Hydromagnetic Flow of Two Immiscible Couple Stress Fluids through Porous Medium in a Cylindrical Pipe with Slip Effect

In this study, the steady hydromagnetic ﬂ ow of two immiscible couple stress ﬂ uids through a uniform porous medium in a cylindrical pipe with slip e ﬀ ect is investigated analytically. Essentially, the ﬂ ow system is divided into two regions, region I and region II, which occupy the core and periphery of the system, respectively. The ﬂ ow is driven by a constant pressure gradient applied in a direction parallel to the cylinder ’ s axis, and an external uniform magnetic ﬁ eld is applied in the direction perpendicular to the direction of ﬂ uid motion. Instead of the classical no-slip condition, the slip velocity along with vanishing couple stress boundary conditions is taken on the surface of the rigid cylinder, and continuity conditions of velocity, vorticity, shear stress, and couple stress are imposed at the ﬂ uid-ﬂ uid interface. The governing equations are modeled using the fully developed ﬂ ow conditions. The resulting di ﬀ erential equations governing the ﬂ ow in the two regions are converted to nondimensional forms using appropriate dimensionless variables. The nondimensional equations are solved analytically, and closed-form expressions for the ﬂ ow velocity, ﬂ ow rate, and stresses are derived in terms of the Bessel functions. The impacts of several parameters pertaining to the ﬂ ow such as the magnetic number, couple stress parameters, Darcy number, viscosity ratio, Reynolds number, and slip parameter on the velocities in respective regions are examined and illustrated through graphs. The ﬂ ow rate ’ s numerical values are also calculated for di ﬀ erent ﬂ uid parameters and displayed in tabular form. It is found that increasing the magnetic number, viscosity ratio, Reynolds number, and slip parameters decreases the velocities of the ﬂ uids whereas increasing the couple stress parameter, Darcy number, and pressure gradient increases ﬂ uid velocities. The results obtained in this paper show an excellent agreement with the already existing results in the literature as limiting cases.


Introduction
Over the last few decades, the study of non-Newtonian fluids has received a lot of attention as these types of fluids frequently occur in many industrial, scientific, and technological processes. Many important complex and real fluids which include molten metals, polymeric liquids, slurries, blood, liquid crystals, lubricants, soaps, greases, gelatin, and paints belong to this family of fluids. This class of fluids does not follow the Newtonian fluid theory as they possess microstructure and a nonsymmetric stress tensor in their fluid structure. For this reason, several new microcontinuum theories [1][2][3] pertaining to non-Newtonian fluids have been developed for describing the rheological behavior of numerous complex fluids. The couple stress fluid, initiated by stokes [4], is one of the popular theories of polar fluids that considers the possibility of polar effects such as the presence of couple stresses and body couples in the fluid medium. The flow behavior of various fluids that contain a substructure such as lubricants with small amounts of additives, polymers, colloidal suspensions, liquid crystals, animal and human blood, polymer-thickened oils, muddy water, and electrorheological and synthetic fluids can be modeled using the couple stress fluid theory [5]. A couple stress fluid model has been successfully employed to study the mechanism of peristalsis [6,7]. The couple stress fluid theory widely used in modeling the flow of biological fluids such as synovial fluids [8][9][10][11] and blood [12][13][14].
In realistic situations, most of the flow problems arising in the industries, manufacturing process, geology, groundwater hydrology, reservoir mechanics, biomechanics, magnetofluid dynamics, geophysics, plasma physics, and so on occur with two or more fluids of different densities/viscosities flowing immiscibly in the same channel or cylindrical pipe. Examples of these systems are the flow of several immiscible oils through the bed of rocks or soils, the flow in the rivers with several industrial fluids, blood flow in the arteries, the flow of air and fuel droplets in combustion chambers, the flow of air and exhaust gases at engine outlets, gas and Petrolia flow in pipes of oil, water-air flows around ship halts, etc. These are referred to as multiphase flows in the literature. Owing to its wide areas of applications, several researchers have studied multiphase fluid flows. Chaturani and Samy [15] as well as Sinha and Singh [16] investigated the effects of couple stresses on blood flow, and many other authors (Valanis and Sun [17], Sharan and Popel [18], and Garcia and Riahi [19]) discussed blood flow considering it as a two-phase flow in which they have assumed blood as a couple stress fluid. Besides its application in blood flow, the study of multiphase flow has several important applications in various fields of engineering and science. Umavathi et al. [20] analytically solved the problem of flow through a horizontal channel with a couple stress fluid sandwiched between viscous fluid layers. They discussed the effects of various flow parameters and concluded that the couple stress parameter influences the flow. Umavathi et al. [21] made a detailed study on the flow and heat transfer of a couple stress fluid in contact with a Newtonian fluid. Abbas et al. [22] analyzed the hydromagnetic mixed convective two-phase flow of couple stress and viscous fluids in an inclined channel. They obtained closed-form solutions of velocity and temperature profiles by using the perturbation method. Devakar et al. [23] studied the unsteady flow of couple stress fluid sandwiched between Newtonian fluids through a channel. There are many other works concerning multiphase fluid flows (see Packham and Shall [24], Rao and Usha [25], Chamkha et. al [26], Umavathi et al. [27,28], and Umavathi and Shekar [29]).
In view of its numerous applications, the researchers at present are engaged in exploring immiscible flows of fluids through a porous medium under various circumstances. Umavathi et al. [30] studied the problem of the convective flow of two immiscible fluids (couple stress and viscous fluids) through a vertical channel. They obtained an approximate solution by using the regular perturbation method. Devakar and Ramgopal [31] presented analytical solutions for the fully developed flows of two immiscible couple stress and Newtonian fluids through a nonporous and porous medium in a horizontal cylinder. Srinivas and Murthy [32] studied the flow of two immiscible couple stress fluids between two permeable beds. They obtained an exact solution to the considered problem. An analysis of the Poiseuille flow of immiscible micropolar-Newtonian fluids through concentric pipes filled with the porous medium was done by Yadav et al. [33]. Other important studies in this direction include Chamkha [34], Harmindar and Singh [35], Singh [36], and Srinivas et al. [37]. Though many researchers have worked on the flow of immiscible fluids through porous channels, the flow of immiscible non-Newtonian fluids through porous cylinders is reasonably underexplored despite its applicability in blood flows, chemical engineering, crude oil extraction, etc. One such type of problem is going to be discussed in this paper.
Magnetohydrodynamics (MHD) is also an interesting and important area of modern engineering sciences and involves the interaction of magnetic forces and electrically conducting fluids. The application of magnetic fields to the flow of immiscible fluids originates from reducing the flows for many medical and industrial purposes. The study of the hydromagnetic flow of moving fluids through a porous medium is currently a subject of great interest owing to plentiful applications in industrial, engineering, and medical devices. Owing to these applications, several studies have been conducted to examine the effect of magnetic fields on the flows of immiscible fluids. In light of this, Vajravelu et al. [38] studied the hydromagnetic unsteady flow of two conducting immiscible fluids between two permeable beds. Malashetty et al. [39] analyzed the magnetohydrodynamic two-fluid convective flow and heat transfer in an inclined composite porous medium. Raju and Nagavalli [40] studied the unsteady two-layered fluid flow and heat transfer of conducting fluids in a channel between parallel porous plates under a transverse magnetic field. Ansari and Deo [41] investigated the effect of a magnetic field on the two immiscible viscous fluids flowing in a channel filled with a porous medium. The influence of an inclined magnetic field on the Poiseuille flow of immiscible micropolar-Newtonian fluids through the horizontal porous channel where the permeability of both the regions of the horizontal porous channel has been taken differently was discussed by Yadav and Jaiswal [42]. In another paper, Jaiswal and Yadav [43] investigated the influence of a magnetic field on the Poiseuille flow of immiscible Newtonian fluids through a highly porous medium. More recently, Kumar and Agrawal [44] studied the magnetohydrodynamic pulsatile flow and heat transfer of two immiscible couple stress fluids in a porous channel.
A majority of the studies regarding the flow of immiscible non-Newtonian fluids quoted above were carried out by imposing the no-slip boundary condition. However, several theoretical and experimental studies [45][46][47][48][49][50] reveal that slip exists at the solid boundary. So, for flows containing fluids through solid boundaries, consideration of a velocity slip is more realistic and appropriate. Recently, Punnamchandar and Fekadu [51] investigated the effects of slip and uniform magnetic field on the flow of immiscible couple stress fluids in a porous medium channel. In another paper, Punnamchandar and Fekadu [52] considered the problem of the effects of slip and inclined magnetic field on the flow of immiscible fluids (couple stress fluid and Jeffrey fluid) in a porous channel. It is observed that the effects of slip and magnetic field on the flow of immiscible couple stress fluids 2 Journal of Applied Mathematics through a porous medium in a cylindrical pipe have not been discussed yet. Keeping this in view the wide potential applications of immiscible couple stress fluids flow and the importance of exact solutions described above, the goal of the current paper is to determine exact solutions for the steady hydromagnetic flow of two immiscible couple stress fluids through a porous medium in a cylindrical pipe with slip effect. The impacts of different flow parameters on the velocity field and flow rate are investigated. The slip factor in fluid flows makes the problem even more realistic and interesting, which motivated us to consider this problem. The practicality and the complexities involved due to the porosity and cylindrical nature of the geometry also make the work presented in this paper novel.

Basic Equations
The basic equations describing the flow of couple stress fluid including a Lorentz force are (Stokes [4,5]) as follows: Continuity equation (conservation of mass): Momentum equation (conservation of momentum): where the scalar quantity ρ is the couple stress fluid density and P is the fluid pressure at any point. The vectors q, f , and c are the velocity, body force per unit mass, and body couple per unit mass, respectively. The term J ! × B ! in equation (2) is the Lorentz force (electromagnetic body force) in which J ! is the electric current density and B ! is the total magnetic field.
The force stress tensor τ ij (Stokes [5]) that arises in the theory of couple stress fluids is given by The couple stress tensor m ij (Stokes [5]) that arises in the theory has the linear constitutive relation In the above, ω i,j is the spin tensor, ρc k is the body couple vector, d ij is the components of the rate of shear strain, δ ij is the Kronecker symbol, e ijk is the Levi-Civita symbol, and comma denotes covariant differentiation.
The material constants λ and μ are the viscosity coefficients, and η and η′ are the couple stress viscosity coefficients satisfying the constraints There is a length parameter l = ffiffiffiffiffiffiffi η/μ p which is a characteristic measure of the polarity of the couple stress fluid, and this parameter is identically zero in the case of nonpolar fluids.

Formulation of the Problem
The physical model concerns an axisymmetric fully developed hydromagnetic flow of two immiscible couple stress fluids flowing through a porous medium in a horizontal circular pipe of radius R 0 . Owing to the fluids' immiscibility, there are two separate regions of fluid flow: region I, or the core region, and region II, or the periphery region. The flow geometry of the problem is depicted in a cylindrical polar coordinate system ðr, θ, zÞ with the origin at the center of the tube and common axis of the cylindrical regions taken as the z-axis, as shown in Figure 1. Region I ð0 ≤ r ≤ RÞ is occupied with couple stress fluid with density ρ 1 , shear viscosity μ 1 , and couple stress viscosity η 1 , comprising the core region of the pipe whereas region II ðR ≤ r ≤ R 0 Þ is occupied by a different couple stress fluid having density ρ 2 , shear viscosity μ 2 , and couple stress viscosity η 2 , comprising the peripheral region of the pipe. The motion of the fluids in both regions is caused by a constant pressure gradient applied in a direction parallel to the cylinder's axis, i.e., z -axis, and an external uniform magnetic field of strength B 0 directed perpendicular to the flow direction is also applied.
To develop the governing equations for the considered model, the following presumptions are taken in the analysis of the current study: (i) The fluids are considered incompressible, and the flow is assumed to be steady, laminar, and fully developed (ii) Both the fluid regions are saturated with the uniform porous media of permeability k (iii) The Lorentz force is the only body force acting on the fluids, with no body couples (iv) The magnetic Reynolds number of the flow is assumed to be very small, and no external voltage is applied so that the induced magnetic field is neglected and the Hall effect of magnetohydrodynamics is assumed to be negligible Under the assumptions made, the vector forms of conservation equations governing the flow of steady, incompressible immiscible couple stress fluids through a porous cylinder in the presence of a transverse magnetic field can be written in the following form (Punnamchandar and Fekadu [51] and Kumar and Agrawal [44]):

Journal of Applied Mathematics
Continuity equations: Momentum equations: where i = 1, 2 denotes distinct fluid regions. The additional term −μ/kq ! i in the governing equation (7) is due to the porous medium (Chamkha [34]) where k is the permeability of the porous medium and q ! i ði = 1, 2Þ is the velocity vector.
The current density J ! i is expressed by Ohm's law (Gold [53]): where σ i ði = 1, 2Þ and E ! stand for electrical conductivity of the fluids for regions I and II and electric field, respectively.
Here, E ! = 0 as there is no external electric field, and jB ! j = B 0 because of our assumption that the induced magnetic field is too less (assumed to be zero) as compared to the external magnetic field. Hence, the Lorentz force is given by Due to the unidirectional and symmetric nature of the flow, the fluid velocity vectors for both regions are to be in the form q ! i = ð0, 0, u i ðrÞÞ where i = 1, 2. These choices of velocities automatically satisfy the continuity equation (6) in respective flow regions. Under the above conditions, equation (7) governing the flow of the couple stress fluids in the respective regions can be written as follows: In region I ð0 ≤ r ≤ RÞ (core region), we have In region II ðR ≤ r ≤ R 0 Þ (peripheral region), we have where ∇ 2 is the differential operator defined as From equations (3) and (4), the force stress tensor τ ij and couple stress tensor m ij of the couple stress fluids are given by To determine u 1 ðrÞ and u 2 ðrÞ, the boundary and interface conditions have to be specified. Instead of the usual no-slip condition, the slip velocity is taken on the surface of the rigid cylinder. In 1823, Navier [54] suggested a general boundary condition that presents the possibility of slipping at the solid boundary. This condition states that the tangential velocity of the fluid relative to the solid at a point on its surface is proportional to the tangential stress acting at that point. The proportionality that characterizes the surface's "slipperiness" is known as the slip length.
In view of the higher-order nature of governing equations, additional boundary conditions are required to find the solution. In addition to the Navier slip boundary condition, we use the Stokes (Stokes [5]) boundary conditions to solve the governing equations of the flow under consideration. The Stokes boundary condition assumes that the couple stresses vanish on the boundary of the solid.
The slip boundary condition along with zero couple stresses on the boundary is not sufficient to find the solution to the problem. A characteristic feature of the two-fluid flow problem is the coupling across the fluid/fluid interface. The fluid layers are mechanically coupled via the transfer of momentum across the interface. By the virtue of coupling of fluid layers at the fluid-fluid interface through momentum transfer, the continuity conditions for the velocity, vorticity, couple stress, and shear stress are adopted at the fluid-fluid interface. Therefore, the following physically realistic and mathematically consistent boundary and interface conditions are used for the considered physical model: (i) The slip condition along with vanishing couple stresses are taken at the boundary of cylindrical pipe r = R 0 Following Punnamchandar and Fekadu [51], the slip condition gives where γ ⋆ s such that ð0 ≤ γ ⋆ s < ∞Þ corresponds to the slip coefficient at the upper boundary (Navier [54]). Note that as γ ⋆ s = 0, the classical no-slip case is recovered (Devakar and Ramgopal [31]). 4 Journal of Applied Mathematics Following Srinivas and Murthy [32], the vanishing of couple stress on the surface of the cylinder leads to (ii) The continuity conditions for the velocity, vorticity, shear stress, and couple stress are adopted at the fluid-fluid interface. Following Kumar and Agrawal [44], this implies (iii) Regularity condition: the axisymmetric flow suggested that the velocity of the fluid is finite on the axis of the cylinder r = 0. Following Devakar and Ramgopal [31], this implies To solve equation (10) and equation (11) under the boundary conditions (equation (15)-equation (21)), we make use of the following nondimensional quantities: where u 0 and R are characteristic velocity and radius for the given flow model, respectively, and i = 1, 2 denotes distinct fluid regions.
Using the dimensionless variables in equations (10) and (11), the nondimensional form of the governing equations (after dropping the stars) is as follows: For the core region I ð0 ≤ r ≤ 1Þ, we have For the peripheral region II ð1 ≤ r ≤ sÞ, we have where In the above equations, G = −∂P/∂z is a constant pressure gradient, Re = ρ 1 UR/μ 1 is the Reynolds number, s 2 i = μ i R 2 /η i is the couple stress parameter, Da = k/R 2 is the Darcy number, n σ = σ 2 /σ 1 is the conductivity ratio, M = B 0 R ffiffiffiffiffiffiffiffiffiffiffi σ 1 /μ 1 p is the magnetic number, and n μ = μ 2 /μ 1 is the viscosity ratio.
From equations (13) and (44), the nondimensional forms of the shear stresses and couple stresses are 4. Solution of the Problem 4.1. Flow Velocity in the Two Regions. The methodology used to get the general solution of the nondimensional differential equations (23) and (24) governing the fluids flow is as follows: finding the complementary solution u c ðrÞ of the homogenous differential equation and then determining the particular solution u p ðrÞ of the nonhomogeneous differential equation. Thus, the general solution can be constructed as Region I ð0 ≤ r ≤ 1Þ: Let 5 Journal of Applied Mathematics Then, the equation (23) governing the fluid in region I can be written as where First, we consider the corresponding homogeneous differential equation The solution of (33) is obtained by using the superposition principle such that where Further, the differential equations (35) and (36) can be reduced to the following modified Bessel differential equations: The solutions to the above two equations (35) and (36), respectively, are Inserting the expressions (38) and (39) into (34), we obtain the general solution of equation (33) as The particular solution of the differential equation (31) can be easily obtained as Therefore, the general solution of the differential equation (31), after substituting (40) and (41) in equation (28), becomes where C 1 , C 2 , C 3 , and C 4 are arbitrary constants. Region-II ð1 ≤ r ≤ sÞ: Let Then, equation (24) governing fluid flow in region II can be written as where for region II, Therefore, similarly solving equation (45) by the method stated above, we get where D 1 , D 2 , D 3 , and D 4 are arbitrary constants. The closed-form solutions of the differential equations (31) and (45) are given by equations (42) and (47) (26), the nondimensional tangential stress of the fluid in region I is given by

Stress in the Two Regions. From equation
Journal of Applied Mathematics From equation (27), the couple stress in region I is given by Similarly, tangential stress in region II given by equation (26) becomes The couple stress of fluid in region II given by equation (26) becomes To obtain a complete solution to the concerned problem, we have to determine the constants C i and D i for i = 1, 2, 3, 4. C i and D i for i = 1, 2, 3, 4 are calculated numerically by solving the algebraic system obtained from the boundary conditions. (15)-(21), we have the following:

Determination of Arbitrary Constants. Nondimensionalizing the boundary conditions
(i) Since the modified Bessel function of the second kind, i.e., K n ðrÞ, is not finite at a singular point r = 0, therefore for finite values of u 1 ðrÞ along the axis of a cylindrical pipe, the coefficient of K n ðα i rÞ for n = 0, 1 and i = 1, 2 should be zero. Thus, we have (ii) The slip and vanishing of couple stress boundary conditions at r = s give where γ s = γ ⋆ s μ 2 /R is the nondimensional slip parameter (iii) Continuity of velocities, vorticites, shear stresses and couple stresses at the fluid-fluid interface r = 1 are as follows: Substituting equation (42) and equations (47)-(51) in equations (52)-(58), the linear system of an algebraic equation with six unknown arbitrary constants C 1 , C 3 , and D i for i = 1, 2, 3, 4 involved in the solution of the problem is Invoking the values of u 1 ðrÞ and u 2 ðrÞ from equations (42) and (47) in equation (59) and integrating, we obtain  Figure 2 illustrates the influence of the magnetic number M on the velocities. It is observed that the fluid velocities in both regions are decreasing with an increment of magnetic number M. This finding suggests that the magnetic field applied to the flow system retards the motion of the fluid. This is consistent with the fact that a strong magnetic field applied to the flow literally increases the Lorentz force, which strongly opposes the fluid's motion and lowers the velocities. This result is validated by the works of Ansari and Deo [41], Kumar and Agarwal [44], and Punnamchandar and Fekadu [51,52]. Further, as M ⟶ 0, the magnetic number loses its properties and behaves as a normal flow in the absence of a magnetic field (Srinivas and Murthy [32]). 8

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The effects of couple stress parameters s 1 and s 2 on the flow are displayed in Figures 3-4. The effect of the couple stress parameter s 1 on the flow velocity profiles is seen in Figure 3. In this case, we notice that a rise in the couple stress parameter s 1 causes the fluid's velocity to increase in both flow areas. Figure 4 shows a similar trend when varied with the couple stress parameter s 2 . Therefore, we draw the conclusion that raising couple stress parameters s i for i = 1 , 2 causes fluid velocities to increase in both flow areas. This result validates our problem with the previous works of Umavathi et al. [20], Devakar et al. [23], Srinivas and Murthy [32], Srinivas et al. [37], and Kumar and Agarwal [44]. Since s 2 i = μ i R 2 /η i , an increase in couple stress viscosities η i for i = 1, 2 corresponds to a decrease in the couple stress parameters s i . As a result, increasing couple stress coefficients η i for i = 1, 2 has a retarding effect on fluid velocities. This indicates that the presence of couple stress in the fluid reduces the velocity of a fluid. This is due to the fact that physically, the couple stresses expend some energy to rotate the particles, which reduces the particles' velocity.  Furthermore, it is to be noted that in the absence of couple stresses, that is, as η i ⟶ 0, the parameter s i ⟶ ∞, the properties of couple stress in the fluid vanish and the case of classical viscous fluid can be obtained from this work ), Abbas et al. [22], Devakar et al. [23], and Devakar and Ramgopal [31]). Therefore, it is understood that the velocity in the case of couple stress fluid is lower than that of a Newtonian fluid.
The effect of Darcy's number Da on the fluid velocities is shown in Figure 5. From this figure, it is noticed that the velocities in both fluid regions increase with the increase of Darcy's number Da. Since Da = k/R 2 , an increase in Darcy's number corresponds to an increase in the permeability (permeable parameter k) of the porous medium, which supports the flow. Lesser permeability causes a slighter fluid velocity to be observed inside the flow medium occupied by the fluid. Thus, it may be concluded that an increase in the Darcy's number enhances fluid velocities. This is due to the reason that the additional flow resistance that the porous structure offers diminishes as Da (permeable parameter k) gradually increases. A similar kind of behavior can be found in Refs. Srinivas and Murthy [32], Srinivas et al. [37], Punnamchandar and Fekadu [51], and Kumar and Agarwal [44]. Figure 6 describes the effect of the ratio of viscosities n μ on velocity profiles. Figure 6 reveals that as the viscosity ratio n μ increases, the velocity of the fluid decreases in both flow regions. This is because as the viscosity ratio n μ increases, greater flow resistance is provided. As a result, velocity drops. Therefore, we conclude that an increase in the ratio of viscosities inhibits fluid motion. A similar view can be found in the works of Umavathi et al. [21], Umavathi et al. [30], Srinivas and Murthy [32], and Punnamchandar and Fekadu [51]. Figure 7 presents the effect of the Reynolds number on the velocity profile. Thereby, we observe that as the Reynolds number Re increases, there is a decrease in the velocities of the fluid in both flow regions. This indicates that velocity is reduced by the increase of the Reynolds number Re and our results well agreed with the results of Devakar et al. [23], Devakar and Ramgopal [31], Srinivas and Murthy [32], and Punnamchandar and Fekadu [51]. Figure 8 represents the velocity profile for the different values of the pressure gradient. It is observed that with the increase in G, velocity is increasing in both the fluid regions. Physically, the more the pressure gradient, the more the fluid is pushed to generate the flow, which results in an increase in fluid velocity. Figure 9 displays the effect of the slip parameter γ s on the fluid flow velocity profiles. Figure 9 shows that increasing the slip parameter reduces fluid velocity in both zones. Obviously, fluid slippage has the opposite impact on fluid motion, and increasing the slip parameter reduces the velocity significantly in both regions. A similar trend was observed in the work of Punnamchandar and Fekadu [51,52]. Furthermore, when the slip parameter is set to zero, the classical case of no slip is recovered as a special case.
The numerical values of the volume flow rate are computed for various pertinent flow parameters and are presented in Tables 1-8. The effect of the magnetic number M on the flow rate is shown in Table 1. From Table 1, we notice that the total flow rate decreases as the magnetic number M increases from 0:5 to 2 for fixed values of Da = 1:0, Re = 2, n σ = 1:0, n μ = 1:1, γ s = 0:1, G = 10, s 1 = 1, s 2 = 1, and s = 2. Tables 2 and 3 shows the nature of flow rates for different values of couple stress parameter s i , i = 1, 2. From the tables, we can see that the total flow rate increases with an increase       Table 4 demonstrates the effect of Darcy's number on the flow rate. From Table 4, we can see that the total flow rate increases with an increase in Darcy's number. Table 5 shows the effect of the viscosity ratio on the flow rate. The flow rate shows a decreasing trend with the growth of the viscosity ratio. Table 6 displays various values of flow rate with respect to the Reynolds number. It is seen from Table 6 that as the Reynolds number increases, the total flow rate decreases. Table 7 represents the flow rate for the different pressure gradient values. From the table, it is observed that increasing the pressure gradient increases the volume flow rate across the pipe cross-section. Table 8 presents the numerical flow rate data with respect to slip parameter γ s . It is observed that the volume flow rate gets decreased with an increase of slip parameter γ s .

Conclusions
The problem of steady, laminar, and fully developed hydromagnetic flow of two immiscible couple stress fluids through a porous medium in a horizontal cylinder under the effect of the Navier slip boundary condition is considered in the present study. The motion is generated by a constant pressure gradient delivered along the axial direction, i.e., z-axis. The resulting set of coupled differential equations associated with the flow of the two fluids subject to the appropriate boundary and interface conditions is solved analytically. Exact solutions are obtained in terms of the modified Bessel functions. The effects of various physical parameters on the velocity profiles and total flow rate are studied. The significant findings of the current investigation are the following: (i) Increasing the magnetic number, viscosity ratio, Reynolds number, and slip parameter reduces fluid velocities (ii) The increment of the couple stress parameters, Darcy number, and pressure gradient enhances the fluid velocity in both flow regions  (iii) Increase in the magnetic number, slip parameter, viscosity ratio, and Reynolds number suppress the volume flow rate (iv) Increase in the couple stress parameter, Darcy number, and pressure gradient promotes volume flow rate This work can be extended to the unsteady flow problem and is also made to include heat transfer/thermal effects. We would like extend this work by taking various fluids like micropolar fluid, or any other non-Newtonian fluid. Velocity vector in regions I and II Q:

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.