MHD micropolar fluid flow in porous media

Abstract

The analysis of mass and heat transfer in magnetohydrodynamic (MHD) flows has significant applications in heat exchangers, cooling nuclear reactors, designing energy systems and casting and injection processes of different types of fluids. On the other hand, extraction of crude oil, the flow of human or animal blood, as well as other polymer fluids or liquid crystals are just some examples of micropolar fluid flows. Due to the broad application spectrum of the theory of micropolar fluid flows, and the significance the impact the external magnetic field has on the flow of these fluids, this paper considers the stationary flow of a micropolar fluid between two plates under the influence of an external magnetic field which is perpendicular to the direction of the flow. Stationary plates are maintained at constant and different temperatures, while the whole problem is considered in the non-inductive approximation. The equation system used to define the physical problem under consideration is reduced to the system of differential equations that have been solved analytically and the solutions of which are of general nature. In addition to the solutions for velocity, microrotation and temperature, the paper gives solutions for shear stress at plates, the Nusselt number and flow rate. The provided solutions have been applied in order to reach some general conclusions about the influence of the magnetic field and physical characteristics of a micropolar fluid and the characteristics of porous media on the nature of micropolar fluid flows in porous media by means of chart analysis. General conclusions, obtained in the result analysis in this paper, give us the opportunity to understand the flows of micropolar fluids and highlight their significance.

Introduction

The analysis of mass and heat transfer in magnetohydrodynamic (MHD) flows has significant applications in heat exchangers, cooling nuclear reactors, designing energy systems and casting and injection processes of different types of fluids. Moreover, electrically conductive fluids are used in many industrial processes. Liquid metals, such as aluminium, mercury or crucible steels are good examples of such fluids. The interaction between the external magnetic field and electrically conductive fluids results in various MHD phenomena that may be applied in different ways. Although from this distance it can be said that the earliest research into MHD flows dates back to the 19th century, it was not until 1960s that the practical usage of magnetic fields in metallurgical industry began. The first occasion when the magnetic field is used in metalworking is in steel casting, where the inspection of the flow in nozzles is performed as steel is injected into moulds, followed by stabilization during the hardening stage in order to obtain a high quality product. Another example of the use of a magnetic field is in the design of semi-conductor crystals, where a high level of crystal purity and quality is achieved by means of electromagnetic control.

Due to the extensive application of MHD flows in industry from the mid-20th century to the present day, we have continued to see the importance of research into electrically conductive fluids under the influence of an external magnetic field. Although the beginnings of magnetic hydrodynamics are associated with the research carried out by Hartmann ¹ and Hartmann and Lazarus, ² the first significant strides were made with the appearance of fusion reactors and the emerging need for their cooling. Two decades later, magnetic fields were routinely used for heating, buoyancy, mixing and levitation melting of liquid metals in metallurgical industry. What is essential here is the fact that Lorentz force enables non-contact manipulation of metal flows. With constant commercial pressure to produce cheaper, better and more consistent materials, MHD provides a unique solution for better control over casting and metalworking processes.

The use of different types of electrically conductive fluids, particularly in industry and other fields (e.g. medicine or nanotechnology) in the 21st century, led to an increased interest in the research on microfluid and nanoparticle flows. Microfluids constitute a distinct class of fluids of this type and can often be encountered in nature (e.g. human or animal blood, muddy waters, lubricants, etc.). Micropolar fluids are non-Newtonian fluids that contain randomly oriented constituents dispersed in a viscous fluid. One of the best established theories of fluids with microstructure is the theory of micropolar fluids put forth by Eringen. ³ The first significant research results were published in 1964 and 1965 by Eringen ⁴ in his papers, but the theory as we know it today was made public in 1966 in the article titled ‘Theory of Micropolar Fluids’. Afterwards, Eringen ⁵ expanded his theory of micropolar fluids to include heat transfer. Willson ⁶ (1970) was the first to consider the theory of a boundary layer in micropolar fluids. That same year (1970), Peddieson and McNitt, ⁷ analysed the flows around the front stagnation point and along an infinite plate, as well as in the boundary layer. Significant research into microcontinuum flows and micropolar fluids was presented by Ariman et al.^8,9 in his papers (1973, 1974). Furthermore, Nath ¹⁰ (1975, 1976) and Ahmadi ¹¹ (1976) suggested similar (automodel) solutions for micropolar fluid flows in the boundary layer.

In the 21st century, the model of a micropolar fluid is widely considered the best model to define the flows of biological fluids, which particularly applies to human blood. Although initial research into the flows of a micropolar fluid as a model of human blood was carried out by Turk et al.^12,13 (1973, 1974), nowadays this kind of research is still the subject of great interest. Thus, Mekheimer and Kot ¹⁴ (2008) applied the model of a micropolar fluid to analyse the flows of blood in a tapered artery with a stenosis. Similar research, relying on the theory of micropolar fluids to represent the flows of blood in an artery, was conducted in the paper by Mohd Zainul Abidin et al. ¹⁵ (2021), the only difference being that the latter took into account the effect of a chemical reaction on solution dispersion. Another notable paper representing the dynamic model of the heart pumping blood aimed at predicting the blood flow and the effect of pressure on the area of blood vessels (arteries, veins, capillaries) was written by Abdullah ¹⁶ (2022). The mathematical model for the cardiovascular system was provided in the paper by Riley ¹⁷ (2017), with the aim to analyse the TGA (Transposition of the Great Arteries) defect which may occur as a congenital heart defect. The model of a micropolar fluid, as stated earlier, may be used to study other relevant processes. Thus, Alshehri ¹⁸ (2021) used the model of a Cattaneo–Christov heat flux to examine the velocity of heat transfer in a hybrid nanofluid which contains a micropolar liquid through a vertical surface. Morever, Salahuddin et al. ¹⁹ (2021), considered the case of a non-stationary flow of a micropolar fluid around a cylindrical object, whereas Umar ²⁰ (2021) analysed heat transfer during the flow of a micropolar fluid in a constricted channel under the influence of heat radiation and the Lorentz force. All these recent studies show the significance that the MHD flows of micropolar fluids have.

On the other hand, a porous medium is a heterogeneous volume of material which has a solid matrix of interconnected voids (pores) relatively small in size compared to the characteristic size of a matrix. Having in mind the definition of a porous medium referred to above, it is easy to conclude that there are natural (rocks, sediments, gravel, sand, clay, wood, the cardiovascular system, human lungs, bile ducts, gall bladder, oesophagus, the reproductive system, etc.) and artificial (liquid composites, reactor insulation, MHD generators and batteries, ground water reservoirs, peristaltic pumps, etc.) porous media. Henry Darcy carried out an experimental study of water flow through sand in 1855 and 1856. Through these experiments he arrived at the conclusion that the velocity of the water flow through sand is proportional to pressure gradient, while the coefficient of proportionality is the permeability of sand whose value depends on fluid characteristics (viscosity, temperature, etc.). The first results were published in the paper by Darcy ²¹ and became known as the Darcy’s law. The expansion of research into heat flow and transfer in porous media took place in the late 20th century. This occurred as a result of an increased presence of MHD flows and heat transfer processes in porous media in broad engineering practice (heat exchangers, metallurgical industry, liquid metal filtration, fusion control, etc.).

Regarding the flow of micropolar fluids in a porous medium, it is worth noting that this type of flow also has a wide range of applications, particularly in the field of medicine, in the flow of biological fluids through tissues, membranes and other porous media. In 2003 Khaled and Vafai ²² considered the flow and heat transfer through biological tissues. Their main concept was transport in porous media using mass diffusion and different models of convective flows such as the Darcy’s and the Brinkman’s model. Based on the obtained results, the authors established that the theory of porous media in biological tissues is the most efficient one for studying this type of flows. Moreover, a notable study and conclusions were those by Mehmood et al. ²³ (2012) arrived at in the analysis of a non-stationary two-dimensional flow of blood in a porous artery with flexible walls. The model of a porous medium was the area containing blood clots, cholesterol and fat deposits, and the results show that the decreased permeability causes the flow to decelerate, whereas the drop in pressure and stress on artery walls increase. Nowadays, we see very wide application of MHD in various ways to treat cancer and control the flow. This was considered in the analysis performed in the paper by Bhatti and Ali Abbas ²⁴ (2016), which presents the slip effects and MHD flows on the peristaltic flow of blood through a porous medium according to Jeffrey’s model.

Recent studies once again show the relevance and topicality of studying the flow of micropolar fluids in a porous medium. Thus Shankar Goud ²⁵ (2020) considers the influence of heat absorption on a stretched permeable surface in MHD flows of a micropolar fluid through a porous medium with variable suction/injection of a fluid. Fatunmbi et al. ²⁶ (2020) considered magnetohydrodynamic flows of micropolar fluids in a porous medium with multiple slip conditions. The results indicate that the impulse and thickness of a heat boundary layer drop as the parameter of non-linear stretching increases, while the opposite happens when the parameter of thermal conductivity increases. In addition, Mishra et al. ²⁷ (2022), compare the two models of fluids, the Williamson liquid and the micropolar fluid, using different values of parameters for shear stress and the Nusselt number. The paper by Xiu et al. ²⁸ (2023) focuses on a combined impact of Lorentz force, microrotation and thermo-migration of particles on the dynamics of micropolar fluids that are exposed to non-linear thermal radiation and the Arrhenius chemical reaction related to activation energy.

The overview of studies given above unequivocally indicates the relevance of research into flow and heat transfer of micropolar fluids in porous media. Numerous papers published in 2023 alone^29–31 suggest that the study of micropolar fluid flows will continue to be of interest in the years to come. Hence, the present paper considers the problem of a stationary flow of a micropolar fluid in a porous medium under the influence of an external magnetic field.

Physical and mathematical model

Micropolar fluids are classified as fluids with non-symmetrical stress tensor widely termed as polar fluids which belong to the class of fluids more generally defined than the ones studied in the classical theory of hydrodynamics. The Navier-Stokes model of classical hydrodynamics has one great limitation – it is not able to describe the fluid flows that contain microstructures. In order to describe the behaviour of such fluids, we need a theory that takes into consideration the internal motion of individual material constituents. In the micropolar theory, each constituent has a finite size and represents a microstructure able to rotate. This type of continuum comprises a continuous sum of such particles of finite size. Additional three degrees of rotation freedom are defined by the vector of microrotation (ω).

The theory and the equations representing fluid flows with couple stress were initially put forth by Stokes. ³² The theory of fluid flow with couple stress has been used by several authors to study non-Newtonian fluids.^33,34 For a non-stationary flow and heat transfer of an incompressible micropolar fluid with non-symmetrical stress tensor, under the influence of conservative forces in a porous medium, the basic equation system is as follows:

$$\frac{\partial \rho }{\partial t}+\mathrm{div}\left(\rho \mathit{v}\right)=0$$

(1)

$$\begin{array}{c}\multicolumn{1}{c}{\rho \frac{D\mathit{v}}{\mathrm{Dt}}=\rho \mathit{f}-\nabla p+(\mathit{j}\times \mathit{B})+\mu \mathrm{\Delta }\mathit{v}+(\epsilon +\mu -\lambda )\nabla \mathrm{div}\mathit{v}}\\ +(\mu +\lambda )\mathrm{\Delta }\mathit{v}+2\lambda \mathrm{rot}\mathbf{\omega }\hfill \end{array}$$

(2)

$$\begin{array}{c}\multicolumn{1}{c}{\rho I\frac{D\mathbf{\omega }}{\mathrm{Dt}}=2\lambda (\mathrm{rot}\mathit{v}-2\mathbf{\omega })+(\gamma +\beta -\alpha )\nabla \mathrm{div}\mathbf{\omega }}\\ +(\alpha +\beta )\mathrm{\Delta }\mathbf{\omega }+\rho \mathit{g}\hfill \end{array}$$

(3)

$$\rho c_p(\frac{\partial T}{\partial t}+\mathit{v}\nabla T)=k{\nabla }^{2}T+\mathrm{\Phi }+\frac{{\mathit{j}}^2}{\sigma }+\rho \mathit{gv}$$

(4)

Where: $\rho$ – fluid density, $\mathit{v}$ – vector of fluid velocity, $\mathit{f}$ – vector of volume forces, $p$ – pressure field, $\mathit{j}$ – density of current, $\mathit{B}$ – magnetic field vector, $\epsilon$ – dielectric constant, $\mu$ – dynamic viscosity, $\alpha ,\beta ,\gamma ,\lambda$ – additional viscosities for micropolar fluids, $\mathbf{\omega }$ – vector of microrotation, $c_p$ – mass heat capacity, $T$ – temperature field, $\sigma$ – electrical conductivity and $\mathit{g}$ – acceleration of gravity.

Solving a system of equations (1–4) for different flow problems is quite difficult due to non-linearity which occurs in partial differential equations. Consequently, a need to use numerical methods when solving MHD flows arises. Many scholars use some of the iterative numerical methods ³⁵ or even high-precision numerical models for solving partial differential equations, or CFD analysis. ³⁶ However, most numerical solutions given in some papers constitute solutions for particular cases, hence the aim of this paper is to present a general and exact analytical solution for solving partial differential equations which could enable us to analyse the influence of all relevant parameters without having to repeat the numerical solving of partial differential equations in each individual case.

A physical problem to be considered in this paper is a stationary flow of a micropolar fluid between two plates in a porous medium under the influence of an external magnetic field (Figure 1). Note that the flow is fully developed in the direction of the x-axis and is therefore treated as a one-dimensional problem. The motion of a micropolar fluid is a consequence of a constant pressure gradient along the x-axis. During the flow, the upper and the lower plates are kept at different constant temperatures T_w1 and T_w2, whereby the difference between the two temperatures is not big. During the flow of a micropolar fluid, the external magnetic field with intensity B has perpendicular action on the direction of the flow (in the direction of the y-axis). The analysis of the flow is considered in the non-inductive approximation and the electrical conductivity of fluid remains constant during the flow. Based on the considered physical model defined above, the velocity vector $\mathit{v}$ and the external magnetic field $\mathit{B}$ are defined by the following expressions:

$$\mathit{v}=u\overrightarrow{i},$$

(5)

$$\mathit{B}=B\overrightarrow{j}.$$

(6)

Figure 1.

Physical model.

By applying the foregoing assumptions, the system of basic equations for the flow of a micropolar fluid in a porous medium, the equations (1–4) may be reduced to the following system:

$$(\mu +\lambda )\frac{d^{2}u}{d{y}^2}+\lambda \frac{d\omega }{\mathrm{dy}}-\sigma B^{2}u-\frac{\mathrm{dp}}{\mathrm{dx}}-\frac{\mu }{\chi }u=0$$

(7)

$$\gamma \frac{d^2\omega }{d{y}^2}-\lambda \frac{\mathrm{du}}{\mathrm{dy}}-2\lambda \omega =0$$

(8)

$$k\frac{d^{2}T}{d{y}^2}+(\mu +\lambda ){\left(\frac{\mathrm{du}}{\mathrm{dy}}\right)}^2+\sigma B^2{u}^2+\frac{\mu }{\chi }{u}^2=0$$

(9)

In the system given above please note that the equation (8) represents an equation for the angular momentum and that this is a characteristic equation for the flow of a micropolar fluid, that is, the fluid with couple stress. Moreover, additional viscosities $\lambda$ and $\gamma$ represent physical characteristics of a micropolar fluid, while $\chi$ stands for the porous medium permeability. From the previous system of equations (7–9),^25,37,38 it can be seen that the impulse equation (7) and the angular momentum equation (8) are interdependent and should therefore be solved together, that is, as a pair of couple equations, which greatly complicates the mathematical model (Figure 1).

In order to solve the system of partial differential equations given above (7–9), it is necessary to introduce the appropriate boundary conditions. For the three physical properties of a fluid (velocity, microrotation and temperature) we define the values of a fluid on the upper and lower plate as boundary conditions. As both plates are fixed, and assuming there is no slipping of the fluid on the plates, the velocity and microrotation in boundary conditions must equal zero ³⁹ :

$$\begin{array}{c}u=0,\omega =0\mathrm{for}y=0\hfill \\ u=0,\omega =0\mathrm{for}y=h.\hfill \end{array}$$

(10)

Regarding the boundary conditions for temperature, the isothermal boundary conditions are taken, that is, the fluid temperature is equal to plate temperature:

$$T=T_{w2}\mathrm{for}y=0\mathrm{and}T=T_{w1}\mathrm{for}y=h$$

(11)

In order to solve the system of partial differential equations (7–9), in addition to boundary conditions (10–11), the appropriate mathematical transformations and dimensionless parameters are introduced thanks to which the equations are reduced to a dimensionless form. The above transformations and dimensionless parameters are given by expressions (12) and are well known in the theory of MHD flows of micropolar fluids ²⁵ :

$$\begin{array}{c}{y}^{*}=\frac{y}{h},u^{*}=\frac{u}{U},{\omega }^{*}=\frac{\omega }{{\omega }_0},{\omega }_0=\frac{U}{h},U=\frac{h^{2}P}{\mu },\hfill \\ P=-\frac{\partial p}{\partial x}=\mathrm{const},\theta =\frac{T-T_{w2}}{T_{w1}-T_{w2}},\hfill \\ \mathrm{Ha}=\mathrm{Bh}\sqrt{\frac{\sigma }{\mu }},\mathrm{Pr}=\frac{\mu c_p}{k},\mathrm{Ec}=\frac{U^2}{c_p(T_{w1}-T_{w2})},\hfill \\ K=\frac{\lambda }{\mu },\Gamma =\frac{\gamma }{\mu h^2},R=\frac{h^2}{\chi }.\hfill \end{array}$$

(12)

The given transformations are: $y^{*},u^{*},{\omega }^{*},\theta$ – non-dimensional coordinate, velocity, microrotation and temperature, respectively, while non-dimensional parameters are: $\mathrm{Ha}$ – Hartmann number, $\mathrm{Pr}$ – Prandtl number, $\mathrm{Ec}$ – Eckert number, $K$ – coupling parameter, $\Gamma$ – spin gradient viscosity parameter, $R$ – porosity parameter.

After applying the introduced transformations and dimensionless parameters, the system of equations (7–9) is reduced to the following system of equations in a dimensionless form:

$$(1+K)\frac{d^2{u}^{*}}{d{y}^{*2}}+K\frac{d\omega }{\mathrm{dy}}-(H{a}^2+R)u+1=0$$

(13)

$$\Gamma \frac{d^2{\omega }^{*}}{d{y}^{*2}}-K\frac{d{u}^{*}}{\mathrm{dy}}-2K{\omega }^{*}=0$$

(14)

$$\frac{d^2\theta }{d{y}^{*2}}+(1+K)\mathrm{PrEc}{\left(\frac{d{u}^{*}}{d{y}^{*}}\right)}^2+\mathrm{PrEc}(H{a}^2+R)u^{*2}=0$$

(15)

while the corresponding boundary conditions (10–11) obtain the following form:

$$\begin{array}{c}{u}^{*}=0,{\omega }^{*}=0,\theta =0\mathrm{for}{y}^{*}=0\hfill \\ u^{*}=0,{\omega }^{*}=0,\theta =1\mathrm{for}{y}^{*}=1\hfill \end{array}$$

(16)

As already mentioned above, when solving the previously transformed dimensionless system of equations, the equations (13) and (14) must be solved together since they are coupled. From the foregoing equations, after the appropriate mathematical transformations and operations are performed, we obtain a fourth-degree differential equation for velocity:

$$\frac{d^4{u}^{*}}{d{y}^{*4}}-a\frac{d^2{u}^{*}}{d{y}^{*2}}+b{u}^{*}-d=0$$

(17)

When obtaining the equation (17), the constants $a,b$ and $d$ were introduced, defined by the following expressions (18):

$$a=B^{*}+E-A{D}^{*},b=E{B}^{*},d=\mathrm{EC},$$

(18)

where:

$$A=\frac{K}{1+K},B^{*}=\frac{H{a}^2+R}{1+K},C=\frac{1}{1+K},D^{*}=\frac{K}{\Gamma },E=\frac{2K}{\Gamma }.$$

(19)

When solving the differential equations (17), we obtain the three possible solutions for velocity which depend on the square root of a characteristic equation. The corresponding particular solution of the equation (17) is:

$$u_p^{*}=\frac{d}{b}=\frac{C}{B^{*}},$$

(20)

while the characteristic equation for the homogenous part of the equation (17) is given in the following mathematical notation:

$${\epsilon }^4-a{\epsilon }^2+b=0,$$

or after the transformation in the following form:

$$\mathrm{for}{\epsilon }^2=r\to r^2-\mathrm{ar}+b=0.$$

(21)

We can now determine square roots of a characteristic homogeneous equation (21), which are given in the following mathematical notation:

$$r_1=\frac{1}{2}(a+\sqrt{a^2-4b}),r_2=\frac{1}{2}(a-\sqrt{a^2-4b})$$

(22)

Further analysis of the solution to the equation (17) is performed based on the square root of the characteristic homogenous equation (22). As the value of the subroot $a^2-4b$ can be higher, lower or equal to zero, this means that the characteristic homogenous equation (21) and consequently the equation (17) have three possible solutions. In this particular case, the expression $a^2-4b$ depends on the intensity of the external magnetic field B and also on the porous medium permeability $\chi$ , the distance between the plates $h$ and finally on the physical characteristics of a micropolar fluid ( $\mu ,\lambda ,\gamma ,\sigma )$ . Any changes in the listed quantities can affect the change in the value of the expression $a^2-4b$ . This can be illustrated with one concrete example. We will observe the flow of a particular micropolar fluid in a porous medium with following characteristics^40,41:

$$\begin{array}{c}\mu =2\times {10}^{-4}[\mathrm{Pa}·s];\lambda =1.8\times {10}^{-4}[\mathrm{Pa}·s];\hfill \\ \gamma =1.57\times {10}^{-9}[N·s];\sigma =0.6[S/m]\hfill \end{array}$$

while we take $B=1\left[T\right]$ to be the intensity of the magnetic field, and $\chi =1\times {10}^{-6}\left[m^2\right]$ and $h=0,01\left[m\right]$ for the values of the porous medium permeability and the distance between the plates, respectively. For the set values, the expression $a^2-4b$ obtains the following value:

$$a^2-4b=5.45$$

Based on the parameters established above we can see that the value of the expression $a^2-4b$ is a positive one. However, if we were to change only the intensity of the magnetic field from $B=1\left[T\right]$ to $B=0.1\left[T\right]$ , the expression $a^2-4b$ has a negative value:

$$a^2-4b=-1.97$$

Finally, for the value of the magnetic field intensity $B=0.523\left[T\right]$ , the expression $a^2-4b$ obtains the value which is close to zero:

$$a^2-4b\approx 0.$$

The foregoing analysis and the discussion of a concrete example bring us to the conclusion that in the analysis of the micropolar fluid flow in a porous medium based on the implemented mathematical model, it is necessary to consider all of the three possible solutions to the differential equation for velocity (17). The solutions are given in the following notation, depending on the values of the expression $a^2-4b$ :

• - in the case: $\sqrt{a^2-4b}>0$

$$\begin{array}{c}\multicolumn{1}{c}{u^{*}=C_1\exp \left({\delta }_1{y}^{*}\right)+C_2\exp \left({\delta }_2{y}^{*}\right)+C_3\exp \left({\delta }_3{y}^{*}\right)}\\ +C_4\exp \left({\delta }_4{y}^{*}\right)+\frac{d}{b},\hfill \end{array}$$

(23)

• - in the case: $\sqrt{a^2-4b}=0$

$$u^{*}=(C_5+C_6{y}^{*})\exp \left({\xi }_1{y}^{*}\right)+(C_7+C_8{y}^{*})\exp \left({\xi }_2{y}^{*}\right)+\frac{d}{b},$$

(24)

• - in the case: $\sqrt{a^2-4b}<0$

$$\begin{array}{c}\multicolumn{1}{c}{u^{*}=[C_9\cos \left({\beta }_1{y}^{*}\right)+C_{10}\sin \left({\beta }_1{y}^{*}\right)]\exp \left({\alpha }_1{y}^{*}\right)}\\ +[C_{11}\cos \left({\beta }_1{y}^{*}\right)+C_{12}\sin \left({\beta }_1{y}^{*}\right)]\exp (-{\alpha }_1{y}^{*})+\frac{d}{b}.\hfill \end{array}$$

(25)

Now, three possible solutions for microrotation are given, respectively:

$$\begin{array}{c}\multicolumn{1}{c}{{\omega }^{*}=C_1{\mathfrak{D}}_1\exp \left({\delta }_1{y}^{*}\right)+C_2{\mathfrak{D}}_2\exp \left({\delta }_2{y}^{*}\right)+C_3{\mathfrak{D}}_3\exp \left({\delta }_3{y}^{*}\right)}\\ +C_4{\mathfrak{D}}_4\exp \left({\delta }_4{y}^{*}\right),\hfill \end{array}$$

(26)

$${\omega }^{*}=(E_{13}+E_{14}{y}^{*})\exp \left({\xi }_1{y}^{*}\right)+(E_{15}+E_{16}{y}^{*})\exp \left({\xi }_2{y}^{*}\right),$$

(27)

$$\begin{array}{c}\multicolumn{1}{c}{{\omega }^{*}=[P_3^{*}\sin \left({\beta }_1{y}^{*}\right)+P_4^{*}\cos \left({\beta }_1{y}^{*}\right)]\exp \left({\alpha }_1{y}^{*}\right)}\\ +[P_5^{*}\sin \left({\beta }_1{y}^{*}\right)+P_6^{*}\cos \left({\beta }_1{y}^{*}\right)]\exp (-{\alpha }_1{y}^{*})\hfill \end{array}$$

(28)

and for non-dimensional temperature, as well:

$$\begin{array}{l}\theta =-PrEc[\frac{1}{4{\delta }_1^2}{}^{1}Cexp\left(2{\delta }_1{y}^{*}\right)+\frac{1}{4{\delta }_2^2}{}^{2}Cexp\left(2{\delta }_2{y}^{*}\right)+\frac{1}{4{\delta }_3^2}{}^{3}Cexp\left(2{\delta }_3{y}^{*}\right)+\frac{1}{4{\delta }_4^2}{}^{4}Cexp\left(2{\delta }_4{y}^{*}\right)\\ +\frac{{}^{2}D}{{\left({\delta }_1+{\delta }_3\right)}^2}exp\left(\left({\delta }_1+{\delta }_3\right)y^{*}\right)+\frac{{}^{3}D}{{\left({\delta }_1+{\delta }_4\right)}^2}exp\left(\left({\delta }_1+{\delta }_4\right)y^{*}\right)\\ +\frac{{}^{1}E}{{\left({\delta }_2+{\delta }_3\right)}^2}exp\left(\left({\delta }_2+{\delta }_3\right)y^{*}\right)+\frac{{}^{2}E}{{\left({\delta }_2+{\delta }_4\right)}^2}exp\left(\left({\delta }_2+{\delta }_4\right)y^{*}\right)\\ +\left(\frac{1}{{\delta }_1^2}{}^{1}Fexp\left({\delta }_1{y}^{*}\right)+\frac{1}{{\delta }_2^2}{}^{2}Fexp\left({\delta }_2{y}^{*}\right)+\frac{1}{{\delta }_3^2}{}^{3}Fexp\left({\delta }_3{y}^{*}\right)+\frac{1}{{\delta }_4^2}{}^{4}Fexp\left({\delta }_4{y}^{*}\right)\right)\\ +\frac{1}{2}{}^{1}D{y}^{*2}+\frac{1}{2}{}^{3}E{y}^{*2}+\frac{1}{2}{}^{5}F{y}^{*2}+{{}^{1}H}_1{y}^{*}+{{}^{1}H}_2],\end{array}$$

(29)

$$\begin{array}{c}\theta =-\mathrm{PrEc}[(({\mathrm{\Omega }}_{28}+{\mathrm{\Omega }}_{29}{y}^{*}+{\mathrm{\Omega }}_{30}{y}^{*2})\exp \left(2{\xi }_1{y}^{*}\right)+({\mathrm{\Omega }}_{31}+{\mathrm{\Omega }}_{32}{y}^{*}+{\mathrm{\Omega }}_{33}{y}^{*2})\exp \left(2{\xi }_2{y}^{*}\right))\hfill \\ +({\mathrm{\Omega }}_{36}^{*}+{\mathrm{\Omega }}_{37}^{*}{y}^{*})\exp \left({\xi }_2{y}^2\right)+{\mathrm{\Omega }}_{38}^{*}{y}^{*2}+{\mathrm{\Omega }}_{39}^{*}{y}^{*3}+{\mathrm{\Omega }}_{40}^{*}{y}^{*4}+{}^2{H}_1{y}^{*}+{}^2{H}_2],\hfill \end{array}$$

(30)

$$\begin{array}{c}\theta =-\mathrm{PrEc}\{\left[(\frac{1}{2{\alpha }_1}{\mathrm{\Omega }}_{45}+\frac{1}{2}({\chi }_1{\mathrm{\Omega }}_{47}-{\chi }_2{\mathrm{\Omega }}_{49})\cos \left(2{\beta }_1{y}^{*}\right)+\frac{1}{2}({\chi }_2{\mathrm{\Omega }}_{47}-{\chi }_1{\mathrm{\Omega }}_{49})\sin \left(2{\alpha }_1{y}^{*}\right))\right]\hfill \\ \exp \left(2{\alpha }_1{y}^{*}\right)+\left[(\frac{1}{2{\alpha }_1}{\mathrm{\Omega }}_{46}-\frac{1}{2}({\chi }_1{\mathrm{\Omega }}_{48}+{\chi }_2{\mathrm{\Omega }}_{50})\cos \left(2{\beta }_1{y}^{*}\right)+\frac{1}{2}({\chi }_2{\mathrm{\Omega }}_{48}-{\chi }_1{\mathrm{\Omega }}_{50})\sin \left(2{\beta }_1{y}^{*}\right))\right]\hfill \\ \exp (-2{\alpha }_1{y}^{*})-\frac{1}{2{\beta }_1}{\mathrm{\Omega }}_{51}\sin \left(2{\beta }_1{y}^{*}\right)-\frac{1}{2{\beta }_1}{\mathrm{\Omega }}_{52}\cos \left(2{\beta }_1{y}^{*}\right)+\hfill \\ [({\mathrm{\Omega }}_{53}{\chi }_1-{\mathrm{\Omega }}_{55}{\chi }_2)\cos \left({\beta }_1{y}^{*}\right)+({\mathrm{\Omega }}_{53}{\chi }_2+{\mathrm{\Omega }}_{55}{\chi }_1)\sin \left({\beta }_1{y}^{*}\right)]\exp \left({\alpha }_1{y}^{*}\right)\hfill \\ +[({\mathrm{\Omega }}_{54}{\chi }_1-{\mathrm{\Omega }}_{56}{\chi }_2)\cos \left({\beta }_1{y}^{*}\right)-({\mathrm{\Omega }}_{54}{\chi }_2-{\mathrm{\Omega }}_{56}{\chi }_1)\sin \left({\beta }_1{y}^{*}\right)]\exp (-{\alpha }_1{y}^{*})+\frac{1}{2}{\mathrm{\Omega }}_{57}{y}^{*2}+{}^3{H}_1{y}^{*}+{}^3{H}_2\}.\hfill \end{array}$$

(31)

In order to obtain a comprehensive analysis of the flow, in addition to the solutions for velocity, microrotation and temperature, this manuscript provides solutions for shear stress at plates, as well as a dimensionless coefficient for heat transfer at plates – the Nusselt number.

As we have already established that the stress tensor in micropolar fluids is not symmetrical, shear stress has a somewhat different definition compared to that of Newtonian fluids. Shear stress is defined in this paper in the following form ⁴² :

$$\tau =(\mu +\lambda )\frac{d{u}^{*}}{d{y}^{*}}+\lambda \omega$$

(32)

Based on the discussion above, as regards the solutions for velocity, microrotation and temperature, the solution for shear stress is:

• - in the case: $\sqrt{a^2-4b}>0$

$$\begin{array}{l}\tau =\left(\mu +\lambda \right)(C_1{\delta }_{1}exp\left({\delta }_1{y}^{\ast }\right)+C_2{\delta }_{2}exp\left({\delta }_2{y}^{\ast }\right)+C_3{\delta }_{3}exp\left({\delta }_3{y}^{\ast }\right)\\ +C_4{\delta }_{4}exp\left({\delta }_4{y}^{\ast }\right))+\lambda (C_1{\mathfrak{D}}_{1}exp\left({\delta }_1{y}^{\ast }\right)+C_2{\mathfrak{D}}_{2}exp\left({\delta }_2{y}^{\ast }\right)\\ +C_3{\mathfrak{D}}_{3}exp\left({\delta }_3{y}^{\ast }\right)+C_4{\mathfrak{D}}_{4}exp\left({\delta }_4{y}^{\ast }\right))\end{array}$$

(33)

• - in the case: $\sqrt{a^2-4b}=0$

$$\begin{array}{l}\tau =\left(\mu +\lambda \right)[C_{6}exp\left({\xi }_1{y}^{\ast }\right)+{\xi }_1\left(C_5+C_6{y}^{\ast }\right)exp\left({\xi }_1{y}^{\ast }\right)\\ +C_{8}exp\left({\xi }_2{y}^{\ast }\right)+\left(C_7+C_8{y}^{\ast }\right)exp\left({\xi }_2{y}^{\ast }\right)]\\ +\lambda \left[\left(E_{13}+E_{14}{y}^{\ast }\right)exp\left({\xi }_1{y}^{\ast }\right)+\left(E_{15}+E_{16}{y}^{\ast }\right)exp\left({\xi }_2{y}^{\ast }\right)\right]\end{array}$$

(34)

• - in the case: $\sqrt{a^2-4b}<0$

$$\begin{array}{l}\tau =\left(\mu +\lambda \right)\{\left({\alpha }_1{C}_{10}-{\beta }_1{C}_9\right)sin\left({\beta }_1{y}^{\ast }\right)+\left({\beta }_1{C}_{10}+{\alpha }_1{C}_9\right)cos\left({\beta }_1{y}^{\ast }\right)]exp\left({\alpha }_1{y}^{\ast }\right).\\ +\left[\left({\beta }_1{C}_{12}-{\alpha }_1{C}_{11}\right)cos\left({\beta }_{1}y\right)-\left({\alpha }_1{C}_{12}+{\beta }_1{C}_{11}\right)sin\left({\beta }_1{y}^{\ast }\right)\right]exp\left(-{\alpha }_{1}y\right)\}\\ +\lambda \left\{\left[P_3^{\ast }sin\left({\beta }_1{y}^{\ast }\right)+P_4^{\ast }cos\left({\beta }_1{y}^{\ast }\right)\right]exp\left({\alpha }_1{y}^{\ast }\right)+\left[P_5^{\ast }sin\left({\beta }_1{y}^{\ast }\right)+P_6^{\ast }cos\left({\beta }_1{y}^{\ast }\right)\right]exp\left(-{\alpha }_1{y}^{\ast }\right)\right\}\end{array}$$

(35)

As the Nusselt number represents the ratio of convective to conductive heat transfer at a boundary in a fluid, it is very important to understand heat transfer on the plates. It is found that the local Nusselt number is higher for micropolar fluids compared to Newtonian fluids. ⁴³ Due to that, the Nusselt number on the plates in the considered problem is:

$$N{u}_{\mathrm{up}}={\frac{d\theta }{d{y}^{*}}|}_{y^{*}=1}\mathrm{and}N{u}_{\mathrm{lp}}={\frac{d\theta }{d{y}^{*}}|}_{y^{*}=0}$$

(36)

and the corresponding solutions are as follows:

• - in the case: $\sqrt{a^2-4b}>0$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Nu}=-\mathrm{PrEc}[\frac{1}{2{\delta }_1}{}^1\mathrm{Cexp}\left(2{\delta }_1{y}^{*}\right)+\frac{1}{2{\delta }_2}{}^2\mathrm{Cexp}\left(2{\delta }_2{y}^{*}\right)}\\ +\frac{1}{2{\delta }_3}{}^3\mathrm{Cexp}\left(2{\delta }_3{y}^{*}\right)+\frac{1}{2{\delta }_4}{}^4\mathrm{Cexp}\left(2{\delta }_4{y}^{*}\right)\hfill \\ +{}^{1}D{y}^{*}\frac{{}^{2}D}{({\delta }_1+{\delta }_3)}\exp \left(({\delta }_1+{\delta }_3)y^{*}\right)+\frac{{}^{3}D}{({\delta }_1+{\delta }_4)}\exp \left(({\delta }_1+{\delta }_4)y^{*}\right)\hfill \\ +\frac{{}^{1}E}{({\delta }_2+{\delta }_3)}\exp \left(({\delta }_2+{\delta }_3)y^{*}\right)+\frac{{}^{2}E}{({\delta }_2+{\delta }_4)}\exp \left(({\delta }_2+{\delta }_4)y^{*}\right)\hfill \\ +{}^{3}E{y}^{*}+\frac{1}{{\delta }_1}{}^1\mathrm{Fexp}\left({\delta }_1{y}^{*}\right)+\frac{1}{{\delta }_2}{}^2\mathrm{Fexp}\left({\delta }_2{y}^{*}\right)+\frac{1}{{\delta }_3}{}^3\mathrm{Fexp}\left({\delta }_3{y}^{*}\right)\hfill \\ +\frac{1}{{\delta }_4}{}^4\mathrm{Fexp}\left({\delta }_4{y}^{*}\right)+{}^{5}F{y}^{*}+{}^1{H}_1]\hfill \end{array}$$

(37)

• - in the case: $\sqrt{a^2-4b}=0$

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Nu}=-\mathrm{PrEc}[{\mathrm{\Omega }}_{15}\exp \left(2{\xi }_1{y}^{*}\right)+{\mathrm{\Omega }}_{16}{y}^{*}\exp \left(2{\xi }_1{y}^{*}\right)}\\ +{\mathrm{\Omega }}_{17}{y}^{*2}\exp \left(2{\xi }_1{y}^{*}\right)+{\mathrm{\Omega }}_{18}\exp \left(2{\xi }_2{y}^{*}\right)+{\mathrm{\Omega }}_{19}{y}^{*}\exp \left(2{\xi }_2{y}^{*}\right)\hfill \\ +{\mathrm{\Omega }}_{20}{y}^{*2}\exp \left(2{\xi }_2{y}^{*}\right)+{\mathrm{\Omega }}_{21}\exp \left(({\xi }_1+{\xi }_2)y^{*}\right)\hfill \\ +{\mathrm{\Omega }}_{22}{y}^{*}\exp \left(({\xi }_1+{\xi }_2)y^{*}\right)+{\mathrm{\Omega }}_{23}{y}^{*2}\exp \left(({\xi }_1+{\xi }_2)y^{*}\right)\hfill \\ +{\mathrm{\Omega }}_{24}\exp \left({\xi }_1{y}^{*}\right)+{\mathrm{\Omega }}_{25}{y}^{*}\exp \left({\xi }_1{y}^{*}\right)+{\mathrm{\Omega }}_{26}\exp \left({\xi }_2\right)]\hfill \end{array}$$

(38)

• - in the case: $\sqrt{a^2-4b}<0$

$$\begin{array}{l}Nu=-PrEc[{\mathrm{\Omega }}_{45}exp\left(2{\alpha }_1{y}^{*}\right)-{\mathrm{\Omega }}_{46}exp\left(-2{\alpha }_1{y}^{*}\right)+{\mathrm{\Omega }}_{47}cos\left(2{\beta }_1{y}^{*}\right)exp\left(2{\alpha }_1{y}^{*}\right)\\ +{\mathrm{\Omega }}_{48}cos\left(2{\beta }_1{y}^{*}\right)exp\left(-2{\alpha }_1{y}^{*}\right)+{\mathrm{\Omega }}_{49}sin\left(2{\beta }_1{y}^{*}\right)exp\left(2{\alpha }_1{y}^{*}\right)\\ +{\mathrm{\Omega }}_{50}sin\left(2{\beta }_1{y}^{*}\right)exp\left(-2{\alpha }_1{y}^{*}\right)-{\mathrm{\Omega }}_{51}cos\left(2{\beta }_1{y}^{*}\right)+{\mathrm{\Omega }}_{52}sin\left(2{\beta }_1{y}^{*}\right)\\ +{\mathrm{\Omega }}_{53}cos\left({\beta }_1{y}^{*}\right)exp\left({\alpha }_1{y}^{*}\right)-{\mathrm{\Omega }}_{54}cos\left({\beta }_1{y}^{*}\right)exp\left(-{\alpha }_1{y}^{*}\right)+{\mathrm{\Omega }}_{55}sin\left({\beta }_1{y}^{*}\right)\\ exp\left({\alpha }_1{y}^{*}\right)+{\mathrm{\Omega }}_{56}sin\left({\beta }_1{y}^{*}\right)exp\left(-{\alpha }_1{y}^{*}\right)+{\mathrm{\Omega }}_{57}{y}^{*}+{}^3{H}_1]\end{array}$$

(39)

At the end, expression for flow rate is given:

$$q={\int }_0^1{u}^{*}\left(y^{*}\right)d{y}^{*}$$

(40)

and the solutions are as follows:

• - in the case: $\sqrt{a^2-4b}>0$

$$\begin{array}{c}\multicolumn{1}{c}{q=\frac{C_1}{{\delta }_1}(\exp \left({\delta }_1\right)-1)+\frac{C_2}{{\delta }_2}(\exp \left({\delta }_2\right)-1)+\frac{C_3}{{\delta }_3}(\exp \left({\delta }_3\right)-1)}\\ +\frac{C_4}{{\delta }_4}(\exp \left({\delta }_4\right)-1)+\frac{d}{b}\hfill \end{array}$$

(41)

• - in the case: $\sqrt{a^2-4b}=0$

$$\begin{array}{c}\multicolumn{1}{c}{q=\frac{b((C_6+C_5){\xi }_1-C_6)\exp \left({\xi }_1\right)+d{\xi }_1^2-C_{5}b{\xi }_1+C_{6}b}{b{\xi }_1^2}}\\ +\frac{((C_8+C_7){\xi }_2-C_8)\exp \left({\xi }_2\right)+C_8}{{\xi }_2^2}-\frac{C_7}{{\xi }_2}\hfill \end{array}$$

(42)

• - in the case: $\sqrt{a^2-4b}<0$

$$\begin{array}{c}q=\exp (-{\alpha }_1)\{(\exp \left({\alpha }_1\right){\beta }_1^2+{\alpha }_1^2\exp \left({\alpha }_1\right))d+\hfill \\ ((C_9\exp \left(2{\alpha }_1\right)+C_{11})b{\beta }_1+(C_{10}{\alpha }_1\exp \left(2{\alpha }_1\right)-C_{12}{\alpha }_1)b)\sin \left({\beta }_1\right)\hfill \\ +((-C_{10}\exp \left(2{\alpha }_1\right)-C_{12})b{\beta }_1+(C_9{\alpha }_1\exp \left(2{\alpha }_1\right)-C_{11}{\alpha }_1)b)\hfill \\ \cos \left({\beta }_1\right)+(C_{12}+C_{10})\exp \left({\alpha }_1\right)b{\beta }_1+(C_{11}-C_9)\exp \left({\alpha }_1\right)b{\alpha }_1\}/\hfill \\ (b({\beta }_1^2+{\alpha }_1^2))\hfill \end{array}$$

(43)

The constants that occur in equations (23)–(43) are given in the Appendix.

Results analysis

In order to gain better understanding of the nature of micropolar fluid flows, this section of the paper deals with the influence the magnetic field (Hartmann number $H_a$ ) has on dimensionless values for velocity, microrotation and temperature, as well as shear stress, the Nusselt number at plates and the flow rate between the plates. In addition, we will consider the influence of physical characteristics of a micropolar fluid, that is, additional viscosities $\lambda$ and $\gamma$ using the coupling parameter ( $K$ ) and the spin gradient viscosity parameter ( $\Gamma$ ). In the end, we will use the charts to show how physical characteristics of fluid flows depend on a porosity parameter ( $R$ ). The analysis provided herein will serve to draw some general conclusions about the nature of micropolar fluid flows in a porous medium.

Before we proceed to further analyse and discuss the problem under consideration using the charts, note that Figure 2(a) and (b) illustrate the comparison between velocity profiles for the problem dealt with in this paper and the results obtained in the paper by Umar. ²⁰ Figure 2(a) shows the dependency of the change in velocity profiles based on the changes in the value of the coupling parameter $K$ , whereas Figure 2(b) compares velocity profiles for values $K=0.1$ and 0.4. The comparisons of dimensionless velocities given in Figure 2(a) and (b), show a very good match. Since the solutions in this paper are given in closed form, they certainly yield the accurate values for velocity, and the comparison confirms the validity of the model as well, that is, the validity of equations used to describe the model.

Figure 2.

(a) Velocity profiles for different values of coupling parameter $K$ and (b) comparison between obtained results and results from a paper by Muhammad Umar.

The first four figures (Figures 3(a), (b), 4(a) and (b)), represent the influence of the magnetic field as the function of the Hartmann number ( $\mathrm{Ha}$ ) and the influence of the porosity parameter ( $R$ ) on dimensionless velocity profiles and the fluid flow.

Figure 3.

(a) Velocity profiles and (b) flow rate for different values of Hartmann number $\mathrm{Ha}$ .

Figure 4.

Influence of porosity parameter $R$ on: (a) velocity profiles and (b) flow rate.

The influence of the magnetic field is seen through the action of the Lorentz force. As it is well known, due to the flow of the electrically conductive fluid under the influence of the external magnetic field the Lorentz force is induced. The Lorentz force acts in the counter direction of that of the flow and it has the tendency to slow down this flow. The intensity of the Lorentz force is linearly dependent on the velocity of fluid flow, and since the flow velocity is the greatest mid-channel, this is where the Lorentz force causes the greatest ‘breaking’ effect compared to other parts of the flow domain. This can clearly be seen in Figure 3(a), where with the increase of the Hartmann number the value of the dimensionless velocity drops along the entire width of the channel, particularly in the mid-channel.^44,45 With this action, the magnetic field tends to flatten out the velocity profile curve along the width between the plates. The consequence of this is the decrease in the fluid flow with the increase in the intensity of the magnetic field, that is, the Hartmann number, as shown in Figure 3(b). We can see that for slight changes in the Hartmann number from 0 to 1 the flow curve has a mild drop, whereas with further increase of the Hartmann number for values from 1 to 7, a steep change in the flow curve occurs, only to return to the mild drop in the end for values between 7 and 10. This is the result of changes in the velocity as the magnetic field intensity increases, which is clear from the trends in the dimensionless velocity as the Hartmann number rises, as shown in Figure 3(a).

The other two charts, Figure 4(a) and (b), represent the influence of material porosity on the velocity profile and the flow rate through a porous medium. Based on the definition of the porosity parameter $R$ in expressions (12), it is evident that, as the value of the porosity parameter rises, the voids in the porous medium get smaller. In consequence to the decrease of the flow domain with the increase in porosity parameter $R$ , the fluid begins to decelerate, as clearly seen in Figure 4(a). This is to be expected and has already been demonstrated by numerous authors.^46,47 Figure 4(b), which illustrates the dependency of the flow on the porosity parameter $R$ , we can see that the flow drops as $R$ rises, and that the dependency between the flow rate $q$ and the porosity parameter $R$ is linear in nature.

Further analysis of results will show the influence of additional viscosities $\lambda$ and $\gamma$ on the nature of micropolar fluid flows. When presenting the mathematical model for the flow of a micropolar fluid, we emphasized that such fluids do not have a symmetrical stress tensor and that the two additional viscosities are taken into account in their consideration. Their influence will be presented using dimensionless parameters: the coupling parameter $K$ and the spin gradient viscosity parameter $\Gamma$ .

The coupling parameter $K$ is characteristic of a coupling between linear and rotational motion, resulting from micro motion of molecules or particles embedded in the fluid (e.g. additives in lubricants ⁴⁸ ). Thus, the coupling parameter $K$ represents the coupling between the Newtonian fluid viscosity and the additional rotational viscosity $\lambda$ . In Figure 5(a) we can see that the increased coupling parameter $K$ results in decreased intensity of a dimensionless velocity. As the rise in the value of the coupling parameter $K$ is a consequence of the increase in the additional viscosity $\lambda$ , that is, the increase in total viscosity of a micropolar fluid, the obtained results are to be expected. For the constant drop in pressure along the direction of the flow, the velocity drops as the total viscosity of a micropolar fluid rises. In addition, in Figure 5(b) it can be seen that with the significant increase in the additional angular viscosity $\gamma$ , that is, the spin gradient viscosity parameter $\Gamma$ , there occurs a drop in the velocity between the plates. This can be accounted for in the way similar to that concerning the influence of additional viscosity $\lambda$ , since decreased velocity is a consequence of the increase in the total fluid viscosity.

Figure 5.

Velocity in function of: (a) coupling parameter $K$ and (b) spin viscosity parameter $\Gamma$ .

The following set of figures shows the influence of the Hartmann number ( $\mathrm{Ha}$ ), the porosity parameter ( $R$ ), the coupling parameter $K$ and the spin gradient viscosity parameter $\Gamma$ on the microrotation vector ${\omega }^{*}$ (Figure 6(a)–(d)).

Figure 6.

Microrotation for different values of: (a) Hartmann number $\mathrm{Ha}$ , (b) porosity parameter $R$ , (c) coupling parameter $K$ and (d) spin viscosity parameter $\Gamma$ .

Regarding the influence of the Hartmann number on the microrotation field (Figure 6(a)) we can see that as the action of the external magnetic field rises, the absolute value of the microrotation vector decreases along the entire channel width. ²⁵ We have mentioned earlier that the microrotation vector ${\omega }^{*}$ , defined by the equation (14), is one of the basic characteristics of micropolar fluids and hence, based on the obtained results, it can be inferred that the magnetic field attenuates the character of a micropolar fluid. The influence of the porosity parameter $R$ on the microrotation vector (Figure 6(b)) is identical to the influence it has on the velocity. In fact, as the flow domain, that is, the permeability of the porous medium decreases, the motion characteristics of the micropolar fluid – velocity and microrotation decrease.

However, in Figure 6(c) we can see that as $K$ rises, so does the absolute value of the microrotation vector close to the plates,^20,49 and is therefore obvious that additional viscosity $\lambda$ emphasizes the characteristics of micropolar fluids. This means that additional viscosity $\lambda$ emphasizes the small particle microrotation, which particles were embedded in the fluid. On the other hand, in Figure 6(d) the results indicate that as the spin gradient viscosity parameter $\Gamma$ rises, the absolute value of the microrotation vector gradually falls, which brings us to the conclusion that the increase in additional angular viscosity $\gamma$ aggravates the microrotation and lessens the generation of vortex in the fluid, which is generally the main characteristic of micropolar fluids in contrast to classic Newtonian fluids. ⁴¹

The influence of the magnetic field and the physical characteristics of a fluid on the dimensionless temperature profile $\theta$ , as a function of Hartmann number and Prandtl number, are shown in Figure 7(a) and (b).

Figure 7.

Dimensionless temperature $\theta$ in function of: (a) Prandtl number $\mathrm{Pr},$ and (b) Hartmann number $\mathrm{Ha}$ .

The values of the Prandtl number are taken based on the values of some real fluids which act as micropolar fluids according to the nature of their flow (e.g. biological fluids like human blood) or the fluids that can be considered to be micropolar when appropriate additives are included (various types of lubricants with additives). In Figure 7(a) we can see that for lower values of the Prandtl number there is only conduction between the plates as the main heat transfer. ²⁵ With higher values of $\mathrm{Pr}$ , the flow of fluid heats up in the middle of the channel, which is mainly the result of the Joule heating process.

Figure 7(b) shows that at lower values of the Hartmann number viscous dissipation causes the increase in the temperature close to the plates, while the Joule heating effect heats up the fluid in the middle of the channel. As we already know, both the viscous dissipation and Joule heating depend on the velocity gradient which is more prominent at lower values of the Hartmann number (Figure 3(a)). However, as the influence of the external magnetic field rises, the temperature field becomes uniform and for the Hartmann number $\mathrm{Ha}=10$ , conduction becomes the dominant heat transfer between the plates.

The final set of figures shows the influence of the Hartmann number ( $\mathrm{Ha}$ ), the porosity parameter ( $R$ ) and the coupling parameter ( $K$ ) on the Nusselt number and shear stress at upper and lower plates.

The first two figures, Figure 8(a) and (b), represent the influence of the Hartmann number ( $\mathrm{Ha}$ ) and the coupling parameter ( $K$ ) on the Nusselt number and shear stress at upper and lower plates. In Figure 8(a) it can be seen that the value of the Nusselt number at both the upper and the lower plate drops as the Hartmann number rises. This is a well-known fact in connection with the Hartmann type of flow, since with the increase of the Hartmann number conduction becomes the dominant heat transfer mechanism (Figure 7(b)). On the other hand, higher values of the coupling parameter $K$ result in a lower heat transfer at upper and lower plates, which is all a consequence of the increased total fluid viscosity (the increased coupling parameter $K$ means increased additional viscosity $\lambda$ , and increased total fluid viscosity). The higher total fluid viscosity means a slower fluid flow (Figure 5(a)), that is, a lower rate of interaction and heat transfer at upper and lower plates.

Figure 8.

Influence of Hartmann number $\mathrm{Ha}$ and coupling parameter $K$ on: (a) Nusselt number at upper and lower plates and (b) shear stress at upper and lower plates.

Figure 8(b) shows that the increased Hartmann number leads to the drop in the absolute value of shear stress at upper and lower plates. As expected, at higher values of the Hartmann number, the Lorentz force reduces the velocity of the fluid flow along the entire width of the channel (Figure 3(a)), and consequently the intensity of shear stress falls at upper and lower plates. Concerning the influence of the coupling parameter $K$ on shear stress, we can see that the value of shear stress at upper and lower plates is higher in fluids that have higher values of additional viscosity $\lambda$ , and the coupling parameter $K$ . ¹⁴ Based on the expression used to represent shear stress (32), the value of shear stress increases as the total fluid viscosity rises.

With higher values of the porosity parameter $R$ , Nusselt number at upper and lower plates drops (Figure 9(a)). It should be noted that with lower values of the porosity parameter $R$ , there is a viscous fluid heating close to the plates, in addition to the Joule increase in fluid temperature in the middle of the channel, resulting from the higher velocity of the fluid flow (Figure 4(a)). In contrast, at higher values of the porosity parameter and decreased velocity of the fluid flow, the values of heat transfer at upper and lower plates drop (Nusselt number). The influence of the coupling parameter $K$ shows the same tendency as in Figure 8(a), verifying the results obtained earlier.

Figure 9.

Influence of porosity parameter $R$ and coupling parameter $K$ on: (a) Nusselt number at upper and lower plates and (b) shear stress at upper and lower plates.

Figure 9(b) shows the influence of the porosity parameter $R$ and the coupling parameter $K$ on shear stress at upper and lower plates. It can be seen that with the increased porosity parameter $R$ , the absolute value of shear stress drops at both upper and lower plates. As we have seen from the analysis above, increased porosity parameter $R$ causes the fluid velocity to decrease (Figure 4(a)), as a consequence of a decreased flow domain in a porous medium. Consequently, due to reducing velocity of the fluid flow, shear stress at upper and lower plates drops. The influence of additional viscosity $\lambda$ by means of coupling parameter acts just like in Figure 8(b), that is, increased additional viscosity $\lambda$ results in the rise of total fluid viscosity and thus the value of shear stress at upper and lower plates is higher in this case (expression 32).

The last two figures, Figure 10(a) and (b), show combined influence of the Hartmann number $\mathrm{Ha}$ and the porosity parameter $R$ on dimensionless heat transfer and shear stress at upper and lower plates.

Figure 10.

(a) Nusselt number at upper and lower plates and (b) shear stress at upper and lower plates, for different values of Hartmann number and porosity parameter $R$ .

The results shown in the last two figures confirm the analysis of the results which was provided in the figures given above. In Figure 10(a) we can see that the value of the Nusselt number drops at upper and lower plates as the intensity of the external magnetic field increases, which is identical to what is shown in Figure 8(a) and is a consequence of the Hartmann type of flow. Moreover, the same figure shows that the increased porosity parameter $R$ results in the decreased heat transfer at upper and lower plates, as illustrated in Figure 9(a). We encounter a similar situation in Figure 10(b), that is, the influence of the Hartmann number $\mathrm{Ha}$ and the porosity parameter $R$ on shear stress at upper and lower plates. In both cases, that is, with the increased Hartmann number and the porosity parameter, the drop in the absolute value of shear stress at upper and lower plates occurs. This influence has already been explained in detail in Figures 8(b) and 9(b).

Conclusion

The physical model of the problem considered in this paper is presented as the flow of a micropolar fluid between two plates at different temperatures in a porous medium. The mathematical model is defined by differential equations which have been solved analytically in a closed form, together with appropriate fluid and thermal boundary conditions. The analysed flow enabled us to grasp the nature of the flow of the micropolar fluid in a porous medium under the influence of an external magnetic field. The results obtained in this paper are presented in the form of an analysis through charts, which represent the influence of the magnetic field, that is, the Hartmann number ( $\mathrm{Ha}$ ), additional viscosities $\lambda$ and $\gamma$ , that is, the coupling parameter ( $K$ ) and the spin viscosity parameter ( $\Gamma$ ), in addition to porosity parameter $R$ and Prandtl number ( $\mathrm{Pr}$ ) on the nature of the micropolar fluid flow. The results obtained for velocity, microrotation, temperature, shear stress and the Nusselt number at upper and lower plates, as well as the flow rate, enabled us to perform a thorough and comprehensive analysis of the problem under consideration.

Based on the analysis performed, we can reach the following general conclusions:

• With the increased intensity of the magnetic field, the Lorentz force tends to reduce the flow rate, while the velocity profile and microrotation between the plates flatten out (‘braking’ effect), whereas the temperature field becomes uniform (conduction as the dominant heat transfer mechanism between the plates).

• Increased values of additional viscosities $\lambda$ and $\gamma$ lead to the increase in the total fluid viscosity and the decrease in the velocity of the micropolar fluid flow.

• On the other hand, while additional viscosity $\lambda$ emphasizes the characteristics of micropolar fluids, that is, the microrotation vector, the rise in additional angular viscosity $\gamma$ , aggravates the occurrence of microrotation and lessens the generation of vortex within a fluid.

• The rise in the Hartmann number $\mathrm{Ha}$ and the porosity parameter $R$ , in addition to fluid ‘braking’ effect and decreased flow, impact and decrease the intensity of heat transfer at upper and lower plates (the Nusselt number), and shear stress at plates.

• Higher values of the coupling parameter $K$ affect the rise in the absolute value of shear stress at upper and lower plates following from the definition of shear stress for polar fluids (expression 32 in the paper).