Investigation of MHD micropolar flow between a stationary and a rotating disc: Keller-box solution

Abstract

In the present investigation, magnetohydrodynamic (MHD) micropolar fluid flow between a stationary disc and a rotating disc with a constant angular velocity is considered. The study investigates the effect of magnetic field and microrotation structure on the flow characteristics. The governing equations of motion are transformed to a system of nonlinear ordinary differential equations (ODEs) in dimensionless form using Von Karman’s similarity transformations. An algorithm based on implicit finite difference method-Keller-box Scheme is employed to solve the resulting similarity equations for various pertinent parameters. Numerical solutions of velocity profiles, pressure gradient and microrotation profiles are discussed, and presented through tables and graphs for various Magnetic parameter. Comparisons are made between the obtained results and previously reported findings in the literature. The successful validation against existing literature supports the effectiveness of the methodology employed in this investigation.

Keywords:

• Magnetohydrodynamics
• micropolar fluids
• finite difference methods
• Keller-box solution
• rotating disc flow
• micropolar fluids
• streamlines

REVIEWING EDITOR:

• D T Pham, University of Birmingham, United Kingdom

SUBJECTS:

• Mathematics & Statistics for Engineers
• Computational Mechanics
• Fluid Mechanics

Introduction

The fundamental theory of micropolar fluid was introduced by Eringen (1966) who established the conservative laws of fluids with microstructure, which cannot be adequately described by the classical Navier–Stokes model of hydrodynamics. The essence of the theory lies in the extension of the constituted equations for Newtonian flow so that more complex fluids which are often made up of materials, possess an internal structure. Given the natural structure of the constituents in these fluids, micropolar fluids serve as an excellent mathematical model for flow through capillaries and microchannels, turbulent shear flow, blood flows, lubricants and porous media. An extensive survey of the literature on micropolar fluids and a useful account of the theory can be found in the book by Lukaszewicz (Lukaszewicz, 1998) and in the articles by Ariman et al. (1968).

The problem of fluid flow through parallel discs/plates has established significant area of study in fluid mechanics. Such flows have a wide range of applications in lubrication, heat and mass transfer, biomechanics, rotating machinery and many other areas. Rasmussen (1970) investigated the problem of steady viscous flow between two porous discs subjected to uniform injection/suction at both discs. Later Elcrat (1976) studied and established the theorems of existence and uniqueness for non-rotational fluid motion between fixed porous discs under uniform injection/suction. Anwar and Guram (1980) investigated steady incompressible micropolar fluid flow between a rotating and stationary disc of infinite length using a numerical method. Later, the problem of the laminar incompressible flow of a micropolar fluid induced by a rotating disc with uniform suction and injection was solved numerically by Guram and Anwar (1981). Desseaux and Kelson (2000) presented the two-dimensional flow of micropolar fluid in a porous channel. Solutions of the equations were evaluated using the perturbation technique. They found the comparison of solutions with velocity profile and variation of wall shear stress. Further Kelson et al. (2003) obtained the solution for micropolar fluid flow in a porous channel with high mass transfer using the perturbation method and approximate method, which was propelled by suction or injection at the channel walls.

Rashidi et al. (2016) evaluated the combined influence of nanoparticles and magnetic field on a micropolar fluid flow between two parallel porous plates with uniform blowing. Two types of nanoparticles were employed to produce the combination of micropolar nanofluids for this study. The problem of magnetohydrodynamic (MHD) micropolar nanofluid flow due to a rotating disk with partial slip effects was investigated numerically by Ramzan et al. (2017). Bhat and Katagi (2020) considered the problem of micropolar fluid flow between porous and non-porous discs with slip conditions at the porous boundary and analysed the influence of slip conditions on velocity and microrotation profiles. Further, they extended their study by incorporating heat exchange and an externally applied magnetic field into the problem. Ibrahim and Zemedu (2021) considered the convection flow of a micropolar nanofluid through a non-isothermal rotating disc and analysed the problem with a magnetic field. Recently, Ganesh and Sridhar (2021) analysed the flow of Casson-type nanofluid in a porous medium over a nonlinear stretching surface with Darcy–Forchheimer relation, chemical reaction, thermophoresis and Brownian diffusion effects.

Mishra et al. extensively worked in the field of Micropolar nanofluids (Mishra et al., 2021; Mohanty et al., 2021; Mathur et al., 2021). Alahmadi et al. (2023) conducted a numerical study on the steady laminar incompressible micropolar nanofluid flow in a channel with stretching and shrinking walls in the presence of a magnetic field and thermal radiation. They employed response surface methodology (RSM) and sensitivity analysis to optimise the analysis. Shamshuddin along with his collaborators (Ram et al., 2023; Shamshuddin et al., 2023), studied the mixed convection micropolar fluid flow with various pertinent parameters.

In the present work, the problem of steady, incompressible micropolar fluid flow between stationary and rotating discs (Anwar and Guram, 1980) is considered. The present work is an attempt to study the effect of a magnetic field on the velocity field, pressure gradient and microrotation. We systematically use an efficient second-order finite difference implicit scheme, the Keller-box method, to solve the governing equations. Among the finite difference schemes, we chose the Keller-box method due to its ability to overcome the difficulty of breakdown for strong nonlinearity problems. Moreover, it exhibits a faster rate of convergence compared to other methods. So, we opted for this method for numerical simulation.

Mathematical formulation

Consider a steady flow of an incompressible micropolar fluid between two infinite co-axial discs in the presence of a magnetic field, which are placed at a finite distance $\mathrm{ }h.$ It is assumed that flow is generated by an infinite rotating disc at the lower region. The lower disc’s rotational speed, the separation between the discs $\left(h\right)$ and the characteristics of the fluid will all have an effect on the velocity field of the flow between the co-axial discs. The bottom disc revolves at constant angular velocity $\mathrm{\omega }.$ Assume $u,v,w$ to be the velocity components and ${\nu }_1,{\nu }_2,{\nu }_3$ to be the micro-rotation components in the $r,\mathrm{\theta },z$ directions, respectively, in cylindrical polar coordinates. The material constants of the micropolar fluid are considered independent of position and body forces, body couples and induced magnetic fields are neglected. The magnetic field with strength $B_0$ is applied normal to the fluid. The geometry and coordinates of the problem is shown in the Figure 1.

Figure 1. Geometry of the problem.

The governing equations for the MHD incompressible steady flow of micropolar fluid are given by Anwar and Guram (1980), (1) $$\nabla \cdot \mathit{u}=0$$(1) (2) $$\left(\lambda +2\mu +\kappa \right)\nabla \left(\nabla \cdot \mathit{u}\right)-\left(\mu +\kappa \right)\nabla \times \nabla \times \mathit{u}+\kappa \nabla \times \nu -\nabla p+\mathit{J}\times \mathit{B}=\rho \dot{\mathit{u}}$$(2) (3) $$\left(\alpha +\beta +\gamma \right)\nabla \left(\nabla \cdot \nu \right)-\gamma \left(\nabla \times \nabla \times \nu \right)+\kappa \nabla \times \mathit{u}-2\kappa \nu =\rho j\dot{\nu }$$(3)

In component form, this equation can be written as, (4) $$\frac{1}{r}\frac{\partial \left(\mathit{\text{ru}}\right)}{\partial r}+\frac{\partial w}{\partial z}=0$$(4) (5) $$\left(\mathrm{\mu }+\mathrm{\kappa }\right)\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{{\partial }^{2}u}{\partial z^2}-\frac{u}{r^2}\right)‐\mathrm{\kappa }\frac{\partial {\mathrm{\nu }}_2}{\partial z}-\frac{\partial p}{\partial r}-{\mathrm{\sigma }}_e{B}_0^{2}u=\mathrm{\rho }\left(w\frac{\partial u}{\partial z}-\frac{v^2}{r}+u\frac{\partial u}{\partial r}\right)$$(5) (6) $$\left(\mathrm{\mu }+\mathrm{\kappa }\right)\left(\frac{{\partial }^{2}v}{\partial r^2}+\frac{1}{r}\frac{\partial v}{\partial r}+\frac{{\partial }^{2}v}{\partial z^2}-\frac{v}{r^2}\right)‐\mathrm{\kappa }\left(\frac{\partial {\mathrm{\nu }}_1}{\partial z}-\frac{\partial {\mathrm{\nu }}_3}{\partial r}\right)-{\mathrm{\sigma }}_e{B}_0^{2}v=\mathrm{\rho }\left(w\frac{\partial v}{\partial z}-\frac{\mathit{\text{uv}}}{r}+u\frac{\partial v}{\partial r}\right)$$(6) (7) $$\left(\mathrm{\mu }+\mathrm{\kappa }\right)\left(\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\frac{\partial w}{\partial r}+\frac{{\partial }^{2}w}{\partial z^2}\right)-\frac{\mathrm{\kappa }}{r}\frac{\partial \left(r{\mathrm{\nu }}_2\right)}{\partial r}-\frac{\partial p}{\partial z}=\mathrm{\rho }\left(u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}\right)$$(7) (8) $$\left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\right)\frac{\partial }{\partial r}\left(\frac{\partial {\mathrm{\nu }}_1}{\partial r}+\frac{{\mathrm{\nu }}_1}{r}+\frac{\partial {\mathrm{\nu }}_3}{\partial z}\right)‐\mathrm{\gamma }\frac{\partial }{\partial z}\left(\frac{\partial {\mathrm{\nu }}_3}{\partial r}-\frac{\partial {\mathrm{\nu }}_1}{\partial z}\right)‐\mathrm{\kappa }\frac{\partial v}{\partial z}-2\mathrm{\kappa }{\mathrm{\nu }}_1=\mathrm{\rho }j\left(u\frac{\partial {\mathrm{\nu }}_1}{\partial r}-\frac{1}{r}v{\mathrm{\nu }}_2+w\frac{\partial {\mathrm{\nu }}_1}{\partial z}\right)$$(8) (9) $$\mathrm{\gamma }\left[\frac{\partial }{\partial r}\left(\frac{\partial {\mathrm{\nu }}_2}{\partial r}+\frac{{\mathrm{\nu }}_2}{r}\right)+\frac{{\partial }^2{\mathrm{\nu }}_2}{\partial z^2}\right]+\mathrm{\kappa }\left(\frac{\partial u}{\partial z}-\frac{\partial w}{\partial r}\right)-2\mathrm{\kappa }{\mathrm{\nu }}_2=\mathrm{\rho }j\left(u\frac{\partial {\mathrm{\nu }}_2}{\partial r}-\frac{1}{r}v{\mathrm{\nu }}_1+w\frac{\partial {\mathrm{\nu }}_2}{\partial z}\right)$$(9) (10) $$\left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }\right)\frac{\partial }{\partial z}\left(\frac{\partial {\mathrm{\nu }}_1}{\partial r}+\frac{{\mathrm{\nu }}_1}{r}+\frac{\partial {\mathrm{\nu }}_3}{\partial z}\right)‐\mathrm{\gamma }\frac{1}{r}\frac{\partial }{\partial r}\left[r\left(\frac{\partial {\mathrm{\nu }}_1}{\partial z}-\frac{\partial {\mathrm{\nu }}_3}{\partial r}\right)\right]-\frac{\mathrm{\kappa }}{r}\frac{\partial }{\partial r}\left(\mathit{\text{rv}}\right)-2\mathrm{\kappa }{\mathrm{\nu }}_3=\mathrm{\rho }j\left(u\frac{\partial {\mathrm{\nu }}_3}{\partial r}+w\frac{\partial {\mathrm{\nu }}_3}{\partial z}\right)$$(10) with respect to boundary conditions, for (11) $$\mathrm{ }z=0;u=0,v=r\mathrm{\omega },w=0,{\mathrm{\nu }}_1={\mathrm{\nu }}_2={\mathrm{\nu }}_3=0,p=0$$(11) for (12) $$z=h;u=0,v=0,{\mathrm{\nu }}_1={\mathrm{\nu }}_2={\mathrm{\nu }}_3=0.$$(12)

The similarity transformation for velocity, microrotation and pressure are given by Anwar and Guram (1980): (13) $$\begin{array}{l}u=r\mathrm{\omega }{F}_1\left(\mathrm{\eta }\right),v=r\mathrm{\omega }{F}_2\left(\mathrm{\eta }\right),w=h\mathrm{\omega }{F}_3\left(\mathrm{\eta }\right)\\ \begin{array}{l}{\mathrm{\nu }}_1=\frac{r\mathrm{\omega }}{h}{G}_1\left(\mathrm{\eta }\right),{\text{ν}}_2=\frac{r\mathrm{\omega }}{h}{G}_2\left(\mathrm{\eta }\right),{\mathrm{\nu }}_3=\mathrm{\omega }{G}_3\left(\mathrm{\eta }\right)\\ p=p\left(z\right)=-\left(\mathrm{\mu }+\mathrm{\kappa }\right)\mathrm{\omega }P\left(\mathrm{\eta }\right)\end{array}\end{array}$$(13) where $\mathrm{\eta }=\frac{z}{h}.$

We substitute Equation (13)(13) $$\begin{array}{l}u=r\mathrm{\omega }{F}_1\left(\mathrm{\eta }\right),v=r\mathrm{\omega }{F}_2\left(\mathrm{\eta }\right),w=h\mathrm{\omega }{F}_3\left(\mathrm{\eta }\right)\\ \begin{array}{l}{\mathrm{\nu }}_1=\frac{r\mathrm{\omega }}{h}{G}_1\left(\mathrm{\eta }\right),{\text{ν}}_2=\frac{r\mathrm{\omega }}{h}{G}_2\left(\mathrm{\eta }\right),{\mathrm{\nu }}_3=\mathrm{\omega }{G}_3\left(\mathrm{\eta }\right)\\ p=p\left(z\right)=-\left(\mathrm{\mu }+\mathrm{\kappa }\right)\mathrm{\omega }P\left(\mathrm{\eta }\right)\end{array}\end{array}$$(13) into Equations (4–10), the dimensionless form of governing equations can be obtained as, (14) $$2F_1+F_3^{\prime }=0$$(14) (15) $$F_1^{″}-{\mathrm{\eta }}_1{G}_2^{\prime }-M{F}_1={\mathrm{\eta }}_2\left(F_3{F}_1^{\prime }-F_2^2+F_1^2\right)$$(15) (16) $$F_2^{″}+{\mathrm{\eta }}_1{G}_1^{\prime }-M{F}_2={\mathrm{\eta }}_2\left(F_3{F}_2^{\prime }-2F_1{F}_2\right)$$(16) (17) $$F_3^{″}+2{\mathrm{\eta }}_1{G}_2+P^{\prime }={\mathrm{\eta }}_2{F}_3{F}_3^{\prime }$$(17) (18) $$G_1^{″}-{\mathrm{\eta }}_3{F}_2^{\prime }-2{\mathrm{\eta }}_3{G}_1={\mathrm{\eta }}_4\left(F_1{G}_1-F_2{G}_2+F_3{G}_1^{\prime }\right)$$(18) (19) $$G_2^{″}+{\mathrm{\eta }}_3{F}_1^{\prime }-2{\mathrm{\eta }}_3{G}_2={\mathrm{\eta }}_4\left(F_1{G}_2+F_2{G}_1+F_3{G}_2^{\prime }\right)$$(19) (20) $$G_3^{″}+2{\mathrm{\eta }}_5{G}_1^{\prime }+2{\mathrm{\eta }}_6\left(F_2-G_3\right)={\mathrm{\eta }}_7{F}_3{G}_3^{\prime }$$(20) where (21) $$\begin{array}{l}{\mathrm{\eta }}_1=\frac{\mathrm{\kappa }}{\mathrm{\mu }+\mathrm{\kappa }},{\mathrm{\eta }}_2=\frac{\mathrm{\rho }\mathrm{\omega }{h}^2}{\mathrm{\mu }+\mathrm{\kappa }},{\mathrm{\eta }}_3=\frac{\mathrm{\kappa }{h}^2}{\mathrm{\gamma }},{\mathrm{\eta }}_4=\frac{\mathrm{\rho }j\mathrm{\omega }{h}^2}{\mathrm{\gamma }}\\ {\mathrm{\eta }}_5=\frac{\mathrm{\alpha }+\mathrm{\beta }}{\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }},{\mathrm{\eta }}_6=\frac{\mathrm{\kappa }{h}^2}{\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }},{\mathrm{\eta }}_7=\frac{\mathrm{\rho }j\mathrm{\omega }{h}^2}{\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }},\\ \mathrm{ }M=\frac{{\mathrm{\sigma }}_0{B}_0^2{h}^2}{\mathrm{\mu }+\mathrm{\kappa }}\end{array}$$(21) are micropolar parameters and magnetic parameter, respectively. The boundary conditions of the above differential equations are, (22) $$\begin{array}{l}\text{at}\mathrm{\eta }=0,F_1=0,F_2=1,F_3=0,G_1=G_2=G_3=0,\\ P=0\\ \text{at}\mathrm{\eta }=0,F_1=F_2=0,G_1=G_2=G_3=0\end{array}$$(22)

Methodology

Equations (14)–(20) subject to boundary conditions (22) are integrated numerically using the Keller-box finite difference Method, which is an implicit second-order method. Keller (1971) introduced this implicit finite difference box method for the solution of diffusion problems, subsequently it has been applied to a broad class of problems. An overview of the Keller Box scheme for solving boundary layer flow problems can be found in the literature (Keller, 1975, 1978). In a recent study, Jamshed et al. (2022) employed the Keller-box method to solve highly nonlinear Navier–Stokes equations. The numerical solution is obtained using the following steps:

• Transform higher-order differential equations to a system of first-order differential equations.

• Convert system of first-order differential equations to system of difference equations using central differences.

• Solve the resulting system of algebraic equations by the N-R method.

• Write a system of linearised equations in the vector-matrix form.

• Solve the linear system of equations in the tri-diagonal form by decomposition method.

The system of non-linear coupled ordinary equations (14–20) subjected to boundary conditions (22) are reduced to first-order simultaneous equations as follows: (23) $$\begin{array}{l}{F}_1=f_1,F_2=f_2,F_3=f_3,G_1=g_1,G_2=g_2,G_3=g_3,\\ \mathrm{ }P=p\\ f_1^{\prime }=u_1,f_2^{\prime }=u_2,f_3^{\prime }=u_3,g_1^{\prime }=h_1,g_2^{\prime }=h_2,g_3^{\prime }=h_3\end{array}$$(23) we get, (24) $$2f_1+u_3=0$$(24) (25) $$u_1^{\prime }-{\mathrm{\eta }}_1{h}_2-M{f}_1={\mathrm{\eta }}_2\left(f_3{u}_1-f_2^2+f_1^2\right)$$(25) (26) $$u_2^{\prime }+{\mathrm{\eta }}_1{h}_1-M{f}_2={\mathrm{\eta }}_2\left(f_3{u}_2-2f_1{f}_2\right)$$(26) (27) $$u_3^{\prime }+2{\mathrm{\eta }}_1{g}_2+p^{\prime }={\mathrm{\eta }}_2{f}_3{u}_3$$(27) (28) $$h_1^{\prime }-{\mathrm{\eta }}_3{u}_2-2{\mathrm{\eta }}_3{g}_1={\mathrm{\eta }}_4\left(f_1{g}_1-f_2{g}_2+f_3{h}_1\right)$$(28) (29) $$h_2^{\prime }+{\mathrm{\eta }}_3{u}_1-2{\mathrm{\eta }}_3{g}_2={\mathrm{\eta }}_4\left(f_1{g}_2+f_2{g}_1+f_3{h}_2\right)$$(29) (30) $$h_3^{\prime }+2{\mathrm{\eta }}_5{h}_1+2{\mathrm{\eta }}_6\left(f_2-g_3\right)={\mathrm{\eta }}_7{f}_3{h}_3$$(30) and the boundary conditions are, (31) $$\begin{array}{c}\text{at}\eta =0,f_1=0,f_2=1,f_3=0,g_1=g_2=g_3=0,p=0\\ \text{at}\eta =1,f_1=f_2=0,g_1=g_2=g_3=0\end{array}$$(31)

We initiate the discretisation process by approximating ordinary differential equations (ODEs) (24–30) using finite differences and then linearised using the following iterates. (32) $$\begin{array}{l}{f}_{1,j}^{\left(i+1\right)}=f_{1,j}^{\left(i\right)}+\delta f_{1,j}^{\left(i\right)},f_{2,j}^{\left(i+1\right)}=f_{2,j}^{\left(i\right)}+\delta f_{2,j}^{\left(i\right)},f_{3,j}^{\left(i+1\right)}=f_{3,j}^{\left(i\right)}+\delta f_{3,j}^{\left(i\right)}\\ u_{1,j}^{\left(i+1\right)}=u_{1,j}^{\left(i\right)}+\delta u_{1,j}^{\left(i\right)},u_{2,j}^{\left(i+1\right)}=u_{2,j}^{\left(i\right)}+\delta u_{2,j}^{\left(i\right)},u_{3,j}^{\left(i+1\right)}=u_{3,j}^{\left(i\right)}+\delta u_{3,j}^{\left(i\right)}\\ g_{1,j}^{\left(i+1\right)}=g_{1,j}^{\left(i\right)}+\delta g_{1,j}^{\left(i\right)},g_{2,j}^{\left(i+1\right)}=g_{2,j}^{\left(i\right)}+\delta g_{2,j}^{\left(i\right)},g_{3,j}^{\left(i+1\right)}=g_{3,j}^{\left(i\right)}+\delta g_{3,j}^{\left(i\right)}\\ h_{1,j}^{\left(i+1\right)}=h_{1,j}^{\left(i\right)}+\delta h_{1,j}^{\left(i\right)},h_{2,j}^{\left(i+1\right)}=h_{2,j}^{\left(i\right)}+\delta h_{2,j}^{\left(i\right)},h_{3,j}^{\left(i+1\right)}=h_{3,j}^{\left(i\right)}+\delta h_{3,j}^{\left(i\right)},\\ p_j^{\left(i+1\right)}=p_j^{\left(i\right)}+\delta p_j^{\left(i\right)}\end{array}$$(32) Substituting the iterates (32) in difference equations and rewriting them, we get system of algebraic equations. (33) $$\delta f_{1,j}-\delta f_{1,j-1}-\frac{{\mathit{\hslash }}_j}{2}\left(\delta u_{1,j}+\delta u_{1,j-1}\right)={\left(r_1\right)}_j$$(33) (34) $$\delta f_{2,j}-\delta f_{2,j-1}-\frac{{\mathit{\hslash }}_j}{2}\left(\delta u_{2,j}+\delta u_{2,j-1}\right)={\left(r_2\right)}_j$$(34) (35) $$\delta f_{3,j}-\delta f_{3,j-1}-\frac{{\mathit{\hslash }}_j}{2}\left(\delta u_{3,j}+\delta u_{3,j-1}\right)={\left(r_3\right)}_j$$(35) (36) $$\delta g_{1,j}-\delta g_{1,j-1}-\frac{{\mathit{\hslash }}_j}{2}\left(\delta h_{1,j}+\delta h_{1,j-1}\right)={\left(r_4\right)}_j$$(36) (37) $$\delta g_{2,j}-\delta g_{2,j-1}-\frac{{\mathit{\hslash }}_j}{2}\left(\delta h_{2,j}+\delta h_{2,j-1}\right)={\left(r_5\right)}_j$$(37) (38) $$\delta g_{3,j}-\delta g_{3,j-1}-\frac{{\mathit{\hslash }}_j}{2}\left(\delta h_{3,j}+\delta h_{3,j-1}\right)={\left(r_6\right)}_j$$(38) (39) $${\left(a_1\right)}_j\mathrm{ }\delta f_{1,j}+{\left(a_2\right)}_j\delta f_{1,j-1}+{\left(a_3\right)}_j\delta f_{3,j}+{\left(a_4\right)}_j\delta f_{3,j-1}={\left(r_7\right)}_j$$(39) (40) $${\left(b_1\right)}_j\delta u_{1,j}+{\left(b_2\right)}_j\delta u_{1,j-1}+{\left(b_3\right)}_j\delta h_{2,j}+{\left(b_4\right)}_j\delta h_{2,j-1}+{\left(b_5\right)}_j\delta f_{3,j}+{\left(b_6\right)}_j\delta f_{3,j-1}+{\left(b_7\right)}_j\delta f_{2,j}+{\left(b_8\right)}_j\delta f_{2,j-1}+{\left(b_9\right)}_j\delta f_{1,j}+{\left(b_{10}\right)}_j\delta f_{1,j-1}={\left(r_8\right)}_j$$(40) (41) $${\left(c_1\right)}_j\delta u_{2,j}+{\left(c_2\right)}_j\delta u_{2,j-1}+{\left(c_3\right)}_j\delta h_{1,j}+{\left(c_4\right)}_j\delta h_{1,j-1}+{\left(c_5\right)}_j\delta f_{3,j}+{\left(c_6\right)}_j\delta f_{3,j-1}+{\left(c_7\right)}_j\delta f_{1,j}+{\left(c_8\right)}_j\delta f_{1,j-1}+{\left(c_9\right)}_j\delta f_{2,j}+{\left(c_{10}\right)}_j\delta f_{2,j-1}={\left(r_9\right)}_j$$(41) (42) $${\left(d_1\right)}_j\delta u_{3,j}+{\left(d_2\right)}_j\delta u_{3,j-1}+{\left(d_3\right)}_j\delta g_{2,j}+{\left(d_4\right)}_j\delta g_{2,j-1}+{\left(d_5\right)}_j\delta p_j+{\left(d_6\right)}_j\delta p_{j-1}+{\left(d_7\right)}_j\delta f_{3,j}+{\left(d_8\right)}_j\delta f_{3,j-1}={\left(r_{10}\right)}_j$$(42) (43) $${\left(e_1\right)}_j\delta h_{1,j}+{\left(e_2\right)}_j\delta h_{1,j-1}+{\left(e_3\right)}_j\delta u_{2,j}+{\left(e_4\right)}_j\delta u_{2,j-1}+{\left(e_5\right)}_j\delta g_{1,j}+{\left(e_6\right)}_j\delta g_{1,j-1}+{\left(e_7\right)}_j\delta f_{1,j}+{\left(e_8\right)}_j\delta f_{1,j-1}+{\left(e_9\right)}_j\delta g_{2,j}+{\left(e_{10}\right)}_j\delta g_{2,j-1}+{\left(e_{11}\right)}_j\delta f_{2,j}+{\left(e_{12}\right)}_j\delta f_{2,j-1}+{\left(e_{13}\right)}_j\delta f_{3,j}+{\left(e_{14}\right)}_j\delta f_{3,j-1}={\left(r_{11}\right)}_j$$(43) (44) $${\left(l_1\right)}_j\delta h_{2,j}+{\left(l_2\right)}_j\delta h_{2,j-1}+{\left(l_3\right)}_j\delta u_{1,j}+{\left(l_4\right)}_j\delta u_{1,j-1}+{\left(l_5\right)}_j\delta g_{2,j}+{\left(l_6\right)}_j\delta g_{2,j-1}+{\left(l_7\right)}_j\delta f_{1,j}+{\left(l_8\right)}_j\delta f_{1,j-1}+{\left(l_9\right)}_j\delta g_{1,j}+{\left(l_{10}\right)}_j\delta g_{1,j-1}+{\left(l_{11}\right)}_j\delta f_{2,j}+{\left(l_{12}\right)}_j\delta f_{2,j-1}+{\left(l_{13}\right)}_j\delta f_{3,j}+{\left(l_{14}\right)}_j\delta f_{3,j-1}={\left(r_{12}\right)}_j$$(44) (45) $${\left(m_1\right)}_j\delta h_{3,j}+{\left(m_2\right)}_j\delta h_{3,j-1}+{\left(m_3\right)}_j\delta h_{1,j}+{\left(m_4\right)}_j\delta h_{1,j-1}+{\left(m_5\right)}_j\delta f_{2,j}+{\left(m_6\right)}_j\delta f_{2,j-1}+{\left(m_7\right)}_j\delta g_{3,j}+{\left(m_8\right)}_j\delta g_{3,j-1}+{\left(m_9\right)}_j\delta f_{3,j}+{\left(m_{10}\right)}_j\delta f_{3,j-1}={\left(r_{13}\right)}_j$$(45) where $${\left(a_1\right)}_j={\left(a_2\right)}_j={\mathrm{\hslash }}_j,{\left(a_3\right)}_j=1,{\left(a_4\right)}_j=-1$$ $${\left(b_1\right)}_j=1-\frac{{\mathit{\hslash }}_j}{2}{\eta }_2{\left(f_3\right)}_{j-\frac{1}{2}},{\left(b_2\right)}_j={\left(b_1\right)}_j-2,{\left(b_3\right)}_j={\left(b_4\right)}_j=-\frac{{\mathit{\hslash }}_j{\eta }_1}{2},{\left(b_5\right)}_j={\left(b_6\right)}_j=-\frac{{\mathit{\hslash }}_j{\eta }_2}{2}{\left(u_1\right)}_{j-\frac{1}{2}},{\left(b_7\right)}_j={\left(b_8\right)}_j={\mathit{\hslash }}_j{\eta }_2{\left(f_2\right)}_{j-\frac{1}{2}},{\left(b_9\right)}_j={\left(b_{10}\right)}_j={‐\mathit{\hslash }}_j{\eta }_2{\left(f_1\right)}_{j-\frac{1}{2}}$$ $${\left(c_1\right)}_j=1-\frac{{\mathit{\hslash }}_j}{2}{\eta }_2{\left(f_3\right)}_{j-\frac{1}{2}},{\left(c_2\right)}_j={\left(c_1\right)}_j-2,{\left(c_3\right)}_j={\left(c_4\right)}_j=\frac{{\mathit{\hslash }}_j{\eta }_1}{2},{\left(c_5\right)}_j={\left(c_6\right)}_j=-\frac{{\mathit{\hslash }}_j{\eta }_2}{2}{\left(u_2\right)}_{j-\frac{1}{2}},{\left(c_7\right)}_j={\left(c_8\right)}_j=-{\mathit{\hslash }}_j{\eta }_2{\left(f_2\right)}_{j-\frac{1}{2}},{\left(c_9\right)}_j={\left(c_{10}\right)}_j={‐\mathit{\hslash }}_j{\eta }_2{\left(f_1\right)}_{j-\frac{1}{2}}$$ $${\left(d_1\right)}_j=1-\frac{{\mathit{\hslash }}_j}{2}{\eta }_3{\left(f_3\right)}_{j-\frac{1}{2}},{\left(d_2\right)}_j={\left(d_1\right)}_j-2,{\left(d_3\right)}_j={\left(d_4\right)}_j={\mathit{\hslash }}_j{\eta }_1,{\left(d_5\right)}_j=1,{\left(d_6\right)}_j=-1,{\left(d_7\right)}_j={\left(d_8\right)}_j=-\frac{{\mathit{\hslash }}_j{\eta }_3}{2}{\left(u_3\right)}_{j-\frac{1}{2}}$$ $${\left(e_1\right)}_j=1-\frac{{\mathit{\hslash }}_j}{2}{\eta }_4{\left(f_3\right)}_{j-\frac{1}{2}},{\left(e_2\right)}_j={\left(e_1\right)}_j-2,{\left(e_3\right)}_j={\left(e_4\right)}_j=-\frac{{\mathit{\hslash }}_j{\eta }_3}{2},{\left(e_5\right)}_j={\left(e_6\right)}_j=-{\mathit{\hslash }}_j{\eta }_3-\frac{{\mathit{\hslash }}_j}{2}{\eta }_4{\left(f_1\right)}_{j-\frac{1}{2}},{\left(e_7\right)}_j={\left(e_8\right)}_j=-\frac{{\mathit{\hslash }}_j}{2}{\eta }_4{\left(g_1\right)}_{j-\frac{1}{2}},{\left(e_9\right)}_j={\left(e_{10}\right)}_j=\frac{{\mathit{\hslash }}_j}{2}{\eta }_4{\left(f_2\right)}_{j-\frac{1}{2}},{\left(e_{11}\right)}_j={\left(e_{12}\right)}_j=\frac{{\mathit{\hslash }}_j}{2}{\eta }_4{\left(g_2\right)}_{j-\frac{1}{2}},{\left(e_{13}\right)}_j={\left(e_{14}\right)}_j=-\frac{{\mathit{\hslash }}_j}{2}{\eta }_4{\left(h_1\right)}_{j-\frac{1}{2}}$$ $${\left(l_1\right)}_j=1-\frac{{\mathit{\hslash }}_j}{2}{\eta }_4{\left(f_3\right)}_{j-\frac{1}{2}},{\left(l_2\right)}_j={\left(l_1\right)}_j-2,{\left(l_3\right)}_j={\left(l_4\right)}_j=\frac{{\mathit{\hslash }}_j{\eta }_3}{2},{\left(l_5\right)}_j={\left(l_6\right)}_j=-{\mathit{\hslash }}_j{\eta }_3-\frac{{\mathit{\hslash }}_j}{2}{\eta }_4{\left(f_1\right)}_{j-\frac{1}{2}},{\left(l_7\right)}_j={\left(l_8\right)}_j=-\frac{{\mathit{\hslash }}_j}{2}{\eta }_4{\left(g_2\right)}_{j-\frac{1}{2}},{\left(l_9\right)}_j={\left(l_{10}\right)}_j=-\frac{{\mathit{\hslash }}_j}{2}{\eta }_4{\left(f\right)}_{j-\frac{1}{2}},{\left(l_{11}\right)}_j={\left(l_{12}\right)}_j=-\frac{{\mathit{\hslash }}_j}{2}{\eta }_4{\left(g_1\right)}_{j-\frac{1}{2}},{\left(l_{13}\right)}_j={\left(l_{14}\right)}_j=-\frac{{\mathit{\hslash }}_j}{2}{\eta }_4{\left(h_2\right)}_{j-\frac{1}{2}}$$ $${\left(m_1\right)}_j=1-\frac{{\mathit{\hslash }}_j}{2}{\eta }_7{\left(f_3\right)}_{j-\frac{1}{2}},{\left(m_2\right)}_j={\left(m_1\right)}_j-2,{\left(m_3\right)}_j={\left(m_4\right)}_j={\mathit{\hslash }}_j{\eta }_5,{\left(m_5\right)}_j={\left(m_6\right)}_j={\mathit{\hslash }}_j{\eta }_6,{\left(m_7\right)}_j={\left(m_8\right)}_j={‐\mathit{\hslash }}_j{\eta }_6,{\left(m_9\right)}_j={\left(m_{10}\right)}_j=-\frac{{\mathit{\hslash }}_j}{2}{\eta }_7{\left(h_3\right)}_{j-\frac{1}{2}}$$

And $${\left(r_1\right)}_j=-\left[f_{1,j}-f_{1,j-1}-{\mathit{\hslash }}_j{\left(u_1\right)}_{j-\frac{1}{2}}\right],$$ $${\left(r_2\right)}_j=-\left[f_{2,j}-f_{2,j-1}-{\mathit{\hslash }}_j{\left(u_2\right)}_{j-\frac{1}{2}}\right],$$ $${\left(r_3\right)}_j=-\left[f_{3,j}-f_{3,j-1}-{\mathit{\hslash }}_j{\left(u_3\right)}_{j-\frac{1}{2}}\right],$$ $${\left(r_4\right)}_j=-\left[g_{1,j}-g_{1,j-1}-{\mathit{\hslash }}_j{\left(h_1\right)}_{j-\frac{1}{2}}\right],$$ $${\left(r_5\right)}_j=-\left[g_{2,j}-g_{2,j-1}-{\mathit{\hslash }}_j{\left(h_2\right)}_{j-\frac{1}{2}}\right],$$ $${\left(r_6\right)}_j=-\left[g_{3,j}-g_{3,j-1}-{\mathit{\hslash }}_j{\left(h_3\right)}_{j-\frac{1}{2}}\right],$$ $${\left(r_7\right)}_j=-\left[f_{3,j}-f_{3,j-1}+2{\mathit{\hslash }}_j{\left(f_1\right)}_{j-\frac{1}{2}}\right],$$ $${(r_8)}_j=-[u_{1,j}-u_{1,j-1}-{{\mathit{\hslash }}_j\eta }_1{(h_2)}_{j-\frac{1}{2}}-{{\mathit{\hslash }}_j\eta }_2[{(f_3{u}_1)}_{j-\frac{1}{2}}-{(f_2)}_{j-\frac{1}{2}}^2+{(f_1)}_{j-\frac{1}{2}}^2]],$$ $${(r_9)}_j=-[u_{2,j}-u_{2,j-1}-{{\mathit{\hslash }}_j\eta }_1{(h_1)}_{j-\frac{1}{2}}-{{\mathit{\hslash }}_j\eta }_2[{(f_3{u}_2)}_{j-\frac{1}{2}}-{({f_{1}f}_2)}_{j-\frac{1}{2}}]],$$ $${(r_{10})}_j=-[u_{3,j}-u_{3,j-1}-{2{\mathit{\hslash }}_j\eta }_1{(g_2)}_{j-\frac{1}{2}}+p_j-p_{j-1}-{{\mathit{\hslash }}_j\eta }_3{(f_3{u}_3)}_{j-\frac{1}{2}}],$$ $${(r_{11})}_j=-[h_{1,j}-h_{1,j-1}-{{\mathit{\hslash }}_j\eta }_3{(u_2)}_{j-\frac{1}{2}}-2{{\mathit{\hslash }}_j\eta }_3{(g_1)}_{j-\frac{1}{2}}-{{\mathit{\hslash }}_j\eta }_4[{(f_1{g}_1)}_{j-\frac{1}{2}}-{(f_2{g}_2)}_{j-\frac{1}{2}}+{(f_3{h}_1)}_{j-\frac{1}{2}}]],$$ $${(r_{12})}_j=-[h_{2,j}-h_{2,j-1}+{{\mathit{\hslash }}_j\eta }_3{(u_1)}_{j-\frac{1}{2}}-2{{\mathit{\hslash }}_j\eta }_3{(g_2)}_{j-\frac{1}{2}}-{{\mathit{\hslash }}_j\eta }_4[{(f_1{g}_2)}_{j-\frac{1}{2}}+{(f_2{g}_1)}_{j-\frac{1}{2}}+{(f_3{h}_2)}_{j-\frac{1}{2}}]],$$ $${(r_{13})}_j=-[h_{3,j}-h_{3,j-1}+2{{\mathit{\hslash }}_j\eta }_5{(h_1)}_{j-\frac{1}{2}}+2{{\mathit{\hslash }}_j\eta }_6[{{(f_2)}_{j-\frac{1}{2}}-(g_3)}_{j-\frac{1}{2}}]-{{\mathit{\hslash }}_j\eta }_7{(f_3{h}_3)}_{j-\frac{1}{2}}]$$

The linearised system of equations $(33-45)$ has a block tri-diagonal structure which is solved with the help of block elimination method. In vector-matrix form, Equations (46–58) can be written as (46) $$A\delta =r$$(46) where (47) $$A=\left[\begin{array}{ccc}\begin{array}{ccc}\left[A_1\right]& \left[C_1\right]& \\ \left[B_2\right]& \left[A_2\right]& \left[C_2\right]\end{array}& & \\ & \begin{array}{c}\ddots \\ \ddots \\ \ddots \end{array}& \\ & & \begin{array}{ccc}\left[B_{J-1}\right]& \left[A_{j-1}\right]& \left[C_{j-1}\right]\\ & \left[B_J\right]& \left[A_J\right]\end{array}\end{array}\right]$$(47) (48) $$\delta ={\left[\begin{array}{ccc}\left[{\delta }_1\right]& \left[{\delta }_2\right]& \begin{array}{ccc}\left[{\delta }_3\right]& \dots & \begin{array}{cc}\left[{\delta }_{J-1}\right]& \left[{\delta }_J\right]\end{array}\end{array}\end{array}\right]}^T$$(48) (49) $$r={\left[\begin{array}{ccc}\left[r_1\right]& \left[r_2\right]& \begin{array}{ccc}\left[r_3\right]& \dots & \begin{array}{cc}\left[r_{J-1}\right]& \left[r_J\right]\end{array}\end{array}\end{array}\right]}^T$$(49) where $$\begin{array}{ccc}[{\delta }_j]=[\begin{array}{c}\delta u_{1,j-1}\\ \delta u_{2,j-1}\\ \begin{array}{c}\delta u_{3,j-1}\\ \delta h_{1,j-1}\\ \begin{array}{c}\delta h_{2,j-1}\\ \delta h_{3,j-1}\\ \begin{array}{c}\delta p_j\\ \delta f_{1,j}\\ \begin{array}{c}\delta f_{2,j}\\ \delta f_{3,j}\\ \begin{array}{c}\delta g_{1,j}\\ \delta g_{2,j}\\ \delta g_{3,j}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}],1\le j\le J-1& \text{and}[{\delta }_J]=[\begin{array}{c}\delta u_{1,J-1}\\ \delta u_{2,J-1}\\ \begin{array}{c}\delta u_{3,J-1}\\ \delta h_{1,J-1}\\ \begin{array}{c}\delta h_{2,J-1}\\ \delta h_{3,J-1}\\ \begin{array}{c}\delta f_{3,J}\\ \delta u_{1,J}\\ \begin{array}{c}\delta u_{2,J}\\ \delta p_J\\ \begin{array}{c}\delta h_{1,J}\\ \delta h_{2,J}\\ \delta h_{3,J}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}];\text{and}[r_j]=[\begin{array}{c}{(r_1)}_j\\ {(r_2)}_j\\ \begin{array}{c}{(r_3)}_j\\ {(r_4)}_j\\ \begin{array}{c}{(r_5)}_j\\ {(r_6)}_j\\ \begin{array}{c}{(r_7)}_j\\ {(r_8)}_j\\ \begin{array}{c}{(r_9)}_j\\ {(r_{10})}_j\\ \begin{array}{c}{(r_{11})}_j\\ {(r_{12})}_j\\ {(r_{13})}_j\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}]& \end{array}$$

The elements of the matrix $A$ are generated using above equations. Then this block tridiagonal matrix is solved using LU-decomposition method. i.e. (50) $$A=\mathit{\text{LU}}$$(50) where $L$ is a lower triangular matrix and $U$ is an upper triangular matrix. where, $$L=\left[\begin{array}{ccccc}\left[{\alpha }_1\right]& & & & \\ & \left[{\alpha }_2\right]& & & \\ & & \ddots & & \\ & & \ddots & & \\ & & & \left[B_J\right]& \left[{\alpha }_J\right]\end{array}\right],u=\left[\begin{array}{ccccc}\left[I\right]& \left[{\Gamma }_1\right]& & & \\ & \left[I\right]& \left[{\Gamma }_2\right]& & \\ & & \ddots & & \\ & & \ddots & & \\ & & & \left[I\right]& \left[{\Gamma }_{J-1}\right]\\ & & & & \left[I\right]\end{array}\right]$$

Here, $[I]$ is unit matrix and $[{\alpha }_i]$ and $[{\Gamma }_i]$ are $13\times 13$ matrices whose elements are determined by the following equations: $$\left[{\alpha }_j\right]=\left[A_1\right]$$ $$\left[A_1\right]\left[{\Gamma }_1\right]=\left[C_1\right]$$ $$\left[{\alpha }_j\right]=\left[A_j\right]-\left[B_j\right]\left[{\Gamma }_{j-1}\right],j=2,3,,J$$ $$\left[{\alpha }_j\right]\left[{\Gamma }_j\right]=\left[C_j\right],j=2,3,,J-1$$

Equation (50)(50) $$A=\mathit{\text{LU}}$$(50) substituted in (46) and we have, (51) $$\mathit{\text{Lu}}\delta =r$$(51)

Defining $u\delta =w,$ we have, $\mathit{\text{Lw}}=r$ where, (52) $$w={\left[\begin{array}{c}\left[w_1\right]\left[w_2\right]\left[w_3\right]\left[w_4\right]\left[w_5\right]\left[w_6\right]\left[w_7\right]\left[w_8\right]\end{array}\right]}^T$$(52) and the $[w_j]$ are $13\times 1$ column matrices. The elements $w$ can be solved as, $$\left[{\alpha }_1\right]\left[w_1\right]=\left[r_1\right]$$ $$\left[{\alpha }_j\right]\left[w_j\right]=\left[r_j\right]-\left[B_j\right]\left[w_{j-1}\right]$$

Once the elements of $w$ are found, solution for $\delta$ can be obtained using relations: $$\left[{\delta }_J\right]=\left[w_J\right]$$ $$\left[{\delta }_j\right]=\left[w_j\right]-\left[{\Gamma }_j\right]\left[{\delta }_{j+1}\right]$$

The solutions are found for $\delta$ using forward and backward sweep. These solution for $\delta$ can be used to find ${(i+1)}^{\mathit{\text{th}}}$ iteration. The MATLAB code was developed using MATLAB_R2022a to implement the Keller-box method. This method proves advantageous over traditional finite difference methods due to its ability to linearise highly nonlinear equations, enhancing computational efficiency and accuracy in solving complex problems.

Results and discussions

This section is devoted to results, discussion and their interpretations presented in graphical and tabular form. The present study demonstrates the magnetic effect and influence of micropolar parameters (Anwar & Guram, 1980) on the flow of fluid contained between a rotating and a stationary disc. An efficient and second-order implicit finite difference scheme - Keller-box method is implemented to solve the governing non-linear system of equations. The convergence criteria, emphasising a 16-decimal accuracy tolerance, has been implemented in the Keller-box method to ensure precise and rigorous computational results.

The solution from Anwar and Guram (1980) for particular micropolar parameters are given in Table 1 for comparison purpose. The results for velocity, microrotation profiles and pressure gradient for different values of $M$ are given in Table 2. Both solutions from Tables 1 and 2 agree, validating the Keller-box solution for $M=0$ case.

Table 1. Velocity profiles, microrotation profiles and pressure gradient for ${\eta }_1=0.3,{ \eta }_2=0.8,{\eta }_3=0.2,{\eta }_4=0.02,{ \eta }_5=0.5,{\eta }_6=0.08,{ \eta }_7=0.01$ from literature (Anwar & Guram, 1980).

Case	η	F1	F2	F3	G1	G2	G3	P
M = 0 (Anwar & Guram, 1980)	0.0	0.0	1.0	0.0	0.0	0.0	0.0	0.0
	0.2	0.0257	0.7974	−0.006	0.0155	0.0002	0.0090	0.0514
	0.4	0.0309	0.5955	−0.0178	0.0231	−0.0002	0.0108	0.0618
	0.6	0.0245	0.3952	−0.0292	0.0230	−0.0006	0.0079	0.0493
	0.8	0.0129	0.1966	−0.0368	0.0154	−0.0005	0.0034	0.0265
	1.0	0	0	−0.0393	0	0	0	0.0008

Table 2. Velocity profiles, microrotation profiles and pressure gradient for various magnetic parameter with ${\eta }_1=0.3,{\mathrm{ }\eta }_2=0.8,{\eta }_3=0.2,{\eta }_4=0.02,{\mathrm{ }\eta }_5=0.5,{\eta }_6=0.08,{\mathrm{ }\eta }_7=0.01.$

Case	η	F1	F2	F3	G1	G2	G3	P
M = 0	0.0	0.0	1.0	0.0	0.0	0.0	0.0	0.0
	0.2	0.0257	0.7974	−0.006	0.0155	0.0002	0.0090	0.0505
	0.4	0.0308	0.5956	−0.0178	0.0231	−0.0002	0.0108	0.0608
	0.6	0.0245	0.3952	−0.0292	0.0230	−0.0006	0.0079	0.0481
	0.8	0.0129	0.1966	−0.0367	0.0154	−0.0005	0.0034	0.0251
	1.0	0	0	−0.0393	0	0	0	−0.0006
M = 3	0.0	0.0	1.0	0.0	0.0	0.0	0.0	0.0
	0.2	0.0147	0.6836	−0.0036	0.0166	2.2E-5	0.0072	0.0290
	0.4	0.0154	0.4502	−0.0099	0.0227	−0.0002	0.0079	0.0304
	0.6	0.0111	0.2720	−0.0153	0.0209	−0.0004	0.0054	0.0218
	0.8	0.0055	0.1277	−0.0186	0.0130	−0.0003	0.0020	0.0107
	1.0	0	0	−0.0197	0	0	0	−0.0003
M = 5	0.0	0.0	1.0	0.0	0.0	0.0	0.0	0.0
	0.2	0.0111	0.6283	−0.0028	0.0170	−1.8E-5	0.0064	0.0219
	0.4	0.0106	0.3844	−0.0074	0.0222	−0.0002	0.0066	0.0210
	0.6	0.0072	0.2191	−0.0109	0.0197	−0.0003	0.0042	0.0141
	0.8	0.0034	0.0990	−0.0131	0.0119	−0.0002	0.0014	0.0066
	1.0	0	0	−0.0138	0	0	0	−0.0002
M = 10	0.0	0.0	1.0	0.0	0.0	0.0	0.0	0.0
	0.2	0.0065	0.5288	−0.0018	0.0174	−5E-5	0.0050	0.0128
	0.4	0.0051	0.2760	−0.0042	0.0211	−0.0001	0.0046	0.0100
	0.6	0.0029	0.1377	−0.0058	0.0176	−0.0002	0.0024	0.0058
	0.8	0.0013	0.0568	−0.0066	0.0100	−0.0001	0.0005	0.0025
	1.0	0	0	−0.007	0	0	0	−6E-5
M = 15	0.0	0.0	1.0	0.0	0.0	0.0	0.0	0.0
	0.2	0.0044	0.4602	−0.0013	0.0174	−5.3E-5	0.0041	0.0087
	0.4	0.0029	0.2102	−0.0028	0.0200	−0.0001	0.0033	0.0058
	0.6	0.0015	0.0931	−0.0037	0.016	−0.0001	0.0014	0.0029
	0.8	0.0006	0.0352	−0.0041	0.0088	−7.3E-5	1.8E-5	0.0011
	1.0	0	0	−0.0042	0	0	0	−2.6E-5
M = 20	0.0	0.0	1.0	0.0	0.0	0.0	0.0	0.0
	0.2	0.0032	0.4086	−0.0010	0.0172	−5.12E-05	0.0035	0.0064
	0.4	0.0018	0.1662	−0.0021	0.0190	−9.61E-05	0.0025	0.0036
	0.6	0.0008	0.0661	−0.0026	0.0147	−8.87E-05	0.0008	0.0016
	0.8	0.0003	0.0230	−0.0028	0.0079	−5.18E-05	−0.0003	0.0006
	1.0	0	0	−0.0028	0	0	0	−1.29E-05

Figures 2–4 represent the influence of magnetic field parameter on velocity profiles $F_1,F_2$ and $F_3.$ From the graph, it is noticed that the profiles are decreasing with the increasing magnetic parameter values. As the magnetic parameter increases, the Lorentz force imposed by the magnetic field becomes more dominant in the flow. This force tends to dampen the fluid motion, affecting the velocity profiles. The stronger the magnetic field, the more pronounced the influence, resulting in decreased velocities throughout the flow domain. The presence of microstructures in micropolar fluids alters the flow characteristics. When combined with the magnetic field, these microstructures could align differently or experience increased resistance due to the magnetic forces, contributing to the observed decrease in velocity profiles. With the exception of $F_3,$ velocity profiles for different magnetic parameter $M$ are close to each other near the lower and upper disks. Velocity profile $F_3$ for various magnetic parameter values $M$ is isolated at the upper disk.

Figure 2. F₁ (η) Graph for various magnetic parameter M.

Figure 3. F₂ (η) Graph for various magnetic parameter M.

Figure 4. F₃ (η) Graph for various magnetic parameter M.

Figure 2 shows the velocity profile $F_1,$ which exhibits the Poiseuille flow. Also, from the figure, it is observed that there is a radial outward flow from the lower disc, whose planes or stationary maxima point keeps decreasing as $M$ increases. i.e. maxima shift towards the lower disc as the magnetic parameter increases. After reaching maxima, it starts approaching zero near the upper disc. Transverse velocity profile $F_2$ and axial velocity profile $F_3$ are given in Figures 3 and 4 for various M values, representing the Couette flow.

Behaviour of pressure $P$ are shown in Figure 5 for different magnetic parameter values $M.$ It is noticed that for increasing magnetic parameter values, pressure decreases. The influence of the magnetic field intensifies as the value of magnetic parameter increases. The fluid motion may experience a resistive force from the magnetic field. This resistance may cause the overall flow velocity to drop, which in turn may increase flow resistance and, as a result, decrease pressure along the flow path. Also, the irregularities in pressure near the rotating disc are decreased for increasing magnetic parameter values. The fluid velocity near the lower disc is higher than near the upper disc because of the increased micro-rotation in this area. Thus, the Lorentz force is greater close to the lower disc. There is an increase in pressure close to the lower rotating disc because of the Lorentz force, which works in the direction opposite to the fluid flow. On the other hand, the fluid velocity is noticeably slower, and the micro-rotation is lower, close to the upper stationary disc. As a result, the Lorentz force is less strong here. There is a decrease in pressure because the lesser Lorentz force has a smaller impact on the pressure distribution close to the upper disc. It is also observed that the rise in the pressure near the lower disc is less compared to the drop in pressure near the upper disc.

Figure 5. P(η) Graph for various magnetic parameter M.

The behaviour of microrotation profiles $G_1,G_2,$ and $G_3$ are shown in Figures 6–8, for various magnetic parameter values $M.$ It is observed that microrotation profile drops with the rise in magnetic parameter values. The magnetic force has a greater impact on the velocity and hence the motion of the fluid. This enhanced impact of the magnetic field is likely to align the micro-rotation of the fluid particles, thereby decreasing the overall magnitude of the microrotation profile. Microrotation profile $G_2$ decreases first, then starts increasing from $0.4\le \eta \le 1,$ as $M$ increases.

Figure 6. G₁ (η) Graph for various M.

Figure 7. G₂ (η) Graph for various M.

Figure 8. G₃ (η) Graph for v arious M.

For flow visualisation, non-dimensional streamlines are plotted in Figures 9–12 for various magnetic parameter.

Figure 9. Streamlines for M = 3 under Case 1.

Figure 10. Streamline for M = 5 under Case 1.

Figure 11. Streamlines for M = 10 under Case 1.

Figure 12. Streamlines for M = 20 under Case 1.

Conclusions

The study employed similarity transformation to convert the nonlinear partial differential equations (PDEs) into seven nonlinear systems of ODEs. These ODEs were effectively solved using the Keller-box method, a second-order implicit finite difference approach. The primary aim was to analyse the impact of the magnetic parameter and characteristic parameters of micropolar elements on various fluid flow properties such as velocity profiles, microrotation profiles, and pressure within the system. Overall, the study demonstrated the intricate relationship between magnetic fields and micropolar fluid behaviour, shedding light on how magnetic parameters influence flow characteristics and providing valuable insights into the complex dynamics of such systems. The important conclusions can be listed as follows:

1. The presence of a magnetic field significantly affected the flow characteristics. It led to alterations in flow patterns, streamlines, vortex formations, and velocity distribution within the micropolar fluid. The magnetic field’s interaction altered the way the micropolar fluid rotated and circulated.

2. The numerical findings showcased that the magnetic field’s presence exerted Lorentz forces on the conducting fluid. This resulted in observable changes in the velocity profiles, microrotation profiles, and pressure distribution. Specifically, an increase in the magnetic parameter led to a decrease in velocity, microrotation and pressure profiles.

3. The decrease observed in velocity, microrotation, and pressure profiles with an increase in the magnetic parameter was primarily attributed to the influence of Lorentz forces exerted on the conducting fluid, highlighting the magnetic field’s impact on fluid behaviour.

4. The obtained numerical results were found to be in good agreement with existing available data, validating the accuracy and reliability of the employed methodology and computations.

Nomenclature
$(r,\theta ,z)$	=	Polar coordinates
$B_0$	=	Magnetic field component $(T)$
$\left(G_1,G_2,G_3\right)$	=	Dimensionless microrotation function
$j$	=	Micro inertia $(m^2)$
$M$	=	magnetic parameter
$P$	=	Dimensionless pressure
$\nu \mathrm{ }\left({\nu }_1,{\nu }_2,{\nu }_3\right)$	=	Microrotation vector $(s^{-1})$
$\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma },\mathrm{\lambda },\mathrm{\kappa }$	=	Material constants (viscosity coefficient)
${\eta }_i,i=1,2,3,4,5,6,7$	=	Micropolar parameter
$\mathrm{\hslash }\mathrm{ }$	=	Step/grid size in Keller-box method
$\mathit{B}$	=	Magnetic field vector $(T)$
$\left(F_1,F_2,F_3\right)$	=	Dimensionless stream function
$\mathit{J}$	=	Current density
$\mathit{h}$	=	height between discs $(m)$
$p\mathrm{ }$	=	Pressure $\left(\mathit{\text{kg}}\mathrm{ }{m}^{-1}\mathrm{ }{s}^{-2}\right)$
u (u,v,w)	=	Velocity vector $(m{s}^{-1})$
$\omega \mathrm{ }$	=	Angular momentum
$\mu \mathrm{ }$	=	Coefficient of viscosity $(\mathit{\text{kg}}\mathrm{ }{m}^{-1}\mathrm{ }{s}^{-1})$
${\sigma }_e\mathrm{ }$	=	Conductivity of the medium $(S{m}^{-1})$

Disclosure statement

No potential conflict of interest was reported by the authors. Param Tangsali-Conceptualization, Methodology, Software; Dr. Ashwini Bhat- Validation, Formal analysis, and writing the original draft; Dr. Nagaraj N. Katagi Writing Review and editing, Visualisation, and Supervision.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.