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MHD heat transfer two-ionized fluids flow between parallel plates with Hall currents
Results in Engineering 4 (2019) 100043
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MHD heat transfer two-ionized fluids flow between parallel plates with Hall currents T. Linga Raju Department of Engineering Mathematics, A.U. Constituent College of Engineering(A), Andhra University, Visakhapatnam, 530 003, India
A R T I C L E I N F O
A B S T R A C T
Keywords: MHD Immiscible fluids Two-fluid flow Plasma Hall effects Insulating and conducting plates
A theoretical attempt is made to investigate the effect of Hall current on temperature distribution in MHD twofluid flow of ionized gases (plasma) through horizontal channel bounded by two parallel plates under the action of an applied transverse magnetic field. It is assumed that the magnetic Reynolds number is small. The fluids in the two regions are considered to be incompressible, immiscible and electrically conducting with different viscosities, electrical and thermal conductivities. The transport properties of the two fluids are taken as constant. The governing momentum and energy equations for two-fluids that flow in two regions are detailed when the temperature of the two plates are prescribed to be the same. The solution is carried out in two cases, that is, (1) when the plates are made of the non-conducting and (2) conducting materials. The profiles for temperature distribution and rate of heat transfer coefficients are plotted for different set of values of the governing parameters. The effect of parameters on the temperature distribution and rate of heat transfer coefficient is discussed. It is observed in the case of non-conducting plates that the temperature distribution is independent of the ratio of electron pressure to the total pressure which relies on this parameter for conducting plates. It is seen in the case of conducting plates that an increase in Hall parameter diminishes the temperature distribution in the two regions for fixed values of the remaining parameters (i.e., Hartmann number, viscosity ratio, height ratio, electrical conductivity ratio and the ratio of thermal conductivities). It is expected that this theoretical study may have some practical application to numerous diversified areas like geophysical flows, aerospace science, in particular, aerodynamic heating and in engineering applications such as MHD generators, Hall accelerators, in thermonuclear power reactors and so forth.
Introduction Modern concern is in employing the magnetohydrodynamic (MHD) principle to explore the effects of magnetic field on the flow of electrically conducting fluids (both liquids and gases) and related heat transfer characteristics in multi-fluid flow systems. The study on two-fluid flow models with varied conditions and various geometrical aspects is a dynamic area of research in many diversified fields due to their rich applications in numerous specialized fields of geophysics, aeronautics and astrophysics as well of various engineering and technological importance as in designing MHD power generators, electromagnetic pumps, plasma jets, fusion machines and so on. Moreover, several issues related to the fields like geophysical fluid dynamics, petroleum industry, and in industrial applications include multi-fluid flow circumstances, in which a stratified two-phase/two-fluid flow frequently occurs. For instance, in the study of geophysical problems, it is fundamental to consider the interaction of the geomagnetic field with the hot springs/or fluids in
geothermal regions. Transportation and extraction of the products of oil are also other different applications using a two-phase/two-fluid framework to obtain the expanded flow rates in an electromagnetic pump etc. The use of liquid metals as heat transfer agents and as working fluid in an MHD power generator and nuclear reactor technology has also created a growing interest in the behavior of liquid metal flows and in particular the nature of interaction with ionized fluids and electromagnetic fields. Background, motivation and objectives of the problem Magnetohydrodynamics (MHD) concerns with the flow of electrically conducting fluids in the presence of magnetic field either externally applied or generated inside the fluid by inductive action. The subject developed rapidly turned out to be an important area under discussion for both scientific and academic community because of its tremendous applications in science, engineering, technology and in various industrial contexts. There are several excellent review articles contributed by numerous researchers [see Refs. 3,5,6,8,22,25] emphasizing the different
E-mail address: [email protected]. https://doi.org/10.1016/j.rineng.2019.100043 Received 21 March 2019; Received in revised form 29 August 2019; Accepted 1 October 2019 2590-1230/© 2019 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
T. Linga Raju
Results in Engineering 4 (2019) 100043
q1m ;
Nomenclature
q2m Complex form of the mean velocities represented for simplicity, q1m ¼ u1m þ iw1m ; q2m ¼ u2m þ iw2m s ¼ pe/p (Ratio of electron pressure to the total pressure) Ti Temperature T1, T2 Temperatures of the fluids in the two regions Twi ði ¼ 1; 2Þ : Tw1 ; Tw2 Constant plate temperatures in the fluid regions ui ði ¼ 1; 2Þ : u1 ; u2 Primary velocity distributions in the two fluid regions (component of velocity field along x-direction) u1m ; u2m Primary mean velocity distributions in the two fluid regions 2 ¼ ∂∂xp hμ1 , the characteristic velocity up
B0 Applied uniform transverse magnetic field B Magnetic flux density cpi ði ¼ 1; 2Þ The specific heat at constant pressure e Electric charge e/c Electron charge Eix ; Eiz Applied electric fields in x- and z-directions respectively Ee Equivalent electric field, Ee ¼ ðc =enÞrpe Ei Electric field h Ratio of the heights of the two regions h1 Height of the channel in the upper region (Region-I) h2 Height of the channel in the lower region (Region-II) Iix ; Iiz Non-dimensionalized current densities I1 ¼ I1x þ iI1z I2 ¼ I2x þ iI2z I1x ; I1z ; I2x ; I2z Dimensionless current densities along x- and zdirections in two fluid regions Ji Current density Jix ; Jiz current densities along x- and z-directions in two fluid regions
i
Vi Fluid velocity wi : w1 ;w2 Secondary velocity distributions in the two fluid regions (component of velocity field along z-direction) w1m ; w2m Secondary mean velocity distributions in the two fluid regions (x, y, z) Space co-ordinates ∂p ∂x
Common constant pressure gradient Ratio of the viscosities Ratio of thermal Conductivities Fluid Viscosity Viscosities of the two fluids Electrical conductivity Electrical conductivities of the two fluids Ratio of the electrical conductivities (¼σσ01 ) 02
α
β
μi μ1 ; μ2 σ 0i σ 01 ; σ 02 σ0 σ 11 ; σ 12 ; σ 21 ; σ 22 Modified conductivities parallel and normal to
2
m λ1 ; λ2 ; λ3 ; λ4 Symbols represented as λ1 ¼ 1 s 1þm λ2 ¼ 2; σ0 σ1 smσ 0 σ 2 m ; λ s 1þm ; λ ¼ 1 s 1 ¼ 2 3 4 1þm2 1þm2
Thermal conductivities of the two fluids in region-I and region-II Ha Hartmann number m Hall parameter mix ; miz Dimensionless electric fields m1x ; m1z ; m2x ; m2z Dimensionless electric field in two fluid regions n Number density of ions ne Density of electron p Pressure pe Electron pressure U1 ; U2 Symbols used for simplicity as U1 ¼ q1 =q1m ; U2 ¼ q2 =q2m U1 ; U2 Complex conjugates of U1 ; U2 q1 ðyÞ; q2 ðyÞ Solutions of the velocity distributions for both fluid regions in complex form ½q1 ðyÞ ¼ u1 ðyÞ þiw1 ðyÞ; q2 ðyÞ ¼ u2 ðyÞ þiw2 ðyÞ K1,K2
σ1 ; σ2 ρ1 ; ρ2 ρ θ1 ;
θ2
the direction of electric field respectively Symbols for the ratios σ 1 ¼ σσ12 ; σ 2 ¼ σσ22 11 21 Densities of the two fluids Ratio of the Densities ¼ ρρ2 1
Non-dimensional form of temperature distributions of the two fluids τ; τe Mean collision time between electron and ion, electron and neutral particles respectively ωe Gyration frequency of electron Subscripts 1, 2 Refers to the quantities in the upper and lower fluid regions respectively
magneto-hydrodynamic (MHD) two-phase/two-fluid flow under various conditions and diverse geometries have been made accessible in the literature in the most recent couple of decades by a number of researchers [See Refs. 1,3,4,10,12,13,14,17,19,20,21,23,26–28]. As it appears from the previously mentioned investigations that a major part of the exploration work has been submitted with single fluid flow framework and very small theoretical work is accounted for in the channel flow regime of two-fluid/or two-phase flow of electrically conducting fluids (liquids and gases) with or without Hall currents aside from a couple of studies as stated above. Yet, a large portion of the issues have been identified in geophysics, astrophysics, aeronautics, magneto-fluid dynamics and in the petroleum industry besides in different industrial applications, they involve multi-fluid flow circumstances in general or two-fluid flow configurations in particular. So, in this article, a theoretical attempt is made to provide a detailed understanding of the flow and heat transfer in MHD two-ionized fluids flow between two parallel plates with effect of Hall currents. The interest in this novel problem arises from its importance in liquid metals, electrolytes and ionized gases. It is anticipated that this theoretical study may be applied in many diversified fields like aero-space science, plasma physics, geophysics and is of great practical concern in a large number of engineering disciplines, including chemical, petroleum and in various power generating industries, in design of MHD control generators, pumps and flow meters, Hall accelerators,
aspects of this rapidly growing field. The investigation on flow problems in the electrically conducting fluids, especially of ionized gases (plasma) are being paid attention since long. If the working fluid is an ionized gas, where the density is low/or the magnetic field is very strong, the presence of magnetic field produces the well-known phenomenon called the Hall effect. The Hall effects are the direct result of Lorentz forces acting on the charges in the current. These Lorentz forces cause secondary flows to develop as the flow progresses down the region of electromagnetic fluid interaction. Numerous model studies on the effect of Hall currents in the MHD flow problems have been carried out by many investigators [see Refs. 2,7,15,16,18]. It is observed that the above mentioned investigations were performed in the flow models of single fluid, but multi-fluid flow with several internal interfaces behave quite differently from that of the flow of a single fluid in several vital aspects. In many issues related to astrophysics, geophysical fluid dynamics, aeronautics, and in petroleum industry, and also in industrial applications, multi-fluid flow situations frequently occur. The advancement of novel technologies based on micro-fluid platforms, like lab-on-chip systems has created abundant applications involving multilayer flows in micro-channels. Multi-fluid flow is also observed in oil production and transportation in the petrochemical industry [see Refs.11], also in the flow inside the micro-channel networks of a lab-on-a-chip device [see Ref.9]. Likewise, the issues on 2
T. Linga Raju
Results in Engineering 4 (2019) 100043
and magnetic fields in two-fluid regions are being formulated based on the fundamental equations for steady MHD flow of neutral fully ionized gas in the presence of Hall currents [see Refs. 7,10,16,18,24].
plasma jets, in the cooling procedure of atomic reactors, and so forth. In addition, it is important to comprehend the dynamics of interface between the fluids and its impact on the transport characteristics of the models.
Fig. 1. Flow model and coordinate scheme.
The governing equations for this investigation are the equations of motion, current and energy along with the boundary and interface conditions. Under the above mentioned assumptions, it is assumed in the fundamental equations that the fluid velocity: Vi ¼ ðui ; 0; wi Þ; magnetic flux intensity: B ¼ ð0; B0 ; 0Þ; the current density: Ji ¼ ðJix ; 0; Jiz Þ; the electric field: Ei ¼ ðEix ; 0; Eiz Þ; and Ji2 ¼ Jix2 þ Jiz2 ; ði ¼ 1; 2Þ for both fluid regions. It is considered that the thermal boundary conditions affect everywhere on the infinite channel plates. Also, it ignores the thermal conduction in the flow direction and electron heating. Hence, the governing equations in the two regions, namely, Region-I and Region-II (that is, for the fluids in upper and lower regions) are simplified and obtained as:
This paper is arranged as follows: The introduction is presented in section Introduction. The background and motivation of the problem is given in section Background, motivation and objectives of the problem. In section Theoretical analysis and formulation of the problem, the theoretical analysis, formulation of the problem - the governing equations of motion, current and energy with relevant boundary, interface conditions, also mathematical analysis of the problem are provided. Section Solutions to the problem deals with the solutions and discussion of the results in two cases of study, that is, when the two side plates comprise non-conducting (insulating) and conducting materials on the basis of profiles drawn (Figs. 2–25) . The conclusion is presented in section Conclusion.
Region-I Theoretical analysis and formulation of the problem
μ1
Consider the steady laminar two-dimensional MHD two-fluid flow of
ionized gases driven by a common constant pressure gradient ∂∂xp in a channel comprising infinitely long parallel plates extending along x– and z– axes in the presence of Hall currents. The x-axis is taken in the direction of hydrodynamic pressure gradient in the plane parallel to the channel plates, but not in the direction of flow. The two bounding plates are maintained at constant temperature Tw (so that Tw1 ¼ Tw2). An external magnetic field of uniform strength B0 is applied in the y–direction, normal to the direction of flow. It is supposed that the magnetic Reynolds number is small, so that the induced magnetic field becomes negligible. Fig. 1, illustrates the flow model and the coordinate scheme preferring the origin halfway between the plates. One of the fluids (fluid-1) occupies the region 0 y h1, and the other fluid (fluid-2) occupies the region –h2 y 0, also these two-fluid regions are designated as Region-I and Region-II respectively. The two regions are occupied by two immiscible electrically conducting incompressible fluids with different densities ρ1 ; ρ2 viscosities μ1 ; μ2 , electrical conductivities σ 01 ,σ 02 and thermal conductivities K1 ; K2 . The channel width is assumed to be very large in comparison with the channel height. All physical quantities except pressure are thought to be the functions of y only because the plates are infinite in extent along x- and z-directions. The interface between the two immiscible fluids is supposed to be flat, stress-free and undisturbed. The boundaries of the channel are rigid. With the assistance of these assumptions and in view of the available investigations, the governing equations for the MHD two-dimensional steady flow between two parallel plates under the transverse electric
∂2 u1 σ 11 1s 1 ∂y2 σ 01
∂p ∂x
þ B0 f σ 11 ðE1z þ u1 B0 Þ þ σ 21 ðE1x w1 B0 Þg ¼ 0
μ1
(3.5)
∂2 w1 σ 21 ∂p þ B0 σ 11 ðE1x w1 B0 Þ þ σ 21 ðE1z þ u1 B0 Þ ¼ 0 þs ∂y2 σ 01 ∂x
K1
∂2 T1 þ μ1 ∂y2
∂u1 ∂y
2
∂w1 ∂y
þ
2 þ
J 21
σ 01
¼ ρ1 μ1 cp1
∂T1 ∂T1 þ ρ1 w1 cp1 ∂x ∂z
(3.6)
(3.7)
J1x ¼
sσ 21 ∂p þ σ 11 E1x B0 σ 11 w1 þ σ 21 E1z þ B0 σ 21 u1 σ 01 B0 ∂x
(3.8)
J1z ¼
s σ 11 ∂p E1z E1x þ σ 11 1 þ u1 σ 21 w1 B0 σ 01 ∂x B0 B0
(3.9)
Region-II
μ2
∂2 u2 σ 12 1s 1 ∂y2 σ 02
μ2
∂ w2 σ 22 ∂p þ B0 þs ∂y2 σ 02 ∂x
∂p þþB0 f σ 12 ðE2z þu2 B0 Þþ σ 22 ðE2x w2 B0 Þg ¼ 0 ∂x (3.10)
K2
3
2
∂2 T2 þ μ2 ∂y2
∂u2 ∂y
σ 12 ðE2x w2 B0 Þ þ σ 22 ðE2z þ u2 B0 Þ ¼ 0
2
þ
∂w2 ∂y
2 þ
J 22
σ 02
¼ ρ2 μ2 cp2
∂T2 ∂T2 þ ρ2 w2 cp2 ∂x ∂z
(3.11) (3.12)
T. Linga Raju
Results in Engineering 4 (2019) 100043
J2x ¼
sσ 22 ∂p þ σ 12 E2x B0 σ 12 w2 þ σ 22 E2z þ B0 σ 22 u2 σ 02 B0 ∂x
(3.13)
J2z ¼
s σ 12 ∂p E2z E2x þ σ 12 1 þ u2 σ 22 w2 B0 σ 02 ∂x B0 B0
(3.14)
where, ωe is the gyration frequency of electron, while τ, τe are the mean collision time between electron and ion, electron and neutral particles respectively. The above expression for Hall parameter ‘m’ which is valid in the case of partially–ionized gas agrees with that of fully–ionized gas when τe approaches infinity. Moreover, Tw1 and Tw2 are constant everywhere on the upper and lower side plates respectively, so ∂∂Tx1 ¼ 0; ∂T1 ∂z
¼ 0; ∂∂Tx2 ¼ 0; ∂∂Tz2 ¼ 0 and T1, T2 are finite everywhere in the two-fluid flow and are functions of 'y' only. Hence, with the assistance of the above mentioned non-dimensional variables (3.23) and for simplicity, ignoring the asterisks, the non-dimensional form of governing equations (3.5)–(3.14) and conditions (3.15)–(3.22) become: Region– I
The subscripts 1 and 2 as mentioned in the above equations refer to the quantities for Region-I and II respectively. The quantities u1, u2 and w1, w2, are the velocity components along x- and z-directions in the twofluid regions respectively. Likewise, these are known as the primary and secondary velocity distributions. The symbols Eix and Eiz , Jix and Jiz ; ði ¼ 1; 2Þare the x- and z-components of electric field, and the current densities respectively, while the quantities T1, T2 are the temperatures. The term: s ¼ pe/p is known as the ratio of electron pressure to the total pressure. The estimation of s is 1/2 for neutral fully–ionized plasma and about zero for a weakly–ionized gas. The symbols σ 11 ; σ 12 and σ 21 , σ 22 are the modified conductivities parallel and normal to the direction of electric field respectively. The boundary conditions on velocity field are assumed to be the no-slip conditions, in addition the fluid velocity and sheer stress are continuous over the interface y ¼ 0. The conditions on temperature field oblige the isothermal boundary conditions at the two plates, also the continuity of temperature and heat flux at the interface y ¼ 0 between the two-fluid regions. The boundary and interface conditions on velocity distributions u1 , w1 and u2 ; w2 for the two-fluid flow become: u1 ðh1 Þ ¼ 0; w1 ðh1 Þ ¼ 0
(3.15)
u2 ðh2 Þ ¼ 0; w2 ðh2 Þ ¼ 0
(3.16)
u1 ð0Þ ¼ u2 ð0Þ; w1 ð0Þ ¼ w2 ð0Þ for h1 ¼ h2
(3.17)
μ1 ¼
∂u1 ∂u2 ∂w1 ∂w2 ¼ μ2 and μ1 ¼ μ2 at y ¼ 0 ∂y ∂y ∂y ∂y
T2 ðh2 Þ ¼ Tw2
(3.20)
T1 ð0Þ ¼ T2 ð0Þfor h1 ¼ h2
(3.21)
∂T1 ∂T2 ¼ K2 at y ¼ 0 K1 ∂y ∂y
(3.22)
μi ,
σ 0i B0 up ;
μ1 μ2 ;
(3.27)
I1x ¼
1 m ðm1x w1 Þ þ ðm1z þ u1 Þ 1 þ m2 1 þ m2
s m H 2a ð1 þ m2 Þ
I1z ¼
1 m s m ðm1z þ u1 Þ ðm1x w1 Þ þ 2 1 2 2 1þm 1þm 1 þ m2 Ha
(3.28)
(3.29)
þ λ3 αh2 ¼0 (3.30) m d 2 w2 1 ðm2z þ u2 Þ ασ 1 h2 Ha 2 ðm2x w2 Þ þ ασ 2 h2 Ha 2 2 2 1þm 1 þ m2 dy þ λ4 αh2 (3.31) d 2 θ2 β þ dy2 α
B20 h2i ð σμ0i i
I2x ¼
Þ; α ðRatio ofthe viscos-
I2z ¼
1
σ0 σ1 1 þ m2
σ1 σ0 1þ
m2
ðm2x w2 Þ þ
ðm2z þ u2 Þ þ
(3.32)
(3.33) mσ 2 σ 0 sσ 2 0 σ 2 m ðm2z þ u2 Þ 2 1 þ m2 H a ð1 þ m2 Þ
(3.34)
mσ 2 σ 0 σ 1 σ 0 sσ 0 ðw2 m2x Þ þ 1 2 1þm 1 þ m2 H 2a
Where,
¼ σσ21 ; m (Hall parameter) ¼ ωe , β (ratio of thermal con01
(3.35)
σ 11 m2 ¼1s ; λ1 ¼ 1 s 1 σ 01 1 þ m2 σ 11 sm ¼ ; λ2 ¼ s σ 01 1 þ m2
1þ 1 τ τe
ductivities of the two-fluids) ¼ KK12 T Twi : ðup 2 μi =Ki Þ
2 2 du2 dw2 þ βσ 0 h2 H 2a I 22 ¼ 0 þ 2 dy dy
I 22 ¼ I 22x þ I 22z
h (ratio of the heights) ¼hh21 ; σ 0 (Ratio of the electrical con 1 , ρ (density ratio) ¼ ρρ2 ; σ 1 ¼ σσ 12 ; σ 2 ¼ σσ22 ; 1þm ductivities) ¼ σσ01 2 ¼ 02 11 21
θi ðtemperature distributionÞ ¼
(3.26)
m d 2 u2 1 ðm2x w2 Þ ασ 1 h2 Ha 2 ðm2z þ u2 Þ þ ασ 2 h2 Ha 2 2 2 1þm 1 þ m2 dy
Iiz ¼ σ0iJBiz0 up ; J 2i ¼ J 2ix þ J 2iz ;
Jix
ði ¼ 1; 2Þ; Ha2 (Hartmann number) ¼
σ 11 m σ 01 ; 1þm2
2 2 du1 dw1 þ þ H 2a I 21 dy dy
¼0
mix ¼ BE0ixup ; miz ¼ BE0izup ; Iix ¼
ities) ¼
(3.25)
Region – II
The following non-dimensional variables are utilized to make the governing equations (3.5)–(3.14) and conditions (3.15)–(3.22) as dimensionless. u*1 ¼ uu1p , u*2 ¼ uu2p , w*1 ¼ wup1 , w*2 ¼ wup2 , y1* ¼ hy1i ; ði ¼ 1; 2Þ; up ¼ ∂∂xp h21
d 2 w1 1 m þ Ha 2 ðm1x w1 Þ þ Ha 2 ðm1z þ u1 Þ þ λ2 ¼ 0 1 þ m2 dy2 1 þ m2
I1 2 ¼ I1x 2 þ I1z 2
The isothermal boundary and interface conditions on temperature for both fluids are given by (3.19)
(3.24)
d 2 θ1 ¼ ∂y2
(3.18)
T1 ðh1 Þ ¼ Tw1
d 2 u1 1 m Ha 2 ðm1z þ u1 Þ þ Ha 2 ðm1x w1 Þ þ λ1 ¼ 0 1 þ m2 dy2 1 þ m2
(3.23)
σ0 σ1 sσ 0 σ 2 m ; λ4 ¼ λ3 ¼ 1 s 1 1 þ m2 1 þ m2 4
(3.36)
T. Linga Raju
Results in Engineering 4 (2019) 100043
iw1m and q2m ¼ u2m þ iw2m : U1 and U2 are the complex conjugates of U1 and U2 respectively; u1 ; u2 and w1 ; w2 are the primary and secondary velocity distributions in the two regions. The quantities q1 and q2 are the solutions of the velocity distributions; qm1 and qm2 are the corresponding mean velocities for both fluid regions in complex notation. The solutions for q1 and q2 are obtained by solving the equations (3.24), (3.25) and (3.30) and 3.31) under the boundary and interface conditions (3.37–3.40). In order to study the effects of the governing parameters appearing in the solution of the problem, the numerical results for the temperature field in the two-fluid regions and the rate of heat transfer coefficient at the two plates are carried out. The numerical results are presented graphically in Figs. 2–9. Since the problem involves many number of non-dimensional parameters, so for sake of simplicity, some of the parameters, specifically σ 1 ¼ 1:2; σ 2 ¼ 1:5; are fixed for all the numerical calculations and it is investigated the effect of other important parameters (such as, the Hall parameter ‘m’, Hartmann number Ha , viscosity ratio α; height ratio ‘h’, electrical conductivity ratio σ 0 and thermal conductivity ratio β on the temperature distribution. The solutions for temperature distribution are all independent of the partial pressure of electron gas 's' (ratio of electron pressure to the total pressure) in case of non-conducting plates, while, these rely upon it for conducting side plates. The effect of varying the Hartmann number on temperature distribution in the two regions (that is for the fluids in the upper and lower regions) is shown in Fig. 2. It is seen that the temperature distribution in region-I increases with increase in Hartmann number up to the estimation Ha ¼ 30 and diminishes for Ha > 30, while it increases in the region–II when all the remaining parameters are fixed. Fig. 3 displays the effect of changing Hall parameter 'm' on temperature distribution. It is seen that an increase in Hall parameter reduces the temperature distribution in the two regions. The greatest temperature distribution in the channel tends to move beneath the channel centerline towards the region-II as Hall parameter increases. The effect of α (ratio of viscosities of the two fluids) is shown in Fig. 4. It is seen that an increase in viscosity ratio α is to diminish the temperature distribution in the two regions. The greatest temperature distribution in the channel tends to move below the channel centerline towards the region-II for viscosity ratio (say for α ¼ 0.05 and 0.1) when all the remaining parameters are fixed. The effect of height ratio ‘h’ is represented in Fig. 5. It is noticed that the temperature distribution in the upper region is diminished, while it increased in the lower region as the height ratio increases. The greatest temperature distribution in the channel tends to move below the channel centerline towards the regionII for height ratio (for h ¼ 0.8) and the most extreme temperature in the channel tends to move over the channel centerline towards the region-I for height ratio h ¼ 0.1. The effect of the electrical conductivity ratio σ 0 on temperature distribution appears in Fig. 6. It is discovered that there is not much critical departure from temperature in the two regions as σ 0 increases. Fig. 7 depicts the varying thermal conductivity ratio β on temperature distribution. It is discovered that the temperature increases in the two regions as the thermal conductivity ratio increases. Additionally, it is seen that the temperature profile in the channel moves over the channel centerline towards region-I. That is, the temperature is higher in the upper region than the lower region for small estimations of the thermal conductivity ratio (say for β ¼ 0.1). The most extreme temperature distribution in the channel tends to move beneath the channel centerline towards region-II for the estimations of thermal conductivity ratio β ¼ 0.5, 1, 1.5 and 2 when all the rest of the parameters are fixed. The rate of heat transfer coefficients against Hartmann number is given in Figs. 8 and 9. It is seen that an increase either in the Hartmann number or in the Hall parameter rises the rate of heat transfer coefficients. Case II - Conducting plates: If the side plates are made of conducting material and short-circuited by an external conductor, then the induced electric current flows out of the channel. In this case no electric potential exists between the plates. When zero electric field is assumed in the
The non-dimensional form of the boundary and interface conditions on velocity and temperature become: u1 ðþ1Þ ¼ 0; w1 ðþ1Þ ¼ 0;
(3.37)
u2 ð1Þ ¼ 0; w2 ð1Þ ¼ 0
(3.38)
u1 ð0Þ ¼ u2 ð0Þ; w1 ð0Þ ¼ w2 ð0Þ;
(3.39)
du1 1 du2 dw1 1 dw2 ¼ and ¼ at y ¼ 0: dy αh dy dy αh dy
(3.40)
θ1 ðþ1Þ ¼ 0
(3.41)
θ2 ð1Þ ¼ 0
(3.42)
θ1 ð0Þ ¼ θ2 ð0Þ
(3.43)
dθ1 dθ2 ¼ ð1 = βhÞ at y ¼ 0 dy dy
(3.44)
Solutions to the problem In order to solve the energy equations (3.26) and (3.32) under the boundary and interface conditions (3.41)–(3.44) for obtaining the temperature distribution, first of all the equations (3.24), (3.25) and (3.30) & 3.31) are solved for the velocity distributions using the conditions (3.37)–(3.40). Consequently, the exact solutions for temperature distribution in the two-fluid flow regions and rate of heat transfer coefficient at the plates are obtained when the side plates are made of the nonconducting and conducting materials. Case I - Non-conducting plates: If the side plates are kept at a large distance in z-direction and are made of the non-conducting (insulating) material, the induced electric current does not go out of the channel but circulates itself in the fluid. Hence, additional conditions for the currents are, Z
1
Z
1
I1z dy ¼ 0 and
0
I2z dy ¼ 0:
(4.1)
0
When the insulation at large x is also assumed, other similar relations are obtained as Z
1
Z
1
I1x dy ¼ 0 and
0
I2x dy ¼ 0:
(4.2)
0
These conditions of insulation at large distances are physically realized for instance when the flow of ionized gas is surrounded by the cold non-conducting gas [see Ref. 18]. Using the above two conditions (4.1) to (4.2), the constants in the solution are determined and then solutions for u1 ; u2 , w1 ; w2 ; I1 and I2 in the two regions are obtained. With the help of these solutions, the following simplified equations are solved under the boundary and interface conditions (3.41)–(3.44) to obtain the temperature distribution in the two-fluid regions. Region I: d 2 θ1 dU1 dU 1 Ha2 þ þ ðU 1Þ U 1 ¼0 1 1 dy2 dy dy 1 þ m2
(4.3)
Region II: d 2 θ2 β dU2 dU2 H 2a hβσ 1 þ þ ðU2 1ÞðU2 1Þ ¼ 0 dy2 α dy dy 1 þ m2 Where U1 ¼ q1 =q1m ; U2 ¼ q2 =q2m ;
q1 ¼ u1 þ iw1 ;
(4.4) q2 ¼ u2 þ iw2 ;
q1 q1 q2 þq2 q2 q2 1 , q1m ¼ u1m þ u1 ðyÞ ¼ q1 þq 2 ; w1 ðyÞ ¼ 2i ; u2 ðyÞ ¼ 2 ; w2 ðyÞ ¼ 2i
5
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Results in Engineering 4 (2019) 100043
x–and z–directions, we have mix ¼ 0; miz ¼ 0; ði ¼ 1; 2Þ[see Ref. 18]. Constants in the solution are determined by these two conditions. The solutions for u1 ; u2 ; w1 ; w2 ; um1 ; um2 ; wm1 ; wm2 ; I1 and I2 in the two-fluid regions are obtained. By utilizing the solution of velocity distributions in this case, the following simplified form of the energy equations are solved under the boundary and interface conditions (3.41)–(3.44) for temperature distribution in both regions and the rate of heat transfer coefficients at the two side plates. Region I:
lower plates are presented in Figs. 16 and 17. It is observed that the rate of heat transfer coefficient at the two plates increases as the Hartmann number Ha increases when all the remaining governing parameters are fixed. Also, it is found that an increase in Hall parameter enhances the rate of heat transfer coefficient at the upper plate. But, the rate of heat transfer coefficient increases up to the estimation ‘m’¼2, thereafter it diminishes with an increase in Hall parameter when the Hartmann number Ha < 23 and beyond this number the rate of heat transfer coefficient increases.
2 d 2 θ1 dU1 dU1 1 1 s 1 is m U1 U1 2 þ H þ ðU U Þ þ 1 þ ¼0 1 1 a 1 þ m2 1 þ m2 H 4a qm1 qm1 H 2a 1 þ m2 qm1 qm1 dy2 dy dy
(4.5)
Region II:
d 2 θ2 β dU2 dU2 σ1 σ 1 s2 1 is σ 2 m U2 U2 2 2 þ H þ h β ðU U Þ þ 1 þ ¼0 2 2 a dy2 α dy dy 1 þ m2 1 þ m2 H 4a qm2 qm2 H 2a 1 þ m2 qm2 qm2
(4.6)
In the case of value s ¼ 1/2, the effect of varying the Hartmann number Ha on temperature distribution in the two regions is shown in Fig. 18. It is observed that an increase in Hartmann number enhances the temperature distribution in the two regions. It is seen that the temperature distribution in the channel moves above the channel centerline towards region-I, that is the temperature distribution is high in the upper region compared to the lower region for higher values of Hartmann number Ha (for Ha ¼ 30 and 50) when all the remaining parameters are kept fixed. Fig. 19 exhibits the effect of varying Hall parameter ‘m’ on temperature distribution in the case of s ¼ 1/2. It is discovered that a rise in the estimation of Hall parameter ‘m’ diminishes the temperature distribution in the two regions. It is found that the temperature distribution is high in the upper region compared to the lower region for small values of Hall parameter (say for m ¼ 0.05 and 1). The effect of viscosity ratio α is shown in Fig. 20. It is noticed that an increase in α diminishes the temperature distributions in the two regions. The effect of height ratio ‘h’ is represented in Fig. 21. It is seen that an increase in ‘h’ increases the temperature distributions in the two regions. Also the most extreme temperature distribution in the channel tends to move over the channel focus line towards region-I as the height ratio ‘h’ increases. The impact of electrical conductivity ratio σ 0 on temperature distribution in the two regions is shown in Fig. 22. It is discovered that there is no critical departure from the temperature distribution as electrical conductivities ratio σ 0 increases. Fig. 23 depicts the cause of thermal conductivity ratio β on the temperature distribution in the two regions. It is observed that an increase in thermal conductivity ratio β enhances the temperature distribution in the two regions. The temperature distribution is high in the upper region compared to the lower region for small estimation of the thermal conductivity ratio (say for β ¼ 0.1) when all the rest of the parameters are fixed. The rate of heat transfer coefficients at the two plates against the Hartmann number is shown in Figs. 24 and 25. It is discovered that an increase in Hartmann number increases the rate of heat transfer coefficient when all the remaining parameters are fixed. Also, it is found that an increase in Hall parameter enhances the rate of heat transfer coefficients at the point when all other governing parameters are fixed.
Numerical results are obtained by the same procedure as the case of non-conducting plates and the profiles are shown in Figs. 10–25. For conducting plates, the velocity and temperature distributions depend on ‘s’ (ratio of electron pressure to the total pressure, that is the ionization parameter). In this context, the graphs for the case [1] when ‘s’ ¼ 0 are depicted in Figs. 10–17 and that for another case [2] when the estimation s ¼ 1/2 are presented in Figs. 18–25. In the case of s ¼ 0, the effect of varying Hartmann number Ha on temperature distribution in the two regions is shown in Fig. 10. It is seen that an increase in Ha increases the temperature distribution in the two regions when all other parameters are fixed. The maximum temperature distribution in the channel tends to move above the channel centerline towards region-I as the Hartmann number increases. That is, the temperature profile is high in the upper region compared to the lower region with an increase in Hartmann number when all the remaining parameters are fixed. Fig. 11 exhibits the effect of varying Hall parameter ‘m’ on temperature distribution in the two regions. It is noticed that an increase in Hall parameter 'm’ reduces the temperature distribution in the two regions. Also, the maximum temperature distribution in the channel tends to move above the channel centerline towards the region-I for small values of the Hall parameter (say for ‘m’ ¼ 0.05 and 1). The impact of α(ratio of viscosities of the two fluids) appears in Fig. 12. It is observed that an increase in viscosity ratio diminishes the temperature distribution in the two regions. The effect of height ratio ‘h’ is represented in Fig. 13. It is seen that an increase in height ratio ‘h’ enhances the temperature distribution in the two regions. It is also discovered that the temperature profile is high in the upper region compared to the lower region with an increase in height ratio when all the rest of the parameters are fixed. The effect of the ratio of electrical conductivity σ 0 on temperature distribution is shown in Fig. 14. It is seen that there is not much noteworthy departure from the temperature distribution as σ 0 increases. Fig. 15 depicts the effect of the thermal conductivity ratio β on temperature distribution in the two-fluid regions. It is found that a rise in thermal conductivity ratio enhances the temperature distribution in the two regions. It is noticed that the temperature profile is high in the upper region compared to the lower region for small estimations of the thermal conductivity ratio (say for β ¼ 0.1). The rate of heat transfer coefficients against Hartmann number at upper and
6
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Fig. 2. Temerature Profiles for different Ha and α ¼ 0.333,σ0 ¼ 2,σ1 ¼ 1.2,σ2 ¼ 1.5, β ¼ 1, m ¼ 2,h ¼ 0.8.
Fig. 3. Temerature Profiles for different m and Ha ¼ 10, α ¼ 0.333,σ0 ¼ 2,σ1 ¼ 1.2,σ2 ¼ 1.5, β ¼ 1, h ¼ 0.8.
Fig. 4. Temperature profiles for different α and Ha ¼ 10, m ¼ 2, σ0 ¼ 2,σ1 ¼ 1.2,σ2 ¼ 1.5, β ¼ 1, h ¼ 0.8. 7
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Fig. 5. Temperature profiles for different h and α ¼ 0.333, Ha ¼ 10, m ¼ 2, σ0 ¼ 2,σ1 ¼ 1.2,σ2 ¼ 1.5, β ¼ 1.
Fig. 6. Temperature profiles for differet σ0 and α ¼ 0.333, Ha ¼ 10, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2, β ¼ 1,s ¼ 0.
Fig. 7. Temperature profiles for different β and Ha ¼ 10, α¼0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2. 8
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Fig. 8. Nusselt Number (Nu1) for different Ha and α ¼ 0.333,σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1 h ¼ 0.8.
Fig. 9. Nusselt Number (Nu2) for different Ha and α ¼ 0.333,σ0 ¼ 2,σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1, h ¼ 0.8.
Fig. 10. Temperature profiles for different Ha and α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, β ¼ 1, m ¼ 2, s ¼ 0. 9
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Results in Engineering 4 (2019) 100043
Fig. 11. Temerature Profiles for different m and Ha ¼ 10, α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1, h ¼ 0.8, s ¼ 0.
Fig. 12. Temperature profiles for different α and Ha ¼ 10, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1, h ¼ 0.8, m ¼ 2, s ¼ 0.
Fig. 13. Temperature profiles for different h and α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, Ha ¼ 10, β¼1, m ¼ 2, s ¼ 0. 10
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Results in Engineering 4 (2019) 100043
Fig. 14. Temperature profiles for differet σ0 and α ¼ 0.333, Ha ¼ 10, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2, β ¼ 1, s ¼ 0.
Fig. 15. Temperature profiles for different β and Ha ¼ 10, α¼0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2, s ¼ 0.
Fig. 16. Nusselt Number (Nu1) for different Ha and α ¼ 0.333,σ0 ¼ 2,σ1 ¼ 1.2,σ2 ¼ 1.5, β¼1, h ¼ 0.8, s ¼ 0. 11
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Results in Engineering 4 (2019) 100043
Fig. 17. Nusselt Number (Nu2) for different Ha and α ¼ 0.333,σ0 ¼ 2,σ1 ¼ 1.2,σ2 ¼ 1.5, β¼1, h ¼ 0.8, s ¼ 0.
Fig. 18. Temperature profiles for differet Ha and α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2, β ¼ 1, s ¼ 0.5.
Fig. 19. Temerature Profiles for different m and Ha ¼ 10, Ha ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β ¼ 1, h ¼ 0.8, s ¼ 0.5. 12
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Results in Engineering 4 (2019) 100043
Fig. 20. Temperature profiles for different α and Ha ¼ 10, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1, h ¼ 0.8, m ¼ 2, s ¼ 0.5.
Fig. 21. Temperature profiles for different h and α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, Ha ¼ 10, β ¼ 1, m ¼ 2, s ¼ 0.5.
Fig. 22. Temperature profiles for differet σ0 and α ¼ 0.333, Ha ¼ 10, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2, β ¼ 1, s ¼ 0.5. 13
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Results in Engineering 4 (2019) 100043
Fig. 23. Temperature profiles for different β and Ha ¼ 10, α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2, s ¼ 0.5.
Fig. 24. Nusselt Number (Nu1) for different Ha and α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1, h ¼ 0.8, s ¼ 0.5.
Fig. 25. Nusselt Number (Nu2) for different Ha and α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1, h ¼ 0.8, s ¼ 0.5. 14
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Conclusion
Acknowledgement
A novel problem in heat transfer aspects of steady MHD two-fluid flow of ionized gases driven by a constant pressure gradient through horizontal channel bounded by two parallel plates under the action of an external magnetic field with effect of Hall currents is considered theoretically. The issue is examined in two cases, that is, when the two side plates are made of the non-conducting and conducting materials. The important findings of this study are summarized as follows: The solutions for temperature distribution in the two-fluid regions are observed to be independent of the ratio of electron pressure to the total pressure only for non-conducting side plates unlike for the conducting plates. In case of non-conducting side plates, it is seen that:
The author would like to thank the anonymous referees for their helpful and critical comments on this manuscript. References [1] M. Abdul, Transient magnetohydrodynamic flow of two immiscible fluids through a horizontal channel, Int. J. of Eng. Research 3 (1) (2014) 13–17. [2] E.M. Aboeldahab, E.M.E. Elbarbary, Hall current effect magnetohydrodynamics free convection flow past a semi-infinite vertical plate with mass transfer, Int. J. Eng. Sci. 39 (2001) 1641–1652. [3] Alireza Setayesh, V. Sahai, Heat transfer in developing magnetohydrodynamic Poiseuille flow and variable transport properties, Int. J. Heat Mass Transf. 33 (8) (1990) 1711. [4] A.J. Chamkha, Hydromagnetic two-phase flow in a channel, Int. J. Eng. Sci. 33 (3) (1995) 437–446. [5] S. Chandrasekhar, Problem of stability in hydrodynamics and hydromagnetics, Mon. Not. R. Astron. Soc. 113 (1953) 667. [6] T.G. Cowling, Magnetohydrodynamics, Rep. Prog. Phys. 25 (1962) 244. [7] Kenneth R. Cramer, Shih-I. Pai, Magnetofluid Dynamics for Engineers and Applied Physicists, McGraw-hill company, 1973. [8] J. Hartmann, Theory of the laminary flow of an electrically conductive liquid in a homogeneous magnetic field, Mat- Fys. Medd. Kgl. Danske vid selskab. 15 (6) (1937) 1–28. [9] S.K. Hussameddine, J.M. Martin, W.J. Sang, Analytical prediction of flow field in magnetohydrodynamic based micro fluidic devices, J. Fluids Eng. 130 (9) (2008) 6. [10] J. Lohrasbi, V. Sahai, Magnetohydrodynamic heat transfer in two phase flow between parallel plates, Appl. Sci. Res. 45 (1988) 53–66. [11] J. Lovick, P. Angeli, Experimental studies on the dual continuous flow pattern in oilwater flows, Int. J. Multiph. Flow 30 (2) (2004) 139–157. [12] M.S. Malashetty, V. Leela, Magnetohydrodynamic heat transfer in two phase flow, Int. J. Eng. Sci. 30 (1992) 371–377. [13] A. Mateen, Transient magnetohydrodynamic flow of two immiscible fluids through a horizontal channel, Int.J.Engg.Res. 3 (1) (2014) 13–17. [14] D. Nikodijevic, D. Milenkovic, Z. Stamenkovic, MHD Couette two-fluid flow and heat transfer in presence of uniform inclined magnetic field, Heat Mass Transf. 47 (12) (2011) 1525–1535. [15] I. Pop, The effect of Hall current on hydromagnetic flow near accelerated plate, J. Math. Phys. Sci. 5 (1971) 375–379. [16] T.L. Raju, V.V. Ramana Rao, Hall effects on temperature distribution in a rotating ionized hydromagnetic flow between parallel walls, Int. J. Eng. Sci. 31 (7) (1993) 1073–1091. [17] T.L. Raju, M. Valli, MHD two-layered unsteady fluid flow and heat transfer through ahorizontal channel between parallel plates in a rotating system, Int. J. Appl. Mech. Eng. 19 (1) (2014) 97–121. [18] H. Sato, The Hall effect in the viscous flow of ionized gas between parallel plates under transverse magnetic field, J. Phys. Soc. Japan 16 (7) (1961) 1427–1433. [19] S. Selimli, Z. Resebli, E. Arcakhoglu, Combined effects of magnetic and electric field on hydrodynamic and thermophysical parameters of magnetoviscous fluid flow, Int. J. Heat Mass Transf. 86 (2015) 426–432. [20] R. Shail, On laminar two-phase flows in magnetohydrodynamics, Int. J. Eng. Sci. 11 (1973) 1103–1109. [21] P.R. Sharma, S. Kalpna, Unsteady MHD two-fluid flow and heat transfer through a horizontal channel, Int. J. Eng. Sci. Invention Research & Development 1 (3) (2014) 65–72. [22] J.A. Shercliff, Thermoelectric magnetohydrodynamics, J.F.M. 91 (2) (1979) 231–251. [23] M. Stamenkovic, Dragisa D. Zivojin, B. Blagojevic, S. Savi, MHD flow and heat transfer of two immiscible fluids between moving plates, Ttransactions of the Canadian Society for Mechanical Eng. 34 (3–4) (2010) 351–372. [24] L. Spitzer Jr., Physics of Fully Ionized Gases, Interscience Publishers, New York, 1956. [25] G.W. Sutton, A. Sherman, Engg. Magnetohydrodynamics, McGraw-Hill, New York, 1965. [26] J.C. Umavathi, Abdul Mateen, A.J. Chamkha, A. Al- Mudhaf, Oscillatory Hartmann two-fluid flow and heat transfer in a horizontal channel, Int. J. Appl. Mech. Eng. 11 (1) (2006) 155–178. [27] Dragisa D. Zivojin, M. Stamenkovic, MHD flow and heat transfer of two immiscible fluids between moving plates, Trans. Can. Soc. Mech. Eng. 34 (3–4) (2010) 351–372. [28] M. Zivojin, D. Stamenkovic, Dragisa, M. Nikodijevic, Milos, D. Kocic & Jelena, MHD flow and heat transfer of two immiscible fluids with induced magnetic field, Therm. Sci.: Int. Scientific J. 16 (2) (2012) S323–S336.
The temperature in the two regions diminished with the rise in Hall parameter. An increase in Hartmann number up to a specific esteem increases the temperature distribution in the region-I (fluid in the upper region) and from that point it diminishes, while the temperature increases in the region-II with an increase in Hartmann number. The temperature distribution in the two regions decreased with an increase in viscosity ratio. An increase in the height ratio diminishes the temperature distribution in the region-I, while it enhances in region-II. There is insignificant variation in temperature distribution with an increase in electrical conductivity ratio. The temperature distribution rises with the increase in thermal conductivity ratio. Also, the maximum temperature distribution in the channel moves over the channel focus line towards the region-I for small estimations of the thermal conductivity ratio. A rise in Hartmann number and Hall parameter, leads to rise in the rate of heat transfer coefficient at the two plates. In case of conducting side plates and for s ¼ 0 (ratio of electron pressure to the total pressure equal to zero), it is observed that: The temperature distribution in the two regions enhanced with increased Hartmann number. A rise in Hall parameter and the viscosity ratio leads to fall in temperature distribution. The temperature distribution increases with an increase in thermal conductivity ratio. There is no variation in the temperature distributions as the electrical conductivity ratio increases. An increase in Hartmann number enhances the rate of heat transfer coefficient at the two plates. Also, the relatively comparable highlights are prevalent for the situation when the ionization parameter is equal to half of its esteem (that is, for the value s ¼ 1/2). Conflict of interest The author declares that there is no conflict of interests regarding the publication of this research paper.
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MHD heat transfer two-ionized fluids flow between parallel plates with Hall currents T. Linga Raju Department of Engineering Mathematics, A.U. Constituent College of Engineering(A), Andhra University, Visakhapatnam, 530 003, India
A R T I C L E I N F O
A B S T R A C T
Keywords: MHD Immiscible fluids Two-fluid flow Plasma Hall effects Insulating and conducting plates
A theoretical attempt is made to investigate the effect of Hall current on temperature distribution in MHD twofluid flow of ionized gases (plasma) through horizontal channel bounded by two parallel plates under the action of an applied transverse magnetic field. It is assumed that the magnetic Reynolds number is small. The fluids in the two regions are considered to be incompressible, immiscible and electrically conducting with different viscosities, electrical and thermal conductivities. The transport properties of the two fluids are taken as constant. The governing momentum and energy equations for two-fluids that flow in two regions are detailed when the temperature of the two plates are prescribed to be the same. The solution is carried out in two cases, that is, (1) when the plates are made of the non-conducting and (2) conducting materials. The profiles for temperature distribution and rate of heat transfer coefficients are plotted for different set of values of the governing parameters. The effect of parameters on the temperature distribution and rate of heat transfer coefficient is discussed. It is observed in the case of non-conducting plates that the temperature distribution is independent of the ratio of electron pressure to the total pressure which relies on this parameter for conducting plates. It is seen in the case of conducting plates that an increase in Hall parameter diminishes the temperature distribution in the two regions for fixed values of the remaining parameters (i.e., Hartmann number, viscosity ratio, height ratio, electrical conductivity ratio and the ratio of thermal conductivities). It is expected that this theoretical study may have some practical application to numerous diversified areas like geophysical flows, aerospace science, in particular, aerodynamic heating and in engineering applications such as MHD generators, Hall accelerators, in thermonuclear power reactors and so forth.
Introduction Modern concern is in employing the magnetohydrodynamic (MHD) principle to explore the effects of magnetic field on the flow of electrically conducting fluids (both liquids and gases) and related heat transfer characteristics in multi-fluid flow systems. The study on two-fluid flow models with varied conditions and various geometrical aspects is a dynamic area of research in many diversified fields due to their rich applications in numerous specialized fields of geophysics, aeronautics and astrophysics as well of various engineering and technological importance as in designing MHD power generators, electromagnetic pumps, plasma jets, fusion machines and so on. Moreover, several issues related to the fields like geophysical fluid dynamics, petroleum industry, and in industrial applications include multi-fluid flow circumstances, in which a stratified two-phase/two-fluid flow frequently occurs. For instance, in the study of geophysical problems, it is fundamental to consider the interaction of the geomagnetic field with the hot springs/or fluids in
geothermal regions. Transportation and extraction of the products of oil are also other different applications using a two-phase/two-fluid framework to obtain the expanded flow rates in an electromagnetic pump etc. The use of liquid metals as heat transfer agents and as working fluid in an MHD power generator and nuclear reactor technology has also created a growing interest in the behavior of liquid metal flows and in particular the nature of interaction with ionized fluids and electromagnetic fields. Background, motivation and objectives of the problem Magnetohydrodynamics (MHD) concerns with the flow of electrically conducting fluids in the presence of magnetic field either externally applied or generated inside the fluid by inductive action. The subject developed rapidly turned out to be an important area under discussion for both scientific and academic community because of its tremendous applications in science, engineering, technology and in various industrial contexts. There are several excellent review articles contributed by numerous researchers [see Refs. 3,5,6,8,22,25] emphasizing the different
E-mail address: [email protected]. https://doi.org/10.1016/j.rineng.2019.100043 Received 21 March 2019; Received in revised form 29 August 2019; Accepted 1 October 2019 2590-1230/© 2019 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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q1m ;
Nomenclature
q2m Complex form of the mean velocities represented for simplicity, q1m ¼ u1m þ iw1m ; q2m ¼ u2m þ iw2m s ¼ pe/p (Ratio of electron pressure to the total pressure) Ti Temperature T1, T2 Temperatures of the fluids in the two regions Twi ði ¼ 1; 2Þ : Tw1 ; Tw2 Constant plate temperatures in the fluid regions ui ði ¼ 1; 2Þ : u1 ; u2 Primary velocity distributions in the two fluid regions (component of velocity field along x-direction) u1m ; u2m Primary mean velocity distributions in the two fluid regions 2 ¼ ∂∂xp hμ1 , the characteristic velocity up
B0 Applied uniform transverse magnetic field B Magnetic flux density cpi ði ¼ 1; 2Þ The specific heat at constant pressure e Electric charge e/c Electron charge Eix ; Eiz Applied electric fields in x- and z-directions respectively Ee Equivalent electric field, Ee ¼ ðc =enÞrpe Ei Electric field h Ratio of the heights of the two regions h1 Height of the channel in the upper region (Region-I) h2 Height of the channel in the lower region (Region-II) Iix ; Iiz Non-dimensionalized current densities I1 ¼ I1x þ iI1z I2 ¼ I2x þ iI2z I1x ; I1z ; I2x ; I2z Dimensionless current densities along x- and zdirections in two fluid regions Ji Current density Jix ; Jiz current densities along x- and z-directions in two fluid regions
i
Vi Fluid velocity wi : w1 ;w2 Secondary velocity distributions in the two fluid regions (component of velocity field along z-direction) w1m ; w2m Secondary mean velocity distributions in the two fluid regions (x, y, z) Space co-ordinates ∂p ∂x
Common constant pressure gradient Ratio of the viscosities Ratio of thermal Conductivities Fluid Viscosity Viscosities of the two fluids Electrical conductivity Electrical conductivities of the two fluids Ratio of the electrical conductivities (¼σσ01 ) 02
α
β
μi μ1 ; μ2 σ 0i σ 01 ; σ 02 σ0 σ 11 ; σ 12 ; σ 21 ; σ 22 Modified conductivities parallel and normal to
2
m λ1 ; λ2 ; λ3 ; λ4 Symbols represented as λ1 ¼ 1 s 1þm λ2 ¼ 2; σ0 σ1 smσ 0 σ 2 m ; λ s 1þm ; λ ¼ 1 s 1 ¼ 2 3 4 1þm2 1þm2
Thermal conductivities of the two fluids in region-I and region-II Ha Hartmann number m Hall parameter mix ; miz Dimensionless electric fields m1x ; m1z ; m2x ; m2z Dimensionless electric field in two fluid regions n Number density of ions ne Density of electron p Pressure pe Electron pressure U1 ; U2 Symbols used for simplicity as U1 ¼ q1 =q1m ; U2 ¼ q2 =q2m U1 ; U2 Complex conjugates of U1 ; U2 q1 ðyÞ; q2 ðyÞ Solutions of the velocity distributions for both fluid regions in complex form ½q1 ðyÞ ¼ u1 ðyÞ þiw1 ðyÞ; q2 ðyÞ ¼ u2 ðyÞ þiw2 ðyÞ K1,K2
σ1 ; σ2 ρ1 ; ρ2 ρ θ1 ;
θ2
the direction of electric field respectively Symbols for the ratios σ 1 ¼ σσ12 ; σ 2 ¼ σσ22 11 21 Densities of the two fluids Ratio of the Densities ¼ ρρ2 1
Non-dimensional form of temperature distributions of the two fluids τ; τe Mean collision time between electron and ion, electron and neutral particles respectively ωe Gyration frequency of electron Subscripts 1, 2 Refers to the quantities in the upper and lower fluid regions respectively
magneto-hydrodynamic (MHD) two-phase/two-fluid flow under various conditions and diverse geometries have been made accessible in the literature in the most recent couple of decades by a number of researchers [See Refs. 1,3,4,10,12,13,14,17,19,20,21,23,26–28]. As it appears from the previously mentioned investigations that a major part of the exploration work has been submitted with single fluid flow framework and very small theoretical work is accounted for in the channel flow regime of two-fluid/or two-phase flow of electrically conducting fluids (liquids and gases) with or without Hall currents aside from a couple of studies as stated above. Yet, a large portion of the issues have been identified in geophysics, astrophysics, aeronautics, magneto-fluid dynamics and in the petroleum industry besides in different industrial applications, they involve multi-fluid flow circumstances in general or two-fluid flow configurations in particular. So, in this article, a theoretical attempt is made to provide a detailed understanding of the flow and heat transfer in MHD two-ionized fluids flow between two parallel plates with effect of Hall currents. The interest in this novel problem arises from its importance in liquid metals, electrolytes and ionized gases. It is anticipated that this theoretical study may be applied in many diversified fields like aero-space science, plasma physics, geophysics and is of great practical concern in a large number of engineering disciplines, including chemical, petroleum and in various power generating industries, in design of MHD control generators, pumps and flow meters, Hall accelerators,
aspects of this rapidly growing field. The investigation on flow problems in the electrically conducting fluids, especially of ionized gases (plasma) are being paid attention since long. If the working fluid is an ionized gas, where the density is low/or the magnetic field is very strong, the presence of magnetic field produces the well-known phenomenon called the Hall effect. The Hall effects are the direct result of Lorentz forces acting on the charges in the current. These Lorentz forces cause secondary flows to develop as the flow progresses down the region of electromagnetic fluid interaction. Numerous model studies on the effect of Hall currents in the MHD flow problems have been carried out by many investigators [see Refs. 2,7,15,16,18]. It is observed that the above mentioned investigations were performed in the flow models of single fluid, but multi-fluid flow with several internal interfaces behave quite differently from that of the flow of a single fluid in several vital aspects. In many issues related to astrophysics, geophysical fluid dynamics, aeronautics, and in petroleum industry, and also in industrial applications, multi-fluid flow situations frequently occur. The advancement of novel technologies based on micro-fluid platforms, like lab-on-chip systems has created abundant applications involving multilayer flows in micro-channels. Multi-fluid flow is also observed in oil production and transportation in the petrochemical industry [see Refs.11], also in the flow inside the micro-channel networks of a lab-on-a-chip device [see Ref.9]. Likewise, the issues on 2
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Results in Engineering 4 (2019) 100043
and magnetic fields in two-fluid regions are being formulated based on the fundamental equations for steady MHD flow of neutral fully ionized gas in the presence of Hall currents [see Refs. 7,10,16,18,24].
plasma jets, in the cooling procedure of atomic reactors, and so forth. In addition, it is important to comprehend the dynamics of interface between the fluids and its impact on the transport characteristics of the models.
Fig. 1. Flow model and coordinate scheme.
The governing equations for this investigation are the equations of motion, current and energy along with the boundary and interface conditions. Under the above mentioned assumptions, it is assumed in the fundamental equations that the fluid velocity: Vi ¼ ðui ; 0; wi Þ; magnetic flux intensity: B ¼ ð0; B0 ; 0Þ; the current density: Ji ¼ ðJix ; 0; Jiz Þ; the electric field: Ei ¼ ðEix ; 0; Eiz Þ; and Ji2 ¼ Jix2 þ Jiz2 ; ði ¼ 1; 2Þ for both fluid regions. It is considered that the thermal boundary conditions affect everywhere on the infinite channel plates. Also, it ignores the thermal conduction in the flow direction and electron heating. Hence, the governing equations in the two regions, namely, Region-I and Region-II (that is, for the fluids in upper and lower regions) are simplified and obtained as:
This paper is arranged as follows: The introduction is presented in section Introduction. The background and motivation of the problem is given in section Background, motivation and objectives of the problem. In section Theoretical analysis and formulation of the problem, the theoretical analysis, formulation of the problem - the governing equations of motion, current and energy with relevant boundary, interface conditions, also mathematical analysis of the problem are provided. Section Solutions to the problem deals with the solutions and discussion of the results in two cases of study, that is, when the two side plates comprise non-conducting (insulating) and conducting materials on the basis of profiles drawn (Figs. 2–25) . The conclusion is presented in section Conclusion.
Region-I Theoretical analysis and formulation of the problem
μ1
Consider the steady laminar two-dimensional MHD two-fluid flow of
ionized gases driven by a common constant pressure gradient ∂∂xp in a channel comprising infinitely long parallel plates extending along x– and z– axes in the presence of Hall currents. The x-axis is taken in the direction of hydrodynamic pressure gradient in the plane parallel to the channel plates, but not in the direction of flow. The two bounding plates are maintained at constant temperature Tw (so that Tw1 ¼ Tw2). An external magnetic field of uniform strength B0 is applied in the y–direction, normal to the direction of flow. It is supposed that the magnetic Reynolds number is small, so that the induced magnetic field becomes negligible. Fig. 1, illustrates the flow model and the coordinate scheme preferring the origin halfway between the plates. One of the fluids (fluid-1) occupies the region 0 y h1, and the other fluid (fluid-2) occupies the region –h2 y 0, also these two-fluid regions are designated as Region-I and Region-II respectively. The two regions are occupied by two immiscible electrically conducting incompressible fluids with different densities ρ1 ; ρ2 viscosities μ1 ; μ2 , electrical conductivities σ 01 ,σ 02 and thermal conductivities K1 ; K2 . The channel width is assumed to be very large in comparison with the channel height. All physical quantities except pressure are thought to be the functions of y only because the plates are infinite in extent along x- and z-directions. The interface between the two immiscible fluids is supposed to be flat, stress-free and undisturbed. The boundaries of the channel are rigid. With the assistance of these assumptions and in view of the available investigations, the governing equations for the MHD two-dimensional steady flow between two parallel plates under the transverse electric
∂2 u1 σ 11 1s 1 ∂y2 σ 01
∂p ∂x
þ B0 f σ 11 ðE1z þ u1 B0 Þ þ σ 21 ðE1x w1 B0 Þg ¼ 0
μ1
(3.5)
∂2 w1 σ 21 ∂p þ B0 σ 11 ðE1x w1 B0 Þ þ σ 21 ðE1z þ u1 B0 Þ ¼ 0 þs ∂y2 σ 01 ∂x
K1
∂2 T1 þ μ1 ∂y2
∂u1 ∂y
2
∂w1 ∂y
þ
2 þ
J 21
σ 01
¼ ρ1 μ1 cp1
∂T1 ∂T1 þ ρ1 w1 cp1 ∂x ∂z
(3.6)
(3.7)
J1x ¼
sσ 21 ∂p þ σ 11 E1x B0 σ 11 w1 þ σ 21 E1z þ B0 σ 21 u1 σ 01 B0 ∂x
(3.8)
J1z ¼
s σ 11 ∂p E1z E1x þ σ 11 1 þ u1 σ 21 w1 B0 σ 01 ∂x B0 B0
(3.9)
Region-II
μ2
∂2 u2 σ 12 1s 1 ∂y2 σ 02
μ2
∂ w2 σ 22 ∂p þ B0 þs ∂y2 σ 02 ∂x
∂p þþB0 f σ 12 ðE2z þu2 B0 Þþ σ 22 ðE2x w2 B0 Þg ¼ 0 ∂x (3.10)
K2
3
2
∂2 T2 þ μ2 ∂y2
∂u2 ∂y
σ 12 ðE2x w2 B0 Þ þ σ 22 ðE2z þ u2 B0 Þ ¼ 0
2
þ
∂w2 ∂y
2 þ
J 22
σ 02
¼ ρ2 μ2 cp2
∂T2 ∂T2 þ ρ2 w2 cp2 ∂x ∂z
(3.11) (3.12)
T. Linga Raju
Results in Engineering 4 (2019) 100043
J2x ¼
sσ 22 ∂p þ σ 12 E2x B0 σ 12 w2 þ σ 22 E2z þ B0 σ 22 u2 σ 02 B0 ∂x
(3.13)
J2z ¼
s σ 12 ∂p E2z E2x þ σ 12 1 þ u2 σ 22 w2 B0 σ 02 ∂x B0 B0
(3.14)
where, ωe is the gyration frequency of electron, while τ, τe are the mean collision time between electron and ion, electron and neutral particles respectively. The above expression for Hall parameter ‘m’ which is valid in the case of partially–ionized gas agrees with that of fully–ionized gas when τe approaches infinity. Moreover, Tw1 and Tw2 are constant everywhere on the upper and lower side plates respectively, so ∂∂Tx1 ¼ 0; ∂T1 ∂z
¼ 0; ∂∂Tx2 ¼ 0; ∂∂Tz2 ¼ 0 and T1, T2 are finite everywhere in the two-fluid flow and are functions of 'y' only. Hence, with the assistance of the above mentioned non-dimensional variables (3.23) and for simplicity, ignoring the asterisks, the non-dimensional form of governing equations (3.5)–(3.14) and conditions (3.15)–(3.22) become: Region– I
The subscripts 1 and 2 as mentioned in the above equations refer to the quantities for Region-I and II respectively. The quantities u1, u2 and w1, w2, are the velocity components along x- and z-directions in the twofluid regions respectively. Likewise, these are known as the primary and secondary velocity distributions. The symbols Eix and Eiz , Jix and Jiz ; ði ¼ 1; 2Þare the x- and z-components of electric field, and the current densities respectively, while the quantities T1, T2 are the temperatures. The term: s ¼ pe/p is known as the ratio of electron pressure to the total pressure. The estimation of s is 1/2 for neutral fully–ionized plasma and about zero for a weakly–ionized gas. The symbols σ 11 ; σ 12 and σ 21 , σ 22 are the modified conductivities parallel and normal to the direction of electric field respectively. The boundary conditions on velocity field are assumed to be the no-slip conditions, in addition the fluid velocity and sheer stress are continuous over the interface y ¼ 0. The conditions on temperature field oblige the isothermal boundary conditions at the two plates, also the continuity of temperature and heat flux at the interface y ¼ 0 between the two-fluid regions. The boundary and interface conditions on velocity distributions u1 , w1 and u2 ; w2 for the two-fluid flow become: u1 ðh1 Þ ¼ 0; w1 ðh1 Þ ¼ 0
(3.15)
u2 ðh2 Þ ¼ 0; w2 ðh2 Þ ¼ 0
(3.16)
u1 ð0Þ ¼ u2 ð0Þ; w1 ð0Þ ¼ w2 ð0Þ for h1 ¼ h2
(3.17)
μ1 ¼
∂u1 ∂u2 ∂w1 ∂w2 ¼ μ2 and μ1 ¼ μ2 at y ¼ 0 ∂y ∂y ∂y ∂y
T2 ðh2 Þ ¼ Tw2
(3.20)
T1 ð0Þ ¼ T2 ð0Þfor h1 ¼ h2
(3.21)
∂T1 ∂T2 ¼ K2 at y ¼ 0 K1 ∂y ∂y
(3.22)
μi ,
σ 0i B0 up ;
μ1 μ2 ;
(3.27)
I1x ¼
1 m ðm1x w1 Þ þ ðm1z þ u1 Þ 1 þ m2 1 þ m2
s m H 2a ð1 þ m2 Þ
I1z ¼
1 m s m ðm1z þ u1 Þ ðm1x w1 Þ þ 2 1 2 2 1þm 1þm 1 þ m2 Ha
(3.28)
(3.29)
þ λ3 αh2 ¼0 (3.30) m d 2 w2 1 ðm2z þ u2 Þ ασ 1 h2 Ha 2 ðm2x w2 Þ þ ασ 2 h2 Ha 2 2 2 1þm 1 þ m2 dy þ λ4 αh2 (3.31) d 2 θ2 β þ dy2 α
B20 h2i ð σμ0i i
I2x ¼
Þ; α ðRatio ofthe viscos-
I2z ¼
1
σ0 σ1 1 þ m2
σ1 σ0 1þ
m2
ðm2x w2 Þ þ
ðm2z þ u2 Þ þ
(3.32)
(3.33) mσ 2 σ 0 sσ 2 0 σ 2 m ðm2z þ u2 Þ 2 1 þ m2 H a ð1 þ m2 Þ
(3.34)
mσ 2 σ 0 σ 1 σ 0 sσ 0 ðw2 m2x Þ þ 1 2 1þm 1 þ m2 H 2a
Where,
¼ σσ21 ; m (Hall parameter) ¼ ωe , β (ratio of thermal con01
(3.35)
σ 11 m2 ¼1s ; λ1 ¼ 1 s 1 σ 01 1 þ m2 σ 11 sm ¼ ; λ2 ¼ s σ 01 1 þ m2
1þ 1 τ τe
ductivities of the two-fluids) ¼ KK12 T Twi : ðup 2 μi =Ki Þ
2 2 du2 dw2 þ βσ 0 h2 H 2a I 22 ¼ 0 þ 2 dy dy
I 22 ¼ I 22x þ I 22z
h (ratio of the heights) ¼hh21 ; σ 0 (Ratio of the electrical con 1 , ρ (density ratio) ¼ ρρ2 ; σ 1 ¼ σσ 12 ; σ 2 ¼ σσ22 ; 1þm ductivities) ¼ σσ01 2 ¼ 02 11 21
θi ðtemperature distributionÞ ¼
(3.26)
m d 2 u2 1 ðm2x w2 Þ ασ 1 h2 Ha 2 ðm2z þ u2 Þ þ ασ 2 h2 Ha 2 2 2 1þm 1 þ m2 dy
Iiz ¼ σ0iJBiz0 up ; J 2i ¼ J 2ix þ J 2iz ;
Jix
ði ¼ 1; 2Þ; Ha2 (Hartmann number) ¼
σ 11 m σ 01 ; 1þm2
2 2 du1 dw1 þ þ H 2a I 21 dy dy
¼0
mix ¼ BE0ixup ; miz ¼ BE0izup ; Iix ¼
ities) ¼
(3.25)
Region – II
The following non-dimensional variables are utilized to make the governing equations (3.5)–(3.14) and conditions (3.15)–(3.22) as dimensionless. u*1 ¼ uu1p , u*2 ¼ uu2p , w*1 ¼ wup1 , w*2 ¼ wup2 , y1* ¼ hy1i ; ði ¼ 1; 2Þ; up ¼ ∂∂xp h21
d 2 w1 1 m þ Ha 2 ðm1x w1 Þ þ Ha 2 ðm1z þ u1 Þ þ λ2 ¼ 0 1 þ m2 dy2 1 þ m2
I1 2 ¼ I1x 2 þ I1z 2
The isothermal boundary and interface conditions on temperature for both fluids are given by (3.19)
(3.24)
d 2 θ1 ¼ ∂y2
(3.18)
T1 ðh1 Þ ¼ Tw1
d 2 u1 1 m Ha 2 ðm1z þ u1 Þ þ Ha 2 ðm1x w1 Þ þ λ1 ¼ 0 1 þ m2 dy2 1 þ m2
(3.23)
σ0 σ1 sσ 0 σ 2 m ; λ4 ¼ λ3 ¼ 1 s 1 1 þ m2 1 þ m2 4
(3.36)
T. Linga Raju
Results in Engineering 4 (2019) 100043
iw1m and q2m ¼ u2m þ iw2m : U1 and U2 are the complex conjugates of U1 and U2 respectively; u1 ; u2 and w1 ; w2 are the primary and secondary velocity distributions in the two regions. The quantities q1 and q2 are the solutions of the velocity distributions; qm1 and qm2 are the corresponding mean velocities for both fluid regions in complex notation. The solutions for q1 and q2 are obtained by solving the equations (3.24), (3.25) and (3.30) and 3.31) under the boundary and interface conditions (3.37–3.40). In order to study the effects of the governing parameters appearing in the solution of the problem, the numerical results for the temperature field in the two-fluid regions and the rate of heat transfer coefficient at the two plates are carried out. The numerical results are presented graphically in Figs. 2–9. Since the problem involves many number of non-dimensional parameters, so for sake of simplicity, some of the parameters, specifically σ 1 ¼ 1:2; σ 2 ¼ 1:5; are fixed for all the numerical calculations and it is investigated the effect of other important parameters (such as, the Hall parameter ‘m’, Hartmann number Ha , viscosity ratio α; height ratio ‘h’, electrical conductivity ratio σ 0 and thermal conductivity ratio β on the temperature distribution. The solutions for temperature distribution are all independent of the partial pressure of electron gas 's' (ratio of electron pressure to the total pressure) in case of non-conducting plates, while, these rely upon it for conducting side plates. The effect of varying the Hartmann number on temperature distribution in the two regions (that is for the fluids in the upper and lower regions) is shown in Fig. 2. It is seen that the temperature distribution in region-I increases with increase in Hartmann number up to the estimation Ha ¼ 30 and diminishes for Ha > 30, while it increases in the region–II when all the remaining parameters are fixed. Fig. 3 displays the effect of changing Hall parameter 'm' on temperature distribution. It is seen that an increase in Hall parameter reduces the temperature distribution in the two regions. The greatest temperature distribution in the channel tends to move beneath the channel centerline towards the region-II as Hall parameter increases. The effect of α (ratio of viscosities of the two fluids) is shown in Fig. 4. It is seen that an increase in viscosity ratio α is to diminish the temperature distribution in the two regions. The greatest temperature distribution in the channel tends to move below the channel centerline towards the region-II for viscosity ratio (say for α ¼ 0.05 and 0.1) when all the remaining parameters are fixed. The effect of height ratio ‘h’ is represented in Fig. 5. It is noticed that the temperature distribution in the upper region is diminished, while it increased in the lower region as the height ratio increases. The greatest temperature distribution in the channel tends to move below the channel centerline towards the regionII for height ratio (for h ¼ 0.8) and the most extreme temperature in the channel tends to move over the channel centerline towards the region-I for height ratio h ¼ 0.1. The effect of the electrical conductivity ratio σ 0 on temperature distribution appears in Fig. 6. It is discovered that there is not much critical departure from temperature in the two regions as σ 0 increases. Fig. 7 depicts the varying thermal conductivity ratio β on temperature distribution. It is discovered that the temperature increases in the two regions as the thermal conductivity ratio increases. Additionally, it is seen that the temperature profile in the channel moves over the channel centerline towards region-I. That is, the temperature is higher in the upper region than the lower region for small estimations of the thermal conductivity ratio (say for β ¼ 0.1). The most extreme temperature distribution in the channel tends to move beneath the channel centerline towards region-II for the estimations of thermal conductivity ratio β ¼ 0.5, 1, 1.5 and 2 when all the rest of the parameters are fixed. The rate of heat transfer coefficients against Hartmann number is given in Figs. 8 and 9. It is seen that an increase either in the Hartmann number or in the Hall parameter rises the rate of heat transfer coefficients. Case II - Conducting plates: If the side plates are made of conducting material and short-circuited by an external conductor, then the induced electric current flows out of the channel. In this case no electric potential exists between the plates. When zero electric field is assumed in the
The non-dimensional form of the boundary and interface conditions on velocity and temperature become: u1 ðþ1Þ ¼ 0; w1 ðþ1Þ ¼ 0;
(3.37)
u2 ð1Þ ¼ 0; w2 ð1Þ ¼ 0
(3.38)
u1 ð0Þ ¼ u2 ð0Þ; w1 ð0Þ ¼ w2 ð0Þ;
(3.39)
du1 1 du2 dw1 1 dw2 ¼ and ¼ at y ¼ 0: dy αh dy dy αh dy
(3.40)
θ1 ðþ1Þ ¼ 0
(3.41)
θ2 ð1Þ ¼ 0
(3.42)
θ1 ð0Þ ¼ θ2 ð0Þ
(3.43)
dθ1 dθ2 ¼ ð1 = βhÞ at y ¼ 0 dy dy
(3.44)
Solutions to the problem In order to solve the energy equations (3.26) and (3.32) under the boundary and interface conditions (3.41)–(3.44) for obtaining the temperature distribution, first of all the equations (3.24), (3.25) and (3.30) & 3.31) are solved for the velocity distributions using the conditions (3.37)–(3.40). Consequently, the exact solutions for temperature distribution in the two-fluid flow regions and rate of heat transfer coefficient at the plates are obtained when the side plates are made of the nonconducting and conducting materials. Case I - Non-conducting plates: If the side plates are kept at a large distance in z-direction and are made of the non-conducting (insulating) material, the induced electric current does not go out of the channel but circulates itself in the fluid. Hence, additional conditions for the currents are, Z
1
Z
1
I1z dy ¼ 0 and
0
I2z dy ¼ 0:
(4.1)
0
When the insulation at large x is also assumed, other similar relations are obtained as Z
1
Z
1
I1x dy ¼ 0 and
0
I2x dy ¼ 0:
(4.2)
0
These conditions of insulation at large distances are physically realized for instance when the flow of ionized gas is surrounded by the cold non-conducting gas [see Ref. 18]. Using the above two conditions (4.1) to (4.2), the constants in the solution are determined and then solutions for u1 ; u2 , w1 ; w2 ; I1 and I2 in the two regions are obtained. With the help of these solutions, the following simplified equations are solved under the boundary and interface conditions (3.41)–(3.44) to obtain the temperature distribution in the two-fluid regions. Region I: d 2 θ1 dU1 dU 1 Ha2 þ þ ðU 1Þ U 1 ¼0 1 1 dy2 dy dy 1 þ m2
(4.3)
Region II: d 2 θ2 β dU2 dU2 H 2a hβσ 1 þ þ ðU2 1ÞðU2 1Þ ¼ 0 dy2 α dy dy 1 þ m2 Where U1 ¼ q1 =q1m ; U2 ¼ q2 =q2m ;
q1 ¼ u1 þ iw1 ;
(4.4) q2 ¼ u2 þ iw2 ;
q1 q1 q2 þq2 q2 q2 1 , q1m ¼ u1m þ u1 ðyÞ ¼ q1 þq 2 ; w1 ðyÞ ¼ 2i ; u2 ðyÞ ¼ 2 ; w2 ðyÞ ¼ 2i
5
T. Linga Raju
Results in Engineering 4 (2019) 100043
x–and z–directions, we have mix ¼ 0; miz ¼ 0; ði ¼ 1; 2Þ[see Ref. 18]. Constants in the solution are determined by these two conditions. The solutions for u1 ; u2 ; w1 ; w2 ; um1 ; um2 ; wm1 ; wm2 ; I1 and I2 in the two-fluid regions are obtained. By utilizing the solution of velocity distributions in this case, the following simplified form of the energy equations are solved under the boundary and interface conditions (3.41)–(3.44) for temperature distribution in both regions and the rate of heat transfer coefficients at the two side plates. Region I:
lower plates are presented in Figs. 16 and 17. It is observed that the rate of heat transfer coefficient at the two plates increases as the Hartmann number Ha increases when all the remaining governing parameters are fixed. Also, it is found that an increase in Hall parameter enhances the rate of heat transfer coefficient at the upper plate. But, the rate of heat transfer coefficient increases up to the estimation ‘m’¼2, thereafter it diminishes with an increase in Hall parameter when the Hartmann number Ha < 23 and beyond this number the rate of heat transfer coefficient increases.
2 d 2 θ1 dU1 dU1 1 1 s 1 is m U1 U1 2 þ H þ ðU U Þ þ 1 þ ¼0 1 1 a 1 þ m2 1 þ m2 H 4a qm1 qm1 H 2a 1 þ m2 qm1 qm1 dy2 dy dy
(4.5)
Region II:
d 2 θ2 β dU2 dU2 σ1 σ 1 s2 1 is σ 2 m U2 U2 2 2 þ H þ h β ðU U Þ þ 1 þ ¼0 2 2 a dy2 α dy dy 1 þ m2 1 þ m2 H 4a qm2 qm2 H 2a 1 þ m2 qm2 qm2
(4.6)
In the case of value s ¼ 1/2, the effect of varying the Hartmann number Ha on temperature distribution in the two regions is shown in Fig. 18. It is observed that an increase in Hartmann number enhances the temperature distribution in the two regions. It is seen that the temperature distribution in the channel moves above the channel centerline towards region-I, that is the temperature distribution is high in the upper region compared to the lower region for higher values of Hartmann number Ha (for Ha ¼ 30 and 50) when all the remaining parameters are kept fixed. Fig. 19 exhibits the effect of varying Hall parameter ‘m’ on temperature distribution in the case of s ¼ 1/2. It is discovered that a rise in the estimation of Hall parameter ‘m’ diminishes the temperature distribution in the two regions. It is found that the temperature distribution is high in the upper region compared to the lower region for small values of Hall parameter (say for m ¼ 0.05 and 1). The effect of viscosity ratio α is shown in Fig. 20. It is noticed that an increase in α diminishes the temperature distributions in the two regions. The effect of height ratio ‘h’ is represented in Fig. 21. It is seen that an increase in ‘h’ increases the temperature distributions in the two regions. Also the most extreme temperature distribution in the channel tends to move over the channel focus line towards region-I as the height ratio ‘h’ increases. The impact of electrical conductivity ratio σ 0 on temperature distribution in the two regions is shown in Fig. 22. It is discovered that there is no critical departure from the temperature distribution as electrical conductivities ratio σ 0 increases. Fig. 23 depicts the cause of thermal conductivity ratio β on the temperature distribution in the two regions. It is observed that an increase in thermal conductivity ratio β enhances the temperature distribution in the two regions. The temperature distribution is high in the upper region compared to the lower region for small estimation of the thermal conductivity ratio (say for β ¼ 0.1) when all the rest of the parameters are fixed. The rate of heat transfer coefficients at the two plates against the Hartmann number is shown in Figs. 24 and 25. It is discovered that an increase in Hartmann number increases the rate of heat transfer coefficient when all the remaining parameters are fixed. Also, it is found that an increase in Hall parameter enhances the rate of heat transfer coefficients at the point when all other governing parameters are fixed.
Numerical results are obtained by the same procedure as the case of non-conducting plates and the profiles are shown in Figs. 10–25. For conducting plates, the velocity and temperature distributions depend on ‘s’ (ratio of electron pressure to the total pressure, that is the ionization parameter). In this context, the graphs for the case [1] when ‘s’ ¼ 0 are depicted in Figs. 10–17 and that for another case [2] when the estimation s ¼ 1/2 are presented in Figs. 18–25. In the case of s ¼ 0, the effect of varying Hartmann number Ha on temperature distribution in the two regions is shown in Fig. 10. It is seen that an increase in Ha increases the temperature distribution in the two regions when all other parameters are fixed. The maximum temperature distribution in the channel tends to move above the channel centerline towards region-I as the Hartmann number increases. That is, the temperature profile is high in the upper region compared to the lower region with an increase in Hartmann number when all the remaining parameters are fixed. Fig. 11 exhibits the effect of varying Hall parameter ‘m’ on temperature distribution in the two regions. It is noticed that an increase in Hall parameter 'm’ reduces the temperature distribution in the two regions. Also, the maximum temperature distribution in the channel tends to move above the channel centerline towards the region-I for small values of the Hall parameter (say for ‘m’ ¼ 0.05 and 1). The impact of α(ratio of viscosities of the two fluids) appears in Fig. 12. It is observed that an increase in viscosity ratio diminishes the temperature distribution in the two regions. The effect of height ratio ‘h’ is represented in Fig. 13. It is seen that an increase in height ratio ‘h’ enhances the temperature distribution in the two regions. It is also discovered that the temperature profile is high in the upper region compared to the lower region with an increase in height ratio when all the rest of the parameters are fixed. The effect of the ratio of electrical conductivity σ 0 on temperature distribution is shown in Fig. 14. It is seen that there is not much noteworthy departure from the temperature distribution as σ 0 increases. Fig. 15 depicts the effect of the thermal conductivity ratio β on temperature distribution in the two-fluid regions. It is found that a rise in thermal conductivity ratio enhances the temperature distribution in the two regions. It is noticed that the temperature profile is high in the upper region compared to the lower region for small estimations of the thermal conductivity ratio (say for β ¼ 0.1). The rate of heat transfer coefficients against Hartmann number at upper and
6
T. Linga Raju
Results in Engineering 4 (2019) 100043
Fig. 2. Temerature Profiles for different Ha and α ¼ 0.333,σ0 ¼ 2,σ1 ¼ 1.2,σ2 ¼ 1.5, β ¼ 1, m ¼ 2,h ¼ 0.8.
Fig. 3. Temerature Profiles for different m and Ha ¼ 10, α ¼ 0.333,σ0 ¼ 2,σ1 ¼ 1.2,σ2 ¼ 1.5, β ¼ 1, h ¼ 0.8.
Fig. 4. Temperature profiles for different α and Ha ¼ 10, m ¼ 2, σ0 ¼ 2,σ1 ¼ 1.2,σ2 ¼ 1.5, β ¼ 1, h ¼ 0.8. 7
T. Linga Raju
Results in Engineering 4 (2019) 100043
Fig. 5. Temperature profiles for different h and α ¼ 0.333, Ha ¼ 10, m ¼ 2, σ0 ¼ 2,σ1 ¼ 1.2,σ2 ¼ 1.5, β ¼ 1.
Fig. 6. Temperature profiles for differet σ0 and α ¼ 0.333, Ha ¼ 10, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2, β ¼ 1,s ¼ 0.
Fig. 7. Temperature profiles for different β and Ha ¼ 10, α¼0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2. 8
T. Linga Raju
Results in Engineering 4 (2019) 100043
Fig. 8. Nusselt Number (Nu1) for different Ha and α ¼ 0.333,σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1 h ¼ 0.8.
Fig. 9. Nusselt Number (Nu2) for different Ha and α ¼ 0.333,σ0 ¼ 2,σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1, h ¼ 0.8.
Fig. 10. Temperature profiles for different Ha and α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, β ¼ 1, m ¼ 2, s ¼ 0. 9
T. Linga Raju
Results in Engineering 4 (2019) 100043
Fig. 11. Temerature Profiles for different m and Ha ¼ 10, α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1, h ¼ 0.8, s ¼ 0.
Fig. 12. Temperature profiles for different α and Ha ¼ 10, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1, h ¼ 0.8, m ¼ 2, s ¼ 0.
Fig. 13. Temperature profiles for different h and α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, Ha ¼ 10, β¼1, m ¼ 2, s ¼ 0. 10
T. Linga Raju
Results in Engineering 4 (2019) 100043
Fig. 14. Temperature profiles for differet σ0 and α ¼ 0.333, Ha ¼ 10, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2, β ¼ 1, s ¼ 0.
Fig. 15. Temperature profiles for different β and Ha ¼ 10, α¼0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2, s ¼ 0.
Fig. 16. Nusselt Number (Nu1) for different Ha and α ¼ 0.333,σ0 ¼ 2,σ1 ¼ 1.2,σ2 ¼ 1.5, β¼1, h ¼ 0.8, s ¼ 0. 11
T. Linga Raju
Results in Engineering 4 (2019) 100043
Fig. 17. Nusselt Number (Nu2) for different Ha and α ¼ 0.333,σ0 ¼ 2,σ1 ¼ 1.2,σ2 ¼ 1.5, β¼1, h ¼ 0.8, s ¼ 0.
Fig. 18. Temperature profiles for differet Ha and α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2, β ¼ 1, s ¼ 0.5.
Fig. 19. Temerature Profiles for different m and Ha ¼ 10, Ha ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β ¼ 1, h ¼ 0.8, s ¼ 0.5. 12
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Results in Engineering 4 (2019) 100043
Fig. 20. Temperature profiles for different α and Ha ¼ 10, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1, h ¼ 0.8, m ¼ 2, s ¼ 0.5.
Fig. 21. Temperature profiles for different h and α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, Ha ¼ 10, β ¼ 1, m ¼ 2, s ¼ 0.5.
Fig. 22. Temperature profiles for differet σ0 and α ¼ 0.333, Ha ¼ 10, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2, β ¼ 1, s ¼ 0.5. 13
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Results in Engineering 4 (2019) 100043
Fig. 23. Temperature profiles for different β and Ha ¼ 10, α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, h ¼ 0.8, m ¼ 2, s ¼ 0.5.
Fig. 24. Nusselt Number (Nu1) for different Ha and α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1, h ¼ 0.8, s ¼ 0.5.
Fig. 25. Nusselt Number (Nu2) for different Ha and α ¼ 0.333, σ0 ¼ 2, σ1 ¼ 1.2, σ2 ¼ 1.5, β¼1, h ¼ 0.8, s ¼ 0.5. 14
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Results in Engineering 4 (2019) 100043
Conclusion
Acknowledgement
A novel problem in heat transfer aspects of steady MHD two-fluid flow of ionized gases driven by a constant pressure gradient through horizontal channel bounded by two parallel plates under the action of an external magnetic field with effect of Hall currents is considered theoretically. The issue is examined in two cases, that is, when the two side plates are made of the non-conducting and conducting materials. The important findings of this study are summarized as follows: The solutions for temperature distribution in the two-fluid regions are observed to be independent of the ratio of electron pressure to the total pressure only for non-conducting side plates unlike for the conducting plates. In case of non-conducting side plates, it is seen that:
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The temperature in the two regions diminished with the rise in Hall parameter. An increase in Hartmann number up to a specific esteem increases the temperature distribution in the region-I (fluid in the upper region) and from that point it diminishes, while the temperature increases in the region-II with an increase in Hartmann number. The temperature distribution in the two regions decreased with an increase in viscosity ratio. An increase in the height ratio diminishes the temperature distribution in the region-I, while it enhances in region-II. There is insignificant variation in temperature distribution with an increase in electrical conductivity ratio. The temperature distribution rises with the increase in thermal conductivity ratio. Also, the maximum temperature distribution in the channel moves over the channel focus line towards the region-I for small estimations of the thermal conductivity ratio. A rise in Hartmann number and Hall parameter, leads to rise in the rate of heat transfer coefficient at the two plates. In case of conducting side plates and for s ¼ 0 (ratio of electron pressure to the total pressure equal to zero), it is observed that: The temperature distribution in the two regions enhanced with increased Hartmann number. A rise in Hall parameter and the viscosity ratio leads to fall in temperature distribution. The temperature distribution increases with an increase in thermal conductivity ratio. There is no variation in the temperature distributions as the electrical conductivity ratio increases. An increase in Hartmann number enhances the rate of heat transfer coefficient at the two plates. Also, the relatively comparable highlights are prevalent for the situation when the ionization parameter is equal to half of its esteem (that is, for the value s ¼ 1/2). Conflict of interest The author declares that there is no conflict of interests regarding the publication of this research paper.
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