Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 132302 | 18 pages | https://doi.org/10.1155/2011/132302

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

Academic Editor: Muhammad R. Hajj
Received17 Dec 2010
Revised12 Apr 2011
Accepted23 May 2011
Published09 Aug 2011

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 [26] 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 0𝑦1 has been occupied by a fluid of viscosity 𝜇1, electrical conductivity 𝜎1, thermal conductivity 𝑘1, and specific heat capacity 𝑐𝑝1, and the region II 2𝑦0 has been filled by a layer of different fluid of viscosity 𝜇2, thermal conductivity 𝑘2, specific heat capacity 𝑐𝑝2, and electrical conductivity 𝜎2.

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 𝐵𝑥 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] are𝐵𝐯=(𝑢(𝑦),0,0),𝐁=𝑥(𝑦)+𝐵01𝜆2,𝐵0,𝜆,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 𝑇𝑤1 and 𝑇𝑤2, 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+𝜇2𝐯+𝐉×𝐁,(2.3) general magnetic induction equation:𝜕𝐁1𝜕𝑡×(𝐯×𝐁)𝜎𝜇𝑒2𝐁=0,(2.4) and an energy equation:𝜌𝑐𝑝𝜕𝑇𝜕𝑡+𝐯𝑇=𝑘2𝐉𝑇+𝜇Φ+2𝜎,(2.5) where:Φ=2𝜕𝑢𝜕𝑥2+𝜕𝑣𝜕𝑦2+𝜕𝑤𝜕𝑧2+𝜕𝑣+𝜕𝑥𝜕𝑢𝜕𝑦2+𝜕𝑤+𝜕𝑦𝜕𝑣𝜕𝑧2+𝜕𝑢+𝜕𝑧𝜕𝑤𝜕𝑥223(𝐯)2.(2.6) In previous equations and in following boundary conditions, used symbols are common for the theory of MHD flows: 𝑡-time, 𝑐𝑝-specific heat capacity, 𝑢-velocity in longitudinal direction, 𝑇-thermodynamic temperature, 𝜇𝑒-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,𝐸𝑧) 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:1𝜌𝑑𝑃+𝜈2𝑢𝑑𝑦2𝜎𝜌𝐵0𝜆𝐸𝑧+𝑢𝐵0𝜆𝐵=0,(2.8)0𝜆𝑑𝑢+1𝑑𝑦𝜎𝜇𝑒𝑑2𝐵𝑥𝑑𝑦2=0,(2.9)𝜌𝑐𝑝𝑢𝜕𝑇𝜕𝜕𝑥=𝑘2𝑇𝜕𝑦2+𝜇𝜕𝑢𝜕𝑦2𝐸+𝜎𝑧+𝑢𝐵0𝜆2,(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 are𝑢11=𝑈0,𝑢22𝑢=0,(2.12)1(0)=𝑢2𝜇(0),(2.13)1𝑑𝑢1𝑑𝑦=𝜇2𝑑𝑢2𝐵𝑑𝑦,𝑦=0,(2.14)𝑥11=0,𝐵𝑥22𝐵=0,(2.15)𝑥1(0)=𝐵𝑥2(10),(2.16)𝜇𝑒1𝜎1𝑑𝐵𝑥1=1𝑑𝑦𝜇𝑒2𝜎2𝑑𝐵𝑥2𝑇𝑑𝑦for𝑦=0,(2.17)11=𝑇𝑤1,𝑇22=𝑇𝑤2𝑇,(2.18)1(0)=𝑇2𝑘(0),(2.19)1𝑑𝑇1𝑑𝑦=𝑘2𝑑𝑇2𝑑𝑦;𝑦=0.(2.20)

3. Velocity and Magnetic Field Distribution

The governing equation for the velocity 𝑢𝑖 in regions I and II can be written as:1𝜌𝑖𝑃+𝜈𝑖𝑑2𝑢𝑖𝑑𝑦2𝜎𝑖𝜌𝑖𝐵0𝜆𝐸𝑧+𝑢𝑖𝐵0𝜆=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 as𝐵0𝜆𝑑𝑢𝑖+1𝑑𝑦𝜎𝑖𝜇𝑒𝑖𝑑2𝐵𝑥𝑖𝑑𝑦2=0.(3.2) It is convenient to transform (3.1) and (3.2) to a nondimensional form. The following transformations have been used:𝑢𝑖=𝑢𝑖𝑈0,𝑦𝑖=𝑦𝑖,𝜇𝛼=1𝜇2,𝛽=12𝜎,𝛾=1𝜎2𝜇,𝛿=𝑒1𝜇𝑒2,𝐺𝑖=𝑃𝜇𝑖𝑈0/2𝑖,𝑏𝑖=𝐵𝑥𝑖𝐵0,𝐸𝐾=𝑧𝑈0𝐵0-loadingparameter,𝐻𝑎𝑖=𝐵0𝑖𝜎𝑖𝜇𝑖-Hartmannnumber,𝑅𝑚𝑖=𝑈0𝑖𝜎𝑖𝜇𝑒𝑖-magneticReynoldsnumber.(3.3) With the above nondimensional quantities, the governing equations become𝑑2𝑢𝑖𝑑𝑦𝑖2𝐻𝑎2𝑖𝐾+𝑢𝑖𝜆𝜆+𝐺𝑖𝑑=0,2𝑏𝑖𝑑𝑦𝑖2+𝜆𝑅𝑚𝑖𝑑𝑢𝑖𝑑𝑦𝑖=0.(3.4) The nondimensional form of the boundary and interface conditions (2.12) to (2.17) becomes𝑢1(1)=1,𝑢2𝑢(1)=0,1(0)=𝑢2(0),𝑑𝑢1𝑑𝑦1=𝛽𝛼𝑑𝑢2𝑑𝑦2for𝑦𝑖𝑏=0;𝑖=1,2,1(1)=0,𝑏2𝑏(1)=0,1(0)=𝑏2(0),𝑑𝑏1𝑑𝑦1=𝛿𝛽𝛾𝑑𝑏2𝑑𝑦2for𝑦𝑖=0,𝑖=1,2.(3.5) The solutions of (3.4) with boundary and interface conditions have the following forms: 𝑢𝑖𝑦𝑖=𝐷1𝑖cosh𝜆𝐻𝑎𝑖𝑦𝑖+𝐷2𝑖sinh𝜆𝐻𝑎𝑖𝑦𝑖+𝐹𝑖,𝑏𝑖𝑦𝑖=𝑅𝑚𝑖𝐻𝑎𝑖𝐷1𝑖sinh𝜆𝐻𝑎𝑖𝑦𝑖+𝐷2𝑖cosh𝜆𝐻𝑎𝑖𝑦𝑖+𝑄1𝑖𝑦𝑖+𝑄2𝑖,(3.6) where 𝐹𝑖=𝐺𝑖𝜆2𝐻𝑎2𝑖𝐾𝜆,𝐷11=1𝐹1𝐻sinh𝜆𝐻𝑎2𝑊𝐿sinh𝜆𝐻𝑎1𝑊,𝐿=𝐹2+𝑆cosh𝜆𝐻𝑎2,𝑊=𝐻cosh𝜆𝐻𝑎1sinh𝜆𝐻𝑎2+cosh𝜆𝐻𝑎2sinh𝜆𝐻𝑎1,𝛼𝐻=𝛽𝐻𝑎1𝐻𝑎2,1𝑆=𝜆2𝐺1𝐻𝑎21𝐺2𝐻𝑎22,𝐷21=1𝐹1cosh𝜆𝐻𝑎2𝑊+𝐿cosh𝜆𝐻𝑎1𝑊,𝐷12=𝑆+𝐷11,𝐷22=𝐻𝐷21,𝑄11=𝑅𝑚1𝜆𝐷11𝑄+𝛿𝛽𝛾12𝜆𝑅𝑚2𝐷12,𝑄21=𝑅𝑚1𝐻𝑎1𝐷11sinh𝜆𝐻𝑎1+𝐷21cosh𝜆𝐻𝑎1𝑄11,𝑄12=𝑀1+𝑀2,𝑀1+𝛿𝛽𝛾1=𝑅𝑚1𝐻𝑎1𝐷11sinh𝜆𝐻𝑎1𝜆𝐻𝑎1+𝐷21cosh𝜆𝐻𝑎1,𝑀12=𝑅𝑚2𝐻𝑎2𝐷12sinh𝜆𝐻𝑎2+𝜆𝛿𝛽𝛾𝐻𝑎2+𝐷221cosh𝜆𝐻𝑎2,𝑄22=𝑅𝑚2𝐻𝑎2𝐷22cosh𝜆𝐻𝑎2𝐷12sinh𝜆𝐻𝑎2+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 by𝑘𝑖𝑑2𝑇𝑖𝑑𝑦2+𝜇𝑖𝑑𝑢𝑖𝑑𝑦2+𝜎𝑖𝐸𝑧+𝑢𝑖𝐵0𝜆2=0.(4.1) In order to nondimensionalize previous equation, the following transformations have been used beside the already introduced (3.3):Θ𝑖=𝑇𝑖𝑇𝑤2𝑇𝑤1𝑇𝑤2𝑘,𝜉=1𝑘2.(4.2) With the above, nondimensional quantities (4.1) for regions I and II becomes:𝑑2Θ𝑖𝑑𝑦𝑖2+Pr𝑖Ec𝑖𝑑𝑢𝑖𝑑𝑦𝑖2+𝐻𝑎2𝑖Pr𝑖Ec𝑖𝐾+𝑢𝑖𝜆2=0,(4.3) wherePr𝑖=𝜇𝑖𝑐𝑝𝑖𝑘𝑖,Ec𝑖=𝑈20𝑐𝑝𝑖𝑇𝑤1𝑇𝑤2.(4.4) In the nondimensional form, the boundary conditions for temperature and heat flux at the interface 𝑦=0 becomes Θ1(1)=1,Θ2Θ(1)=0,1(0)=Θ2(0),𝑑Θ1𝑑𝑦1||||0=𝛽𝜉𝑑Θ2𝑑𝑦2||||0,𝑦𝑖=0.(4.5) The solution of (4.3) with boundary and interface conditions has the following form: Θ𝑖𝑦𝑖=Pr𝑖Ec𝑖𝜆𝐷4𝜆21𝑖+𝐷22𝑖cosh2𝜆𝐻𝑎𝑖𝑦𝑖+8𝐷2𝑖𝐶𝑖sinh𝜆𝐻𝑎𝑖𝑦𝑖+2𝐷1𝑖𝐷2𝑖𝜆sinh2𝜆𝐻𝑎𝑖𝑦𝑖+8𝐷1𝑖𝐶𝑖cosh𝜆𝐻𝑎𝑖𝑦𝑖2𝜆2𝐷3𝑖+2𝐷4𝑖𝑦𝑖𝐻𝑎𝑖2𝐶2𝑖𝑦𝑖2,(4.6) where𝐶𝑖=𝐾+𝜆𝐹𝑖=𝐺𝑖𝜆𝐻𝑎2𝑖𝐷,𝑖=1,2,31=14𝜆1𝐷41,𝐷41=1𝛽4𝜆(1+(𝛽/𝜉))𝜉1𝑁2𝛽𝜉3+4,𝐷32=14𝜆𝑁131𝑁𝐷41,𝐷42=𝐷41𝑁44𝜆𝑁,1𝐷=𝜆211+𝐷221cosh2𝜆𝐻𝑎1+8𝐷21𝐶1sinh𝜆𝐻𝑎1+2𝐷11𝐷21𝜆sinh2𝜆𝐻𝑎1+8𝐷11𝐶1cosh𝜆𝐻𝑎1+2𝜆𝐻𝑎12𝐶21+4𝜆Pr1Ec1,2𝐷=𝜆212+𝐷222cosh2𝜆𝐻𝑎28𝐷22𝐶2sinh𝜆𝐻𝑎22𝐷12𝐷22𝜆sinh2𝜆𝐻𝑎2+8𝐷12𝐶2cosh𝜆𝐻𝑎2+2𝜆𝐻𝑎22𝐶22,3𝐷=𝜆211+𝐷221𝐷𝜆𝑁212+𝐷222+8𝐷11𝐶18𝐷12𝐶2𝑁,4=8𝜆𝐻𝑎1𝐷21𝐶1+4𝜆2𝐷11𝐷21𝐻𝑎18𝜆𝐻𝑎2𝐷22𝐶2𝑁4𝜆2𝐷12𝐷22𝐻𝑎2𝑁,𝑁=Pr2Ec2Pr1Ec1,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.

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.

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 𝑦𝑖. This ratio has tendency to change the sign while 𝜆 decreases and 𝑦𝑖 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.

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.

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 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.

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, 𝐻𝑎𝑖, and 𝐾 in order to obtain a precise flow and heat transfer process control.

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.

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 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.

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Copyright © 2011 Dragiša Nikodijević et al. 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.


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