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

Hydromagnetic Flow of Two Immiscible Couple Stress Fluids through Porous Medium in a Cylindrical... Hindawi Journal of Applied Mathematics Volume 2023, Article ID 1902844, 13 pages https://doi.org/10.1155/2023/1902844 Research Article Hydromagnetic Flow of Two Immiscible Couple Stress Fluids through Porous Medium in a Cylindrical Pipe with Slip Effect 1 2 Punnamchandar Bitla and Yitagesu Daba Kore Department of Mathematics, Anurag University, Telangana, India Department of Mathematics, Wollega University, Nekemte, Ethiopia Correspondence should be addressed to Punnamchandar Bitla; punnam.nitw@gmail.com Received 23 November 2022; Revised 23 February 2023; Accepted 18 April 2023; Published 3 May 2023 Academic Editor: Oluwole D. Makinde Copyright © 2023 Punnamchandar Bitla and Yitagesu Daba Kore. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, the steady hydromagnetic flow of two immiscible couple stress fluids through a uniform porous medium in a cylindrical pipe with slip effect is investigated analytically. Essentially, the flow system is divided into two regions, region I and region II, which occupy the core and periphery of the system, respectively. The flow is driven by a constant pressure gradient applied in a direction parallel to the cylinder’s axis, and an external uniform magnetic field is applied in the direction perpendicular to the direction of fluid 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 fluid-fluid interface. The governing equations are modeled using the fully developed flow conditions. The resulting differential equations governing the flow 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 flow velocity, flow rate, and stresses are derived in terms of the Bessel functions. The impacts of several parameters pertaining to the flow 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 flow rate’s numerical values are also calculated for different fluid 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 fluids whereas increasing the couple stress parameter, Darcy number, and pressure gradient increases fluid velocities. The results obtained in this paper show an excellent agreement with the already existing results in the literature as limiting cases. 1. Introduction theories [1–3] pertaining to non-Newtonian fluids have been developed for describing the rheological behavior of numer- Over the last few decades, the study of non-Newtonian fluids ous complex fluids. The couple stress fluid, initiated by has received a lot of attention as these types of fluids fre- stokes [4], is one of the popular theories of polar fluids that quently occur in many industrial, scientific, and technologi- considers the possibility of polar effects such as the presence cal processes. Many important complex and real fluids of couple stresses and body couples in the fluid medium. The which include molten metals, polymeric liquids, slurries, flow behavior of various fluids that contain a substructure blood, liquid crystals, lubricants, soaps, greases, gelatin, such as lubricants with small amounts of additives, poly- and paints belong to this family of fluids. This class of fluids mers, colloidal suspensions, liquid crystals, animal and does not follow the Newtonian fluid theory as they possess human blood, polymer-thickened oils, muddy water, and microstructure and a nonsymmetric stress tensor in their electrorheological and synthetic fluids can be modeled using fluid structure. For this reason, several new microcontinuum the couple stress fluid theory [5]. A couple stress fluid model 2 Journal of Applied Mathematics has been successfully employed to study the mechanism of concentric pipes filled with the porous medium was done peristalsis [6, 7]. The couple stress fluid theory widely used by Yadav et al. [33]. Other important studies in this direc- in modeling the flow of biological fluids such as synovial tion include Chamkha [34], Harmindar and Singh [35], fluids [8–11] and blood [12–14]. Singh [36], and Srinivas et al. [37]. Though many In realistic situations, most of the flow problems arising researchers have worked on the flow of immiscible fluids in the industries, manufacturing process, geology, ground- through porous channels, the flow of immiscible non- water hydrology, reservoir mechanics, biomechanics, mag- Newtonian fluids through porous cylinders is reasonably netofluid dynamics, geophysics, plasma physics, and so on underexplored despite its applicability in blood flows, chem- occur with two or more fluids of different densities/viscosi- ical engineering, crude oil extraction, etc. One such type of ties flowing immiscibly in the same channel or cylindrical problem is going to be discussed in this paper. pipe. Examples of these systems are the flow of several Magnetohydrodynamics (MHD) is also an interesting immiscible oils through the bed of rocks or soils, the flow and important area of modern engineering sciences and in the rivers with several industrial fluids, blood flow in the involves the interaction of magnetic forces and electrically arteries, the flow of air and fuel droplets in combustion conducting fluids. The application of magnetic fields to chambers, the flow of air and exhaust gases at engine outlets, the flow of immiscible fluids originates from reducing the flows for many medical and industrial purposes. The study gas and Petrolia flow in pipes of oil, water-air flows around ship halts, etc. These are referred to as multiphase flows in of the hydromagnetic flow of moving fluids through a the literature. Owing to its wide areas of applications, several porous medium is currently a subject of great interest researchers have studied multiphase fluid flows. Chaturani owing to plentiful applications in industrial, engineering, and Samy [15] as well as Sinha and Singh [16] investigated and medical devices. Owing to these applications, several the effects of couple stresses on blood flow, and many other studies have been conducted to examine the effect of mag- authors (Valanis and Sun [17], Sharan and Popel [18], and netic fields on the flows of immiscible fluids. In light of Garcia and Riahi [19]) discussed blood flow considering it this, Vajravelu et al. [38] studied the hydromagnetic as a two-phase flow in which they have assumed blood as a unsteady flow of two conducting immiscible fluids between couple stress fluid. Besides its application in blood flow, two permeable beds. Malashetty et al. [39] analyzed the the study of multiphase flow has several important applica- magnetohydrodynamic two-fluid convective flow and heat tions in various fields of engineering and science. Umavathi transfer in an inclined composite porous medium. Raju et al. [20] analytically solved the problem of flow through a and Nagavalli [40] studied the unsteady two-layered fluid horizontal channel with a couple stress fluid sandwiched flow and heat transfer of conducting fluids in a channel between viscous fluid layers. They discussed the effects of between parallel porous plates under a transverse magnetic various flow parameters and concluded that the couple stress field. Ansari and Deo [41] investigated the effect of a mag- parameter influences the flow. Umavathi et al. [21] made a netic field on the two immiscible viscous fluids flowing in a detailed study on the flow and heat transfer of a couple stress channel filled with a porous medium. The influence of an fluid in contact with a Newtonian fluid. Abbas et al. [22] inclined magnetic field on the Poiseuille flow of immiscible analyzed the hydromagnetic mixed convective two-phase micropolar-Newtonian fluids through the horizontal flow of couple stress and viscous fluids in an inclined chan- porous channel where the permeability of both the regions nel. They obtained closed-form solutions of velocity and of the horizontal porous channel has been taken differently temperature profiles by using the perturbation method. was discussed by Yadav and Jaiswal [42]. In another paper, Devakar et al. [23] studied the unsteady flow of couple stress Jaiswal and Yadav [43] investigated the influence of a mag- fluid sandwiched between Newtonian fluids through a chan- netic field on the Poiseuille flow of immiscible Newtonian fluids through a highly porous medium. More recently, nel. There are many other works concerning multiphase fluid flows (see Packham and Shall [24], Rao and Usha Kumar and Agrawal [44] studied the magnetohydrody- [25], Chamkha et. al [26], Umavathi et al. [27, 28], and namic pulsatile flow and heat transfer of two immiscible Umavathi and Shekar [29]). couple stress fluids in a porous channel. In view of its numerous applications, the researchers at A majority of the studies regarding the flow of immisci- present are engaged in exploring immiscible flows of fluids ble non-Newtonian fluids quoted above were carried out by through a porous medium under various circumstances. imposing the no-slip boundary condition. However, several Umavathi et al. [30] studied the problem of the convective theoretical and experimental studies [45–50] reveal that slip flow of two immiscible fluids (couple stress and viscous exists at the solid boundary. So, for flows containing fluids fluids) through a vertical channel. They obtained an approx- through solid boundaries, consideration of a velocity slip is imate solution by using the regular perturbation method. more realistic and appropriate. Recently, Punnamchandar Devakar and Ramgopal [31] presented analytical solutions and Fekadu [51] investigated the effects of slip and uniform for the fully developed flows of two immiscible couple stress magnetic field on the flow of immiscible couple stress fluids and Newtonian fluids through a nonporous and porous in a porous medium channel. In another paper, Punnam- medium in a horizontal cylinder. Srinivas and Murthy [32] chandar and Fekadu [52] considered the problem of the studied the flow of two immiscible couple stress fluids effects of slip and inclined magnetic field on the flow of between two permeable beds. They obtained an exact solu- immiscible fluids (couple stress fluid and Jeffrey fluid) in a tion to the considered problem. An analysis of the Poiseuille porous channel. It is observed that the effects of slip and flow of immiscible micropolar-Newtonian fluids through magnetic field on the flow of immiscible couple stress fluids Journal of Applied Mathematics 3 through a porous medium in a cylindrical pipe have not The material constants λ and μ are the viscosity coeffi- been discussed yet. ′ cients, and η and η are the couple stress viscosity coeffi- Keeping this in view the wide potential applications of cients satisfying the constraints immiscible couple stress fluids flow and the importance of exact solutions described above, the goal of the current paper ′ ′ μ ≥ 0, 3λ +2μ ≥ 0,jj η ≥ η , η ≥ 0: ð5Þ is to determine exact solutions for the steady hydromagnetic flow of two immiscible couple stress fluids through a porous pffiffiffiffiffiffiffi There is a length parameter l = η/μ which is a charac- medium in a cylindrical pipe with slip effect. The impacts of teristic measure of the polarity of the couple stress fluid, different flow parameters on the velocity field and flow rate and this parameter is identically zero in the case of nonpolar are investigated. The slip factor in fluid flows makes the fluids. problem even more realistic and interesting, which moti- vated us to consider this problem. The practicality and the 3. Formulation of the Problem complexities involved due to the porosity and cylindrical nature of the geometry also make the work presented in this The physical model concerns an axisymmetric fully devel- paper novel. oped hydromagnetic flow of two immiscible couple stress fluids flowing through a porous medium in a horizontal cir- 2. Basic Equations cular pipe of radius R . Owing to the fluids’ immiscibility, there are two separate regions of fluid flow: region I, or the The basic equations describing the flow of couple stress fluid core region, and region II, or the periphery region. The flow including a Lorentz force are (Stokes [4, 5]) as follows: geometry of the problem is depicted in a cylindrical polar Continuity equation (conservation of mass): coordinate system ðr, θ, zÞ with the origin at the center of the tube and common axis of the cylindrical regions taken ∂ρ as the z-axis, as shown in Figure 1. Region I ð0 ≤ r ≤ RÞ is +∇: ρq =0: ð1Þ ∂t occupied with couple stress fluid with density ρ , shear vis- cosity μ , and couple stress viscosity η , comprising the core 1 1 Momentum equation (conservation of momentum): region of the pipe whereas region II ðR ≤ r ≤ R Þ is occupied by a different couple stress fluid having density ρ , shear vis- Dq 1 ! ! cosity μ , and couple stress viscosity η , comprising the 2 2 ρ = ρ f + ∇ × ρ c −∇P − μ∇ × ∇ × q − η∇ Dt 2 peripheral region of the pipe. The motion of the fluids in ð2Þ ! ! ! ! both regions is caused by a constant pressure gradient × ∇ × ∇ × ∇ × q +ðÞ λ + μ ∇∇·q + J × B, applied in a direction parallel to the cylinder’s axis, i.e., z -axis, and an external uniform magnetic field of strength B directed perpendicular to the flow direction is also 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 applied. To develop the governing equations for the considered c are the velocity, body force per unit mass, and body couple ! ! model, the following presumptions are taken in the analysis per unit mass, respectively. The term J × B in equation (2) of the current study: is the Lorentz force (electromagnetic body force) in which ! ! J is the electric current density and B is the total magnetic (i) The fluids are considered incompressible, and the field. flow is assumed to be steady, laminar, and fully The force stress tensor τ (Stokes [5]) that arises in the ij developed theory of couple stress fluids is given by (ii) Both the fluid regions are saturated with the uni- form porous media of permeability k τ = −P + λ∇:q δ +2μd + ε m +4ηω + ρc : ½Š ij ij ij ijk ,k k,rr k (iii) The Lorentz force is the only body force acting on the fluids, with no body couples ð3Þ (iv) The magnetic Reynolds number of the flow is The couple stress tensor m (Stokes [5]) that arises in the ij assumed to be very small, and no external voltage theory has the linear constitutive relation is applied so that the induced magnetic field is neglected and the Hall effect of magnetohydrody- namics is assumed to be negligible m = mδ +4η ω +4ηω : ð4Þ ij ij j,i i,j Under the assumptions made, the vector forms of con- servation equations governing the flow of steady, incom- In the above, ω is the spin tensor, ρc is the body cou- i,j k pressible immiscible couple stress fluids through a porous ple vector, d is the components of the rate of shear strain, ij cylinder in the presence of a transverse magnetic field can δ is the Kronecker symbol, e is the Levi-Civita symbol, be written in the following form (Punnamchandar and ij ijk and comma denotes covariant differentiation. Fekadu [51] and Kumar and Agrawal [44]): 4 Journal of Applied Mathematics Continuity equations: From equations (3) and (4), the force stress tensor τ ij and couple stress tensor m of the couple stress fluids are ij ∇:q =0: ð6Þ given by Momentum equations: Âà ð13Þ τ = μ u − η ∇ u , i =1,2, rzðÞ i i i i i dr −∇P − μ ∇ × ∇ × q − η ∇ × ∇ × ∇ × ∇ i i i ð7Þ ! ! ! ! d u η ′ du × q + J × B − q =0, i i i i i i ð14Þ k m = η − , i =1,2: rθðÞ i i dr r dr where i =1,2 denotes distinct fluid regions. To determine u ðrÞ and u ðrÞ, the boundary and inter- ! 1 2 The additional term −μ/kq in the governing equation i face conditions have to be specified. (7) is due to the porous medium (Chamkha [34]) where k 3.1. Boundary and Interface Conditions. The description and is the permeability of the porous medium and q ði =1, 2Þ mathematical form of the boundary conditions are pre- is the velocity vector. ! sented in this section. The current density J is expressed by Ohm’s law (Gold Instead of the usual no-slip condition, the slip velocity is [53]): 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 condi- J = σ E + q × B , ð8Þ i i i tion states that the tangential velocity of the fluid relative to the solid at a point on its surface is proportional to the tan- gential stress acting at that point. The proportionality that where σ ði =1, 2Þ and E stand for electrical conductivity of characterizes the surface’s “slipperiness” is known as the slip length. the fluids for regions I and II and electric field, respectively. ! ! In view of the higher-order nature of governing equa- Here, E =0 as there is no external electric field, and jBj tions, additional boundary conditions are required to find = B because of our assumption that the induced magnetic the solution. In addition to the Navier slip boundary condi- field is too less (assumed to be zero) as compared to the tion, we use the Stokes (Stokes [5]) boundary conditions to external magnetic field. Hence, the Lorentz force is given by solve the governing equations of the flow under consider- ation. The Stokes boundary condition assumes that the cou- ! ! ! ! ! ! ! ple stresses vanish on the boundary of the solid. F = J × B = σ q × B × B = −σ B q : ð9Þ i i i i i 0 i 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 Due to the unidirectional and symmetric nature of the problem is the coupling across the fluid/fluid interface. The flow, the fluid velocity vectors for both regions are to be in fluid layers are mechanically coupled via the transfer of the form q = ð0, 0, u ðrÞÞ where i =1,2. These choices of i i momentum across the interface. By the virtue of coupling velocities automatically satisfy the continuity equation (6) of fluid layers at the fluid-fluid interface through momentum in respective flow regions. Under the above conditions, transfer, the continuity conditions for the velocity, vorticity, equation (7) governing the flow of the couple stress fluids couple stress, and shear stress are adopted at the fluid-fluid in the respective regions can be written as follows: interface. Therefore, the following physically realistic and In region I ð0 ≤ r ≤ RÞ (core region), we have mathematically consistent boundary and interface condi- tions are used for the considered physical model: μ ∂P 4 2 1 2 −η ∇ u + μ ∇ u − + σ B u = : ð10Þ 1 1 1 1 1 0 1 (i) The slip condition along with vanishing couple k ∂z stresses are taken at the boundary of cylindrical pipe r = R In region II ðR ≤ r ≤ R Þ (peripheral region), we have Following Punnamchandar and Fekadu [51], the slip condition gives μ ∂P 4 2 2 2 −η ∇ u + μ ∇ u − + σ B u = , ð11Þ 2 2 2 2 2 2 0 k ∂z u ðÞ R =±γ τ ðÞ R , ð15Þ 2 0 rz 2 0 s ðÞ ⋆ ⋆ where ∇ is the differential operator defined as where γ such that ð0 ≤ γ < ∞Þ corresponds to the slip s s coefficient at the upper boundary (Navier [54]). Note that d 1 d as γ =0, the classical no-slip case is recovered (Devakar ∇ = + : ð12Þ dr r dr and Ramgopal [31]). Journal of Applied Mathematics 5 Following Srinivas and Murthy [32], the vanishing of For the peripheral region II ð1 ≤ r ≤ sÞ, we have couple stress on the surface of the cylinder leads to 1 s ReG 4 2 2 2 2 2 ∇ u − s ∇ u + s + M u = , ð24Þ 2 2 2 2 1 2 m ðÞ R =0: ð16Þ rθ 2 0 ðÞ Da n where (ii) The continuity conditions for the velocity, vorticity, μ R 2 i shear stress, and couple stress are adopted at the s = , fluid-fluid interface. Following Kumar and Agrawal i 2 2 [44], this implies B R σ 2 0 1 M = , u r = u r at r = R, ð17Þ ðÞ ðÞ 1 2 sffiffiffiffiffi M = M , du r du r ðÞ ðÞ 1 2 ð18Þ ð25Þ = at r = R, dr dr n = , τ r = τ r at r = R, ðÞ ðÞ ð19Þ rzðÞ 1 rzðÞ 2 m r = m r at r = R: n = , ðÞ ðÞ ð20Þ rθðÞ 1 rθðÞ 2 μ (iii) Regularity condition: the axisymmetric flow sug- Da = : gested that the velocity of the fluid is finite on the axis of the cylinder r =0. Following Devakar and In the above equations, G = −∂P/∂z is a constant pres- Ramgopal [31], this implies sure gradient, Re = ρ UR/μ is the Reynolds number, s = 1 1 i 2 2 μ R /η is the couple stress parameter, Da = k/R is the Darcy u ðÞ r is finite at r =0: ð21Þ 1 i i number, n = σ /σ is the conductivity ratio, M = B R σ 2 1 0 pffiffiffiffiffiffiffiffiffiffiffi σ /μ is the magnetic number, and n = μ /μ is the vis- 1 1 μ 2 1 To solve equation (10) and equation (11) under the cosity ratio. boundary conditions (equation (15)–equation (21)), we From equations (13) and (44), the nondimensional make use of the following nondimensional quantities: forms of the shear stresses and couple stresses are r = , u μ d 1 0 i 2 τ = u − ∇ u , i =1,2, ð26Þ rzðÞ i i i R dr s z = , "# u η d u η 1 du ⋆ i 0 i i i i u = , m = − , i =1,2: ð27Þ rθðÞ i 2 2 dr η r dr 0 R ð22Þ P = , 4. Solution of the Problem ρ u 1 0 R 4.1. Flow Velocity in the Two Regions. The methodology used s = , to get the general solution of the nondimensional differential ⋆ equations (23) and (24) governing the fluids flow is as fol- γ μ s 2 γ = , lows: finding the complementary solution u ðrÞ of the s c homogenous differential equation and then determining the particular solution u ðrÞ of the nonhomogeneous differ- where u and R are characteristic velocity and radius for the ential equation. Thus, the general solution can be con- given flow model, respectively, and i =1,2 denotes distinct structed as fluid regions. Using the dimensionless variables in equations (10) and u r = u r + u r : ð28Þ ðÞ ðÞ ðÞ (11), the nondimensional form of the governing equations i c p (after dropping the stars) is as follows: For the core region I ð0 ≤ r ≤ 1Þ,wehave Region I ð0 ≤ r ≤ 1Þ: Let 4 2 2 2 2 2 ∇ u − s ∇ u + s + M u = s ReG: ð23Þ 1 1 1 1 1 1 2 2 2 ð29Þ Da α + α = s , 1 2 1 6 Journal of Applied Mathematics Therefore, the general solution of the differential equa- 2 2 2 2 α α = s + M : ð30Þ 1 2 1 tion (31), after substituting (40) and (41) in equation (28), Da becomes Then, the equation (23) governing the fluid in region I can be written as u ðÞ r = C IðÞ α r + C K ðÞ α r + C IðÞ α r 1 1 0 1 2 0 1 3 0 2 ÀÁÀÁ 2 2 2 2 2 ð42Þ ReGs ∇ − α ∇ − α u = ReGs , ð31Þ 1 1 2 1 1 + C K α r + , ðÞ 4 0 2 2 2 α α 1 2 where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where C , C , C , and C are arbitrary constants. 2 4 2 2 1 2 3 4 s ± s − 4s 1/Da + M ðÞ ðÞ 2 2 1 1 1 α , α = : ð32Þ Region-II ð1 ≤ r ≤ sÞ: 1 2 Let First, we consider the corresponding homogeneous dif- 2 2 2 ð43Þ β + β = s , ferential equation 1 2 2 ÀÁÀÁ 2 2 2 2 ∇ − α ∇ − α u =0: ð33Þ 1 1 2 2 2 2 2 β β = s + M : ð44Þ 1 2 2 Da The solution of (33) is obtained by using the superposi- Then, equation (24) governing fluid flow in region II can tion principle such that be written as ðÞ 1 u r = y r + y r , ð34Þ ðÞ ðÞ ðÞ c 1 2 ÀÁÀÁ Re Gs 2 2 2 2 ∇ − β ∇ − β u = , ð45Þ 1 2 where ÀÁ 2 2 ð35Þ ∇ − α y =0, where for region II, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ 2 2 2 4 2 2 ∇ − α y =0: ð36Þ s ± s − 4sðÞ ðÞ 1/Da + M 2 2 2 2 2 2 2 1 β , β = : ð46Þ 1 2 Further, the differential equations (35) and (36) can be reduced to the following modified Bessel differential equa- Therefore, similarly solving equation (45) by the method tions: stated above, we get d y 1 dy 2 i i 2 2 u ðÞ r = D IðÞ β r + D K ðÞ β r + D IðÞ β r r + − α r y =0, i =1, 2: ð37Þ 2 1 0 2 0 3 0 1 1 2 i i dr r dr ReGs ð47Þ + D K β r + , ðÞ 4 0 2 2 2 n β β μ 1 2 The solutions to the above two equations (35) and (36), respectively, are where D , D , D , and D are arbitrary constants. 1 2 3 4 ð38Þ y ðÞ r = C IðÞ α r + C K ðÞ α r , 1 1 0 1 2 0 1 The closed-form solutions of the differential equations (31) and (45) are given by equations (42) and (47) contain ð39Þ y ðÞ r = C IðÞ α r + C K ðÞ α r : modified Bessel functions. Here, I ðα rÞ, I ðα rÞ, I ðβ rÞ, 2 3 0 2 4 0 2 0 1 0 2 0 and I ðβ rÞ and K ðα rÞ, K ðα rÞ, K ðβ rÞ, and K ðβ rÞ 0 2 0 1 0 2 0 1 0 2 Inserting the expressions (38) and (39) into (34), we are the first kind modified Bessel’s functions of zero order obtain the general solution of equation (33) as and the second kind modified Bessel’s functions of zero order, respectively. u r = C I α r + C K α r + C I α r + C K α r : ðÞ ðÞ ðÞ ðÞ ðÞ c 1 0 1 2 0 1 3 0 2 4 0 2 4.2. Stress in the Two Regions. From equation (26), the non- ð40Þ dimensional tangential stress of the fluid in region I is given by The particular solution of the differential equation (31) μ u α can be easily obtained as 1 0 1 τ = α − I rα C − K rα C ðÞ ðÞ ðÞ rzðÞ 1 1 1 1 1 1 1 2 R s ð48Þ ReGs α 1 2 y ðÞ r = : ð41Þ + α −ðÞ IðÞ rα C − K ðÞ rα C : 2 1 2 3 1 2 4 2 2 2 α α s 1 2 1 Journal of Applied Mathematics 7 Region-II: Couple stress fluid Ro Region-I: Couple stress fluid Pressure gradient (G) Magnetic field of intensity Bo Figure 1: Geometrical configuration. From equation (27), the couple stress in region I is given To obtain a complete solution to the concerned problem, by we have to determine the constants C and D for i =1, 2,3, i i 4. C and D for i =1,2,3,4 are calculated numerically by solv- i i " ! ! ing the algebraic system obtained from the boundary η u α η 0 2 1 1 1 conditions. m = α I rα − 1+ I rα C ðÞ ðÞ rθðÞ 1 1 0 1 1 1 1 r η ! ! 4.3. Determination of Arbitrary Constants. Nondimensiona- α η 2 1 1 lizing the boundary conditions (15)–(21), we have the + α K rα + 1+ K rα C ðÞ ðÞ 1 0 1 1 1 2 r η following: ! ! α η (i) Since the modified Bessel function of the second kind, 2 2 1 + α I rα − 1+ I rα C ðÞ ðÞ 2 0 2 1 2 3 i.e., K ðrÞ,isnot finite at a singular point r =0,there- r η ! ! # fore for finite values of u ðrÞ along the axis of a cylin- α η drical pipe, the coefficient of K ðα rÞ for n =0,1 and 2 2 n i + α K ðÞ rα + 1+ K ðÞ rα C : 2 0 2 1 2 4 r η i =1,2 should be zero. Thus, we have ð49Þ C = C =0: ð52Þ 2 4 Similarly, tangential stress in region II given by equation (ii) The slip and vanishing of couple stress boundary (26) becomes conditions at r = s give " ! u s =±γ τ s , μ u β ðÞ ðÞ ð53Þ 2 s rzðÞ 2 2 0 1 τ = β −ðÞ IðÞ rα D − K ðÞ rβ D rz 2 1 1 1 1 2 ðÞ 1 1 R s ð50Þ ! # m ðÞ s =0, ð54Þ rθðÞ 2 + β − I rβ D − K rβ D : ðÞ ðÞ ðÞ ⋆ 2 1 2 3 1 2 4 2 where γ = γ μ /R is the nondimensional slip s s 2 parameter (iii) Continuity of velocities, vorticites, shear stresses The couple stress of fluid in region II given by equation and couple stresses at the fluid-fluid interface r =1 (26) becomes are as follows: " ! ! ð55Þ u ðÞ r = u ðÞ r , η u β η 1 2 0 2 2 1 2 m = β IðÞ rβ − 1+ IðÞ rβ D rθ 2 0 1 1 ðÞ 2 1 1 1 R r η du r du r ðÞ ðÞ ! ! 1 2 ð56Þ = , À ′ β η dr dr 1 2 + β K ðÞ rβ + 1+ K ðÞ rβ D 1 0 1 1 1 2 r η τ ðÞ 1 = τ ðÞ 1 , ð57Þ rz 1 rz 2 ðÞ ðÞ ! ! β η 2 2 m 1 = m 1 : + β I rβ − 1+ I rβ D ðÞ ðÞ ð58Þ ðÞ ðÞ rθðÞ 1 rθðÞ 2 2 0 2 1 2 3 r η ! ! # β η Substituting equation (42) and equations (47)–(51) in 2 2 + β K ðÞ rβ + 1+ K ðÞ rβ D : 0 1 4 2 2 2 equations (52)–(58), the linear system of an algebraic equa- r η tion with six unknown arbitrary constants C , C , and D 1 3 i ð51Þ for i =1,2, 3,4 involved in the solution of the problem is 8 Journal of Applied Mathematics 7 7 6 6 Region-II Region-I Region-I Region-II 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 r r M = 0.5 M = 1.5 s = 1 s = 3 1 1 s = 2 s = 4 M = 1 M = 2 1 1 Figure 2: Variations of u ðrÞði =1,2Þ against M when Da =1:0, Figure 3: Variations of u ðrÞði =1, 2Þ with s when Da =1:0, M =1, i 1 Re = 2, n =1:0, n =1:1, γ =0:1, G =10, s =1, s =1, and s =2. σ μ 1 2 Re = 2, n =1:0, n =1:1, γ =0:1, G =10, s =1, and s =2. σ μ 2 formed. Using Mathematica software, all constants C , C , 1 3 and D for i =1,2, 3,4 have been evaluated uniquely using the above boundary conditions. Owing to the lengthy expressions of these constants, they are not presented here. 4.4. Total Flow Rate. The nondimensional volumetric flow rate across the whole cross-section of a porous cylinder is Region-I given by (Devakar and Ramgopal [31]) Region-II ð ð ð 2π 1 2 Q = ru dr + ru dr dθ: ð59Þ 1 2 0 0 1 0.5 1.0 1.5 2.0 Invoking the values of u ðrÞ and u ðrÞ from equations 1 2 (42) and (47) in equation (59) and integrating, we obtain s = 1 s = 3 2 2 s = 2 s = 4 I ðÞ α I ðÞ α ReDaG 1 1 1 2 Q =2π C + C +ÀÁ Figure 4: Variations of u ðrÞði =1, 2Þ with s when Da =1:0, M =1, 1 3 i 2 α α 21+ DaM 1 2 Re = 2, n =1:0, n =1:1, γ =0:1, G =10, s =1, and s =2. σ μ s 1 D D 1 2 ÁðÞ sI ðÞ sβ − IðÞ sβ −ðÞ sK ðÞ sβ − K ðÞ sβ 1 1 1 1 1 1 1 1 β β 1 1 for each case, when a particular parameter is varied, are D D 3 4 obtained by keeping Da =1:0, M =1, Re = 2, n =1:0, n = + sI sβ − I sβ − sK sβ − K sβ σ μ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 1 2 1 2 1 2 1 2 β β 2 2 1:1, γ =0:1, G =10, s =1, s =1, and s =2. s 1 2 ÀÁ The variations of velocity profiles for different flow ReGDa s − 1 + ÀÁ : parameters are shown graphically through Figures 2–9. 2n 1+ DaM Figure 2 illustrates the influence of the magnetic number ð60Þ M on the velocities. It is observed that the fluid velocities in both regions are decreasing with an increment of mag- netic number M. This finding suggests that the magnetic 5. Results and Discussion field applied to the flow system retards the motion of the fluid. This is consistent with the fact that a strong magnetic Analytical solutions for the steady, laminar hydromagnetic flow of two immiscible and incompressible couple stress field applied to the flow literally increases the Lorentz fluids through porous medium in a horizontal cylinder have force, which strongly opposes the fluid’s motion and lowers the velocities. This result is validated by the works of been obtained. The numerical evaluation of the analytical expressions for velocity profile and flow rate are done for Ansari and Deo [41], Kumar and Agarwal [44], and Pun- different flow parameters values, such as the magnetic num- namchandar and Fekadu [51, 52]. Further, as M⟶ 0, ber, couple stress parameter, Reynolds number, Darcy num- the magnetic number loses its properties and behaves as ber, ratio of viscosities, slip parameter, and pressure gradient a normal flow in the absence of a magnetic field (Srinivas and Murthy [32]). using Mathematica software package. The numerical values u (r) u (r) u (r) i i Journal of Applied Mathematics 9 Region-I Region-I Region-II Region-II 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Da = 0.1 Da = 0.5 R = 0.5 R = 1.5 Da = 0.3 Da = 0.8 R = 1 R = 2 Figure 5: Variations of u ðrÞði =1, 2Þ with Da when M =1, Re = 2, Figure 7: Variations of u ðrÞði =1, 2Þ with Re when Da =1:0, M n =1:0, n =1:1, γ =0:1, G =10, s =1, s =1, and s =2. σ μ 1 2 =1, n =1:0, n =1:1, γ =0:1, G =10, s = s =1, and s =2. σ μ s 1 2 Region-I Region-II Region-I Region-II 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 n = 1.5 n = 0.5 𝜇 G = 15 G = 5 n = 1.0 n = 2.0 𝜇 𝜇 G = 20 G = 10 Figure 6: Variations of u ðrÞði =1, 2Þ with n when Da =1:0, M Figure 8: Variations of u ðrÞði =1, 2Þ with G when Da =1:0, M =1, i μ i Re = 2, n =1:0, n =1:1, γ =0:1, s =1, s =1, and s =2. =1, Re = 2, n =1:0, γ =0:1, G =10, s =1, s =1, and s =2. σ μ s 1 2 σ s 1 2 The effects of couple stress parameters s and s on the 1 2 flow are displayed in Figures 3–4. The effect of the couple stress parameter s on the flow velocity profiles is seen in 1 Region-II Figure 3. In this case, we notice that a rise in the couple stress parameter s causes the fluid’s velocity to increase in Region-I both flow areas. Figure 4 shows a similar trend when varied with the couple stress parameter s . Therefore, we draw the conclusion that raising couple stress parameters s 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 –1 Murthy [32], Srinivas et al. [37], and Kumar and Agarwal 2 2 0.0 0.5 1.0 1.5 2.0 [44]. Since s = μ R /η , an increase in couple stress viscosi- i i i ties η for i =1,2 corresponds to a decrease in the couple stress parameters s . As a result, increasing couple stress γ = 0.15 γ = 0.05 i s coefficients η for i =1, 2 has a retarding effect on fluid veloc- γ = 0.2 γ = 0.1 i s ities. This indicates that the presence of couple stress in the fluid reduces the velocity of a fluid. This is due to the fact Figure 9: Variations of u ðrÞði =1, 2Þ with γ when Da =1:0, M =1 i s that physically, the couple stresses expend some energy to , Re = 2, n =1:0, n =1:1, G =10, s =1, s =1, and s =2. σ μ 1 2 rotate the particles, which reduces the particles’ velocity. u (r) u (r) u (r) u (r) u (r) i i 10 Journal of Applied Mathematics Table 1: Variations of Q ði =1, 2Þ and Q with respect to M. Table 4: Variations of Q ði =1, 2Þ, Q with respect to Da. i i MQ1 Q2 Q Da Q1 Q2 Q 0. 19.6805 20.0851 39.7655 0.1 5.58574 7.23685 12.8226 1 15.9975 16.9066 32.904 0.3 10.9187 12.3188 23.2375 1.5 12.9213 14.2132 27.1345 0.5 13.3591 14.5399 27.899 2. 10.8443 12.4318 23.2761 0.8 15.2487 16.2378 31.4865 Table 2: Variations of Q ði =1,2Þ, Q with respect to s . Table 5: Variations of Q ði =1, 2Þ, Q with respect to n . i 1 i μ s Q1 Q2 Q 1 n Q1 Q2 Q 1 15.9975 16.9066 32.904 0.5 22.9166 27.7223 50.6389 2 18.0884 18.7352 36.8237 1. 16.7912 17.9853 34.7765 3 18.5571 19.2242 37.7813 1.5 13.5173 13.7398 27.2571 4 18.7069 19.4358 38.1428 2. 11.3849 11.2239 22.6088 Table 3: Variations of Q ði =1,2Þ, Q with respect to s . Table 6: Variations of Q ði =1,2Þ, Q with respect to Re. i 2 s Q1 Q2 Q Re Q1 Q2 Q 1 15.9975 16.9066 32.904 1 16.7912 17.9853 34.7765 2 18.0883 23.0374 41.1257 2 11.3849 11.2239 22.6088 3 18.6223 25.6862 44.3085 3 8.71848 8.28735 17.0058 4 18.8456 26.9956 45.8412 4 7.09974 6.59498 13.6947 Furthermore, it is to be noted that in the absence of couple Fekadu [51]. Figure 7 presents the effect of the Reynolds stresses, that is, as η ⟶ 0, the parameter s ⟶ ∞, the number on the velocity profile. Thereby, we observe that as properties of couple stress in the fluid vanish and the case the Reynolds number Re increases, there is a decrease in of classical viscous fluid can be obtained from this work the velocities of the fluid in both flow regions. This indicates (Umavathi et al. ([28–30]), Abbas et al. [22], Devakar et al. that velocity is reduced by the increase of the Reynolds num- [23], and Devakar and Ramgopal [31]). Therefore, it is ber Re and our results well agreed with the results of Deva- understood that the velocity in the case of couple stress fluid kar et al. [23], Devakar and Ramgopal [31], Srinivas and is lower than that of a Newtonian fluid. Murthy [32], and Punnamchandar and Fekadu [51]. The effect of Darcy’s number Da on the fluid velocities is Figure 8 represents the velocity profile for the different shown in Figure 5. From this figure, it is noticed that the values of the pressure gradient. It is observed that with the velocities in both fluid regions increase with the increase of increase in G, velocity is increasing in both the fluid regions. Darcy’s number Da. Since Da = k/R , an increase in Darcy’s Physically, the more the pressure gradient, the more the fluid number corresponds to an increase in the permeability (per- is pushed to generate the flow, which results in an increase in meable parameter k) of the porous medium, which supports fluid velocity. Figure 9 displays the effect of the slip param- the flow. Lesser permeability causes a slighter fluid velocity eter γ on the fluid flow velocity profiles. Figure 9 shows that to be observed inside the flow medium occupied by the fluid. increasing the slip parameter reduces fluid velocity in both Thus, it may be concluded that an increase in the Darcy’s zones. Obviously, fluid slippage has the opposite impact on number enhances fluid velocities. This is due to the reason fluid motion, and increasing the slip parameter reduces the that the additional flow resistance that the porous structure velocity significantly in both regions. A similar trend was offers diminishes as Da (permeable parameter k) gradually observed in the work of Punnamchandar and Fekadu [51, increases. A similar kind of behavior can be found in Refs. 52]. Furthermore, when the slip parameter is set to zero, Srinivas and Murthy [32], Srinivas et al. [37], Punnamchan- the classical case of no slip is recovered as a special case. dar and Fekadu [51], and Kumar and Agarwal [44]. The numerical values of the volume flow rate are com- Figure 6 describes the effect of the ratio of viscosities n puted for various pertinent flow parameters and are pre- on velocity profiles. Figure 6 reveals that as the viscosity ratio sented in Tables 1–8. The effect of the magnetic number M n increases, the velocity of the fluid decreases in both flow on the flow rate is shown in Table 1. From Table 1, we notice regions. This is because as the viscosity ratio n increases, 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, greater flow resistance is provided. As a result, velocity n =1:0, n =1:1, γ =0:1, G =10, s =1, s =1, and s =2. drops. Therefore, we conclude that an increase in the ratio σ μ s 1 2 of viscosities inhibits fluid motion. A similar view can be Tables 2 and 3 shows the nature of flow rates for different found in the works of Umavathi et al. [21], Umavathi et al. values of couple stress parameter s , i =1, 2. From the tables, [30], Srinivas and Murthy [32], and Punnamchandar and we can see that the total flow rate increases with an increase Journal of Applied Mathematics 11 Table 7: Variations of Q ði =1, 2Þ, Q with respect to G. i (iii) Increase in the magnetic number, slip parameter, viscosity ratio, and Reynolds number suppress the GQ1 Q2 Q volume flow rate 5 7.99873 8.45329 16.452 (iv) Increase in the couple stress parameter, Darcy num- 10 15.9975 16.9066 32.904 ber, and pressure gradient promotes volume flow 15 23.9962 25.3599 49.3561 rate 20 31.9949 33.8132 65.8081 This work can be extended to the unsteady flow problem and is also made to include heat transfer/thermal effects. We Table 8: Variations of Q ði =1, 2Þ, Q with respect to γ . would like extend this work by taking various fluids like micropolar fluid, or any other non-Newtonian fluid. γ Q1 Q2 Q 0.05 16.3244 18.5806 34.905 Nomenclature 0.1 15.9975 16.9066 32.904 0.15 15.656 15.1585 30.8145 B : Magnetic field intensity 0.2 15.2991 13.3312 28.6303 C , D , ði =1,2, 3,4Þ: Arbitrary constants i i Da: Darcy number G: Pressure gradient of couple stress parameters s , i =1, 2. Table 4 demonstrates Current density the effect of Darcy’s number on the flow rate. From J : Table 4, we can see that the total flow rate increases with M: Magnetic number an increase in Darcy’s number. Table 5 shows the effect of m : Couple stress ij the viscosity ratio on the flow rate. The flow rate shows a P: Fluid pressure at any point decreasing trend with the growth of the viscosity ratio. ! Velocity vector in regions I and II q ði =1, 2Þ: Table 6 displays various values of flow rate with respect to Q: Total volumetric flow rate the Reynolds number. It is seen from Table 6 that as the Q ði =1,2Þ: Flow rate in regions I and II Reynolds number increases, the total flow rate decreases. R: Radius of the inner cylindrical region Table 7 represents the flow rate for the different pressure Re: Reynolds number gradient values. From the table, it is observed that increasing R : Radius of the cylinder the pressure gradient increases the volume flow rate across s: = R /R, radius ratio the pipe cross-section. Table 8 presents the numerical flow s ði =1,2Þ: Couple stress parameters rate data with respect to slip parameter γ . It is observed that u ði =1,2Þ: Velocity components the volume flow rate gets decreased with an increase of slip 2 2 ∇ : The operator d /dr + ð1/rÞðd/drÞ parameter γ . r, θ, z: Cylindrical coordinates I ð:Þ, K ð:Þ: Modified Bessel functions n n ′ Couple stress viscosity coefficients η , η : 6. Conclusions i i γ : Nondimensional slip parameter γ : Slip coefficient The problem of steady, laminar, and fully developed hydro- μ ði =1, 2Þ: Dynamic viscosity coefficients magnetic flow of two immiscible couple stress fluids through n : Ratio of viscosities a porous medium in a horizontal cylinder under the effect of the Navier slip boundary condition is considered in the pres- ρ ði =1,2Þ: Density of fluid in regions I and II ent study. The motion is generated by a constant pressure σ ði =1,2Þ: Electrical conductivity gradient delivered along the axial direction, i.e., z-axis. 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[41] I. A. Ansari and S. Deo, “Effect of magnetic field on the two immiscible viscous fluids flow in a channel filled with porous medium,” National Academy Science Letters, vol. 40, no. 3, pp. 211–214, 2017. [42] P. K. Yadav and S. Jaiswal, “Influence of an inclined magnetic field on the Poiseuille flow of immiscible micropolar- Newtonian fluids in a porous medium,” Canadian Journal of Physics, vol. 96, no. 9, pp. 1016–1028, 2018. [43] S. Jaiswal and P. K. Yadav, “Influence of magnetic field on the Poiseuille flow of immiscible Newtonian fluids through highly porous medium,” Journal of the Brazilian Society of Mechani- cal Sciences and Engineering, vol. 42, no. 4, pp. 1–15, 2020. [44] D. Kumar and M. Agarwal, “MHD pulsatile flow and heat transfer of two immiscible couple stress fluids between perme- able beds,” Kyungpook Mathematical Journal, vol. 61, pp. 323– 351, 2021. [45] A. B. Basset, A Treatise on Hydrodynamics, vol. 2, Dover, New York, NY, USA, 1961. [46] B. S. Bhatt and N. C. 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Fekadu, “Effects of slip and uni- form magnetic field on flow of immiscible couple stress fluids in a porous medium channel,” The International Journal of Engineering and Science, vol. 1, pp. 1–8, 2020. [52] B. Punnamchandar and Y. S. Fekadu, “Effects of slip and inclined magnetic field on the flow of immiscible fluids (couple stress fluid and Jeffrey fluid) in a porous channel,” Journal of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied Mathematics Hindawi Publishing Corporation

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

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10.1155/2023/1902844
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Hindawi Journal of Applied Mathematics Volume 2023, Article ID 1902844, 13 pages https://doi.org/10.1155/2023/1902844 Research Article Hydromagnetic Flow of Two Immiscible Couple Stress Fluids through Porous Medium in a Cylindrical Pipe with Slip Effect 1 2 Punnamchandar Bitla and Yitagesu Daba Kore Department of Mathematics, Anurag University, Telangana, India Department of Mathematics, Wollega University, Nekemte, Ethiopia Correspondence should be addressed to Punnamchandar Bitla; punnam.nitw@gmail.com Received 23 November 2022; Revised 23 February 2023; Accepted 18 April 2023; Published 3 May 2023 Academic Editor: Oluwole D. Makinde Copyright © 2023 Punnamchandar Bitla and Yitagesu Daba Kore. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, the steady hydromagnetic flow of two immiscible couple stress fluids through a uniform porous medium in a cylindrical pipe with slip effect is investigated analytically. Essentially, the flow system is divided into two regions, region I and region II, which occupy the core and periphery of the system, respectively. The flow is driven by a constant pressure gradient applied in a direction parallel to the cylinder’s axis, and an external uniform magnetic field is applied in the direction perpendicular to the direction of fluid 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 fluid-fluid interface. The governing equations are modeled using the fully developed flow conditions. The resulting differential equations governing the flow 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 flow velocity, flow rate, and stresses are derived in terms of the Bessel functions. The impacts of several parameters pertaining to the flow 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 flow rate’s numerical values are also calculated for different fluid 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 fluids whereas increasing the couple stress parameter, Darcy number, and pressure gradient increases fluid velocities. The results obtained in this paper show an excellent agreement with the already existing results in the literature as limiting cases. 1. Introduction theories [1–3] pertaining to non-Newtonian fluids have been developed for describing the rheological behavior of numer- Over the last few decades, the study of non-Newtonian fluids ous complex fluids. The couple stress fluid, initiated by has received a lot of attention as these types of fluids fre- stokes [4], is one of the popular theories of polar fluids that quently occur in many industrial, scientific, and technologi- considers the possibility of polar effects such as the presence cal processes. Many important complex and real fluids of couple stresses and body couples in the fluid medium. The which include molten metals, polymeric liquids, slurries, flow behavior of various fluids that contain a substructure blood, liquid crystals, lubricants, soaps, greases, gelatin, such as lubricants with small amounts of additives, poly- and paints belong to this family of fluids. This class of fluids mers, colloidal suspensions, liquid crystals, animal and does not follow the Newtonian fluid theory as they possess human blood, polymer-thickened oils, muddy water, and microstructure and a nonsymmetric stress tensor in their electrorheological and synthetic fluids can be modeled using fluid structure. For this reason, several new microcontinuum the couple stress fluid theory [5]. A couple stress fluid model 2 Journal of Applied Mathematics has been successfully employed to study the mechanism of concentric pipes filled with the porous medium was done peristalsis [6, 7]. The couple stress fluid theory widely used by Yadav et al. [33]. Other important studies in this direc- in modeling the flow of biological fluids such as synovial tion include Chamkha [34], Harmindar and Singh [35], fluids [8–11] and blood [12–14]. Singh [36], and Srinivas et al. [37]. Though many In realistic situations, most of the flow problems arising researchers have worked on the flow of immiscible fluids in the industries, manufacturing process, geology, ground- through porous channels, the flow of immiscible non- water hydrology, reservoir mechanics, biomechanics, mag- Newtonian fluids through porous cylinders is reasonably netofluid dynamics, geophysics, plasma physics, and so on underexplored despite its applicability in blood flows, chem- occur with two or more fluids of different densities/viscosi- ical engineering, crude oil extraction, etc. One such type of ties flowing immiscibly in the same channel or cylindrical problem is going to be discussed in this paper. pipe. Examples of these systems are the flow of several Magnetohydrodynamics (MHD) is also an interesting immiscible oils through the bed of rocks or soils, the flow and important area of modern engineering sciences and in the rivers with several industrial fluids, blood flow in the involves the interaction of magnetic forces and electrically arteries, the flow of air and fuel droplets in combustion conducting fluids. The application of magnetic fields to chambers, the flow of air and exhaust gases at engine outlets, the flow of immiscible fluids originates from reducing the flows for many medical and industrial purposes. The study gas and Petrolia flow in pipes of oil, water-air flows around ship halts, etc. These are referred to as multiphase flows in of the hydromagnetic flow of moving fluids through a the literature. Owing to its wide areas of applications, several porous medium is currently a subject of great interest researchers have studied multiphase fluid flows. Chaturani owing to plentiful applications in industrial, engineering, and Samy [15] as well as Sinha and Singh [16] investigated and medical devices. Owing to these applications, several the effects of couple stresses on blood flow, and many other studies have been conducted to examine the effect of mag- authors (Valanis and Sun [17], Sharan and Popel [18], and netic fields on the flows of immiscible fluids. In light of Garcia and Riahi [19]) discussed blood flow considering it this, Vajravelu et al. [38] studied the hydromagnetic as a two-phase flow in which they have assumed blood as a unsteady flow of two conducting immiscible fluids between couple stress fluid. Besides its application in blood flow, two permeable beds. Malashetty et al. [39] analyzed the the study of multiphase flow has several important applica- magnetohydrodynamic two-fluid convective flow and heat tions in various fields of engineering and science. Umavathi transfer in an inclined composite porous medium. Raju et al. [20] analytically solved the problem of flow through a and Nagavalli [40] studied the unsteady two-layered fluid horizontal channel with a couple stress fluid sandwiched flow and heat transfer of conducting fluids in a channel between viscous fluid layers. They discussed the effects of between parallel porous plates under a transverse magnetic various flow parameters and concluded that the couple stress field. Ansari and Deo [41] investigated the effect of a mag- parameter influences the flow. Umavathi et al. [21] made a netic field on the two immiscible viscous fluids flowing in a detailed study on the flow and heat transfer of a couple stress channel filled with a porous medium. The influence of an fluid in contact with a Newtonian fluid. Abbas et al. [22] inclined magnetic field on the Poiseuille flow of immiscible analyzed the hydromagnetic mixed convective two-phase micropolar-Newtonian fluids through the horizontal flow of couple stress and viscous fluids in an inclined chan- porous channel where the permeability of both the regions nel. They obtained closed-form solutions of velocity and of the horizontal porous channel has been taken differently temperature profiles by using the perturbation method. was discussed by Yadav and Jaiswal [42]. In another paper, Devakar et al. [23] studied the unsteady flow of couple stress Jaiswal and Yadav [43] investigated the influence of a mag- fluid sandwiched between Newtonian fluids through a chan- netic field on the Poiseuille flow of immiscible Newtonian fluids through a highly porous medium. More recently, nel. There are many other works concerning multiphase fluid flows (see Packham and Shall [24], Rao and Usha Kumar and Agrawal [44] studied the magnetohydrody- [25], Chamkha et. al [26], Umavathi et al. [27, 28], and namic pulsatile flow and heat transfer of two immiscible Umavathi and Shekar [29]). couple stress fluids in a porous channel. In view of its numerous applications, the researchers at A majority of the studies regarding the flow of immisci- present are engaged in exploring immiscible flows of fluids ble non-Newtonian fluids quoted above were carried out by through a porous medium under various circumstances. imposing the no-slip boundary condition. However, several Umavathi et al. [30] studied the problem of the convective theoretical and experimental studies [45–50] reveal that slip flow of two immiscible fluids (couple stress and viscous exists at the solid boundary. So, for flows containing fluids fluids) through a vertical channel. They obtained an approx- through solid boundaries, consideration of a velocity slip is imate solution by using the regular perturbation method. more realistic and appropriate. Recently, Punnamchandar Devakar and Ramgopal [31] presented analytical solutions and Fekadu [51] investigated the effects of slip and uniform for the fully developed flows of two immiscible couple stress magnetic field on the flow of immiscible couple stress fluids and Newtonian fluids through a nonporous and porous in a porous medium channel. In another paper, Punnam- medium in a horizontal cylinder. Srinivas and Murthy [32] chandar and Fekadu [52] considered the problem of the studied the flow of two immiscible couple stress fluids effects of slip and inclined magnetic field on the flow of between two permeable beds. They obtained an exact solu- immiscible fluids (couple stress fluid and Jeffrey fluid) in a tion to the considered problem. An analysis of the Poiseuille porous channel. It is observed that the effects of slip and flow of immiscible micropolar-Newtonian fluids through magnetic field on the flow of immiscible couple stress fluids Journal of Applied Mathematics 3 through a porous medium in a cylindrical pipe have not The material constants λ and μ are the viscosity coeffi- been discussed yet. ′ cients, and η and η are the couple stress viscosity coeffi- Keeping this in view the wide potential applications of cients satisfying the constraints immiscible couple stress fluids flow and the importance of exact solutions described above, the goal of the current paper ′ ′ μ ≥ 0, 3λ +2μ ≥ 0,jj η ≥ η , η ≥ 0: ð5Þ is to determine exact solutions for the steady hydromagnetic flow of two immiscible couple stress fluids through a porous pffiffiffiffiffiffiffi There is a length parameter l = η/μ which is a charac- medium in a cylindrical pipe with slip effect. The impacts of teristic measure of the polarity of the couple stress fluid, different flow parameters on the velocity field and flow rate and this parameter is identically zero in the case of nonpolar are investigated. The slip factor in fluid flows makes the fluids. problem even more realistic and interesting, which moti- vated us to consider this problem. The practicality and the 3. Formulation of the Problem complexities involved due to the porosity and cylindrical nature of the geometry also make the work presented in this The physical model concerns an axisymmetric fully devel- paper novel. oped hydromagnetic flow of two immiscible couple stress fluids flowing through a porous medium in a horizontal cir- 2. Basic Equations cular pipe of radius R . Owing to the fluids’ immiscibility, there are two separate regions of fluid flow: region I, or the The basic equations describing the flow of couple stress fluid core region, and region II, or the periphery region. The flow including a Lorentz force are (Stokes [4, 5]) as follows: geometry of the problem is depicted in a cylindrical polar Continuity equation (conservation of mass): coordinate system ðr, θ, zÞ with the origin at the center of the tube and common axis of the cylindrical regions taken ∂ρ as the z-axis, as shown in Figure 1. Region I ð0 ≤ r ≤ RÞ is +∇: ρq =0: ð1Þ ∂t occupied with couple stress fluid with density ρ , shear vis- cosity μ , and couple stress viscosity η , comprising the core 1 1 Momentum equation (conservation of momentum): region of the pipe whereas region II ðR ≤ r ≤ R Þ is occupied by a different couple stress fluid having density ρ , shear vis- Dq 1 ! ! cosity μ , and couple stress viscosity η , comprising the 2 2 ρ = ρ f + ∇ × ρ c −∇P − μ∇ × ∇ × q − η∇ Dt 2 peripheral region of the pipe. The motion of the fluids in ð2Þ ! ! ! ! both regions is caused by a constant pressure gradient × ∇ × ∇ × ∇ × q +ðÞ λ + μ ∇∇·q + J × B, applied in a direction parallel to the cylinder’s axis, i.e., z -axis, and an external uniform magnetic field of strength B directed perpendicular to the flow direction is also 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 applied. To develop the governing equations for the considered c are the velocity, body force per unit mass, and body couple ! ! model, the following presumptions are taken in the analysis per unit mass, respectively. The term J × B in equation (2) of the current study: is the Lorentz force (electromagnetic body force) in which ! ! J is the electric current density and B is the total magnetic (i) The fluids are considered incompressible, and the field. flow is assumed to be steady, laminar, and fully The force stress tensor τ (Stokes [5]) that arises in the ij developed theory of couple stress fluids is given by (ii) Both the fluid regions are saturated with the uni- form porous media of permeability k τ = −P + λ∇:q δ +2μd + ε m +4ηω + ρc : ½Š ij ij ij ijk ,k k,rr k (iii) The Lorentz force is the only body force acting on the fluids, with no body couples ð3Þ (iv) The magnetic Reynolds number of the flow is The couple stress tensor m (Stokes [5]) that arises in the ij assumed to be very small, and no external voltage theory has the linear constitutive relation is applied so that the induced magnetic field is neglected and the Hall effect of magnetohydrody- namics is assumed to be negligible m = mδ +4η ω +4ηω : ð4Þ ij ij j,i i,j Under the assumptions made, the vector forms of con- servation equations governing the flow of steady, incom- In the above, ω is the spin tensor, ρc is the body cou- i,j k pressible immiscible couple stress fluids through a porous ple vector, d is the components of the rate of shear strain, ij cylinder in the presence of a transverse magnetic field can δ is the Kronecker symbol, e is the Levi-Civita symbol, be written in the following form (Punnamchandar and ij ijk and comma denotes covariant differentiation. Fekadu [51] and Kumar and Agrawal [44]): 4 Journal of Applied Mathematics Continuity equations: From equations (3) and (4), the force stress tensor τ ij and couple stress tensor m of the couple stress fluids are ij ∇:q =0: ð6Þ given by Momentum equations: Âà ð13Þ τ = μ u − η ∇ u , i =1,2, rzðÞ i i i i i dr −∇P − μ ∇ × ∇ × q − η ∇ × ∇ × ∇ × ∇ i i i ð7Þ ! ! ! ! d u η ′ du × q + J × B − q =0, i i i i i i ð14Þ k m = η − , i =1,2: rθðÞ i i dr r dr where i =1,2 denotes distinct fluid regions. To determine u ðrÞ and u ðrÞ, the boundary and inter- ! 1 2 The additional term −μ/kq in the governing equation i face conditions have to be specified. (7) is due to the porous medium (Chamkha [34]) where k 3.1. Boundary and Interface Conditions. The description and is the permeability of the porous medium and q ði =1, 2Þ mathematical form of the boundary conditions are pre- is the velocity vector. ! sented in this section. The current density J is expressed by Ohm’s law (Gold Instead of the usual no-slip condition, the slip velocity is [53]): 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 condi- J = σ E + q × B , ð8Þ i i i tion states that the tangential velocity of the fluid relative to the solid at a point on its surface is proportional to the tan- gential stress acting at that point. The proportionality that where σ ði =1, 2Þ and E stand for electrical conductivity of characterizes the surface’s “slipperiness” is known as the slip length. the fluids for regions I and II and electric field, respectively. ! ! In view of the higher-order nature of governing equa- Here, E =0 as there is no external electric field, and jBj tions, additional boundary conditions are required to find = B because of our assumption that the induced magnetic the solution. In addition to the Navier slip boundary condi- field is too less (assumed to be zero) as compared to the tion, we use the Stokes (Stokes [5]) boundary conditions to external magnetic field. Hence, the Lorentz force is given by solve the governing equations of the flow under consider- ation. The Stokes boundary condition assumes that the cou- ! ! ! ! ! ! ! ple stresses vanish on the boundary of the solid. F = J × B = σ q × B × B = −σ B q : ð9Þ i i i i i 0 i 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 Due to the unidirectional and symmetric nature of the problem is the coupling across the fluid/fluid interface. The flow, the fluid velocity vectors for both regions are to be in fluid layers are mechanically coupled via the transfer of the form q = ð0, 0, u ðrÞÞ where i =1,2. These choices of i i momentum across the interface. By the virtue of coupling velocities automatically satisfy the continuity equation (6) of fluid layers at the fluid-fluid interface through momentum in respective flow regions. Under the above conditions, transfer, the continuity conditions for the velocity, vorticity, equation (7) governing the flow of the couple stress fluids couple stress, and shear stress are adopted at the fluid-fluid in the respective regions can be written as follows: interface. Therefore, the following physically realistic and In region I ð0 ≤ r ≤ RÞ (core region), we have mathematically consistent boundary and interface condi- tions are used for the considered physical model: μ ∂P 4 2 1 2 −η ∇ u + μ ∇ u − + σ B u = : ð10Þ 1 1 1 1 1 0 1 (i) The slip condition along with vanishing couple k ∂z stresses are taken at the boundary of cylindrical pipe r = R In region II ðR ≤ r ≤ R Þ (peripheral region), we have Following Punnamchandar and Fekadu [51], the slip condition gives μ ∂P 4 2 2 2 −η ∇ u + μ ∇ u − + σ B u = , ð11Þ 2 2 2 2 2 2 0 k ∂z u ðÞ R =±γ τ ðÞ R , ð15Þ 2 0 rz 2 0 s ðÞ ⋆ ⋆ where ∇ is the differential operator defined as where γ such that ð0 ≤ γ < ∞Þ corresponds to the slip s s coefficient at the upper boundary (Navier [54]). Note that d 1 d as γ =0, the classical no-slip case is recovered (Devakar ∇ = + : ð12Þ dr r dr and Ramgopal [31]). Journal of Applied Mathematics 5 Following Srinivas and Murthy [32], the vanishing of For the peripheral region II ð1 ≤ r ≤ sÞ, we have couple stress on the surface of the cylinder leads to 1 s ReG 4 2 2 2 2 2 ∇ u − s ∇ u + s + M u = , ð24Þ 2 2 2 2 1 2 m ðÞ R =0: ð16Þ rθ 2 0 ðÞ Da n where (ii) The continuity conditions for the velocity, vorticity, μ R 2 i shear stress, and couple stress are adopted at the s = , fluid-fluid interface. Following Kumar and Agrawal i 2 2 [44], this implies B R σ 2 0 1 M = , u r = u r at r = R, ð17Þ ðÞ ðÞ 1 2 sffiffiffiffiffi M = M , du r du r ðÞ ðÞ 1 2 ð18Þ ð25Þ = at r = R, dr dr n = , τ r = τ r at r = R, ðÞ ðÞ ð19Þ rzðÞ 1 rzðÞ 2 m r = m r at r = R: n = , ðÞ ðÞ ð20Þ rθðÞ 1 rθðÞ 2 μ (iii) Regularity condition: the axisymmetric flow sug- Da = : gested that the velocity of the fluid is finite on the axis of the cylinder r =0. Following Devakar and In the above equations, G = −∂P/∂z is a constant pres- Ramgopal [31], this implies sure gradient, Re = ρ UR/μ is the Reynolds number, s = 1 1 i 2 2 μ R /η is the couple stress parameter, Da = k/R is the Darcy u ðÞ r is finite at r =0: ð21Þ 1 i i number, n = σ /σ is the conductivity ratio, M = B R σ 2 1 0 pffiffiffiffiffiffiffiffiffiffiffi σ /μ is the magnetic number, and n = μ /μ is the vis- 1 1 μ 2 1 To solve equation (10) and equation (11) under the cosity ratio. boundary conditions (equation (15)–equation (21)), we From equations (13) and (44), the nondimensional make use of the following nondimensional quantities: forms of the shear stresses and couple stresses are r = , u μ d 1 0 i 2 τ = u − ∇ u , i =1,2, ð26Þ rzðÞ i i i R dr s z = , "# u η d u η 1 du ⋆ i 0 i i i i u = , m = − , i =1,2: ð27Þ rθðÞ i 2 2 dr η r dr 0 R ð22Þ P = , 4. Solution of the Problem ρ u 1 0 R 4.1. Flow Velocity in the Two Regions. The methodology used s = , to get the general solution of the nondimensional differential ⋆ equations (23) and (24) governing the fluids flow is as fol- γ μ s 2 γ = , lows: finding the complementary solution u ðrÞ of the s c homogenous differential equation and then determining the particular solution u ðrÞ of the nonhomogeneous differ- where u and R are characteristic velocity and radius for the ential equation. Thus, the general solution can be con- given flow model, respectively, and i =1,2 denotes distinct structed as fluid regions. Using the dimensionless variables in equations (10) and u r = u r + u r : ð28Þ ðÞ ðÞ ðÞ (11), the nondimensional form of the governing equations i c p (after dropping the stars) is as follows: For the core region I ð0 ≤ r ≤ 1Þ,wehave Region I ð0 ≤ r ≤ 1Þ: Let 4 2 2 2 2 2 ∇ u − s ∇ u + s + M u = s ReG: ð23Þ 1 1 1 1 1 1 2 2 2 ð29Þ Da α + α = s , 1 2 1 6 Journal of Applied Mathematics Therefore, the general solution of the differential equa- 2 2 2 2 α α = s + M : ð30Þ 1 2 1 tion (31), after substituting (40) and (41) in equation (28), Da becomes Then, the equation (23) governing the fluid in region I can be written as u ðÞ r = C IðÞ α r + C K ðÞ α r + C IðÞ α r 1 1 0 1 2 0 1 3 0 2 ÀÁÀÁ 2 2 2 2 2 ð42Þ ReGs ∇ − α ∇ − α u = ReGs , ð31Þ 1 1 2 1 1 + C K α r + , ðÞ 4 0 2 2 2 α α 1 2 where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where C , C , C , and C are arbitrary constants. 2 4 2 2 1 2 3 4 s ± s − 4s 1/Da + M ðÞ ðÞ 2 2 1 1 1 α , α = : ð32Þ Region-II ð1 ≤ r ≤ sÞ: 1 2 Let First, we consider the corresponding homogeneous dif- 2 2 2 ð43Þ β + β = s , ferential equation 1 2 2 ÀÁÀÁ 2 2 2 2 ∇ − α ∇ − α u =0: ð33Þ 1 1 2 2 2 2 2 β β = s + M : ð44Þ 1 2 2 Da The solution of (33) is obtained by using the superposi- Then, equation (24) governing fluid flow in region II can tion principle such that be written as ðÞ 1 u r = y r + y r , ð34Þ ðÞ ðÞ ðÞ c 1 2 ÀÁÀÁ Re Gs 2 2 2 2 ∇ − β ∇ − β u = , ð45Þ 1 2 where ÀÁ 2 2 ð35Þ ∇ − α y =0, where for region II, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ 2 2 2 4 2 2 ∇ − α y =0: ð36Þ s ± s − 4sðÞ ðÞ 1/Da + M 2 2 2 2 2 2 2 1 β , β = : ð46Þ 1 2 Further, the differential equations (35) and (36) can be reduced to the following modified Bessel differential equa- Therefore, similarly solving equation (45) by the method tions: stated above, we get d y 1 dy 2 i i 2 2 u ðÞ r = D IðÞ β r + D K ðÞ β r + D IðÞ β r r + − α r y =0, i =1, 2: ð37Þ 2 1 0 2 0 3 0 1 1 2 i i dr r dr ReGs ð47Þ + D K β r + , ðÞ 4 0 2 2 2 n β β μ 1 2 The solutions to the above two equations (35) and (36), respectively, are where D , D , D , and D are arbitrary constants. 1 2 3 4 ð38Þ y ðÞ r = C IðÞ α r + C K ðÞ α r , 1 1 0 1 2 0 1 The closed-form solutions of the differential equations (31) and (45) are given by equations (42) and (47) contain ð39Þ y ðÞ r = C IðÞ α r + C K ðÞ α r : modified Bessel functions. Here, I ðα rÞ, I ðα rÞ, I ðβ rÞ, 2 3 0 2 4 0 2 0 1 0 2 0 and I ðβ rÞ and K ðα rÞ, K ðα rÞ, K ðβ rÞ, and K ðβ rÞ 0 2 0 1 0 2 0 1 0 2 Inserting the expressions (38) and (39) into (34), we are the first kind modified Bessel’s functions of zero order obtain the general solution of equation (33) as and the second kind modified Bessel’s functions of zero order, respectively. u r = C I α r + C K α r + C I α r + C K α r : ðÞ ðÞ ðÞ ðÞ ðÞ c 1 0 1 2 0 1 3 0 2 4 0 2 4.2. Stress in the Two Regions. From equation (26), the non- ð40Þ dimensional tangential stress of the fluid in region I is given by The particular solution of the differential equation (31) μ u α can be easily obtained as 1 0 1 τ = α − I rα C − K rα C ðÞ ðÞ ðÞ rzðÞ 1 1 1 1 1 1 1 2 R s ð48Þ ReGs α 1 2 y ðÞ r = : ð41Þ + α −ðÞ IðÞ rα C − K ðÞ rα C : 2 1 2 3 1 2 4 2 2 2 α α s 1 2 1 Journal of Applied Mathematics 7 Region-II: Couple stress fluid Ro Region-I: Couple stress fluid Pressure gradient (G) Magnetic field of intensity Bo Figure 1: Geometrical configuration. From equation (27), the couple stress in region I is given To obtain a complete solution to the concerned problem, by we have to determine the constants C and D for i =1, 2,3, i i 4. C and D for i =1,2,3,4 are calculated numerically by solv- i i " ! ! ing the algebraic system obtained from the boundary η u α η 0 2 1 1 1 conditions. m = α I rα − 1+ I rα C ðÞ ðÞ rθðÞ 1 1 0 1 1 1 1 r η ! ! 4.3. Determination of Arbitrary Constants. Nondimensiona- α η 2 1 1 lizing the boundary conditions (15)–(21), we have the + α K rα + 1+ K rα C ðÞ ðÞ 1 0 1 1 1 2 r η following: ! ! α η (i) Since the modified Bessel function of the second kind, 2 2 1 + α I rα − 1+ I rα C ðÞ ðÞ 2 0 2 1 2 3 i.e., K ðrÞ,isnot finite at a singular point r =0,there- r η ! ! # fore for finite values of u ðrÞ along the axis of a cylin- α η drical pipe, the coefficient of K ðα rÞ for n =0,1 and 2 2 n i + α K ðÞ rα + 1+ K ðÞ rα C : 2 0 2 1 2 4 r η i =1,2 should be zero. Thus, we have ð49Þ C = C =0: ð52Þ 2 4 Similarly, tangential stress in region II given by equation (ii) The slip and vanishing of couple stress boundary (26) becomes conditions at r = s give " ! u s =±γ τ s , μ u β ðÞ ðÞ ð53Þ 2 s rzðÞ 2 2 0 1 τ = β −ðÞ IðÞ rα D − K ðÞ rβ D rz 2 1 1 1 1 2 ðÞ 1 1 R s ð50Þ ! # m ðÞ s =0, ð54Þ rθðÞ 2 + β − I rβ D − K rβ D : ðÞ ðÞ ðÞ ⋆ 2 1 2 3 1 2 4 2 where γ = γ μ /R is the nondimensional slip s s 2 parameter (iii) Continuity of velocities, vorticites, shear stresses The couple stress of fluid in region II given by equation and couple stresses at the fluid-fluid interface r =1 (26) becomes are as follows: " ! ! ð55Þ u ðÞ r = u ðÞ r , η u β η 1 2 0 2 2 1 2 m = β IðÞ rβ − 1+ IðÞ rβ D rθ 2 0 1 1 ðÞ 2 1 1 1 R r η du r du r ðÞ ðÞ ! ! 1 2 ð56Þ = , À ′ β η dr dr 1 2 + β K ðÞ rβ + 1+ K ðÞ rβ D 1 0 1 1 1 2 r η τ ðÞ 1 = τ ðÞ 1 , ð57Þ rz 1 rz 2 ðÞ ðÞ ! ! β η 2 2 m 1 = m 1 : + β I rβ − 1+ I rβ D ðÞ ðÞ ð58Þ ðÞ ðÞ rθðÞ 1 rθðÞ 2 2 0 2 1 2 3 r η ! ! # β η Substituting equation (42) and equations (47)–(51) in 2 2 + β K ðÞ rβ + 1+ K ðÞ rβ D : 0 1 4 2 2 2 equations (52)–(58), the linear system of an algebraic equa- r η tion with six unknown arbitrary constants C , C , and D 1 3 i ð51Þ for i =1,2, 3,4 involved in the solution of the problem is 8 Journal of Applied Mathematics 7 7 6 6 Region-II Region-I Region-I Region-II 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 r r M = 0.5 M = 1.5 s = 1 s = 3 1 1 s = 2 s = 4 M = 1 M = 2 1 1 Figure 2: Variations of u ðrÞði =1,2Þ against M when Da =1:0, Figure 3: Variations of u ðrÞði =1, 2Þ with s when Da =1:0, M =1, i 1 Re = 2, n =1:0, n =1:1, γ =0:1, G =10, s =1, s =1, and s =2. σ μ 1 2 Re = 2, n =1:0, n =1:1, γ =0:1, G =10, s =1, and s =2. σ μ 2 formed. Using Mathematica software, all constants C , C , 1 3 and D for i =1,2, 3,4 have been evaluated uniquely using the above boundary conditions. Owing to the lengthy expressions of these constants, they are not presented here. 4.4. Total Flow Rate. The nondimensional volumetric flow rate across the whole cross-section of a porous cylinder is Region-I given by (Devakar and Ramgopal [31]) Region-II ð ð ð 2π 1 2 Q = ru dr + ru dr dθ: ð59Þ 1 2 0 0 1 0.5 1.0 1.5 2.0 Invoking the values of u ðrÞ and u ðrÞ from equations 1 2 (42) and (47) in equation (59) and integrating, we obtain s = 1 s = 3 2 2 s = 2 s = 4 I ðÞ α I ðÞ α ReDaG 1 1 1 2 Q =2π C + C +ÀÁ Figure 4: Variations of u ðrÞði =1, 2Þ with s when Da =1:0, M =1, 1 3 i 2 α α 21+ DaM 1 2 Re = 2, n =1:0, n =1:1, γ =0:1, G =10, s =1, and s =2. σ μ s 1 D D 1 2 ÁðÞ sI ðÞ sβ − IðÞ sβ −ðÞ sK ðÞ sβ − K ðÞ sβ 1 1 1 1 1 1 1 1 β β 1 1 for each case, when a particular parameter is varied, are D D 3 4 obtained by keeping Da =1:0, M =1, Re = 2, n =1:0, n = + sI sβ − I sβ − sK sβ − K sβ σ μ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 1 2 1 2 1 2 1 2 β β 2 2 1:1, γ =0:1, G =10, s =1, s =1, and s =2. s 1 2 ÀÁ The variations of velocity profiles for different flow ReGDa s − 1 + ÀÁ : parameters are shown graphically through Figures 2–9. 2n 1+ DaM Figure 2 illustrates the influence of the magnetic number ð60Þ M on the velocities. It is observed that the fluid velocities in both regions are decreasing with an increment of mag- netic number M. This finding suggests that the magnetic 5. Results and Discussion field applied to the flow system retards the motion of the fluid. This is consistent with the fact that a strong magnetic Analytical solutions for the steady, laminar hydromagnetic flow of two immiscible and incompressible couple stress field applied to the flow literally increases the Lorentz fluids through porous medium in a horizontal cylinder have force, which strongly opposes the fluid’s motion and lowers the velocities. This result is validated by the works of been obtained. The numerical evaluation of the analytical expressions for velocity profile and flow rate are done for Ansari and Deo [41], Kumar and Agarwal [44], and Pun- different flow parameters values, such as the magnetic num- namchandar and Fekadu [51, 52]. Further, as M⟶ 0, ber, couple stress parameter, Reynolds number, Darcy num- the magnetic number loses its properties and behaves as ber, ratio of viscosities, slip parameter, and pressure gradient a normal flow in the absence of a magnetic field (Srinivas and Murthy [32]). using Mathematica software package. The numerical values u (r) u (r) u (r) i i Journal of Applied Mathematics 9 Region-I Region-I Region-II Region-II 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Da = 0.1 Da = 0.5 R = 0.5 R = 1.5 Da = 0.3 Da = 0.8 R = 1 R = 2 Figure 5: Variations of u ðrÞði =1, 2Þ with Da when M =1, Re = 2, Figure 7: Variations of u ðrÞði =1, 2Þ with Re when Da =1:0, M n =1:0, n =1:1, γ =0:1, G =10, s =1, s =1, and s =2. σ μ 1 2 =1, n =1:0, n =1:1, γ =0:1, G =10, s = s =1, and s =2. σ μ s 1 2 Region-I Region-II Region-I Region-II 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 n = 1.5 n = 0.5 𝜇 G = 15 G = 5 n = 1.0 n = 2.0 𝜇 𝜇 G = 20 G = 10 Figure 6: Variations of u ðrÞði =1, 2Þ with n when Da =1:0, M Figure 8: Variations of u ðrÞði =1, 2Þ with G when Da =1:0, M =1, i μ i Re = 2, n =1:0, n =1:1, γ =0:1, s =1, s =1, and s =2. =1, Re = 2, n =1:0, γ =0:1, G =10, s =1, s =1, and s =2. σ μ s 1 2 σ s 1 2 The effects of couple stress parameters s and s on the 1 2 flow are displayed in Figures 3–4. The effect of the couple stress parameter s on the flow velocity profiles is seen in 1 Region-II Figure 3. In this case, we notice that a rise in the couple stress parameter s causes the fluid’s velocity to increase in Region-I both flow areas. Figure 4 shows a similar trend when varied with the couple stress parameter s . Therefore, we draw the conclusion that raising couple stress parameters s 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 –1 Murthy [32], Srinivas et al. [37], and Kumar and Agarwal 2 2 0.0 0.5 1.0 1.5 2.0 [44]. Since s = μ R /η , an increase in couple stress viscosi- i i i ties η for i =1,2 corresponds to a decrease in the couple stress parameters s . As a result, increasing couple stress γ = 0.15 γ = 0.05 i s coefficients η for i =1, 2 has a retarding effect on fluid veloc- γ = 0.2 γ = 0.1 i s ities. This indicates that the presence of couple stress in the fluid reduces the velocity of a fluid. This is due to the fact Figure 9: Variations of u ðrÞði =1, 2Þ with γ when Da =1:0, M =1 i s that physically, the couple stresses expend some energy to , Re = 2, n =1:0, n =1:1, G =10, s =1, s =1, and s =2. σ μ 1 2 rotate the particles, which reduces the particles’ velocity. u (r) u (r) u (r) u (r) u (r) i i 10 Journal of Applied Mathematics Table 1: Variations of Q ði =1, 2Þ and Q with respect to M. Table 4: Variations of Q ði =1, 2Þ, Q with respect to Da. i i MQ1 Q2 Q Da Q1 Q2 Q 0. 19.6805 20.0851 39.7655 0.1 5.58574 7.23685 12.8226 1 15.9975 16.9066 32.904 0.3 10.9187 12.3188 23.2375 1.5 12.9213 14.2132 27.1345 0.5 13.3591 14.5399 27.899 2. 10.8443 12.4318 23.2761 0.8 15.2487 16.2378 31.4865 Table 2: Variations of Q ði =1,2Þ, Q with respect to s . Table 5: Variations of Q ði =1, 2Þ, Q with respect to n . i 1 i μ s Q1 Q2 Q 1 n Q1 Q2 Q 1 15.9975 16.9066 32.904 0.5 22.9166 27.7223 50.6389 2 18.0884 18.7352 36.8237 1. 16.7912 17.9853 34.7765 3 18.5571 19.2242 37.7813 1.5 13.5173 13.7398 27.2571 4 18.7069 19.4358 38.1428 2. 11.3849 11.2239 22.6088 Table 3: Variations of Q ði =1,2Þ, Q with respect to s . Table 6: Variations of Q ði =1,2Þ, Q with respect to Re. i 2 s Q1 Q2 Q Re Q1 Q2 Q 1 15.9975 16.9066 32.904 1 16.7912 17.9853 34.7765 2 18.0883 23.0374 41.1257 2 11.3849 11.2239 22.6088 3 18.6223 25.6862 44.3085 3 8.71848 8.28735 17.0058 4 18.8456 26.9956 45.8412 4 7.09974 6.59498 13.6947 Furthermore, it is to be noted that in the absence of couple Fekadu [51]. Figure 7 presents the effect of the Reynolds stresses, that is, as η ⟶ 0, the parameter s ⟶ ∞, the number on the velocity profile. Thereby, we observe that as properties of couple stress in the fluid vanish and the case the Reynolds number Re increases, there is a decrease in of classical viscous fluid can be obtained from this work the velocities of the fluid in both flow regions. This indicates (Umavathi et al. ([28–30]), Abbas et al. [22], Devakar et al. that velocity is reduced by the increase of the Reynolds num- [23], and Devakar and Ramgopal [31]). Therefore, it is ber Re and our results well agreed with the results of Deva- understood that the velocity in the case of couple stress fluid kar et al. [23], Devakar and Ramgopal [31], Srinivas and is lower than that of a Newtonian fluid. Murthy [32], and Punnamchandar and Fekadu [51]. The effect of Darcy’s number Da on the fluid velocities is Figure 8 represents the velocity profile for the different shown in Figure 5. From this figure, it is noticed that the values of the pressure gradient. It is observed that with the velocities in both fluid regions increase with the increase of increase in G, velocity is increasing in both the fluid regions. Darcy’s number Da. Since Da = k/R , an increase in Darcy’s Physically, the more the pressure gradient, the more the fluid number corresponds to an increase in the permeability (per- is pushed to generate the flow, which results in an increase in meable parameter k) of the porous medium, which supports fluid velocity. Figure 9 displays the effect of the slip param- the flow. Lesser permeability causes a slighter fluid velocity eter γ on the fluid flow velocity profiles. Figure 9 shows that to be observed inside the flow medium occupied by the fluid. increasing the slip parameter reduces fluid velocity in both Thus, it may be concluded that an increase in the Darcy’s zones. Obviously, fluid slippage has the opposite impact on number enhances fluid velocities. This is due to the reason fluid motion, and increasing the slip parameter reduces the that the additional flow resistance that the porous structure velocity significantly in both regions. A similar trend was offers diminishes as Da (permeable parameter k) gradually observed in the work of Punnamchandar and Fekadu [51, increases. A similar kind of behavior can be found in Refs. 52]. Furthermore, when the slip parameter is set to zero, Srinivas and Murthy [32], Srinivas et al. [37], Punnamchan- the classical case of no slip is recovered as a special case. dar and Fekadu [51], and Kumar and Agarwal [44]. The numerical values of the volume flow rate are com- Figure 6 describes the effect of the ratio of viscosities n puted for various pertinent flow parameters and are pre- on velocity profiles. Figure 6 reveals that as the viscosity ratio sented in Tables 1–8. The effect of the magnetic number M n increases, the velocity of the fluid decreases in both flow on the flow rate is shown in Table 1. From Table 1, we notice regions. This is because as the viscosity ratio n increases, 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, greater flow resistance is provided. As a result, velocity n =1:0, n =1:1, γ =0:1, G =10, s =1, s =1, and s =2. drops. Therefore, we conclude that an increase in the ratio σ μ s 1 2 of viscosities inhibits fluid motion. A similar view can be Tables 2 and 3 shows the nature of flow rates for different found in the works of Umavathi et al. [21], Umavathi et al. values of couple stress parameter s , i =1, 2. From the tables, [30], Srinivas and Murthy [32], and Punnamchandar and we can see that the total flow rate increases with an increase Journal of Applied Mathematics 11 Table 7: Variations of Q ði =1, 2Þ, Q with respect to G. i (iii) Increase in the magnetic number, slip parameter, viscosity ratio, and Reynolds number suppress the GQ1 Q2 Q volume flow rate 5 7.99873 8.45329 16.452 (iv) Increase in the couple stress parameter, Darcy num- 10 15.9975 16.9066 32.904 ber, and pressure gradient promotes volume flow 15 23.9962 25.3599 49.3561 rate 20 31.9949 33.8132 65.8081 This work can be extended to the unsteady flow problem and is also made to include heat transfer/thermal effects. We Table 8: Variations of Q ði =1, 2Þ, Q with respect to γ . would like extend this work by taking various fluids like micropolar fluid, or any other non-Newtonian fluid. γ Q1 Q2 Q 0.05 16.3244 18.5806 34.905 Nomenclature 0.1 15.9975 16.9066 32.904 0.15 15.656 15.1585 30.8145 B : Magnetic field intensity 0.2 15.2991 13.3312 28.6303 C , D , ði =1,2, 3,4Þ: Arbitrary constants i i Da: Darcy number G: Pressure gradient of couple stress parameters s , i =1, 2. Table 4 demonstrates Current density the effect of Darcy’s number on the flow rate. From J : Table 4, we can see that the total flow rate increases with M: Magnetic number an increase in Darcy’s number. Table 5 shows the effect of m : Couple stress ij the viscosity ratio on the flow rate. The flow rate shows a P: Fluid pressure at any point decreasing trend with the growth of the viscosity ratio. ! Velocity vector in regions I and II q ði =1, 2Þ: Table 6 displays various values of flow rate with respect to Q: Total volumetric flow rate the Reynolds number. It is seen from Table 6 that as the Q ði =1,2Þ: Flow rate in regions I and II Reynolds number increases, the total flow rate decreases. R: Radius of the inner cylindrical region Table 7 represents the flow rate for the different pressure Re: Reynolds number gradient values. From the table, it is observed that increasing R : Radius of the cylinder the pressure gradient increases the volume flow rate across s: = R /R, radius ratio the pipe cross-section. Table 8 presents the numerical flow s ði =1,2Þ: Couple stress parameters rate data with respect to slip parameter γ . It is observed that u ði =1,2Þ: Velocity components the volume flow rate gets decreased with an increase of slip 2 2 ∇ : The operator d /dr + ð1/rÞðd/drÞ parameter γ . r, θ, z: Cylindrical coordinates I ð:Þ, K ð:Þ: Modified Bessel functions n n ′ Couple stress viscosity coefficients η , η : 6. Conclusions i i γ : Nondimensional slip parameter γ : Slip coefficient The problem of steady, laminar, and fully developed hydro- μ ði =1, 2Þ: Dynamic viscosity coefficients magnetic flow of two immiscible couple stress fluids through n : Ratio of viscosities a porous medium in a horizontal cylinder under the effect of the Navier slip boundary condition is considered in the pres- ρ ði =1,2Þ: Density of fluid in regions I and II ent study. The motion is generated by a constant pressure σ ði =1,2Þ: Electrical conductivity gradient delivered along the axial direction, i.e., z-axis. The n : Ratio of electrical conductivity resulting set of coupled differential equations associated with τ : Shear stress. ij the flow of the two fluids subject to the appropriate bound- ary and interface conditions is solved analytically. Exact Data Availability solutions are obtained in terms of the modified Bessel func- tions. The effects of various physical parameters on the No data were used to support this study. velocity profiles and total flow rate are studied. The signifi- cant findings of the current investigation are the following: Conflicts of Interest (i) Increasing the magnetic number, viscosity ratio, The authors declare that they have no conflicts of interest. Reynolds number, and slip parameter reduces fluid velocities References (ii) The increment of the couple stress parameters, Darcy number, and pressure gradient enhances the [1] A. C. 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