Heat and mass t ransfer on MHD flow of Non -Newtonian ... · 1 Heat and mass t ransfer on MHD flow of Non -Newtonian fluid over an infiniteverticalporous plate with Hall effects z

1

Heat and mass transfer on MHD flow ofNon-Newtonian

fluid over an infiniteverticalporous plate with Hall effects

z

D.Dastagiri Babu1, S. Venkateswarlu2, E.Keshava Reddy3 1Research Scholar of JNTUA Anantapuramu,A.P. India

2Dept. of Mathematics, RGM College of Engg. and Technology, Nandyal,Kurnool, A.P, India 3Dept.of Mathematics,JNTUA College of Engg.Anantapuramu, A.P. India

1corresponding author:[email protected] [email protected]

[email protected]

Abstract

We have considered the boundary-layer flow of a heat-absorbing MHD

non-Newtonian fluid along a semi-infinite vertical porous moving plate in the

presence of thermal buoyancy effect and taking hall current effects into account.

The dimensionless governing equations are solved analytically using two-term

harmonic and non-harmonic functions. Computational analysis of the results is

presented with a view to revealfor the velocity, temperature and concentration

profiles within the boundary layer. The Skin friction, Nusselt number and

Sherwood number are also examined with the reference to governing

parameters.

Keywords:MHD flows,Non-Newtonian flow, Porous mediumandunsteady flows

Nomenclature:

In this paperx, z and t are the dimensional distance along and

perpendicular to the plate and dimensional time, respectively u and v are the

components of dimensional velocities along x and z directions respectively is

the fluid density, is the kinematic velocity, pC is the specific heat at constant

pressure, is the fluid electrical conductivity, 0B is the magnetic induction, k is

the permeability of the porous medium. T is the dimensional temperature, 0Q is

the dimensional heat observation co-efficient, is the thermal diffusivity, g is

the gravitational acceleration and T is the coefficient of volumetric thermal

expansion, C isvolumetric expansion coefficient for concentration and 1 is the

kinematic visco-elasticity. pU , wT are the wall dimensional velocity temperature,

respectively ,U T are the free steam dimensional velocity, temperature

respectively, 0 andu are constants. 0

2

kwK

v is the permeability of the porous

medium, PrpC

k

is the prandtl number,

22 0

2

0

BM

w

is the magnetic field

International Journal of Pure and Applied MathematicsVolume 119 No. 15 2018, 87-103ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/

87

2

parameter,

3

0

GrT g T T

w

is the grashof number, Sc

v

D is the Schmidth

number, 2

1 0

2

wRm

is the dimensionless form visco-elasticity parameter of the

Rivlin-Ericksen Fluid.

1. Introduction: The phenomenon of flows through porous medium has been a subject of

interest of many researchers because of its wide range of application in different

fields such as petroleum engineering, chemical engineering etc. In petroleum

engineering, it is dealt with the movement of natural gas and oil through

reservoirs. Further, the study on underground water resources, seepage of water

in river bed is also related to the flow through porous medium. The free

convection flow past vertical plate in the presence of viscous dissipates heat

using the perturbation method was discussed by Gupta et al. [1]. Kafousias and

Raptis [2] extended the work with mass transfer effects. Chambre and Young [3]

analysed the problem of the first-order chemical reactions over a horizontal

plate. Soundalgekar et al. [4] discussed the mass transfer effects on the flow past

a vertical plate impulsively started with variable temperature and constant heat

flux. Elbashbeshy [5] discussed the effects of a magnetic field in mass transfer

along a vertical plate. The thermal, mass diffusion, magnetic field and hall

current effects were discussed by Takhar et.al [6] . Investigation of the

significance of step change in wall temperature is meaningful and it was done in

the fabrication of thin-film photovoltaic devices by Chandran et.al[7]

.Muthucumaraswamy and Muralidharan [8] discussed the effects of thermal

radiation on a linearly accelerated vertical plate with variable temperature and

uniform mass flux. Double dispersion convection effects on the combined heat

and mass transfer in a non-Newtonian fluid-saturated porous medium has been

discussed by Kari and Murthy [9]. Free convective power-law fluid flow past a

vertical plate in a non-Darcian porous medium in the presence of a homogeneous

chemical reaction was studied by Chamkha et. al [10]. The internal heat

generation or absorption is important in problems involving chemical reactions

where heat may be generated or absorbed in the course of such reactions and it

was investigated by Rao et. al [11]. Rajput and Kumar [12] investigated

radiation effects on magnetic field flow of an impulsively started vertical plate

with non-uniform heat and mass transfer. Siva raj and Rushi Kumar [13]

visualized the flow of viscoelastic fluid over a moving vertical plate and flat plate

with variable electric conductivity. Fluid flows past an infinite plate are of much

importance due to its large practical applications, such as motions due to wall

shear stress and it was studied by Seth et. al [14]. Several authors have carried

out their research works for the importance of the chemical reaction effects, on

some mass transfer flow problems. Poornima et. al [15] studied the thermal

radiation and chemical reaction effects on free Convective flow past a Semi-

infinite vertical porous moving plate in the presence of magnetic field. Okedoye

[16] reported the non-uniform heat source/sink in controlling the heat transfer in

the boundary layer region. Rashad et al. [17] have studied the viscous dissipation

effects on the free convective heat transfer of nanofluids. Singh and Makinde [18]

International Journal of Pure and Applied Mathematics Special Issue

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analysed the combined convection slip flow with temperature variation along a

moving plate in free stream. Sheikholeslami and Ganji [19] discussed heat

Transfer effects in Nano fluid flow between parallel plates.Recently, Krishna et

al. [20-23] discussed the MHD flows of an incompressible and electrically

conducting fluid in planar channel.

Motivated by the above studies, in this paper we have consideredMHD

flow of non-Newtonian (Rivlin-Ericksen) fluid past a semi-infinite moving porous

plate.

2.Formulation and Solution of the problem:

We consider the MHD flow of anelectrically conducting and heat absorbing Non-

Newtonian fluid (Rivlin-Ericksen type)over a semi-infinite vertical permeable

moving plate embedded in a porous medium with a uniform transverse magnetic

field and taking hall currents into account(Fig.1).It is assumed that there is no

applied voltage which implies the absence of an electrical field the transversely

applied magnetic field and magnetic Reynolds number are assumed to be very

small to that the induced magnetic field and the hall effects are negligible.A

consequence of the small magnetic Reynolds number is a the uncoupling of the

Navier-stokes equations from Maxwell’s equations the governing equations for

this investigation are based on the balances of mass,linear momentum made

above,these equations can be written in Cartesian frame of reference as follows:

x g

Momentum boundary layer

Thermal boundary layer

Porous medium

0 1 ntw w Ae Concentration boundary layer

0

z

Fig. 1 Physical configuration of the Problem

0w

z

(1)

2 3 3

1 02 2 3

1y T C

u u p u u uw v B J u T T C C

t z x z t z z k

(2)

2 3 3

1 02 2 3

1x

v v v v v+w = B J v

t z y z z t z k

(3)


89

4

2

0

2

p

QT T T+w T T

t z z C

(4)

2

2

C CD

t z

(5)

The magnetic and viscous dissipations are neglected in this study. It is

assumed that the permeable plate moves with aconstant velocity in the direction

of fluid flow and the free steam velocity follows the exponentially increasing

small perturbation law. Inaddition, it is assumed that the temperature at the

wall as well as the suction velocity is exponentially varying with time. Under

then assumptions the appropriate conditions for the velocity, temperature fields

are

0 at 0nt nt

p w w w wu U , v , T T T T e ,C C C C e z ,

, 0 , ,0 1 , asntu U v u e T T C C z (6) It is clear

from equation(1) that the suction velocity at the plate surface is a function of

time only assuming that it takes the following exponential form

0 1 ntw w Ae (7)

When the strength of the magnetic field is very large, the generalized

Ohm’s law is modified to include the hall current so that

1e e

e

O e

J J B E V B PB e

(8)

The ion-slip and thermo electric effects are not included. Further it is

assumed that ee ~ 0 (1) and ,1ii where i and i are the cyclotron

frequency and collision time for ions respectively. In the equation (8) the electron

pressure gradient, the ion-slip and thermo-electric effects are neglected. We also

assume that the electric field E=0 under assumptions reduces to

x y 0J m J σB v

(9)

y x 0J m J σB u

(10)

Where e em is the Hall parameter.

On solving equations (9) and (10) we obtain,

( )1

0x 2

σBJ v mu

m

(11)

( )1

0y 2

σBJ mv u

m

(12)

Substituting the equations (11) and (12) in (3) and (2) respectively, we obtain

22 3 3

012 2 3 2

1( )

1

T C

σBu u p u u uw v mv u u

t z x z t z z m k

T T C C

(13)

22 3 3

01 22 2 3

1( )

1

σBv v v v vv mu+w = v

mt z y z z t z k

(14)


90

5

Combining equations (13) and (14), let andq u iv x iy , weobtain,

22 3 3

01 22 2 3

1

1( )

C

Bq q p q q q vw q

mt z z z t z k

g g C C

(15)

Where A is a real positive constants, and are small less than unity, and 0w is

a scale of suction velocity which has non-zero positive constant outside the

boundary layer equation(2) gives

2

0

1 Up= U B U

t k

(16)

We introduced the following non-dimensional variables 2

* * * * * 10 0

0 0 0 0

, , , , , , ,p

p

w w

uw z twU C Cu vu v z U U t

U w v U U C C

Making use of non-dimensional variables the governing equations reduces

to (Droppingaskerisks) .

2 2 3

2 2 2

3

3

11 Gr Gm

1

1

nt

nt

Uq q q M ue U q Rm

t z t z m K t z

qe

z

(17)

2

2

11

Pr

nte = Qt z z

(18)

2

2

11

Sc

ntet z z

(19)

The dimensionless form boundary conditions equations become

1 1 1 at 0

0 0 0 at

nt nt nt

pq U , e , e ,U e z

q U , , ,U z

(20)

Equation (10) and (11) represent a set of partial differential equations that

cannot be solved in closed form however if can be reduced to a setoff ordinary

differential equations in dimensionless form that can be solved analytically this

can be done representing the velocity and temperature as,

2

0 1

ntq q z e q z O (21) 2

0 1

ntz e z O (22)

2

0 1

ntz e z O (23)

Substituting the equations (21), (22) and (23) into equation (17), (18) and

(19),equating the harmonic and non-harmonic terms, the neglecting and higher

order terms of 2O ,one obtains the following pairs of equations

0 0 0 1 1 1, , and , ,q q ,

3 2 2 2

0 0 00 03 2 2 2

1 1Gr

1 1

d q d q dq M MRm q

dz dz dz m K m K

(24)


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6

3 2 2

1 1 11 13 2 2

32

0 01 2 3

11

1

1

1

d q d q dq MRm nRm q nq

dz dz dz m K

d q dqMn Gr RmA A

m K dz dz

(25)

2

0 002

Pr Pr 0d d

Qd z dz

(26)

2

01 11 12

Pr Pr Pr Prdd d

n Q Ad z dz dz

(27)

2

0 0

2Sc 0

d d

d z dz

(28)

2

01 112

Sc Sc Scdd d

n Adz dz dz

(29)

The corresponding boundary conditions can be written as

0 1 0 1 0 10 1 1 1 1 at 0pq U ,q , , , , z

0 1 0 1 0 11 1 0 0 0 0 atq , q , , , , z (30)

Without going into detail,the solution of equation (24)-(29) subject to

conditions(30) can be show to be 1

0

m zq e

(31)

3 1

1 2 1

m z m zq a e a e

(32)

Equations (24) and (25) are third degree order differential equations when

0Rm and we have two boundary conditions so we follows bears and Walter’s as

2

0 01 02q q Rmq O Rm (33)

2

1 11 12q q Rmq O Rm (34)

Substituting equations (33) and (34) into (24) and (25), equating different powers

of Rm and neglecting 2O Rm are,

1 1

2 2 2

01 01012 2 2

1 1Gr Gm

1 1

m z n zd q dq M Mq e e

dz dz m K m K

(35)

3 2 2

01 02 02023 2 2

10

1

d q d q dq Mq

z dz dz m K

(36)

31

31

2 2

11 1111 1 22 2

2

011 2 2

1Gr

1

1Gm

1

m zm z

n zn z

d q dq Mn q n a e a e

dz dz m K

dqMb e b e A

m K dz

(37)

33 2 2 2

01 0211 12 11 12123 2 2 2 3

1

1

d q dqd q d q d q dq Mn n q

dz dz dz dz m K dz dz

(38)

The corresponding boundary conditions are

01 02 11 120 0 0 at 0pq u ,q ,q ,q z

01 02 11 121 0 1 0 asq ,q ,q ,q , z (39)

We get zeroth order and first order solutions of Rm


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7

6 61 1

01 4 3 4 3 1m z m zm z n z

q a e a e b e b e

(40)

8 6 8 6 1 1

02 7 5 7 5 6 6

m z m z m z m z m z n zq a e a e b e b e a e b e

(41)

10 3 3 61 1

11 16 9 15 5 10 12 14

m z m z n z m zm z n zq a e a e a e b e b e a e a

(42)

6 8 10 312 1

12 27 22 23 24 25 26

m z m z m z m zm z m zq a e a e a e a e a e a e

(43)

In view of the above solutions, the velocity and temperature

distributions in the boundary layer become

0 1, ntq z t q z e q z

6 61 1

8 6 8 6 1 1

10 3 3 61 1

6 812 1 1

4 3 4 3

7 5 7 5 6 6

16 9 15 8 12 12 14

27 22 23 16 24

1m z m zm z n z

m z m z m z m z m z n z

m z m z n z m zm z n znt

m z m zm z m z n z

a e a e b e b e

Rm a e a e b e b e a e b e

e a e a e a e b e b e a e a

Rm a e a e a e b e a e

10 3 3

25 26 17

m z m z n za e a e b e

0 1, ntz t z e z

31 1

2 1

m zm z m znte e a e a e (44)

0 1, ntz t z e z

31 1

2 1

n zn z n znte e b e be (45)

The skin friction co-efficient, Nusselt number and Sherwood number are

important physical parameters for this type of boundarylayer flow.These

parameters can be defined and determined as follows.

28 29 30 31

0

nt

z

qa Rma e a Rma

z

(46)

32 33

0

nt

z

Nu a e az

(47)

33 35

0

nt

z

Sh a e az

(48)

3.Results and Discussion:

We noticed that, the velocity component u and v reduces with increasing

the intensity of the magnetic field or Hartmann number M. The similar

behaviour is observed for the resultant velocity Figs.(2). It is obvious that the

effect of increasing values of the magnetic field parameter M results in a

decreasing velocity components u and v across the boundary layer. Figs.(3)

depicts the effect of permeability of the porous medium parameter (K) on velocity

distribution profiles for u and v and it is obvious that as permeability parameter

(K) increases, the velocity components for u and v increases along the boundary

layer thickness which is expected since when the holes of porous medium become

larger, the resistive of the medium may be neglected. Similar behaviour is

observed with increasing permeability parameter K for the resultant velocity.

Figs. (4)illustrate the variation in velocity components u and v with span wise

coordinate n for several values of Rm. We observed that both u reduces and v

increases with increasing visco-elastic fluid parameter of the Rivlin-Ericksen

fluid Rm. It was found that an increase in Rm leads to a decrease in the

resultant velocity distribution across the boundary layer. Figs.(5) illustrate the


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velocity profiles for u and v for different values of the Grashof number Gr. It can

be seen that an increase in Gr leads to a rise in velocity u and v profiles. The

resultant velocity is also enhances throughout the fluid region with increasing

the thermal Grashof number Gr. We noticed that from the Figs. (6), the

magnitude of the velocity component u reduces and v enhances with increasing

Suction parameter A throughout the fluid region. The resultant velocity is

reduces with increasing Suction parameter A. Figs. (7) presents the velocity

distribution profiles for different values of the Prandtl number (Pr). The results

show that the effect of increasing values of the prandtl number results in

anincrease in u and in a decrease in the velocity componentv. The resultant

velocity is also reduces with Prandtl number Pr.Figs. (8) shows the velocity

profiles for u and v for different values of dimensionless heat absorption

coefficient Q. Clearly as Q increase the pack values of velocity components u

increase and v tends to decrease. The resultant velocity reduces with increasing

heat absorption coefficient Q, physically the presence of heat absorption

coefficient has the tendency to reduce the fluid temperature. This causes the

thermal buoyancy effects to decrease resulting in a net reduction in the fluid

velocity.The Figs. (9) shows the velocity profiles for u and v against span wise

direction for different values of the scalar constant . It was found that an

increase in the value of leads to an increase in the resultant velocity

distribution across the boundary layer. The velocity component u diminishes first

and then experiences enhancement andwhereasv increases with increasing the

scalar constant . Figs. (10) Depicts the effect of the frequency of oscillation n on

the velocity distribution. The primary velocity component u and secondary

velocity v reducewith increasing the frequency of oscillation n . The resultant

velocity reduces with increasing the frequency of oscillation throughout the fluid

medium. Finally, the

Figs. (11) Shows the velocity profiles for u and v against hall parameter m. It

was found that an increase in the value of m leads to an increase in the

resultant velocity distribution across the boundary layer. The velocity component

u and v enhances with increasing the hall parameter m.

We noticed that from the Figs. (12), the temperature reduces with increasing

Prandtl number Pr or the frequency of oscillation n or suction velocity A or heat

absorption coefficient Q.The result display that an increase in the value of Q

results in decrease in the temperature profiles as expected. The temperature

profiles with span wise coordinate n for various scalar constant . The numerical

results show that the effect of increase value of e results in an increase thermal

boundary layer thickness and more uniform temperature distribution across the

boundary layer. Similar behaviour is observed in entire fluid region with

increasing time.

We noticed from the Figs.(13), the temperature reduces with increasing

Schmidt number Sc or the frequency of oscillation n or suction velocity A. The

Concentration profiles with span wise coordinate n for various scalar constant .

The numerical results show that the effect of increase value of results in an

increase concentration boundary layer thickness and more uniform concentration

distribution across the boundary layer.


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9

We have also shown Table (1) of the surface skin friction coefficient

against some parameters. The stress components x and the magnitude of y are

reduces with increasing the intensity of the magnetic field M, Schmidt number

Sc or the suction velocity A. The reversal behaviour is observed throughout the

region with increasing hall parameter mGrashof number Gr or heat absorption

coefficient Q or scalar constant , because an increase in Gr influences the

buoyancy that results in skin friction. The stress component x enhances and the

magnitude of y reduces with increasing permeability parameter K, whereas x

decreases and the magnitude of y boost up with increasing visco-elastic fluid

parameter of the Rivlin-Ericksen fluid Rm or Prandtl number Pr or frequency of

oscillationn. The rate of heat transfer (Nu) is shown in the Table (2) with reference to

all governing parameters. The magnitude of the Nusselt number rise up

throughout the fluid region with increasing scalar constant , suction velocity A,

Prandtl number Pr, heat absorption parameter Q and the frequency of oscillation

n.

The rate of mass transfer (Sh) is shown in the Table (3). The magnitude of

the Sherwood number enhances throughout the fluid region with increasing

scalar constant , suction velocity A, Schmidth number Sc, and the frequency of

oscillation n.

Figures 2. The velocity Profiles for andu v against M

1 0 01 0 5 0 5 0 5 Gr 3 0 1 Rm 0 01 Pr = 0.71m , . ,n . ,A . ,K . , ,Q . , . ,

Figures 3. The velocity Profiles for andu v against K

1 0 5 0 01 0 5 0 5 Gr 3 0 1 Rm 0 01 Pr = 0.71m ,M . , . ,n . ,A . , ,Q . , . ,


95

10

Figures 4. The velocity Profiles for andu v against Rm

1 0 5 0 01 0 5 0 5 0 5 Gr 3 0 1 Pr = 0.71m ,M . , . ,n . ,A . ,K . , ,Q . ,

Figures 5. The velocity Profiles for andu v against Gr

1 0 5 0 01 0 5 0 5 0 5 0 1 Rm 0 01 Pr = 0.71m ,M . , . ,n . ,A . ,K . ,Q . , . ,

Figures 6. The velocity Profiles for andu v against A

1 0 5 0 01 0 5 0 5 Gr 3 0 1 Rm 0 01 Pr = 0.71m ,M . , . ,n . , K . , ,Q . , . ,

Figures 7. The velocity Profiles for andu v against Pr

1 0 5 0 01 0 5 0 5 0 5 Gr 3 0 1 Rm 0 01m ,M . , . ,A . ,K . ,n . , ,Q . , .

Figures 8. The velocity Profiles for andu v against Q

1 0 5 0 01 0 5 0 5 0 5 Gr 3 Rm 0 01 Pr = 0.71m ,M . , . ,n . ,A . ,K . , , . ,


96

11

Figures 9. The velocity Profiles for andu v against

1 0 5 0 5 0 5 0 5 Gr 3 0 1 Rm 0 01 Pr = 0.71m ,M . ,n . ,A . ,K . , ,Q . , . ,

Figures 10. The velocity Profiles for andu v against n

1 0 5 0 01 0 5 0 5 Gr 3 0 1 Rm 0 01 Pr = 0.71m ,M . , . ,A . ,K . , ,Q . , . ,

Figures 11. The velocity Profiles for andu v against m

0 5 0 01 0 5 0 5 Gr 3 0 1 Rm 0 01 Pr = 0.71M . , . ,A . ,K . , ,Q . , . ,

Figures 12. Temperature Profiles for against Pr, Q, , t, n and A


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Figures 13. Concentration Profiles for against Sc, ,nand A

Table 1: Skin friction coefficient

M K m Gr Rm A Pr Sc Q n x y

0.5 0.5 1 3 0.01 0.5 0.71 0.22 0.1 0.01 0.5 3.85856 -

0.039855

1 2.91101 -

0.037115

2 2.09415 -

0.027485

1 5.17811 -

0.037811

1.5 7.16410 -

0.015221

2 4.58859 -

0.048857

3 5.41125 -

0.056632

4 5.14110 -

0.057118

5 6.41744 -

0.064714

0.03 3.825812 -

0.037488

0.05 3.78781 -

0.038884

1 3.87845 -

0.017485

1.5 3.77841 0.014100


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3 2.14811 0.048555

7 0.64778 0.051454

0.3 3.55781 -0.03411

0.6 3.29663 -

0.036355

0.3 5.38020 -

0.068956

0.5 9.17815 -

0.071524

0.03 3.97111 -

0.117841

0.05 4.03326 -

0.198442

1 3.81966 -

0.042744

1.5 2.57854 -

0.094855

Table 2: Nusselt number (Nu)

A Pr Q n Nu

0.01 0.5 0.71 0.01 0.5 -0.733466

0.03 -0.760672

0.05 -0.787877

1 -0.735992

1.5 -0.738518

3 -3.059900

7 -7.123040

0.03 -0.752536

0.05 -0.770710

1 -0.736135

1.5 -0.738739

Table 3: Sherwood number (Sh)

A Sc n Sh

0.01 0.5 0.22 0.5 -0.225384

0.03 -0.236151

0.05 -0.246918

1 -0.225937

1.5 -0.226491

0.3 -0.306780

0.6 -0.611766

1 -0.226992

1.5 -0.228478


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4. Conclusions

The conclusions are

1. When scalar constant increase the velocity increase, whereas when

dimensionless visco-elasticity parameter of the Rivlin–Ericksen fluid Rm

and dimensionless heat absorption coefficient Q, increase the velocity

decreases.

2. The resultant velocity reduces with increasing Hartmann number M and

enhances with increasing hall parameter m.

3. Heat absorption coefficient Q increase results a decrease in temperature

but, a reverse case is noticed in the presence of a constant .

4. It is recognized that there are many other methods thatcould be

considered in order to describe some reasonable solution for this particular

type of problem.

5. For better understanding of the thermal and concentrationbehavior of this

work, however, it may benecessary to perform the experimental works.

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Heat and mass t ransfer on MHD flow of Non -Newtonian ... · 1 Heat and mass t ransfer on MHD flow of Non -Newtonian fluid over an infiniteverticalporous plate with Hall effects z