# On the MHD flow and heat transfer of a micropolar fluid in a rectangular duct under the effects of the induced magnetic field and slip boundary conditions

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## Abstract

The heat and mass transfer due to the steady laminar and incompressible micropolar fluid flow through a rectangular duct with the slip flow and convective boundary conditions are numerically calculated. The fluid moves under an external magnetic field applied on a plane perpendicular to the axis of the duct. The governing nonlinear partial differential equations of momentum, microrotation, induction, and the energy are solved simultaneously by the finite difference method. The effect of various numbers and parameters such as Reynolds, magnetic Reynolds, Hartmann, coupling, Brinkman numbers, the slip flow and convective parameters are presented in graphs. Some comparisons with previous works are included.

## Keywords

Magnetohydrodynamics Heat transfer Micropolar fluid Slip flow Convection Rectangular duct## 1 Introduction

In the last years, several investigators have studied the fluid flow and heat transfer inside a rectangular duct which has received considerable attention in engineering applications. This type of fluid flow is observed in several mechanical types of the equipment and the heat exchangers. Eringen [1, 2, 3] studied the theory of generalized continuum configuration of micropolar fluids which exhibit the microrotational effects and microrotational inertia. Subba et al. [4] studied the nonsimilar boundary-layer solutions for mixed convective micropolar fluid flow around a rotating cone, they provided the microrotation boundary conditions and their influence on the gyration, velocity, and heat transfer fields. Vantieghem [5] used the numerical simulation for steady flows of both laminar and turbulent in the quasi-static MHD flow through a toroidal duct of square cross-section with insulating Hartmann and conducting sidewalls, they presented a comprehensive analysis of the secondary flow and a comparison between MHD and hydrodynamic flows. Chou et al. [6] used the numerical solutions for combined free and forced laminar convection through a horizontal rectangular duct by a vorticity-velocity method, the walls are heated with a uniform heat flux without the assumptions of the small Grashof number and large Prandtl number. The numerical study of Aung et al. [7] made a combination between the free and forced laminar convection through vertical parallel plates with asymmetric wall heating at the uniform heat flux (UHF). Mahaney et al. [8] used a numerical technique to solve the momentum and energy equations and studied the effects of buoyancy-induced secondary fluid flow on forced flow with uniform bottom heating through a horizontal rectangular duct. Huang et al. [9] investigated numerically the mixed convection heat and mass transfer with film evaporation and condensation along the wetted wall with different temperatures and aspect ratios through the vertical ducts. Sayed et al. [10] investigated the laminar fully developed MHD flow and heat transfer of a viscous incompressible electrically conducting of a Bingham fluid in a rectangular duct, they took into consideration the constant pressure gradient, external uniform magnetic field and Hall effect. Rarnakrishna et al. [11] investigated the laminar natural convection of air for constant temperature and constant heat flux through a vertical square duct opened at both ends, they assumed that the velocity of air entering at the bottom of the duct is uniform at atmospheric pressure. Abd-Alla et al. [12] studied numerically the effect of magnetic field, rotation and initial stress on the motion of a micropolar fluid through a circular cylindrical flexible tube with small values of amplitude ratio, they considered that the wall properties is elastic or viscoelastic. Srnivasacharya et al. [13] investigated the steady flow of an incompressible and electrically conducting micropolar fluid flow with Hall and ionic effects in a rectangular duct, the government partial differential equations are solved numerically by the finite difference method. Pandey et al. [14] studied analytically the MHD flow of a micropolar fluid through a porous medium by sinusoidal peristaltic waves moving down the channel walls, the low Reynolds number and long wavelength approximations are applied to solve the nonlinear problem. Janardhana et al. [15] investigated numerically the thermal radiation heat transfer effect on the unsteady MHD flow of micropolar fluid over a uniformly heated vertical hollow cylinder using Bejan’s heat function concept. Ayano et al. [16] investigated numerically of mixed convection flow through a rectangular duct under the transversely applied magnetic field with at least one of the sidewalls of the duct being isothermal, the governing differential equations have been transformed into a system of nondimensional differential equations and are solved numerically such as the velocity, temperature, and microrotation component profiles are displayed graphically. Miroshnichenko et al. [17] investigated numerically the laminar mixed convection of micropolar fluid through a horizontal wavy channel by the finite difference method, they solved the system of equations of dimensionless stream function, vorticity and temperature then, studied the effects of Reynolds, Rayleigh, Prandtl numbers, vortex viscosity parameter and undulation number onto streamlines, isotherms, vorticity isolines as well as horizontal velocity and temperature profiles. Shit et al. [18] investigated the effect of the slip velocity on the peristaltic transport of a physiological fluid through a porous non-uniform channel under low-Reynolds number and long wavelength, the flow characteristics of incompressible, viscous, electrically conducting micropolar fluid has been derived analytically. Bhattacharyya et al. [19] investigated the combined influence between magnetic field and its dissipation on convective heat and mass transfer of a viscous chemically reacting fluid in the concentric cylindrical annulus, the inner cylinder is maintained at constant temperature and concentration while, the outer cylinder is maintained under constant heat flux. Sheremet et al. [20] investigated numerically the natural convection of a micropolar fluid in the triangular cavity, the system of micropolar equations of dimensionless stream function, vorticity and temperature have been solved by the finite difference method of the second-order accuracy under the initial and boundary conditions. Gupta et al. [21] studied numerically for the steady mixed convection (MHD) flow of micropolar fluid over a porous shrinking sheet, they assumed that the magnetic field and velocity of shrinking sheet are varied as a power functions of the distance from the origin. All the above studies are on the fluid flow and heat transfer under the magnetic field effect.

Our effort in this paper is dedicated to study the behavior of the micropolar fluid under the effect of *induced magnetic fields*. The micropolar fluid is flowing through a rectangular duct subjected to an applied magnetic field with the inclusion of both the effects of the induced magnetic field and the slip conditions.

## 2 The physical problem and mathematical modeling

*The governing equations*

The governing equations in a dimension of the flow of an incompressible and electrically conducting micropolar fluid [2] with the induced magnetic field and without of body force and body couple are

*Continuity equation*

*Momentum equation*

*Microrotation equation*

*Energy equation*

*Induction equation*

*Momentum equation*

*Microrotation equation in X-direction*

*Microrotation equation in Y-direction*

*Energy equation*

## 3 Results and discussion

## 4 Conclusion

The velocity increases with the increase Reynolds number and slip flow parameter, but decreases with the increase in magnetic Reynolds, Hartmann and coupling numbers. It is not affected by the slip convection parameter and Brinkman number.

The induced magnetic field increases with the increase of Reynolds and magnetic Reynolds numbers, but decreases with the increase of Hartmann, coupling numbers and the slip flow parameter. It is not affected by the slip convection parameter and Brinkman number.

The microrotation in x-axis increases with the increases of Reynolds number, but decreases with the increase in magnetic Reynolds, Hartmann, coupling numbers and the slip flow parameter. It is not affected by the slip convection parameter and Brinkman number.

The microrotation in the y-axis increase with the increases of Reynolds number and the slip flow parameter, but decreases with the increase of magnetic Reynolds, Hartmann and coupling numbers. It is not affected by the slip convection parameter and Brinkman number.

The temperature increases with the increase of Reynolds, Brinkman numbers and the slip convection parameter, but decreases with the increase of magnetic Reynolds, Hartmann, coupling numbers and the slip flow parameter.

## Notes

### Acknowledgements

The authors are grateful to the anonymous referee for his suggestions, which have greatly improved the presentation of the paper.

### Authors’ contribution

The author has made an equal contribution. The author read and approved the final manuscript.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no competing interests.

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