FreeFEM++ based heat transfer analysis of an electrically induced magnetic flow within the framework of micropolar continuum

Abstract

In this article, we aim to analyze electrically induced magnetic incompressible flow and heat transfer through a channel within the framework of micropolar continuum. Finite element method is employed to compute the numerical solution of the governing dynamics formulated in the form of PDEs along with associated boundary conditions on the channel. The weak form is presented and variational problem is implemented in FreeFem++ language. To validate, the correct implementation of the problem and obtained numerical results, a reduced micropolar magnetohydrodynamic model is taken into consideration. The analytical solution is presented. Computed numerical results are compared with the exact solution in case of reduced micropolar model and a good agreement is achieved. An excellent convergence of the numerical solution to the exact solution is shown through Tables where $L^2$-norm and $H^1$-errors are computed at refined meshes. Effect of different physical parameters such as magnetic Reynolds number ($R_{\mathrm{m}}$), Hartmann number (Ha), micorpolar constants (m and $l_1^2$), micropolar coupling number (N), Prandtl number (Pr) and Brinkmann number (Br) are studied and discussed in detail. The development of thermal, translational and micro-rotational velocity profiles over the cross-sectional domain of the channel is shown for varying values of these material parameters. Some interesting and new findings in this investigation are presented and discussed. It is observed that the maximum temperature in the medium shifts from the center of the domain toward the boundaries if the micropolar coupling constant N is increased. Maximum micro-rotations are always found near the channel’s boundary where it is evidenced that the particle’s micro-rotations have a direct proportionality with the micropolar constant m. Moreover, the counter rotations of the continuum particles are evidenced at the channel’s boundary. Furthermore, the micorpolar coupling constant $l_1^2$ is found to resist the micro motions of the particles in the channel. The successful implementation of one and higher dimensional magnetohydrodynamic model in FreeFEM++ in general and within the context of micorpolar continuum in particular is delineated. This shows that it provides an efficient platform to simulate magnetohydrodynamic problems in higher order continuum which has applications in designing magneto-rheological airbags, aircraft take-off gear, mechanical heart valves and cooling systems.

Keywords:

• Finite element
• FreeFEM++
• magnetic flow
• mHD
• micropolar
• numerical heat transfer