associated boundary conditions. Sparse linear system is thus obtained with the application of finite element method. The resulting system is then solved by sparse solver available in unsymmetric multifrontal package (UMFPACK) through FreeFem++. The variational form of the model is presented and implemented in FreeFem++. Pertinent to physical significance different parameters i.e., magnetic Reynolds number, Hartmann number, micropolar coupling constant, micropolar parameter are studied and results are shown through simulations and graphs.It is observed that with an increase in micropolar coupling parameter the magnetic induction as well as the microrotational velocity of particles decreases in the medium. The maximum magnitude of microrotational velocities is observed near walls of the channel. The microrotational spin in the channel flow slows down due to an increase in induced magnetic field. Moreover, it is found that direct solver for the asymmetric sparse linear system arise in this case does not lead to stable solution. This instability in the solution arises when the Hartmann number in simulations increases than a value of 1.5. Therefore, iterative solver based on generalized minimal residual method (GMRES) should be adopted to weed out the instabilities due to fine scale oscillations of translational as well as microrotational motions of the fluid particles in case of large Hartmann number.