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

Consider the steady laminar and incompressible flow of an electrically conducting micropolar fluid moving through a rectangular duct at a constant pressure gradient \(\partial P/\partial Z\) under transverse external magnetic flux density \(B_{0}\) applied in the X-direction. The fluid flow is generated due to the pressure gradient along the Z-direction of the duct. Choosing the coordinate system along the Z-axis to be the length of the duct and (X and Y) axis to be the cross-sectional area of the duct, as in Fig. 1. Each wall of the duct is kept at a uniform temperature (\(T_{w}\)). Assume that the slip flow condition for the velocity in the walls is parallel to X-axis and the slip convective conditions are applied to the walls, which is parallel to Y-axis. Assuming that the velocity vector of the micropolar fluid is \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} = V_{z } \left( {X,Y} \right)\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {k}\) and the microrotation vector \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } = \omega_{x} \left( {X,Y} \right)\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {i} + \omega_{y } \left( {X,Y} \right)\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {j}\). The fluid has a large magnetic Reynolds number (\(R_{m}\)) so that the induced magnetic field is produced significantly inside the fluid.

Fig. 1
figure 1

Geometrical model of the duct flow

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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

$$\frac{\partial \rho }{\partial t} + \nabla \cdot (\rho \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} ) = 0$$
(1)

Momentum equation

$$\rho \frac{{D\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} }}{Dt} = - \nabla P - \left( {\mu + k} \right)\nabla \times \nabla \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} + k\nabla \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {J} \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B}$$
(2)

Microrotation equation

$$\rho j^{*} \frac{{D\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } }}{Dt} = k\nabla \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} - 2k \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } - \gamma \nabla \times \nabla \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } + \left( {\tilde{\alpha } + \beta + \gamma } \right)\nabla \left( {\nabla \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } } \right)$$
(3)

Energy equation

$$\begin{aligned} \rho C_{p} \frac{DT}{Dt} & = K_{f} \nabla^{2} T + \lambda \left( {\nabla \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} } \right)^{2} + 2\mu \left( {D:D} \right) + 4k\left( {\frac{1}{2}\nabla \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } } \right)^{2} + \tilde{\alpha }\left( {\nabla \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } } \right)^{2} \\ & \quad + \gamma \left( {\nabla \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } } \right):\left( {\nabla \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } } \right) + \beta \left( {\nabla \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } } \right):\left( {\nabla \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\omega } } \right)^{T} + \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {J}^{ 2} }}{\sigma } \\ \end{aligned}$$
(4)

where \(\rho\), \(P\) and \(T\) are the fluid density, fluid pressure and temperature, respectively with \(j^{*}\),\(K_{f}\), \(C_{p}\), \(\sigma\) are the microgyration parameter, thermal conductivity, specific heat at constant pressure and the electrical conductivity, respectively, while \(D\) is the deformation tensor \(\left(D = 0.5\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{V_{ij} }} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{V_{ji} }} } \right)\right)\), \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{ J}\) is the electric current density vector, \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B}\) is the total magnetic field \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B} = \left( {B_{0} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {i} + B_{{z \left( {X,Y} \right)}} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {k} } \right)\) and \(B_{z}\) is the induced magnetic field and \(\mu , k, \tilde{\alpha }, \beta\) and \(\gamma\) are the material constants (viscosity coefficients) which are satisfying the following inequalities;

$$k \ge 0,\quad 2\mu + k \ge 0, \quad 3\tilde{\alpha } + \beta + \gamma \ge 0, \quad \gamma \ge \left| \beta \right|$$
(5)

The induction equation, Ohm’s law and Maxwell’s equations related to the magnetic field \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B}\), electric current density \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {J}\) and electric field density vector \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E}\) are given by;

$$\frac{{\partial \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B} }}{\partial t} = \nabla \times \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B} } \right) + \frac{1}{{\sigma \mu_{m} }}\nabla^{2} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B} ,\quad \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {J} = \sigma \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V} \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B} } \right),\quad \nabla \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} = 0,\quad \nabla \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B} = \mu_{m} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {J} , \quad \nabla \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} = \frac{{\partial \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B} }}{\partial t},\quad \nabla \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B} = 0$$
(6)

where \(\upmu_{\text{m}}\) is the magnetic permeability. Due to that the system does not apply polarization voltage (each wall is an electrical insulator), the electric field density vector \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{E }\) is neglected. Under the above assumptions, the MHD equations become

Induction equation

$$B_{0} \left[ {\frac{{\partial V_{z} }}{\partial X}} \right] + \frac{1}{{\sigma \mu_{m} }}\left[ {\frac{{\partial^{2} B_{z} }}{{\partial X^{2} }} + \frac{{\partial^{2} B_{z} }}{{\partial Y^{2} }}} \right] = 0,$$
(7)

Momentum equation

$$- \left[ {\frac{\partial P}{\partial Z}} \right] + \left( {\mu + k} \right)\left[ {\frac{{\partial^{2} V_{z} }}{{\partial X^{2} }} + \frac{{\partial^{2} V_{z} }}{{\partial Y^{2} }}} \right] + k\left[ {\frac{{\partial \omega_{y} }}{\partial X} - \frac{{\partial \omega_{x} }}{\partial Y}} \right] + \frac{{B_{0} }}{{\mu_{m} }}\left[ {\frac{{\partial B_{z} }}{\partial X}} \right] = 0,$$
(8)

Microrotation equation in X-direction

$$- 2k \omega_{x} + k\left[ {\frac{{\partial V_{z} }}{\partial Y}} \right] - \gamma \frac{\partial }{\partial Y}\left[ {\frac{{\partial \omega_{y} }}{\partial X} - \frac{{\partial \omega_{x} }}{\partial Y}} \right] + \left( {\tilde{\alpha } + \beta + \gamma } \right)\frac{\partial }{\partial X}\left[ {\frac{{\partial \omega_{x} }}{\partial X} + \frac{{\partial \omega_{y} }}{\partial Y}} \right] = 0,$$
(9)

Microrotation equation in Y-direction

$$- 2k \omega_{y} - k\left[ {\frac{{\partial V_{z} }}{\partial X}} \right] + \gamma \frac{\partial }{\partial X}\left[ {\frac{{\partial \omega_{y} }}{\partial X} - \frac{{\partial \omega_{x} }}{\partial Y}} \right] + \left( {\tilde{\alpha } + \beta + \gamma } \right)\frac{\partial }{\partial Y}\left[ {\frac{{\partial \omega_{x} }}{\partial X} + \frac{{\partial \omega_{y} }}{\partial Y}} \right] = 0,$$
(10)

Energy equation

$$\begin{aligned} & k_{f} \left[ {\frac{{\partial^{2} T}}{{\partial X^{2} }} + \frac{{\partial^{2} T}}{{\partial Y^{2} }}} \right] + \left( {\mu + k} \right)\left[ {\left( {\frac{{\partial V_{z} }}{\partial X}} \right)^{2} + \left( {\frac{{\partial V_{z} }}{\partial Y}} \right)^{2} } \right] + \tilde{\alpha }\left[ {\frac{{\partial \omega_{x} }}{\partial X} + \frac{{\partial \omega_{y} }}{\partial Y}} \right]^{2} + 2k\left[ {\omega_{x}^{2} + \omega_{y}^{2} - \omega_{x} \frac{{\partial V_{z} }}{\partial Y} + \omega_{y} \frac{{\partial V_{z} }}{\partial X}} \right] \\ & \quad + \gamma \left[ {\left( {\frac{{\partial \omega_{x} }}{\partial X}} \right)^{2} + \left( {\frac{{\partial \omega_{x} }}{\partial Y}} \right)^{2} + \left( {\frac{{\partial \omega_{y} }}{\partial X}} \right)^{2} + \left( {\frac{{\partial \omega_{y} }}{\partial Y}} \right)^{2} } \right] + \beta \left[ {\left( {\frac{{\partial \omega_{x} }}{\partial X}} \right)^{2} + \left( {\frac{{\partial \omega_{y} }}{\partial Y}} \right)^{2} + 2\left( {\frac{{\partial \omega_{x} }}{\partial Y}\frac{{\partial \omega_{y} }}{\partial X}} \right)} \right] \\ & \quad + \frac{1}{{\sigma \mu_{m}^{2} }}\left( {\frac{{\partial^{2} B_{z} }}{{\partial X^{2} }} + \frac{{\partial^{2} B_{z} }}{{\partial Y^{2} }}} \right) = 0. \\ \end{aligned}$$
(11)

The boundary conditions are classified as follows [22];

$$B = 0,\quad V_{z} = 0,\quad \omega_{x} = 0,\quad \omega_{y} = 0\quad at\quad X = 0 \quad and\quad X = a;$$
(12a)
$$k_{f} \frac{dT}{dX} - h\left( {T - T_{f} } \right) = 0\quad at\quad X = 0;$$
(12b)
$$k_{f} \frac{dT}{dX} + h\left( {T - T_{w} } \right) = 0\quad at\quad X = a;$$
(12c)
$$B = 0,\quad T = T_{w} ,\quad \omega_{x} = 0,\quad \omega_{y} = 0,\quad at\quad Y = 0\quad and\quad Y = b;$$
(12d)
$$V_{z} = \alpha^{*} \frac{{dV_{z} }}{dY}\quad at\quad Y = 0;$$
(12e)
$$V_{z} = \alpha^{*} \frac{{dV_{z} }}{dY}\quad at\quad Y = b;$$
(12f)

where \(T_{f}\) is the initial temperature of the fluid, while \(\alpha^{*}\) and \(h\) are the slip flow coefficient and the slip convection coefficient, respectively. The dimensionless variables are introduced as follows;

$$x = \frac{X}{a},\quad y = \frac{Y}{a},\quad v = \frac{{V_{z} }}{{V_{0} }},\quad \omega_{1} = \frac{{\omega_{x} a}}{{V_{0} }},\quad \omega_{2} = \frac{{\omega_{y} a}}{{V_{0} }},\quad \theta = \frac{{T - T_{w} }}{{T_{f} - T_{w} }},\quad B = \frac{{B_{z} }}{{B_{0} }},\quad \frac{\partial P}{\partial Z} = - \frac{{\rho V_{0}^{2} }}{a}P_{l} .$$
(13)

Substituting the Eqs. (13) into Eqs. (7)–(11), we get;

$$\left[ {\frac{\partial v}{{\partial {\text{x}}}}} \right] + \frac{1}{{R_{m} }}\left[ {\frac{{\partial^{2} B}}{{\partial x^{2} }} + \frac{{\partial^{2} B}}{{\partial y^{2} }}} \right] = 0$$
(14)
$$R_{e} P_{l} + \left( {\frac{1}{1 - N}} \right)\left[ {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }}} \right] + \left( {\frac{N}{1 - N}} \right)\left[ {\frac{{\partial \omega_{2} }}{\partial x} - \frac{{\partial \omega_{1} }}{\partial y}} \right] + \frac{{Ha^{2} }}{{R_{m} }}\left[ {\frac{\partial B}{\partial x}} \right] = 0$$
(15)
$$- \omega_{1} + \left( {\frac{1}{2}} \right)\left[ {\frac{\partial v}{\partial y}} \right] + \left( {\frac{2 - N}{{2 m^{2} }}} \right)\left[ {\frac{{\partial^{2} \omega_{1} }}{{\partial y^{2} }}} \right] + \left( {\frac{1}{{l^{2} }}} \right)\left[ {\frac{{\partial^{2} \omega_{1} }}{{\partial x^{2} }}} \right] + \left( {\frac{{2 m^{2} - l^{2} \left( {2 - N} \right)}}{{2 m^{2} l^{2} }}} \right)\left[ {\frac{{\partial^{2} \omega_{2} }}{\partial x \partial y}} \right] = 0$$
(16)
$$- \omega_{2} - \left( {\frac{1}{2}} \right)\left[ {\frac{\partial v}{\partial x}} \right] + \left( {\frac{2 - N}{{2 m^{2} }}} \right)\left[ {\frac{{\partial^{2} \omega_{2} }}{{\partial x^{2} }}} \right] + \left( {\frac{1}{{l^{2} }}} \right)\left[ {\frac{{\partial^{2} \omega_{2} }}{{\partial y^{2} }}} \right] + \left( {\frac{{2 m^{2} - l^{2} \left( {2 - N} \right)}}{{2 m^{2} l^{2} }}} \right)\left[ {\frac{{\partial^{2} \omega_{1} }}{\partial x \partial y}} \right] = 0$$
(17)
$$\begin{aligned} & \left[ {\frac{{\partial^{2} \theta }}{{\partial x^{2} }} + \frac{{\partial^{2} \theta }}{{\partial y^{2} }}} \right] + Br\left[ {\left( {\frac{1}{1 - N}} \right)\left[ {\left( {\frac{\partial v}{\partial x}} \right)^{2} + \left( {\frac{\partial v}{\partial y}} \right)^{2} } \right]} \right. + \left( {\frac{2 N}{1 - N}} \right)\left[ {\omega_{1}^{2} + \omega_{2}^{2} - \omega_{1} \frac{\partial v}{\partial y} + \omega_{2} \frac{\partial v}{\partial x}} \right] \\ & \quad + A\left[ {\frac{{\partial \omega_{1} }}{\partial x} + \frac{{\partial \omega_{2} }}{\partial y}} \right]^{2} + \left( {\frac{{N\left( {2 - N} \right)}}{{m^{2} \left( {1 - N} \right)}}} \right)\left[ {\left( {\frac{{\partial \omega_{1} }}{\partial x}} \right)^{2} + \left( {\frac{{\partial \omega_{1} }}{\partial y}} \right)^{2} + \left( {\frac{{\partial \omega_{2} }}{\partial x}} \right)^{2} + \left( {\frac{{\partial \omega_{2} }}{\partial y}} \right)^{2} } \right] \\ & \quad + \left. {C\left[ {\left( {\frac{{\partial \omega_{1} }}{\partial x}} \right)^{2} + \left( {\frac{{\partial \omega_{2} }}{\partial y}} \right)^{2} + 2\left( {\frac{{\partial \omega_{1} }}{\partial y}\frac{{\partial \omega_{2} }}{\partial x}} \right)} \right]} \right] + \frac{{Ha^{2} Br}}{{R_{m}^{2} }}\left( {\frac{{\partial^{2} B}}{{\partial x^{2} }} + \frac{{\partial^{2} B}}{{\partial y^{2} }}} \right) = 0 \\ \end{aligned}$$
(18)

where \(m^{2} = \frac{{a^{2} k\left( {2\mu + k} \right)}}{{\gamma \left( {\mu + k} \right)}}, A = \frac{{\tilde{\alpha }}}{{\mu a^{2} }}\) and \(C = \frac{\beta }{{\mu a^{2} }}\) are the micropolar parameters, \(N = \frac{k}{\mu + k}\) is the coupling number, \(l^{2} = \frac{{2a^{2} k}}{{\tilde{\alpha } + \beta + \gamma }}\) is dimensionless parameter, \(R_{e} = \rho V_{0} a/\mu\) is the Reynolds number, \(R_{m} = \sigma \mu_{m} V_{0} a\) is the magnetic Reynolds number, \(Ha = B_{0} a\sqrt {\sigma /\mu } = \sqrt {R_{e} R_{m} /Al^{2} }\) is the Hartmann number, \(Al = V_{0} /\sqrt {B_{0}^{2} /\mu_{m} \rho }\) is the Alfven number, \(Br = \frac{{\mu V_{0}^{2} }}{{k_{f} \left( {T_{f} - T_{w} } \right)}}\) is the Brinkman number and the inverted pressure gradient \(P_{l} = 1/\left( {\frac{\partial P}{\partial Z}} \right)\).

The boundary conditions in dimensionless form are.

$$B = 0,\quad v = 0, \quad \omega_{1} = 0,\quad \omega_{2} = 0\quad {\text{at}}\quad x = 0 \quad and\quad x = 1;$$
(19a)
$$\frac{d\theta }{dX} - Bi\left( {\theta - 1} \right) = 0\quad at\quad x = 0;$$
(19b)
$$\frac{d\theta }{dX} - Bi \theta = 0\quad at\quad x = 1;$$
(19c)
$$B = 0,\quad \theta = 0,\quad \omega_{1} = 0, \quad \omega_{2} = 0,\quad at\quad y = 0\quad and\quad y = yo;$$
(19d)
$$v = \alpha \frac{dv}{dy}\quad at\quad y = 0,$$
(19e)
$$v = \alpha \frac{dv}{dy}\quad at\quad y = yo;$$
(19f)

where \(yo = \frac{b}{a}\) is the aspect ratio, \(\alpha = \frac{{\alpha^{*} }}{a}\) is the slip flow parameter and \(Bi = \frac{a h}{{k_{f} }}\) is the slip convection parameter.

3 Results and discussion

In this study, the mass and heat transfer of the MHD of a micropolar fluid through a rectangular duct have been solved numerically by the finite difference method with 41 × 41 mesh points in both directions. The dimensionless temperature Eq. (18) and the coupled induction, velocity and microrotation Eqs. (14)–(17) are solved subject to the boundary conditions (19a)–(19f). The effect of various parameters \(R_{m}\), \(R_{e}\), \(Ha\), \(N\), \(\alpha\), \(Bi\) and \(Br\) are presented graphically. Figure 2 show the profiles of the velocity, magnetic field, microrotation in x- and y-directions and temperature at \(R_{m}\) = 10, \(R_{e}\) = 4, \(Ha\) = 5, \(N\) = 0.4, \(\alpha\) = 0.1, \(Bi\) = 1, \(Br\) = 1, \(P_{l}\) = 1, \(m\) = 1, \(l\) = 0.5, \(A\) = 1 and \(C\) = 0.1. It is clear from Fig. 2a that the maximum magnitude of the velocity at the center and decreases at the boundaries of the duct [13]. In Fig. 2b, it is observed that the maximum value of the induced magnetic field inside the fluid between the vertical boundaries and the center of the duct, and equal zero at both of the center and the boundaries of the rectangular duct.

Fig. 2
figure 2

3-Dimensional profiles of a velocity, b magnetic field, c microrotation in x-direction, d microrotation in y-direction, e temperature

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It is clear from Fig. 2c that the maximum values of the microrotation (\(\omega_{1}\)) are situated between the horizontal boundaries and the center of the duct, and it is zero at each wall of the duct. But at the center, the value is approaching zero. Also, it is observed that the rotation in the region below the center of the duct in the positive x-direction. But above the center, it rotates in the negative x-direction, while the coupling between the rotational and Newtonian viscosity is represented by the coupling number \(N\). The vanishing to zero in the figures of microrotation means that the fluid is non-polar and the micropolarity is lost (\(k\) → 0 and \(N\) → 0). In Fig. 2d it is shown that the values of microrotation(\(\omega_{2}\)) are maximum between the vertical walls and the center of the duct, but at the center, the value is approaching zero and tends to zero at the walls of the duct. Also, it is observed that the fluid rotates in the negative y-direction at the left region of the duct. But in the right region, it rotates about the positive y-direction [13, 16]. Figure 2e presents the maximum magnitude of temperature along x-axis at the center of the left vertical wall and decreasing at the center of the duct, it increases by a smaller rate when approaching to the center of the right vertical wall, while the maximum value of temperature along y-axis at the center and the rate is decreased when it is near to the horizontal walls. The profiles of velocity in Fig. 3a, magnetic microrotations in Fig. 3c, d and temperatures Fig. 3e, f decrease with the increase of the magnetic Reynolds number in the entire rectangular duct, while the profiles of the induced magnetic field in Fig. 3b increase when the increase of the magnetic Reynolds number. Figure 4 show; the velocity (a), microrotations (c) and (d), induced magnetic field (b) and temperature profiles (e) and (f) are increased with increase Reynolds number.

Fig. 3
figure 3

The profiles of a velocity, b magnetic field, c microrotation in y-direction, d microrotation in x-direction, e, f temperatures in x and y with various values of \(R_{m}\) at the centerline of the duct

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Fig. 4
figure 4

The profiles of a velocity, b magnetic field, c microrotation in x-direction, d microrotation in y-direction, e, f temperatures in x and y with various values of \(R_{e}\) at the centerline of the duct

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Figure 5 show; the velocity (a), microrotations (b) and (c) decrease with the increase of the Hartmann number but, the induced magnetic field is increasing the delay the flow (decrease the velocity) compared with that occurred in [13]. Also, the induced magnetic field in Fig. 5f and temperature in Fig. 5d, e decrease with the increase of the Hartmann number. Figure 6 show; the velocity (a), microrotations (b) and (c) decrease with the increase of the coupling number but, the induced magnetic field causes to increase the delay the flow (decrease the velocity) compared with that occurred in [13]. Also, the induced magnetic field in Fig. 6f and temperature in Fig. 6d, e decrease with the increase in the coupling number. Figure 7a, d show the velocity and microrotation in y-direction are increased with the increase of the slip flow, but Fig. 7b, c, e show the induced magnetic field, microrotation in x-direction and temperature decrease with increase the slip flow.

Fig. 5
figure 5

The profiles of a velocity, b Microrotation in x-direction, c microrotation in y-direction, d, e temperatures in x and y and f magnetic fields with various values of \(Ha\) at the centerline of the duct

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Fig. 6
figure 6

The profiles of a velocity, b Microrotation in x-direction, c microrotation in y-direction, d, e temperatures in x and y and f magnetic fields with various values of \(N\) at the centerline of the duct

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Fig. 7
figure 7

The profiles of a velocity, b magnetic field, c microrotation in x-direction, d microrotation in y-direction, e temperatures in y direction with various values of \(\alpha\) at the centerline of the duct

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Figure 8 present that the profile of temperature increases with increased the Brinkman number in both directions. It is clear that the Brinkman number does not exist in Eqs. (14), (15), (16) and (17) so that each of the velocity, microrotations and induced magnetic field do not enter into a discussion of Brinkman number. Figure 9 present that the profile of temperature increases with increase slip convection parameter. But does not exist in each Eqs. (19a), (19c) and (19d) so that each of the velocity, microrotations and induced magnetic field do not enter into a discussion of the slip convection parameter.

Fig. 8
figure 8

The profiles of a temperatures in x direction and b in y direction with various values of \(Br\) at the centerline of the duct

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Fig. 9
figure 9

The profiles of a temperatures in x-direction and b in y-direction with various values of \(Bi\) at the centerline of the duct

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4 Conclusion

In this article, we presented the effect induced magnetic field into the micropolar fluid. Numerical method is used to solve the MHD equations concerning the mass and heat transfers of the steady, incompressible micropolar fluid through a rectangular duct with the effect of the induced magnetic field and slip conditions. The analysis of the micropolar fluid flow and heat transfer have been conducted by the various values of the magnetic Reynolds, Reynolds, Hartmann, coupling and Brinkman numbers, slip flow and convection parameters. Based on the obtained results, we can conclude that:

  • 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.