Flow and Heat Transfer of Two Immiscible Fluids in the Presence of Uniform Inclined Magnetic Field

Abstract

The magnetohydrodynamic (MHD) Couette flow of two immiscible fluids in a horizontal channel with isothermal walls in the presence of an applied electric and inclined magnetic field has been investigated in the paper. Both fluids are electrically conducting, while the channel plates are electrically insulated. The general equations that describe the discussed problem under the adopted assumptions are reduced to ordinary differential equations, and closed-form solutions are obtained in both fluid regions of the channel. Separate solutions with appropriate boundary conditions for each fluid have been obtained, and these solutions have been matched at the interface using suitable matching conditions. The analytical results for various values of the Hartmann number, the angle of magnetic field inclination, loading parameter, and the ratio of fluid heights have been presented graphically to show their effect on the flow and heat transfer characteristics.

1. Introduction

The flow and heat transfer of electrically conducting fluids in channels and circular pipes under the effect of a transverse magnetic field occurs in magnetohydrodynamic (MHD) generators, pumps, accelerators, and flowmeters and have applications in nuclear reactors, filtration, geothermal systems, and others.

The interest in the outer magnetic field effect on heat-physical processes appeared seventy years ago. Blum et al. [1] carried out one of the first works in the field of mass and heat transfer in the presence of a magnetic field. The flow and heat transfer of a viscous incompressible electrically conducting fluid between two infinite parallel insulating plates have been studied by many researchers [2–6] due to its important applications in the further development of MHD technology. Also convective heat transfer in channels has been an important research topic for the last few decades because of its applications in solar technology, safety aspects of gas cooled reactors and crystal growth in liquids, and so forth.

Yang and Yu [7] studied the problem of convective magnetohydrodynamic channel flow between two parallel plates subjected simultaneously to an axial temperature gradient and a pressure gradient numerically. The problem of an unsteady two-dimensional flow of a viscous incompressible and electrically conducting fluid between two parallel plates in the presence of a uniform transverse magnetic field has been analyzed by Bodosa and Borkakati [8] for the case of isothermal plates and one isothermal and other adiabatic. The MHD fully developed flow and heat transfer of an electrically conducting fluid between two parallel plates with temperature-dependent viscosity is studied in [9, 10]. An analytical solution to the problem of steady and unsteady hydromagnetic flow of viscous incompressible electrically conducting fluid under the influence of constant and periodic pressure gradient in presence of inclined magnetic field has been obtained exactly by Ghosh [11]. Borkakati and Chakrabarty [12] investigated the unsteady free convection MHD flow between two heated vertical parallel plates in induced magnetic field. Analytical investigation of laminar heat convection in a Couette-Poiseuille flow between two parallel plates with a simultaneous pressure gradient and an axial movement of the upper plate was carried out by Aydin and Avci [13]. Recently, Singha [14] gave an analytical solution to the problem of MHD free convective flow of an electrically conducting fluid between two heated parallel plates in the presence of an induced magnetic field.

All the mentioned studies pertain to a single-fluid model. Most of the problems relating to the petroleum industry, geophysics, plasma physics, magneto-fluid dynamics, and so forth involve multifluid flow situations. Hartmann flow of a conducting fluid and a non-conducting fluid layer contained in a channel has been studied by Shail [15]. His results predicted that an increase of the order 30% can be achieved in the flow rate for suitable ratios of heights and viscosities of the two fluids. Lohrasbi and Sahai [16] studied two-phase MHD flow and heat transfer in a parallel plate channel with the fluid in one phase being conducting. These studies are expected to be useful in understanding the effect of the presence of a slag layer on heat transfer characteristics of a coal-fired MHD generator.

There have been some experimental and analytical studies on hydrodynamic aspects of the two-fluid flow reported in the recent literature. Following the ideas of Alireza and Sahai [17], Malashetty et al. [18, 19] have studied the two fluid MHD flow and heat transfer in an inclined channel, and flow in an inclined channel containing porous and fluid layer. Umavathi et al. [20, 21] have presented analytical solutions of an oscillatory Hartmann two-fluid flow and heat transfer in a horizontal channel and an unsteady two-fluid flow and heat transfer in a horizontal channel. Recently, Umavathi et al. [22] have analysed the magnetohydrodynamic Poiseuille-Couette flow and heat transfer of two immiscible fluids between inclined parallel plates.

Recent studies show that magnetohydrodynamic (MHD) flows can also be a viable option for transporting weakly conducting fluids in microscale systems, such as flow inside the microchannel networks of a lab-on-a-chip device [23, 24]. In microfluidic devices, multiple fluids may be transported through a channel for various reasons. For example, increase in mobility of a fluid may be achieved by stratification of a highly mobile fluid or mixing of two or more fluids in transit may be designed for emulsification or heat and mass transfer applications. In that regard, magnetic field-driven micropumps are in increasing demand due to their long-term reliability in generating flow, absence of moving parts, low power requirement, flow reversibility, feasibility of buffer solution manipulation, and mixing efficiency [25, 26].

MHD flows inside channels can be propelled in many different ways, for example, in electromagnetohydrodynamics (EMHDs) axial flow along a channel is generated by the interaction between the magnetic field and an electric field acting normal to it. Regardless of the purpose of a multifluid EMHD flow, it is important to understand the dynamics of interfaces between the fluids and its effect on the transport characteristics of the system. Keeping in view the wide area of practical importance of multifluid flows as mentioned above, it is the objective of the present study to investigate the MHD Couette flow and heat transfer of two immiscible fluids in a parallel-plate channel in the presence of applied electric and inclined magnetic fields.

2. Mathematical Model

As mentioned in the introduction, the problem of the EMHD Couette two fluid flow has been considered in this paper. The fluids in the two regions have been assumed immiscible and incompressible, and the flow has been steady, one-dimensional, and fully developed. Furthermore, the two fluids have different kinematic viscosities 𝜈₁ and 𝜈₂ and densities 𝜌₁ and 𝜌₂. The transport properties of the two fluids have been taken to be constant. The analytical solutions for velocities, magnetic field, and temperature distributions have been obtained and computed for different values of the characteristic parameters. The physical model, shown in Figure 1, consists of two infinite parallel plates extending in the 𝑥 and 𝑧-direction. The upper plate moves with constant velocity 𝑈₀ in longitudinal direction. The region I 0≤𝑦≤ℎ₁ has been occupied by a fluid of viscosity 𝜇₁, electrical conductivity 𝜎₁, thermal conductivity 𝑘₁, and specific heat capacity 𝑐_𝑝1, and the region II −ℎ₂≤𝑦≤0 has been filled by a layer of different fluid of viscosity 𝜇₂, thermal conductivity 𝑘₂, specific heat capacity 𝑐_𝑝2, and electrical conductivity 𝜎₂.

Figure 1 

Physical model and coordinate system.

A uniform magnetic field of the strength 𝐵₀ has been applied in the direction making an angle 𝜃 to the vertical line and due to the fact that the fluid motion magnetic field of the strength 𝐵_𝑥 has been induced along the lines of motion.

The fluid velocity, treating the problem as a monodimensional, and the magnetic field distributions for the case of inclined and induced magnetic field [8, 11, 27, 28] are𝐁=𝐵_𝑥𝐯=(𝑢(𝑦),0,0),(𝑦)+𝐵₀√1−𝜆²,𝐵₀𝜆,0,(2.1) where 𝐁 is magnetic field vector and 𝜆=cos𝜃. The upper and lower plates have been kept at the two constant temperatures 𝑇_𝑤1 and 𝑇_𝑤2, respectively, and the plates are electrically insulated. The described MHD two fluid flow problem is mathematically presented with a continuity equation:∇𝐯=0,(2.2) momentum equation:𝜌𝜕𝐯𝜕𝑡+(𝐯∇)𝐯=−∇𝑝+𝜇∇²𝐯+𝐉×𝐁,(2.3) general magnetic induction equation:𝜕𝐁𝜕𝑡−∇×(𝐯×𝐁)−𝜎1𝜇_𝑒∇²𝐁=0,(2.4) and an energy equation:𝜌𝑐_𝑝𝜕𝑇𝜕𝑡+𝐯∇𝑇=𝑘∇²𝑇+𝜇Φ+𝐉²𝜎,(2.5) where:Φ=2𝜕𝑢𝜕𝑥²+𝜕𝑣𝜕𝑦²+𝜕𝑤𝜕𝑧²+𝜕𝜕𝑥𝑣+𝜕𝜕𝑦𝑢²+𝜕𝑤𝜕𝑦+𝜕𝜕𝑣𝑧²+𝜕𝑢𝜕𝑧+𝜕𝜕𝑤𝑥²−23(∇𝐯)².(2.6) In previous equations and in following boundary conditions, used symbols are common for the theory of MHD flows: 𝑡-time, 𝑐_𝑝-specific heat capacity, 𝑢-velocity in longitudinal direction, 𝑇-thermodynamic temperature, 𝜇_𝑒-magnetic permeability and Φ-dissipation function. The third term on the right hand side of (2.3) is the magnetic body force, and 𝐉 is the current density vector due to the magnetic field and electric field defined by𝐉=𝜎(𝐄+𝐯×𝐁),(2.7) where 𝐄=(0,0,𝐸_𝑧) is the vector of the applied electric field.

Finally the continuity, momentum, and induction equation written in the classic quasi-static low magnetic Reynolds number approximation [29, 30] takes the following form:1𝜌𝑃+𝜈𝑑²𝑢𝑑𝑦²−𝜎𝜌𝐵⁰𝐵𝜆𝐸_𝑧+𝑢𝐵₀𝜆=0,(2.8)₀𝜆𝑑𝑢𝑑𝑦+𝜎𝜇1_𝑒𝑑²𝐵_𝑥𝑑𝑦²𝜌𝑐_𝑝=0,(2.9)𝑢𝜕𝑇𝜕𝑥=𝑘𝜕²𝑇𝜕𝑦²+𝜇𝜕𝑢𝜕𝑦²+𝜎𝐸_𝑧+𝑢𝐵₀𝜆²,(2.10) where𝑃=−𝜕𝑝𝜕𝑥.(2.11)

The fluid and thermal boundary conditions have been unchanged by the addition of electromagnetic fields. The no-slip conditions require that the fluid velocities are equal to the plate’s velocities, and boundary conditions on temperature are isothermal conditions. In addition, the fluid velocity, sheer stress, induced magnetic field, induced magnetic flux (induced currents at the interface of conductors [29]), temperature, and heat flux must be continuous across the interface 𝑦=0. Equations which represent these conditions are𝑢₁ℎ₁=𝑢𝑈₀,𝑢₂−ℎ₂=0,(2.12)₁𝜇(0)=𝑢₂(0),(2.13)₁𝑑𝑢₁𝑑𝑦=𝜇²𝑑𝑢₂𝐵𝑑𝑦,𝑦=0,(2.14)_𝑥1ℎ₁𝐵=0,𝐵_𝑥2−ℎ₂=0,(2.15)_𝑥11(0)=𝐵_𝑥2(0),(2.16)𝜇_𝑒1𝜎₁𝑑𝐵_𝑥1𝑑𝑦=𝜇1_𝑒2𝜎₂𝑑𝐵_𝑥2𝑇𝑑𝑦for𝑦=0,(2.17)₁ℎ₁=𝑇𝑇_𝑤1,𝑇₂−ℎ₂=𝑇_𝑤2,(2.18)₁𝑘(0)=𝑇₂(0),(2.19)₁𝑑𝑇₁𝑑𝑦=𝑘²𝑑𝑇₂𝑑𝑦;𝑦=0.(2.20)

3. Velocity and Magnetic Field Distribution

The governing equation for the velocity 𝑢_𝑖 in regions I and II can be written as:1𝜌_𝑖𝑃+𝜈_𝑖𝑑²𝑢_𝑖𝑑𝑦²−𝜎^𝑖𝜌_𝑖𝐵₀𝜆𝐸_𝑧+𝑢_𝑖𝐵₀𝜆=0,(3.1) where suffix 𝑖 (𝑖=1,2) represents the values for regions I and II, respectively. The equation for the magnetic field induction in the regions I and II can be written as𝐵₀𝜆𝑑𝑢^𝑖𝑑𝑦+𝜎_𝑖1𝜇_𝑒𝑖𝑑²𝐵_𝑥𝑖𝑑𝑦²=0.(3.2) It is convenient to transform (3.1) and (3.2) to a nondimensional form. The following transformations have been used:𝑢^∗𝑖=𝑢^𝑖𝑈₀,𝑦^∗𝑖=𝑦ℎ_𝑖𝛼=𝜇₁,𝜇₂,𝛽=ℎ¹ℎ₂,𝛾=𝜎¹𝜎₂,𝛿=𝜇^𝑒1𝜇_𝑒2𝐺_𝑖,=𝑃𝜇_𝑖𝑈₀/ℎ^2𝑖,𝑏^𝑖=𝐵^𝑥𝑖𝐵₀𝐾=𝐸_𝑧,𝑈₀𝐵₀𝐻𝑎_𝑖-loadingparameter,=𝐵₀ℎ_𝑖𝜎_𝑖𝜇_𝑖𝑅𝑚_𝑖-Hartmannnumber,=𝑈₀ℎ_𝑖𝜎_𝑖𝜇_𝑒𝑖-magneticReynoldsnumber.(3.3) With the above nondimensional quantities, the governing equations become𝑑²𝑢^∗𝑖𝑑𝑦^∗𝑖2−𝑑𝐻𝑎^2𝑖𝐾+𝑢^∗𝑖𝜆𝜆+𝐺_𝑖=0,₂𝑏_𝑖𝑑𝑦^𝑖∗2+𝜆𝑅𝑚_𝑖𝑑𝑢^∗𝑖𝑑𝑦^∗𝑖=0.(3.4) The nondimensional form of the boundary and interface conditions (2.12) to (2.17) becomes𝑢^∗1(1)𝑢=1,𝑢^∗2(−1)=0,_∗1𝑑𝑢_∗1(0)=𝑢^∗2(0),𝑑𝑦^∗1=𝛽𝛼𝑑𝑑𝑦𝑢^∗2∗2𝑏for𝑦^∗𝑖=0;𝑖=1,2,₁(1)𝑏=0,𝑏₂(−1)=0,₁𝑑𝑏₁(0)=𝑏₂(0),𝑑𝑦^∗1=𝛿𝛽𝛾𝑑𝑏²𝑑𝑦^∗2for𝑦^∗𝑖=0,𝑖=1,2.(3.5) The solutions of (3.4) with boundary and interface conditions have the following forms: 𝑏_𝑖𝑢^∗𝑖𝑦^∗𝑖=𝐷_1𝑖cosh𝜆𝐻𝑎_𝑖𝑦^∗𝑖+𝐷_2𝑖sinh𝜆𝐻𝑎_𝑖𝑦^∗𝑖+𝐹_𝑖,𝑦^∗𝑖=−𝑅𝐻𝑚𝑎^𝑖_𝑖𝐷_1𝑖sinh𝜆𝐻𝑎_𝑖𝑦^∗𝑖+𝐷_2𝑖cosh𝜆𝐻𝑎_𝑖𝑦^∗𝑖+𝑄_1𝑖𝑦^∗𝑖+𝑄_2𝑖,(3.6) where 𝐹_𝑖=𝐺^𝑖𝜆²𝐻𝑎^2𝑖−𝐾𝐷₁₁𝜆,=1−𝐹¹𝐻sinh𝜆𝐻𝑎₂𝑊−𝐿sinh𝜆𝐻𝑎¹𝐿=𝐹₂𝑊,𝑊=𝐻cosh𝜆𝐻𝑎₁+𝑆cosh𝜆𝐻𝑎₂,sinh𝐻𝜆=𝐻𝑎𝛼₂+cosh𝜆𝐻𝑎₂sinh𝜆𝐻𝑎₁,𝛽𝐻𝐻𝑎𝑎¹₂𝑆=1,𝜆²𝐻𝐺𝑎¹²¹−𝐺²𝐷₂₁𝐻𝑎²²,=1−𝐹¹cosh𝜆𝐻𝑎₂𝑊+𝐿cosh𝜆𝐻𝑎¹𝐷₁₂𝑊,𝐷₂₂=𝑆+𝐷₁₁,𝑄₁₁=𝐻𝐷₂₁,𝑄₂₁=𝑅𝑚₁𝜆𝐷₁₁+𝛿𝛽𝛾𝑄₁₂−𝜆𝑅𝑚₂𝐷₁₂,=𝑅𝑚¹𝐻𝑎₁𝐷₁₁sinh𝑄₁₂𝜆𝐻𝑎₁+𝐷₂₁cosh𝜆𝐻𝑎₁−𝑄₁₁,=𝑀¹+𝑀₂𝑀1+𝛿𝛽𝛾₁,=𝑅𝑚¹𝑀𝐻𝑎₁𝐷₁₁sinh𝜆𝐻𝑎₁−𝜆𝐻𝑎₁+𝐷₂₁cosh𝜆𝐻𝑎₁−1₂,=𝑅𝑚²𝐻𝑄𝑎₂𝐷₁₂sinh𝜆𝐻𝑎₂+𝜆𝛿𝛽𝛾𝐻𝑎₂+𝐷₂₂1−cosh𝜆𝐻𝑎₂₂₂,=𝑅𝑚²𝐻𝑎₂𝐷₂₂cosh𝜆𝐻𝑎₂−𝐷₁₂sinh𝜆𝐻𝑎₂+𝑄₁₂.(3.7)

4. Temperature Distribution

Once the velocity distributions were known, the temperature distributions for the two regions have been determined by solving the energy equation subject to the appropriate boundary and interface conditions (2.18)–(2.20). In the present problem, it has been assumed that the two walls have been maintained at constant temperatures. The term involving 𝜕𝑇/𝜕𝑥=0 in the energy equation (2.10) drops out for such a condition. The governing equation for the temperatures 𝑇₁ and 𝑇₂ in region I and II is then given by𝑘_𝑖𝑑²𝑇_𝑖𝑑𝑦²+𝜇_𝑖𝑑𝑑𝑢𝑦^𝑖²+𝜎_𝑖𝐸_𝑧+𝑢_𝑖𝐵₀𝜆²=0.(4.1) In order to nondimensionalize previous equation, the following transformations have been used beside the already introduced (3.3):Θ_𝑖=𝑇^𝑖−𝑇_𝑤2𝑇_𝑤1−𝑇_𝑤2,𝜉=𝑘¹𝑘₂.(4.2) With the above, nondimensional quantities (4.1) for regions I and II becomes:𝑑²Θ_𝑖𝑑𝑦^∗𝑖2+Pr_𝑖Ec_𝑖𝑑𝑑𝑦𝑢^{∗𝑖∗𝑖}²+𝐻𝑎^2𝑖Pr_𝑖Ec_𝑖𝐾+𝑢^∗𝑖𝜆²=0,(4.3) wherePr_𝑖=𝜇^𝑖𝑐_𝑝𝑖𝑘_𝑖,Ec_𝑖=𝑈²⁰𝑐_𝑝𝑖𝑇_𝑤1−𝑇_𝑤2.(4.4) In the nondimensional form, the boundary conditions for temperature and heat flux at the interface 𝑦=0 becomes Θ₁(1)Θ=1,Θ₂(−1)=0,₁𝑑Θ₁(0)=Θ₂(0),𝑑𝑦^∗1||||₀=𝛽𝜉𝑑𝑑Θ𝑦^2∗2||||₀,𝑦^∗𝑖=0.(4.5) The solution of (4.3) with boundary and interface conditions has the following form: Θ_𝑖𝑦^∗𝑖=−Pr^𝑖Ec_𝑖4𝜆𝜆𝐷^21𝑖+2𝐷_1𝑖+𝐷^22𝑖cosh2𝜆𝐻𝑎_𝑖𝑦^∗𝑖+8𝐷_2𝑖𝐶_𝑖sinh𝜆𝐻𝑎_𝑖𝑦^∗𝑖−2𝜆2𝐷𝐷_2𝑖3𝑖𝜆sinh2𝜆𝐻𝑎_𝑖𝑦^∗𝑖+8𝐷_1𝑖𝐶_𝑖cosh𝜆𝐻𝑎_𝑖𝑦^∗𝑖+2𝐷_4𝑖𝑦^∗𝑖−𝐻𝑎_𝑖2𝐶^2𝑖𝑦^𝑖∗2,(4.6) where𝐶_𝑖=𝐾+𝜆𝐹_𝑖=𝐺^𝑖𝜆𝐻𝑎^2𝑖𝐷,𝑖=1,2,₃₁=14𝜆ℑ¹𝐷₄₁−𝐷₄₁,=14𝜆(1+(𝛽/𝜉))𝛽𝜉ℑ¹−𝑁^∗ℑ₂−𝛽𝜉ℑ³+ℑ₄𝐷₃₂,=14𝜆𝑁ℑ¹−ℑ₃−1𝑁𝐷⁴¹𝐷₄₂,=𝐷⁴¹𝑁^∗−ℑ⁴4𝜆𝑁^∗ℑ₁,=+𝜆2𝐷𝐷²¹¹₁₁+𝐷²²¹cosh2𝜆𝐻𝑎₁+8𝐷₂₁𝐶₁sinh𝜆𝐻𝑎₁𝐷₂₁𝜆sinh2𝜆𝐻𝑎₁+8𝐷₁₁𝐶₁cosh𝜆𝐻𝑎₁+2𝜆𝐻𝑎₁₂𝐶²¹+4𝜆Pr₁Ec₁ℑ₂,=−𝜆2𝐷𝐷²¹²₁₂+𝐷²²²cosh2𝜆𝐻𝑎₂−8𝐷₂₂𝐶₂sinh𝜆𝐻𝑎₂ℑ₃𝐷₂₂𝜆sinh2𝜆𝐻𝑎₂+8𝐷₁₂𝐶₂cosh𝜆𝐻𝑎₂+2𝜆𝐻𝑎₂₂𝐶²²,ℑ=𝜆𝐷²¹¹+𝐷²²¹−𝜆𝑁𝐷²¹²+𝐷²²²+8𝐷₁₁𝐶₁−8𝐷₁₂𝐶₂𝑁,₄=8𝜆𝐻𝑎₁𝐷₂₁𝐶₁+4𝜆²𝑁𝐷=₁₁𝐷P₂₁r₂𝐻𝑎₁−8𝜆𝐻𝑎₂𝐷₂₂𝐶₂𝑁^∗−4𝜆²𝐷₁₂𝐷₂₂𝐻𝑎₂𝑁^∗,Ec₂Pr₁Ec₁,𝑁^∗=𝛽𝜉𝑁.(4.7)

5. Results and Discussion

Recent technological trends show that the use of external fields to generate the flow inside channels, such as electrohydrodynamic, MHD, and electrokinetic flows, can be more advantageous in many microscale applications. In order to show the results of the considered MHD Couette flow problem graphically, two fluids important for technical practice (selected for the development of MHD pump under the project TR35016) have been chosen, and the parameters 𝛼, 𝜉 and 𝛾 take the values of 0.677; 0.0647 and 0.025, respectively. Fluids Prandtl number is Pr₁=7.43 and Pr₂=0.25, while Eckert number is equal to Ec₁=0.0017 and Ec₂=0.005 for all the results given in Figures 2 to 13 and except in Figure 14 where it takes different values. The part of obtained results has been presented graphically in Figures 2 to 13 to elucidate the significant features of the hydrodynamic and thermal state of the flow.

Figure 2 

Velocity profiles for different values of inclination angle (𝐻𝑎1=3;𝐻𝑎2=15;𝐾=0).

Figures 2 to 4 show the effect of the magnetic field inclination angle on the distribution of velocity, temperature, and the ratio of the applied and induced magnetic field.

Figure 2 shows the effect of the angle of inclination on velocity which predicts that the velocity increases as the inclination angle increases. These results are expected because the application of a transverse magnetic field normal to the flow direction has a tendency to create a drag-like Lorentz force which has a decreasing effect on the flow velocity. Dimensionless temperature in function of angle of inclination of applied magnetic field is shown in Figure 3. In region II containing higher electrical conductivity fluid, the viscous heating is less pronounced and the influence of applied magnetic field is more expressed.

Figure 3 

Temperature profiles for different values of inclination angle (𝐻𝑎1=3;𝐻𝑎2=15;𝐾=0).

Figure 4 

Ratio of an induced and externally imposed magnetic field (𝐻𝑎1=3;𝐻𝑎2=15;𝐾=0).

It can be seen from Figures 2 and 3 that the magnetic field flattens out the velocity and temperature profiles and reduces the flow energy transformation as the inclination angle decreases.

Figure 4 shows that the ratio of an induced and externally imposed magnetic field increases as the inclination angle of an applied field increases, for negative values of 𝑦^∗𝑖. This ratio has tendency to change the sign while 𝜆 decreases and 𝑦^∗𝑖 have positive values.

Figures 5 to 7 depict the effect of the Hartmann number, while the electric loading factor 𝐾 is equal to zero (so-called short-circuited case). The influence of the Hartmann number on the velocity profiles was more pronounced in the channel region II containing the fluid with greater electrical conductivity. Figure 5 illustrates the effect of the Hartmann number on the velocity field. It was found that for large values of Hartmann number, flow can be almost completely stopped in the region II, while in region I velocity decrease is significant.

Figure 5 

Velocity profiles for different values of Hartmann numbers 𝐻𝑎𝑖(𝜆=1).

The effect of increasing the Hartmann number on temperature profiles (Figure 6) in both of the parallel-plate channel regions was in equalizing the fluid temperatures.

Figure 6 

Temperature profiles for different values of Hartmann numbers 𝐻𝑎𝑖(𝜆=1).

Figure 7 

Ratio of an induced and externally imposed magnetic field for different values of Hartmann numbers 𝐻𝑎𝑖(𝜆=1).

The influence of the Hartmann number had quite similar effect on the ratio of induced and externally applied magnetic field as shown in Figure 7.

The influence of the induced magnetic field in the considered case is not so important, but in similar flow problems where the transversal component velocity is present, the knowledge of the imposed and induced field ratio can have great significance.

Of particular significance is the analysis when the loading factor 𝐾 is different from zero (value of loading factor 𝐾 defines the system as generator, flowmeter, or pump). Figure 8 illustrates that with the increase in the absolute value of loading factor 𝐾 the temperature in both regions increases. In region I, viscous heating decreases while Joule heating increases, and, in region II, viscous heating increases near the lower plate and towards the middle of the channel Joule heating is more pronounced.

Figure 8 

Temperature profiles for different values of loading factor (𝐻𝑎1=2;𝐻𝑎2=10;𝜆=0.75).

Figure 9 shows the effect of the loading factor on velocity, which predicts the possibility to change the flow direction. For negative 𝐾 values, the flow rate increases.

Figure 9 

Velocity profiles for different values of loading factor(𝐻𝑎1=2;𝐻𝑎2=10;𝜆=0.75).

The obtained results show that different values of the inclination angle, the Hartmann number, and the loading factor are a convenient control method for heat and mass transfer processes.

The ratio of an induced and externally imposed magnetic field had a considerable change when the loading parameter was different from zero, especially in region II.

Figure 10 also shows a direction change of the induced field in some areas of regions I and II. This property can be used together with the change of parameters 𝜆, 𝐻𝑎_𝑖, and 𝐾 in order to obtain a precise flow and heat transfer process control.

Figure 10 

Ratio of an induced and externally imposed magnetic field for different values of loading factor (𝐻𝑎1=2;𝐻𝑎2=10;𝜆=0.75).

The effect of the ratio of heights of the two regions on the velocity field is shown in Figure 11. It is interesting to note that decreasing of 𝛽 flattens out velocity profiles and for small values, even change curves shape.

Figure 11 

Velocity profiles for different values of height ratio 𝛽(𝐻𝑎1=1;𝐻𝑎2=5;𝜆=1).

The effect of ratio of the heights of the two regions on temperature field is same as its effect on velocity field, which is evident from Figure 12. It is found that the effect of decreasing 𝛽 is to decrease the temperature field. It is also interesting to note that for small 𝛽, the ratio of induced and externally imposed magnetic field become negligible small.

Figure 12 

Temperature profiles for different values of height ratio 𝛽(𝐻𝑎1=1;𝐻𝑎2=5;𝜆=1).

Figure 13 

Ratio of an induced and externally imposed magnetic field for different values of height ratio 𝛽(𝐻𝑎1=1;𝐻𝑎2=5;𝜆=1).

Figure 14 

Temperature profiles for different values of Eckert number(𝐻𝑎1=2;𝐻𝑎2=10;𝜆=1).

Figure 14 demonstrates the temperature distribution for different values of Eckert number Ec. It is observed that increasing values of Eckert number is to increase the temperature distribution in the flow region. Increase in Eckert number enhances the temperature because the heat energy is stored in the liquid due to the frictional heating. Temperature change is more pronounced in the region 1, while in region 2 a linear change is observed.

6. Conclusion

The problem of MHD Couette flow and heat transfer of two immiscible fluids in a horizontal parallel-plate channel in the presence of applied electric and inclined magnetic fields was investigated analytically. Both fluids were assumed to be Newtonian, electrically conducting, and have constant physical properties. Separate closed form solutions for velocity, temperature, and magnetic induction of each fluid were obtained taking into consideration suitable interface matching conditions and boundary conditions. The results were numerically evaluated and presented graphically for two fluids important for technical practice. Only part of the results are presented for various values of the magnetic field inclination angle, Hartmann number, loading parameter, and ratio of fluid heights in region I and II.

Furthermore, it was concluded that the flow and heat transfer aspects of two immiscible fluids in a horizontal channel with insulating walls can be controlled by considering different fluids having different viscosities and conductivities and also by varying the heights of regions. The obtained results show also that different values of the inclination angle, the Hartmann number, and the loading factor are a convenient control method for heat and mass transfer processes.

Acknowledgments

This paper is supported by the Serbian Ministry of Sciences and Technological development (Project no. TR 35016; Research of MHD flow in the channels, around the bodies and application in the development of the MHD pump). The authors wish to thank the reviewer for his careful, unbiased, and constructive suggestions that significantly improved the quality of this paper.