sink on MHD flow between vertical alternate conducting walls with Hall effect

Journal Pre-proof Influence of heat source/sink on MHD flow between vertical alternate conducting walls with Hall Effect Dileep Kumar, A.K. Singh, Devendra Kumar

PII: DOI: Reference:

S0378-4371(19)31984-3 https://doi.org/10.1016/j.physa.2019.123562 PHYSA 123562

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Physica A

Received date : 24 May 2019 Revised date : 25 September 2019 Please cite this article as: D. Kumar, A.K. Singh and D. Kumar, Influence of heat source/sink on MHD flow between vertical alternate conducting walls with Hall Effect, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123562. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

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Highlights (for review)

Highlights

 Effects of Hall current on the fully developed free convective flow are discussed.

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 The transformed system of equations is solved analytically.  The expressions of the induced current density, skin friction and mass flow rates are

calculated.

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 Influence of various physical parameters is discussed graphically.

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*Manuscript Click here to view linked References

1

Influence of heat source/sink on MHD flow between vertical alternate conducting walls with Hall Effect 1

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Dileep Kumar1, A. K. Singh2 and Devendra Kumar3,* Department of Mathematics, VSS University of Technology, Burla, Sambalpur-768018,India 1 &2 3

Department of Mathematics, Banaras Hindu University, Varanasi-221005, India

Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India

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*

Corresponding author Email: [email protected]

Abstract

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The key object of this article is to present the impact of heat source/sink and Hall current on the completely advanced natural convection flow of a viscous incompressible and electrically conducting fluid through a vertical channel with the condition that one wall is conducting and other is non-conducting. The resulting system of non-dimensional linear equations has been examined with the help of theory of simultaneous differential equations. Finally, we have found the expressions for the fluid velocity, induced magnetic field and temperature field in the compact form. Also, we have derived the induced current density from induced magnetic field and with the help of velocity we found skin friction and rates of mass flow. The obtained results are presented through graphs for distinct values of the Hartmann number, Hall current and heat source/sink parameters. It is noted that the impact of the Hartmann number is to reduce the parts of the velocity and induced current density but improve the constituents of induced magnetic field. Further, the increase of the Hall current parameter leads to enhancement and reduces the primary and secondary constituents of the velocity and induced magnetic field respectively. The Hall current gives rise to a cross flow and the variable fluid properties have strong effects on the shear stress and the Nusselt number. Hall current is applicable in Hall accelerators, Hall effect sensors and constrictions of turbines, etc.

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Keywords: Heat source/sink; Natural convection; Magnetohydrodynamics; Induced magnetic field; Hall current Nomenclature

Heat capacity at fixed pressure

Cp

d

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E

Distance apart two walls Electrical field in vector form

E x , E y

Electrical field in x - and y  -direction

g

Acceleration due to gravity

H

Magnetic field in vector form

)

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2

Hartmann number

H x , H y

Induced magnetic field in x  - direction and y  -direction

Hx , Hy

Induced magnetic field in non-dimensional form in x- direction and ydirection

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Ha

Induced current density field in vector form

J x , J y

Induced current density field towards the x  -direction and y  -direction

Jx ,Jy

Induced current density field in non-dimensional form in x- and ydirections

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J

Hall current parameter

S

Heat source/sink parameter

TL , TR

Left and right vertical wall temperature

T

Fluid temperature

T

Fluid temperature in non-dimensional form

U

Characteristic velocity

V

Velocity field in vector form

Vx  , Vy

Velocity in x  - and y -direction

Vx , Vy

Velocity components in non-dimensional form in x- and y- directions

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mω τ e e

Greek Symbol

κ

μ μe

ν

ρ σ

Electrical permittivity Thermal conductivity Coefficient of viscosity

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ε

Magnetic permeability Kinematic viscosity of the fluid Density of the fluid Electrical conductivity

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3 τ0

Left wall skin-friction coefficient

τ1

Right wall skin-friction coefficient

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1. Introduction

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There are numerous natural incident and engineering processes supposed to be magnetohydrodynamic flow of an electrically conducting fluid can be achieved theoretical and experimental. Such types of problems have received broad spectrum of its uses in the scope of aerodynamics, engineering, astrophysics and different branches of science and technology like as processing of electromagnetic materials, measuring the ship’s speed by flow meters, designing communications and radar systems, design of MHD power generator and controlling the melt convection during crystal growth. The applications of MHD flow have magnetized in the plasma study, magnetic drug targeting, blood flow adjusting during surgery, nuclear reactor and magnetohydrodynamic jet flow etc. The impact of induced magnetic field (IMF) has been considered in the present problem which is obtained due to an electrically conducting fluid moves in a magnetic field. Usually, we have neglected induced magnetic field in the study of hydromagnetic flows due to make easy the mathematical representation of the problem. Many works have been conducted in MHD without considering the induced magnetic field such as Sparrow and Cess [1], Cramer and Pai [2], Geograntopoulos and Nanousis [3], Raptis and Singh [4], Jha et al. [5], Chandran et al. [6], Singh and Singh [7], Hamza and his colleagues [8] andMa et al [9, 10, 11]have analyzed the influence of magnetic field for various flow problems. Due to importance of induced magnetic field Globe [12], Verma and Mathur [13], Arora and Gupta [14], Guria et al [15], Singh and his co-authors [16] have taken the induced magnetic field in their mathematical problems of magnetohydrodynamic flows. Also, Singh and Singh [17], Kumar and Singh [18] have analytically studied the consequence of IMF on natural convection through vertical concentric annuli for different boundary conditions. The analysis of MHD natural convection flow of an electrically conducting in a vertical channel with symmetric heating and induced magnetic field has been done by Jha and Sani [19]. Sarveshanand and Singh [20] have executed an exact analysis of MHD natural convection flow across vertical walls by taking into assumption the IMF. Currently, Kumar and Singh [21] have performed analytical study of the influences of the Newtonian heating/cooling and IMF on free convective flow via vertical concentric annuli. The impact of IMF on MHD free convective flow via a vertical micro channel having electrically non-conducting infinite equidistant plates has been analyzed by Jha and Aina [22]. Edwin Hall (1879) discovered that because of the motion of an electrically conducting fluid in a magnetic field the electromagnetic force, which is taken upright to both the fields i.e. electric and magnetic fields, generated in the flow. Due to presence of this force the charged molecules to transpose in its own direction and generate an electrical current density termed as the Hall current. Normally, due to little and moderate values of the magnetic field the impact of Hall current is disregarded in enforcing the Ohm’s law whereas the influence of Hall current cannot be ignored for strong magnetic field. Datta and Jana [23], Singh [24], Seth and Ghosh [25] have studied different magnetohydrodynamic free convective flow mathematical problems with taking the impact of the Hall current into account. Ghosh et al.[26] has performed an analytical study of magnetohydrodynamic flow via a rotating vertical channel having heat transfer and induced magnetic field (IMF). The numerical study of hydromagnetic free convective flow with strong

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cross magnetic field and the Hall current has been done by Siddiqa et al. [27]. Currently, Seth and Singh [28] have been analytically studied of hybrid convection magnetohydrodynamic flow via a rotating channel by assuming the impacts of Hall current and wall conductance. In very recent work, Kumar and Singh [29, 30] have performed the influence of the Hall current on MHD free convective flow between vertical Walls with taking different boundary conditions on IMF. Also, since the volumetric heat creation or absorption term employs powerful effect on the heat transfer and the fluid flow in the presence of huge temperature differences. Due to this the impact of internal heat source or heat sink has exalted both physical and theoretical importance in hydromagnetic fluid flow. The heat source or sink which are temperature dependent has its importance in combustion process, spraying of fluid into the medium or after shutdown, cooling problem associated with nuclear reaction. Also, the effects of source and sink have magnetized utilization in neurobiology to the study of brain function. The consequence of heat source/sink on magnetohydrodynamic flow has been considered by Chamkha et al. [31], Sharma and Singh [32], Bhattacharyya [33] in their problems. Recently, Kumar and Singh [34] have reported the influence of temperature dependent heat source/sink on free convection in vertical concentric annular cylinders. Some more applications of magnetohydrodynamic flows and heat transfer have been found in [35-50]. To the best knowledge of the authors, in the current work, the impression of Hall current and temperature dependent heat source/sink on free convective magnetohydrodynamic flow of an electrically conducting and viscous incompressible fluid via a vertical channel by taking IMF into assumption. The dimensionless ruling differential equations have been solved by utilizing the dimensionless boundary conditions (BCs) to find the compact form of the velocity, IMF and temperature field. With the help of these equations, we have calculated the primary and secondary terms of the velocity, IMF, induced current density (ICD), skin-friction and mass flow rates. Lastly, we observed the influence of the Hall current parameter, heat source/sink parameter and the Hartmann number on these components of the velocity, IMF and ICD with the help of graphs and the numerical values are presented in table for the primary term and secondary term of skin-friction and rate of mass flow.

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2. Mathematical equations and physical model Here, we have used the basic MHD and Maxwell’s electromagnetic equations for the steady, natural convection and an electrically conducting fluid flow, given in the following manner: Continuity equation .V  0,

(1)

Momentum equation

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ρ (V.)V  P  μ( 2 V)  (J  B)  ρ g,

(2)

Generalized Ohm’s Law J  (ω e τ e /H 0 )(J  H )  σ(E  μ e V  H ),

(3)

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Energy equation (V.)T  (κ / ρ C p )  2 T  (Q0 /ρC p ),

for the presence of heat source and

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Note: Here, we have considered the presence of heat sink.

(4)

Maxwell’s equations

(5)

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. E  ρ/ε , . B  0,   E  0,   H  J.

for

The physical meaning of notations is given in nomenclature.

The steady magnetohydrodynamic natural convection of a viscous, incompressible and electrically conducting fluid across2 infinite long alternate conducting vertical walls in the presence of Hall current and heat source/sink by taking into assumption the IMF is considered. In the formulation of the present problem it is assumed that:

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 The applied magnetic field H 0 is too large that the contribution of buoyancy force, which is comparable with Lorentz force. Hence, magnetic effects are quite intense and have dominant effect within the boundary layer region.  One of the plates is conducting and other is non-conducting.  The IMF as well as the Hall current has been considered due to large magnetic Reynolds.  Joule heating and viscous dissipation are neglected.

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A Cartesian coordinate system is assumed as, the x   axis is along one of the vertical walls measured upward direction, the z   axis is upright to both walls and y   axis is considered as upright to the x  z   plane. The wall located at z  0 is considered as non-conducting whereas the wall located at z   h is considered as conducting. A fixed magnetic field of strength H 0 is exerted upright to both walls i.e. in z   direction. The temperature of vertical walls is assumed as T0 and Th such that T0  Th . The physical configuration for the considered model is shown in Fig.1. The transverse velocity of the considered fluid is zero due to steady and fully developed nature. Also, as both the walls are of unlimited length in x  - and y   directions so the variables which describes the flow model will depend on a single variable z  except the density.

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Figure-1: Physical Model. By the Faraday’s law (i.e.   E  0) in steady flow, we have

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dE x /dz   0, dE y /dz   0.

(6)

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Eqn. (6) shows that the components E x  and E y can be taken as constant surrounded by the fluid and on the interfaces of the walls. Thus for the considered mathematical model, with the Boussinesq approximation, the governing equations of velocity, momentum, generalized Ohm’s law and energy with heat source/sink, Hall current and IMF are acquired in the subsequent manner [Seth and Singh (2016)]: (7)

ν(d2 Vy /dz 2 )  (μ e H0 /ρ ) (dHy /dz)  0,

(8)

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ν(d 2 Vx /dz 2 )  (μ e H0 /ρ )(dHx /dz)  g β (T  Th )  0,

m(dHx /dz)  (dHy /dz)  σ(Ex  μ e H0 Vy ),

(9)

m(dHy /dz)  (dHx /dz)  σ(Ey  μ e H0 Vx ),

(10)

2 2 ( /ρ C v ) (d T/dz )  (Q 0 /ρ C v ) (T  Th )  0.

(11)

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Vx   Vy   0, Hx   Hy  0, T  T0 at z  0,

Vx  Vy  0, (d Hx /dz)  (dHy /dz)  0, T  Th at

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Boundary conditions, which show the conducting and non-conducting behavior of inner and outer walls respectively, for the velocity, IMF and temperature fields are given in the subsequent manner:

z  h.

(12) (13)

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By differentiating Eqs. (9) and (10) with respect to z  and using   E  0 , we have m(d 2 Hx /dz 2 )  (d 2 Hy /dz 2 )  σμ e H0 (dVy /dz),

(14)

m(d 2 Hy /dz 2 )  (d 2 Hx /dz  σμ e H0 (dVx /dz).

(15)

Now combining Eqs. (7) with (8) and (14) with (15) and using V  Vx  iVy , H  Hx  iHy and E  Ex  iEy , we obtain

(16)

(d H/dz )  {(σ( e H0 )/(1  im)}(d V/dz)  0.

(17)

2

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ν(d 2 V/dz 2 )  (μ e H0 /ρ ) dH/dz)  g β (T  Th )  0,

BCs in the are given in the subsequent manner V   0,

H   0,

V   0, (dH /dz )  0,

T   T0

at

z   0,

(18)

T   Th

at

z   h.

(19)

On using the subsequent dimensionless variables, it yields

V  (V /U), z  (z /h), H  (H / σ μ e H 0 Uh), T  {(T   Th )/(T0  Th )}, Ha  μ e H 0 h σ/μ ,

U  (g β h 2 / ν )(T0  Th ), S  (Q 0 h 2 )/(  ).

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(20)

The governing Eqs.(16), (17) and (11) in non-dimensional form have taken the form of (21)

(d 2 H/dz 2 )  {1/(1  im)}(dV/dz )  0,

(22)

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(d 2 V/dz 2 )  Ha 2 (dH/dz)  T  0,

(d 2 T/dz 2 )  ST  0.

(23)

with conditions (18) and (19) in the dimensionless form as H  0,

T  1 at

z  0,

(24)

V  0,

(dH/dz)  0,

T  0 at

z  1.

(25)

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V  0,

3. Analytical Solutions 3.1 In the absent of heat source or sink is (i.e.

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Here, we have solved the associated system of linear differential equations presented by equations (21)-(23) with appropriate boundary conditions Eqs. (24) and (25) when . Thus,

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the dimensionless expressions for the velocity, the IMF and the temperature field are presented in the subsequent manner: (26)

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V(z)  Vx (z)  iVy (z)  E1cosh(λ z)  E 2sinh(λ z)  (1  z)/λ 2 ,

H(z)  H x (z)  iH y (z)  E 3 - {E1 λsinh(λ z)  E 2 λcosh(λ z)  (z  z 2 /2)}/Ha 2 , (27)

The expression for the ICD is given by:

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T(z)  1  z. (28)

J  J x (z)  iJ y (z)  (dH/dz)  {(E1λ 2 cosh(λ z)  E 2 λ 2sinh(λ z)  (1  z)}/Ha 2 . (29)

, we get u y (z)  h y (z)  J y (z)  0 and u x (z)  u(z),

In the absence of Hall current

h x (z)  h(z), J x (z)  J(z) are same as obtained by Sarveshanand and Singh (2015).

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The expressions for the skin friction on both walls (at z  0 and at z  1 ) and the mass flow rates Q x and Q y , in dimensionless form, are given as: (τ ) z0  {τ x (z)  i τ y (z)}z0  (dV/dz) z0  E 2 λ  1/λ 2 ,

(30)

(τ ) z1  {τ x (z)  i τ y (z)}z1  (dV/dz) z1  {E1λsinh(λ)  E 2 λcos(λ)  1/λ 2 },

(31)

Q  Q x  iQ y 

z 1  V(z)dz z 0

 {E1sinh λ  E 2 (cosh λ - 1)}/λ  1/2λ 2 .

(32)

3.2 In the present of heat source (i.e. S > 0)

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The exact solution of Eqs. (21)-(23) with boundary conditions (24)-(25), in the presence of temperature dependent heat source, is given as: V(z)  Vx (z)  iVy (z)  E 4 cosh(λ z)  E 5sinh(λ z)  (cos Sz  c1sin Sz)/(λ 2  S),

(33) (34)

T(z)  cos Sz  c1sin Sz.

(35)

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H(z)  H x (z)  iH y (z)  E 6 - {E 4 λsinh(λ z)  E 5 λcosh(λ z)}/Ha 2  {(sin Sz  c1cos Sz)λ 2 / S Ha 2 (λ 2  S)},

The expressions for the ICD, the skin friction and the rate of mass flow in dimensionless form are acquired as:

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J  J x (z)  iJ y (z)  (dH/dz)  {(E 4 λ 2 cosh(λ z)  E 5 λ 2 sinh(λ z)}/Ha 2  {cos Sz  c1sin Sz)λ 2 /Ha 2 (S  λ 2 )},

(36)

(τ ) z0  {τ x (z)  i τ y (z)}z0  (dV/dz) z0  E 5 λ  (c1 S )/(S  λ 2 ), (37)

(τ ) z 1  {τ x (z)  i τ y (z)}z 1  (dV/dz) z 1  {E 4 λsinh(λ)  E 5 λcosh(λ)}  ( sin S  c1 cos S ) S/(S  λ 2 ).

Q  Q x  iQ y 

z 1  V(z)dz z 0

 {E 4 sinh λ  E 5 (cosh λ - 1)}/λ

 {sin S  c1 (cos S - 1)}/ S (S  λ 2 ).

(38)

(39)

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3.3 In the present of heat sink (i.e. S < 0)

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Finally, we have solved the equations (21)-(23) surrounding the BCs (24)-(25) in presence of the temperature dependent heat sink to get the representations for the velocity field, the induced magnetic and the temperature fields and they have obtained as follows: (40)

H(z)  H x (z)  iH y (z)  E 9 - {E 7 λsinh(λ z)  E 8 λcosh(λ z)}/Ha 2  (sinh Sz  c 2 cosh Sz)λ 2 / S Ha 2 (λ 2  S),

(41)

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V(z)  Vx (z)  iVy (z)  E 7 cosh(λ z)  E 8sinh(λ z)  (cosh Sz  c 2 sinh Sz)/(λ 2  S),

T(z)  cosh Sz  c 2 sinh Sz. (42)

In the attendance of heat sink, the expressions of dimensionless ICD, skin frictions and the rate of mass flow are found as: (43)

(τ ) z0  {τ x (z)  i τ y (z)}z0  (dV/dz) z0  E 8 λ  c 2 S/(λ 2 - S),

(44)

(τ ) z 1  {τ x (z)  i τ y (z)}z 1  (dV/dz) z 1  {E 7 λsinh(λ)  E 8 λcos(λ)}  (sinh S  c 2 cosh S ) S/(λ 2 - S).

(45)

Q  Q x  iQ y 

z 1  V(z)dz z 0

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J  J x (z)  iJ y (z)  (dH/dz)  {(E 7 λ 2 cosh(λ z)  E 8 λ 2 sinh(λ z)}/Ha 2  (cosh Sz  c 2 sinh Sz)λ 2 /{Ha 2 (λ 2 - S)},

 {E 7 sinh λ  E 8 (cosh λ - 1)}/λ

 {sinh S  c 2 (cosh S - 1)}/ S (λ 2 - S).

(46)

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4. Numerical simulation

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where λ  α  i β, α, β  Ha{( 1  m 2  1)/2(1  m 2 )} 2 , E 1 , E 2 , E 4 , E 5 , E 6 , E 8 , E 9 , c1 and c 2 are all constants, comes in up above equations, explained in the form of appendix A.

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In the present study, we have analyzed the influences of the Hall current parameter , temperature dependent heat source/sink parameter and the Hartmann number on the primary constituent and secondary constituent of the velocity profile, IMF and ICD, in the presence and absence of heat source/sink, by graphs.

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The effect of on the primary constituent and secondary constituent of the velocity profile is shown in Figs. 2(a)-2(b) for heat source and sink ( S    0 ). It can be seen that as the value of increases, then both the primary constituent and secondary constituent of the velocity profiles decrease in all three cases of heat source/sink, because the electromagnetic force (Lorentz force) produces in the fluid which retards the fluid flow. It is observed that as the S rises the primary component of the velocity profile rises but the secondary component of the velocity profile declines because with enhancement in the strength of the heat source parameter the thermal boundary layer increases. Also it is understandable that the heat sink parameter is to decline the primary component of the velocity profiles while it has opposite impact in case of secondary component of the velocity profiles due to reduction in thermal boundary layer. It also clears from the velocity profiles that the primary components of the velocity is more than the secondary component of the velocity in all three cases of heat source/sink because of Hall

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and

on the primary component of the velocity profiles at

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Fig. 2(a) Effects of

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current. The appearance of the primary part of the velocity is parabolic towards a higher level direction but secondary component has reverse direction and it reduces to flatter for higher values of the Hartmann number in all three occasions of heat source/sink.

Fig. 2(b)Impacts of

and

on the secondary component of the velocity at

.

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Fig. 2(c) shows that the effect of Hartmann number on the velocity profiles by taking Hall current parameter which is just similar to the results of reference Sarveshanand and Singh (2015).

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Fig. 2(c) Effect of Ha on the velocity profiles at

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The impact of the on the primary part and secondary part of the IMFs is presented via Figs. 3(a)-3(b). It is found that both the primary and secondary components of the IMF profiles decrease with increasing the Hartmann number in all three cases of temperature dependent heat source/sink due to presence of the Lorentz force. It is revealed that both the primary and secondary parts of the IMF profiles decreases in presence of heat source but both components have just reverse effect in presence of heat sink. This is because the thermal boundary layer increases in case of heat source and decreases in heat sink. The profiles of primary component of the IMF are more than the secondary component of the IMF.

Fig. 3(a) Effects of .

and on the primary component of the induced magnetic field at

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and on the secondary component of the IMF at

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Fig.3(b) Effects of

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Figs 4(a)-4(b) exhibit the impact of the on profiles of the primary and secondary parts of the ICD in without the heat source/sink, with the of heat source and with the heat sink. It shows that both the primary and secondary parts of the ICD profiles decrease with enhancing the due to attendance of electromagnetic force. It is observed from Figs. 4(a)-4(b) that both the primary and secondary parts of the ICD profiles increase on enhancing the heat source parameter while these have reverse behavior on raising the heat sink parameter. In all three cases of temperature dependent heat source/sink, the profiles of the primary constituent of ICD are more than the secondary constituent of ICD. We can see in Figs 4(a)-4(b) that the appearance of the primary part and secondary part of ICD portrayals are parabolic in upwards direction.

Fig.4(a) Effects of

and on the primary component of ICD at

.

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and on the secondary component of ICD at

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Fig. 4(b) Effects of

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Figs. 5(a)-5(b) show that the impact of the hall current parameter on the primary and secondary parts of velocity profiles in all cases of the heat source/sink. It can be observed form these figures that as the increasing, the primary part of the velocity profiles increase but the secondary component of the velocity profiles has reverse effect in all cases of heat source/sink. This shows that the Hall current enhance the velocity profiles towards the primary fluid flow and retard the velocity profiles in towards the secondary fluid flow because the Hall current generates secondary flow into the flow field. It clearly shows that the impact of the Hall current on the primary part of the velocity profiles is more than the secondary part of the velocity profiles.

Fig. 5(a)Impact of source/sink at

on the primary component of the velocity profiles in all cases of heat .

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on the secondary component of the velocity profiles in all cases of heat .

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Fig. 5(b) Impact of source/sink at

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Figs. 6(a)-6(c) illustrate the impact of the on the primary part and secondary part of the IMF profiles in all the three cases of heat source/sink parameter. It can be noticed from these Figs. that on enhancing the , the primary part of the IMF profiles increase in the region of the channel but the secondary component of the IMF profiles have reverse tendency due to presence of the Hall current. It is exposed that both the primary and secondary components of the IMF profiles to reduce in the impendence of heat source, but both components have just inverted effect in the impendence of heat sink. This is because the thermal boundary layer enhances in case of heat source and reduces in heat sink. Such effects of source and sink have attracting applications in neurobiology to the study of brain function.

Fig. 6(a)Impact of source/sink at

on the primary component of the IMF profiles in all cases of heat .

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on thesecondary component of the IMF profiles in all cases of heat .

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Fig. 6(b)Impact of source/sink at

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The influence of the on primary component and secondary component of the ICD is shown in shown in Figs 7(a)-7(c) for the cases when heat source/sink is absent and present. We observed that the primary part of the ICD profiles decrease on increasing the Hall current parameter while reverse the secondary constituent of the ICD descriptions. This implies that the Hall current tends to reduce the primary constituent of the generated current and enhance the secondary constituent of the ICD.

Fig. 7(a) Impact of source/sink when

on the primary component of the ICD field profiles in all cases of heat .

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on the primary component of the ICD field profiles in all cases of heat .

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Fig. 7(b)Impact of source/sink at

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The impacts of the and the on the primary part and secondary part of the skin-friction and mass flux at the surface of both walls without the heat source/sink are shown in table 1. It is clear that the effect of the is to decrease both the primary part and secondary part of the skinfriction at surface of the walls and rate of mass flow. It is also clear that with increasing value of the , the primary component of the skin-friction at surface of the walls and rate of mass flow increase but the secondary component of the skin-friction at surface of the walls and rate of mass flow has reverse effect.

al

Table 1 Variation of the parts of the skin-friction and rate of mass flow with without the heat source/sink. Ha

(τ ) z 0 x

(τ ) z 0 y

(τ ) z 1 x

(τ ) z1 y

1.0 0.25 2.0 3.0 1.0 0.50 2.0 3.0 1.0 0.75 2.0 3.0

0.31404 0.27076 0.22605 0.31654 0.27631 0.23200 0.31952 0.28363 0.24043

-0.004406 -0.011443 -0.015012 -0.007671 -0.020827 -0.028240 -0.009463 -0.027191 -0.038634

0.14993 0.11374 0.07916 0.15206 0.11804 0.08295 0.15462 0.12387 0.08866

-0.003781 -0.009239 -0.010925 -0.006601 -0.016992 -0.020965 -0.008169 -0.022460 -0.029408

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m

Q

x

0.038068 0.030165 0.022363 0.038529 0.031135 0.023293 0.039081 0.032434 0.024659

and Q

on

y

-0.000816 -0.002047 -0.002528 -0.001424 -0.003749 -0.004812 -0.001760 -0.004930 -0.006681

Table 2 demonstrates the numerical effect Hall current, Hartmann number on the primary component and secondary component of the skin-friction and rate of mass flow on the surface of both walls the present of heat source. It is noted that, in this case the effect of the is to reduce both the primary component and secondary component of the skin-friction at both the walls and rate of mass flow.

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Table 2 Variation of the parts of the skin-friction and rate of mass flow with Hartmann number, Hall current parameter and the heat source parameter. (τ ) z 0 x

(τ ) z 0 y

(τ ) z 1 x

(τ ) z1 y

1.0 .25 2.0 3.0 1.0 1.0 .50 2.0 3.0 1.0 .75 2.0 3.0 1.0 .25 2.0 3.0 1.0 2.0 .50 2.0 3.0 1.0 .75 2.0 3.0

0.33658 0.28882 0.23964 0.33934 0.29493 0.24614 0.34263 0.30298 0.25536 0.36449 0.31113 0.25639 0.36758 0.31793 0.26356 0.37126 0.32691 0.27377

-0.004868 -0.012608 -0.016467 -0.008476 -0.022959 -0.031004 -0.010458 -0.029991 -0.042461 -0.005446 -0.014064 -0.018281 -0.009483 -0.025623 -0.034451 -0.011702 -0.033490 -0.047236

0.16969 0.12919 0.09039 0.17207 0.13402 0.09467 0.17493 0.14054 0.10111 0.19468 0.14876 0.10465 0.19738 0.15425 0.10955 0.20062 0.16166 0.11691

-0.004228 -0.010351 -0.012283 -0.007380 -0.019031 -0.023554 -0.009133 -0.025145 -0.033013 -0.004790 -0.011751 -0.013994 -0.008360 -0.021596 -0.026817 -0.010344 -0.028523 -0.037554

Q

Q

x

0.042299 0.033506 0.024827 0.042812 0.034586 0.025862 0.043426 0.036030 0.027381 0.047603 0.037695 0.027916 0.048181 0.038912 0.029082 0.048874 0.040539 0.030793

p ro

Pr e-

m

y

of

Ha

S

-0.0009089 -0.0022777 -0.0028126 -0.0015847 -0.0041709 -0.0053535 -0.0019586 -0.0054850 -0.0074324 -0.0010242 -0.0025665 -0.0031688 -0.0017858 -0.0046999 -0.0060316 -0.0022072 -0.0061807 -0.0083742

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Form above table, we can observe that the influence of the is to increase, the primary component of the skin-friction at both the walls and rate of mass flow but the secondary component of the skin-friction at both the walls and mass flow rate has reverse effect with increasing the Hall current parameter. The impact of the S is to enhance the primary component of the skin-friction at both the walls and mass flow rate while it is opposite in case of the secondary component of the skin-friction at both the walls and mass flow.

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Table 3 shows the influence of the , and the on the both parts of skin-friction at the surface of both walls and mass rate of flow. Like Hartmann number enhances then both the parts of skin-friction, on the surface of walls, and mass flow rate reduces. The consequence of the is to enhance the primary part of the skin-friction at both surfaces of the walls and rate of mass flow but the secondary component of the skin-friction at both surfaces of walls and rate of mass flow is just reverse.

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Table 3 Variation of the parts of the skin-friction and rate of mass flow with Hartmann number, Hall current parameter and heat sink parameter. (τ ) z 0 x

(τ ) z 0 y

(τ ) z 1 x

(τ ) z1 y

1.0 .25 2.0 3.0 1.0 -1.0 .50 2.0 3.0 1.0 .75 2.0 3.0 1.0 .25 2.0 3.0 1.0 -2.0 .50 2.0 3.0 1.0 .75 2.0 3.0

0.29540 0.25580 0.21474 0.29769 0.26090 0.22025 0.30041 0.26760 0.22803 0.27970 0.24316 0.20516 0.28181 0.24788 0.21029 0.28432 0.25408 0.21752

-0.004028 -0.010488 -0.013817 -0.007011 -0.019081 -0.025971 -0.008648 -0.024898 -0.035494 -0.003713 -0.009690 -0.012817 -0.006462 -0.017622 -0.024073 -0.007969 -0.022983 -0.032869

0.13395 0.10127 0.07011 0.13587 0.10514 0.07350 0.13818 0.11039 0.07863 0.12078 0.09100 0.06269 0.12254 0.09453 0.06575 0.12464 0.09930 0.07039

-0.003417 -0.008335 -0.009823 -0.005966 -0.015333 -0.018862 -0.007385 -0.020276 -0.026479 -0.003116 -0.007586 -0.008912 -0.005440 -0.013959 -0.017122 -0.006734 -0.018465 -0.024055

5. Conclusions

Q

x

0.034615 0.027437 0.020351 0.035034 0.028319 0.021197 0.035535 0.029498 0.022437 0.031742 0.025169 0.018677 0.032126 0.025976 0.019452 0.032585 0.027056 0.020589

p ro

Pr e-

m

of

Ha

S

Q

y

-0.000741 -0.001859 -0.002296 -0.001293 -0.003405 -0.004371 -0.001598 -0.004477 -0.006068 -0.000679 -0.001703 -0.002104 -0.001184 -0.003118 -0.004004 -0.001464 -0.004101 -0.005559

    

urn



Both components of the velocity, IMF and ICD decrease due to enhance with the Hartmann number. The increase of the leads to enhance and reduce both parts of the velocity and IMF respectively but the ICD has reverse effect to the velocity and IMF. Both parts of the skin-friction at the surface of walls reduce with the impact of the . The influence of the is to increase the primary part of the skin-friction but it is just opposite for the secondary component of the skin-friction. The increase of the Hartmann number is to minimize both parts of rate of mass flow. The influence of the Hall current is to increase the first component of mass flow rate but it is vice-versa for the secondary components. The impact of the temperature dependent heat source parameter is to rise the primary component and decrease the secondary component of the velocity, decrease both of the parts of the IMF and increase both parts of the ICD while all these fields has opposite effect with the heat sink.

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

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In this analysis, the main impacts of the heat source/sink parameter, Hall current and IMF on MHD natural convection flow in perpendicular channel is investigated. The effects of varying parameters like S, Ha and the mhave been tested on the constituent of the velocity IMF, ICD, skin-friction, and rate of mass flow. The following inferences can be observed from this analysis:

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

The source parameter is to increase the primary parts and decrease the secondary part of skin-friction at both the walls but has opposite nature with heat sink. The primary rate of mass flow increases and the secondary rate of mass flow decrease with the heat source parameter but it the heat sink parameter has opposite impact.

of



Appendix-A: 2

E λ coth λ  1) 12 1 , E  2 , , E  } , c  cot S , c  coth S , E   2 1 2 1 2 3 2 2 2 2(1  m ) λ λ Ha

( 1 m

p ro

λ  α  i β, α, β  Ha{

2 c λ {(cosh λ  cos S )  c sin S} 1 1 1 1 ,E  {E λ , E  E  , E  , 5 6 7 4 2 5 2 (S  λ 2 )sinh λ S λ2 S  λ2 Ha S (S  λ ) 1

1 Ha

2

{E 8 λ 

2 c λ 2 }. 2 S (λ - S)

Pr e-

{(cosh λ  cosh S )  c sinh S} 2 E  , E  8 9 (λ 2 - S)sinh λ

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