Heat Transfer Analysis On MHD Rotating Jeffery Fluid Flow ...iaetsdjaras.org/gallery/3-may-729.pdf · Heat Transfer Analysis On MHD Rotating Jeffery Fluid Flow Past A Vertical Plate

Heat Transfer Analysis On MHD Rotating Jeffery Fluid Flow Past A

Vertical Plate Filled In Porous Medium

NVN Babu1, Murali.G2, S.M.Bhati2

1Department of Engineering Science, Sanjivani College of Engineering, Kopargaon, India. 2Department of Mathematics, Sandip University, India.

Corresponding Email addresses: [email protected]

Abstract This paper presents the study of heat transfer analysis of magnetohydrodynamic free convection flow past a

electrically, incompressible and viscous non-Newtonian Jeffrey fluid flow past a vertical plate in presence of Hall current

and transverse magnetic field. The basic governing non-linear coupled partial differential equations are reduced to linear

partial differential equations with the aid of non-dimensional transformation, which are then solved numerically using an

explicit finite difference scheme. The effects of some of the embedded parameters, such as Grashof number for heat

transfer, Magnetic field parameter, Hall parameter, Permeability parameter, Prandtl number, Eckert number and Ekmann

number on the flow and heat transfer characteristics, are given in forms of graphs. The expressions for Shear stresses and

Nusselt number are also evaluated for different values of emerging parameters. A comparative study with the existing

published work is provided in order to verify the present results. An excellent agreement is found.

Keywords Heat transfer; MHD; Hall Current; Rotation; Jeffrey fluid; Explicit Finite difference method;

Nomenclature:

List of variables:

z Coordinate axis normal to the plate (m )

oH Transverse magnetic field (1mA )

g Acceleration due to gravity (2sm )

x Coordinate axis along the plate (m )

Z Dimensionless coordinate axis normal to the plate (m )

vu , Non-dimensional components of velocities along and perpendicular to the plate respectively, x direction

(m/s)

U Primary velocity component in x direction (m/s)

V Secondary velocity component in y direction (m/s)

T Fluid temperature (K )

K Porosity (permeability) parameter ( 2m )

T Fluid temperature at free stream ( K )

wT Hot fluid temperature at the wall (K )

pC Specific heat at constant pressure ( KKgJ 1)

ow Dimensional suction velocity (1sm )

2M Magnetic field parameter

Pr Prandtl number

Ec Eckert number

IAETSD JOURNAL FOR ADVANCED RESEARCH IN APPLIED SCIENCES

VOLUME 5, ISSUE 5, MAY/2018

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http://iaetsdjaras.org/37

kE Ekmann number

Gr Grashof number for heat transfer

O Origin

uN Rate of heat transfer (or) Nusselt number

k Dimensional Porosity (permeability) parameter ( 2m )

m Hall parameter

t Dimensional time (sec )

H Induced magnetic field (1mA )

J Current density

E Electric field

zyx jjj ,, Components of J

ep Electron pressure

en Number of electrons

e Electron

Greek Symbols

Coefficient of thermal expansion ( 1K )

Fluid density (3mKg )

Angular velocity

Kinematic viscosity (12 sm )

x Shear stress due to primary velocity profiles ( pascal)

y Shear stress due to secondary velocity profiles ( pascal)

Time ( sec )

Dimensionless temperature (K )

Electrical conductivity (1mS )

Thermal conductivity (11 KmW )

Jeffrey fluid parameter

Subscripts

Free stream condition

p Plate

w Wall condition

1. Introduction Due to its great range of applications in various fields, the investigation of convective heat transfer in fluid-saturated

porous media has become a subject of interest, especially in geothermal energy recovery, food processing, fibre and

granular insulation, design of packed bed reactors and dispersion of chemical contaminants in various processes in the

chemical industry and environment1. Comprehensive studies can be found in Vafai2, Nield and Bejan3 and Vadasz4. There

is an abundance of literature available which discusses fluid flow over stretching surfaces in porous medium. Some of

them are Gbadeyan et al.5 who investigated the effects of thermal diffusion and diffusion thermos effects on combined

heat and mass transfer on mixed convection boundary layer flow over a stretching vertical sheet in a porous medium filled

with a viscoelastic fluid in the presence of magnetic field,



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Imran et al.6 studied the analysis of an unsteady mixed convection flow of a fluid saturated porous medium adjacent to

heated/cooled semi-infinite stretching vertical sheet in the presence of heat source and Aly and Ebaid7 investigated the

mixed convection boundary-layer nanofluids flow along an inclined plate embedded in a porous medium using both

analytical and numerical approaches. Dessie and Kishan8 examined the MHD boundary layer flow and heat transfer of a

fluid with variable viscosity through a porous medium towards a stretching sheet along with viscous dissipation and heat

source/sink effects. Narayana9 carried out a study on the effects of radiation and first-order chemical reaction on unsteady

mixed convection flow of a viscous incompressible electrically conducting fluid through a porous medium of variable

permeability between two long vertical non conducting wavy channels in the presence of heat generation, and to name a

few. Mansur and Ishak10 studied numerically magnetohydrodynamic (MHD) boundary layer flow of a nanofluid past a

stretching/shrinking sheet with velocity, thermal, and solutal slip boundary conditions. Siddheshwar and Mahabaleshwar11

examined analytically MHD flow of micropolar fluid over linear stretching sheet using regular perturbation technique and

Ahmed et al.12 applied the successive linearization method to study the effects of radiation and viscous dissipation on

MHD boundary layer convective heat transfer with low pressure gradient in porous media.

The study of rotating fluids has gained much interest nowadays since it has been encountered in many important

problems such as cosmic and geophysical flows. The effect of Coriolis force helps us to understand the phenomena of

earth’s rotation, the behaviour of ocean circulation and galaxies formation better. Therefore, many mathematical models

including numerical and analytical studies are presented to study the effect of Coriolis force on the fluid flows. Saleem et

al.13 analyzed buoyancy and metallic particle effects on unsteady water-based fluid flow along a vertically rotating cone with

the help of Runge-Kutta-Fehlberg method. Hosseini et al.14 used Homotopy perturbation method to investigate nanofluid

flow and heat transfer between two horizontal plates in a rotating system. Sheikholeslami et al.15 extended the idea of14, by

taking into account the magnetohydrodynamic (MHD) effect and solved the problem numerically using fourth-order

Runge-Kutta method. Besides that, Hussain et al.16 examined the flow and heat transfer effects of both single and multiple

wall carbon nanotubes within the base fluid (water) in a rotating channel. In these papers, the authors have used the

nanofluid model proposed by Tiwari and Das17. After that, Sheikholeslami and Ganji18 extended the previous work of15

and analyzed three dimensional nanofluid flow and heat transfer in a rotating system in the presence of magnetic field

using Buongiorno’s model19. Very recently, Khan et al.20 studied numerically three dimensional squeezing flow of

nanofluid in a rotating channel with lower stretching wall suspended by carbon nanotubes with the aid of Runge-Kutta-

Fehlberg method. Few other attempts in this direction are those made by Nadeem et al.21, Sree Ranga Vani and Prasada

Rao22, Das23, Raza et al.24, Mohyud-Din et al.25 and Satya Narayana et al.26.

Materials that do not obey the Newtonian law of viscosity are non-Newtonian fluids such as apple sauce, drilling muds,

certain oils, ketchup and colloidal and suspension solution. The study of non-Newtonian fluids has gained interest because

of their extensive industrial and technological applications. However, the Navier Stokes equations are no longer valid to

precisely describe the rheological behaviour of all non-Newtonian fluids. In view of their differences with Newtonian

fluids, several models of non-Newtonian fluids have been proposed. The most common and simplest model of non-

Newtonian fluids is Jeffrey fluid which has time derivative instead of convicted derivative27. Hussain et al.28 developed a

model to examine the radiative hydromagnetic flow of Jeffrey nanofluid by an exponentially stretching sheet. Hayat et al.29

investigated analytically three dimensional flow of Jeffrey nanofluid with a new mass flux condition using Homotopy

analysis method. Later on, Shehzad et al.30, Hayat et al.31, Dalir et al.32 and Naramgari et al.33 studied a magnetic field effect

on the flow of Jeffrey nanofluid under various aspects. An analysis of the boundary layer flow and heat transfer in a Jeffrey

fluid containing nanoparticles due to a stretchable cylinder was reported by Hayat et al.34. Nadeem and Saleem35 presented

the unsteady mixed convection flow on a rotating cone in a rotating Jeffrey nanofluid and solved analytically with the help

of optimal homotopy analysis method. Very recently, Raju et al.36 investigated numerically the influence of thermal

radiation and chemical reaction on the boundary layer flow of a magnetohydrodynamic Jeffrey nanofluid over a permeable

cone in the presence of thermophoresis and Brownian effects using Runge-Kutta fourth order with shooting technique.



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Jeffrey fluid is a type of non-Newtonian fluid that uses a relatively simpler linear model using time derivatives instead

of convicted derivatives, which are used by most fluid models. Recently, this model of fluid has prompted active

discussion. Some of the studies can be found in Shehzad et al.37, Nallapu and Radhakrishnamacharya38, Ahmad and Ishak39

and Prasad et al.40. In view of the above discussions, the aim of this paper is to investigate the effect of Hall current on an

unsteady MHD free convection flow of a viscous, incompressible, electrically conducting Jeffrey fluid past an vertical plate

in presence of porous medium, heat transfer and rotation. The model of the Jeffrey fluid flow is presented mathematically

and has been solved numerically using an explicit finite difference scheme.

2. Mathematical Formulation

An unsteady MHD free convective flow of an electrically conducting incompressible viscous fluid past an infinite

vertical porous plate with the effects of Hall current is considered. Let the fluid rotate with uniform angular velocity Ω

about the z - axis normal to the plate. It is assumed that there is a constant suction velocity. The flow is also assumed to be

in the x - axis that is taken along the plate in the upward direction and z - axis is normal to it. At time t > 0 , the

temperature at the plate is constantly raised from Tw to T∞ which is thereafter maintained constant. Where Tw and T∞ are

the temperatures at the wall and outside the plate respectively. A uniform magnetic field Ho is imposed along the z - axis

and the plate is taken to be electrically non-conducting. It is assumed that the induced magnetic field is negligible so that H

= (0, 0, Ho). This assumption is justified when the magnetic Reynolds number is very small. The equation of conservation

of electric charge ∇. J = 0 gives jz = constant, where J = (jx, jy, jz). This constant is assumed to be zero at the non

conducting plate, therefore jz = 0 everywhere in the flow. Since the plate is infinite in extent, all the physical variables

except pressure depend on z and t only. Hence the equation of continuity ∇. q = 0 gives w = − wo (> 0) , where q = (u, v, w)

. The generalized Ohm’s law including the effect of Hall current (Cowling) is

e

e

e

o

ee pen

HqEHJH

J .1

(1)

It has been assumed that the ion slip and thermoelectric effect is negligible. Further it is considered that the electric

field E = 0 and electron pressure have

been neglected. Under this assumption equation (1) gives

umvm

Hjmuv

m

Hj oe

yoe

x

22 1

&1

(2)

The Cauchy stress tensor, S , of a Jeffrey’s non-Newtonian fluid41 takes the form as follows:

1

1S

(3)

where is the dynamic viscosity, 1 is the ratio of relaxation to retardation times, dot above a quantity denotes the

material time derivative and is the shear rate. The Jeffrey’s model provides an elegant formulation for simulating

retardation and relaxation effects arising in non-Newtonian polymer flows. The shear rate and gradient of shear rate are

further defined in terms of velocity vector, V , as follows:

where TVV

(4)



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and

.Vdt

d

(5)

Thus, accordance with the above assumptions relevant to the problem and under the electromagnetic Boussinesq's

approximation, in a rotating frame the

basic boundary layer equations43 are given by

Momentum Equation:

uk

umvm

HvTTg

z

u

z

uw

t

u oeo

2

2

2

2

12

1

1

(6)

vk

muvm

Hu

z

v

z

vw

t

v oeo

2

2

2

2

12

(7)

Equation of Conservation of Energy:

22

2

2

z

v

z

u

Cy

T

Cy

Tw

t

T

pp

o

(8)

The corresponding boundary conditions are

zasTTvu

zatTTvwut

zallforTTvwut

wo

o

,0,0

0,0,:0

,0,:0

(9)

To obtain the governing equations and the boundary conditions in dimensionless form, the following non-dimensional

quantities are introduced as

TTC

wEc

C

kwK

w

HM

w

TTgGr

wE

TT

TTtw

w

vV

w

uU

zwZ

wp

op

o

o

oe

o

w

o

k

w

o

oo

o

2

2

2

22

32

2

,Pr

,,,,,,,,,

(10)

Substituting the above relations in equations (6)-(8) and the boundary conditions (9) are

Momentum Equation:

K

UUmV

m

MVEGr

Z

U

Z

UUk

2

2

2

2

12

1

1

(11)

K

VmUV

m

MUE

Z

V

Z

VVk

2

2

2

2

12

(12)

Equation of Conservation of Energy:

22

2

2

Pr

1

Z

V

Z

UEc

ZZ

(13)

The corresponding boundary conditions are



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ZasVU

ZatVU

ZallforVU

0,0,0

01,0,1:0

0,0,0:0

(14)

3. Shear Stresses And Nusselt Number

From the velocity field, the effects of various parameters on the plate shear stresses have been investigated. The

primary shear stress is in the x-direction 0

z

xz

u and the secondary shear stress is in the y-direction,

0

z

yz

v which are proportional to

0

ZZ

Uand

0

ZZ

Vrespectively. From the temperature field, the effects

of various parameters on heat transfer coefficient (Nusselt number) have been calculated. Nusselt number,

0

z

uz

TN which is proportional to

0

ZZ

.

4. Method Of Solution By Explicit Finite Difference Method

In this section, the governing second order non-linear coupled dimensionless partial differential equations with initial and

boundary conditions have been solved. The explicit finite difference method has been used to solve equations (11) to (13)

subject to the boundary conditions (14). The region within the boundary layer is divided by some perpendicular line of Z-

axis, where Z-axis is normal to the medium as shown in the Fig. 1. It is assumed that the maximum length of the boundary

layer is Zmax = 1 i. e. Z varies from 0 to ∞ and the number of grid spacing in Z direction is n = 50. Hence the constant

mesh size along Z-axis becomes ΔZ = 0.001 (0 ≤ Z < ∞)with smaller time Δτ = 0.00001. Let U', V' and θ' denote the

values U, V and θ at the end of a time step respectively. Using the explicit finite difference method the system of partial

differential equations (11) to (14) is obtained an appropriate set of finite difference equations.

K

UUmV

m

MVEGr

Z

UUU

Z

UUUU nin

ini

nik

ni

ni

ni

ni

ni

ni

ni

ni

2

2

2111

1

12

2

1

1

(15)

K

VmUV

m

MUE

Z

VVV

Z

VVVV nin

ini

nik

ni

ni

ni

ni

ni

ni

ni

2

2

2111

1

12

2

(16)

2

1

2

12

1111 2

Pr

1

Z

VV

Z

UUEc

ZZ

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

(17)

and the initial and boundary conditions with the finite difference scheme are

LasVU

LatVU

LallforVU

nL

nL

nL

nL

nL

nL

nL

nL

nL

0,0,0

01,0,1:0

0,0,0:0

(18)



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Fig. 1. Finite Difference Space Grid

5. Comparison With Previous Research Work

Table-1. Comparison of present results with publsiehd results of Guchhait et al.42 and Abdul and Alam43

Increased Parameter Previous results given by Guchhait et al.42

U V θ

m Increasing Decreasing No Effect

M 2 Decreasing Increasing No Effect

Gr Increasing Decreasing No Effect

Pr Decreasing Decreasing Decreasing

Increased Parameter Previous results given by Abdul and Alam et al.43

U V θ

m Increasing Minor Decreasing No Effect

M 2 Decreasing Increasing No Effect

Gr Minor Increasing Minor Decreasing No Effect

Pr Decreasing Decreasing Decreasing

Increased Parameter Present results

U V θ

m Minor Increasing Decreasing No Effect

M 2 Decreasing Minor Increasing No Effect

Gr Increasing Minor Decreasing No Effect

Pr Decreasing Minor Decreasing Decreasing

Finally, a qualitative comparison of the present results with the published results of Guchhait et al.42 and Abdul

and Alam43 in absence of jeffrey fluid is presented in the following table. The present results are qualitatively as well as

quantitatively quite different in case of some flow parameters.



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6. Results And Discussions

In the present study we adopted the following default parameter values of explicit finite difference computations:

Gr = 1.0, Pr = 0.71, M 2 = 1.0, K = 1.0, m = 1.0, Ec = 0.001, Ek = 1.0, λ = 0.5 and τ = 1.0. The effect of

Grashof numbers for heat transfer on primary velocity profiles is illustrated in Fig. 2 (a). The Grashof number for heat

transfer signifies the relative effect of the thermal buoyancy force to the viscous hydrodynamic force in the boundary layer.

As expected, it is observed that there was a rise in the primary velocity due to the enhancement of thermal buoyancy force.

Also, as Gr increases, the peak values of the primary velocity increases rapidly near the porous plate and then decays

smoothly to the free stream velocity.

(a)

(b)



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(c)

Fig. 2. Primary velocity profiles

Fig. 2 (a) display the effect of magnetic field parameter or Hartmann number (M 2) on primary velocity profiles. It is

seen from this figure that, the primary velocity falls when M 2 increases. That is the primary velocity fluid motion is

retarded due to application of transverse magnetic field. This phenomenon clearly agrees to the fact that Lorentz force that

appears due to interaction of the magnetic field and fluid velocity resists the fluid motion. The influence of hall parameter

m on primary velocity profiles is as shown in the Fig. 2 (a). It is observed from this figure that, the primary velocity profiles

increase with an increase in hall parameter m . This is because, in general, the Hall currents reduce the resistance offered by

the Lorentz force. This means that Hall currents have a tendency to increase the fluid primary velocity component. In Fig.

2 (b), it has shown that the primary velocity profile decreases drastically for the increase of Prandtl number (Pr), while the

opposite nature of primary velocity profiles is observed for the increase of K which is shown in Fig. 2 (b). The primary

velocity increases with the increase of Eckert number (Ec) which has been observed in Fig. 2 (b). The primary velocity

profiles in the Fig. 2 (c) show that rate of motion is significantly reduced with increasing of Jeffrey fluid parameter. In Fig.

2 (c), it has shown that the primary velocity profile decreases drastically for the increase of Ekmann number (Ek). The

influence of heat transfer coefficient (Grashof numbers for heat transfer) on secondary velocity profiles is as shown in Fig.

3 (a). As heat transfer coefficient increases, this secondary velocity component increases as well. From Fig. 3 (a) depicts the

effect of magnetic field parameter or Hartmann number (M 2) on the secondary velocity of the flow field. Here, the

secondary velocity profiles are drawn against Z for three different values of M 2. The Hartmann number is found to

accelerate the secondary velocity of the flow field at all points.



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(a)

(b)



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(c)

Fig. 3. Secondary velocity profiles

The effect of hall parameter m on the secondary velocity flow field is presented in the Fig. 3 (a). Here, the secondary

velocity profiles are drawn against Z for three different values of m. The hall parameter is found to decreases the secondary

velocity of the flow field at all points. The effect of Eckert number on the secondary velocity flow field is presented in Fig.

3 (b). Here, the secondary velocity profiles are drawn against Z for three different values of Ec. The Eckert number is

found to accelerate the secondary velocity of the flow field at all points. It is observed in Fig. 3 (b) that the secondary

velocity profiles have a decreasing effect with the increase of Prandtl number (Pr) and having increasing effect in case of

permeability parameter (K). The secondary velocity profiles in the Fig. 3 (c) show that rate of motion is significantly

reduced with increasing of Jeffrey fluid parameter. Also, it is observed from this figure, the boundary layer momentum

thickness decreases as increase of Jeffrey fluid parameter. It has been shown in Fig. 3 (c) that the increasing values of the

Ekmann number (Ek) decreases the secondary velocity profiles. From Fig. 4 it is shown that the increasing values of the

Eckert number (Ec) increases the temperature distribution. Fig. 4 depicts the effect of Prandtl number on the temperature

field. It is observed that an increase in the Prandtl number leads to decrease in the temperature field. Also, temperature

field falls more rapidly for water in comparison to air and the temperature curve is exactly linear for mercury, which is

more sensible towards change in temperature. From this observation it is conclude that mercury is most effective for

maintaining temperature differences and can be used efficiently in the laboratory. Air can replace mercury, the

effectiveness of maintaining temperature changes are much less than mercury. However, air can be better and cheap

replacement for industrial purpose. This is because, either increase of kinematic viscosity or decrease of thermal

conductivity leads to increase in the value of Prandtl number. Hence temperature decreases with increasing of Prandtl

number. The numerical values of shear stresses and Nusselt number are presented in table 2 for variations of Grashof

number for heat transfer, Magnetic field parameter, Prandtl number, Jeffrey fluid parameter, Eckert number, Ekmann

number, Permeability parameter and Hall parameter. From this table, it is observed that the shear stress due to primary

velocity profiles are increasing with Grashof number for heat transfer, Eckert number, Permeability parameter and Hall

parameter and reverse effect is observed that with increasing of Magnetic field parameter, Prandtl number, Jeffrey fluid

parameter, and Ekmann number. From this same table, it is observed that, the shear stress due to secondary velocity

profiles are increasing with Grashof number for heat transfer, Magnetic field parameter, Eckert number, Permeability

parameter and Hall parameter and decreasing with increasing of Prandtl number, Jeffrey fluid parameter, and Ekmann

number. In table 2, it is observed that, the rate of heat transfer coefficient or Nusselt number due to temperature profiles

is increasing with Eckert number and decreasing with increasing values of Prandtl number.



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Also, it is observed that, Nusselt number has no effect for variations of remaining parameters Grashof number for heat

transfer, Magnetic field parameter, Jeffrey fluid parameter, Ekmann number, Permeability parameter and Hall parameter.

Fig. 4. Temperature profiles

Table-2. Numerical values of Shear stresses and Nusselt number for variations of Gr, M 2, Pr, λ, Ec, Ek, K and m

Gr M 2 Pr λ Ec Ek K m x y uN

1.0 1.0 0.71 0.5 0.001 1.0 1.0 1.0 0.8852165926 - 0.0522186296 0.6511893265

2.0 1.0 0.71 0.5 0.001 1.0 1.0 1.0 1.0233154896 - 0.0211659864 0.6511893265

1.0 2.0 0.71 0.5 0.001 1.0 1.0 1.0 0.7511249621 - 0.0422018517 0.6511893265

1.0 1.0 7.00 0.5 0.001 1.0 1.0 1.0 0.8011246921 - 0.0711589215 0.5612059845

1.0 1.0 0.71 1.0 0.001 1.0 1.0 1.0 0.7930154332 - 0.0622183047 0.6511893265

1.0 1.0 0.71 0.5 1.000 1.0 1.0 1.0 0.9022175186 - 0.0412065841 0.6721589215

1.0 1.0 0.71 0.5 0.001 2.0 1.0 1.0 0.8115244762 - 0.0622159324 0.6511893265

1.0 1.0 0.71 0.5 0.001 1.0 2.0 1.0 0.9150036485 - 0.0322156482 0.6511893265

1.0 1.0 0.71 0.5 0.001 1.0 1.0 2.0 0.9231665842 - 0.0412558921 0.6511893265

7. Conclusions

In this study, the numerical solutions using explicit finite difference scheme of unsteady MHD free convective

Jeffrey fluid flow through a porous vertical plate in presence of Hall current in a rotating system is investigated. The

obtained results are graphically presented for the variations of associated parameters. Some of the important findings of

this observation are given below.

1. The primary velocity increases with the increase of Grashof number for heat transfer, Eckert number, Permeability

parameter, Hall parameter and

while it decreases with the increase of Magnetic field parameter, Prandtl number, Jeffrey fluid parameter and Ekmann

number.

2. The secondary velocity increases with the increase of Grashof number for heat transfer, Magnetic field parameter, Eckert

number, Permeability parameter, Hall parameter and while it decreases with the increase of Prandtl number, Jeffrey fluid

parameter and Ekmann number.

3. The fluid temperature is increasingly affected by Eckert number and decreasingly affected by Prandtl number.



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4. The shear stress due to primary velocity profiles increases with the increase of Grashof number for heat transfer, Eckert

number, Permeability parameter, Hall parameter and while it decreases with the increase of Magnetic field parameter,

Prandtl number, Jeffrey fluid parameter and Ekmann number.

5. The shear stress due to secondary velocity increases with the increase of Grashof number for heat transfer, Magnetic field

parameter, Eckert number, Permeability parameter, Hall parameter and while it decreases with the increase of Prandtl

number, Jeffrey fluid parameter and Ekmann number.

6. The rate of heat transfer coefficient due to temperature profiles is increasing with increasing values of Eckert number and

decreasing with increasing values of Prandtl number.

7. Furthermore, comparisons are found to be good with available bench mark solutions.

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