Hall effects on Unsteady MHD Rotating flow of Second grade ...xajzkjdx.cn/gallery/386-mar2020.pdf · Hall effects on Unsteady MHD Rotating flow of Second grade fluid through Porous

Hall effects on Unsteady MHD Rotating flow of Second grade

fluid through Porous medium between two vertical plates

B.V. Swarnalathamma1, D.M. Praveen Babu2 and M. Veera Krishna3*

1Department of Science and Humanities, JB institute of Engineering & Technology, Moinabad, Hyderabad,

Telangana-500075, India. Email: [email protected] 2Department of Basic Science and Humanities, Vaagdevi Institute of Technology and Science, Proddatur,

Kadapa, Andhra Pradesh - 516360, India – Email: [email protected] 3Department of Mathematics, Rayalaseema University, Kurnool, Andhra Pradesh - 518007, India

Email: [email protected]

Abstract

In this paper, a probe on the impact of Hall effects on the flow of an oscillatory convective

MHD viscous, incompressible, radiating and electrically-conducting second-grade fluid in a vertical

porous rotating channel in a slip flow regime is carried out. The fluid is assumed to be gray, absorbing

and emitting radiation in a non-scattering medium. It is presupposed that MHD flow is laminar and

fully developed. In order to use the regular perturbation technique, the solutions of velocity and

temperature distributions for the governing flow are obtained. The profiles of velocity and

temperature are presented graphically. On the other hand, the skin friction coefficient and the Nusselt

number are evaluated numerically and presented in the form of tables, examined and discussed for

various values of the flow parameters that govern it.

Keywords: Hall Effect, radiating fluid, MHD oscillatory flow, rotating channel, slip flow

regime.

Nomenclature:

(u, v) the velocity components along x and y directions

p the modified pressure

t time

pC Specific heat

B0 electromagnetic induction

g Acceleration due to gravity

K1 thermal conductivity

k permeability of the porous medium

0Q Heat absorption

T temperature of the fluid

Journal of Xi'an University of Architecture & Technology

Volume XII, Issue III, 2020

Issn No : 1006-7930

Page No: 4290

mailto:[email protected]



1f Maxwell’s reflection co-efficient

Co-efficient of viscosity

L mean free path

0T meantemperature

0w suction velocity

1q radiative heat

q Complex velocity

M Hartmann number

K Permeability parameter

S second grade fluid parameter

E Ekman number

Gr thermal Grashof number

Pr Prandtl number

m Hall parameter

Greek symbols

the mean variation absorption coefficient

1 normal stress moduli

Dimension less temperature

Ω Angular velocity

Electrical conductivity of the fluid

Density of the fluid

Kinematic viscosity

the frequency of oscillation

The coefficient of volume expansion

e the electron frequency

e the electron charge

Sub scripts

e electron



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1. Introduction

Rapid efforts have been made to study the effects of various fields in porous media by

both experimentally and theoretically. Flows through porous medium are of principal interest

in the fields of agricultural engineering, underground water resources and seepage of water in

river beds, filtration and purification processes in ocean engineering, petroleum technology to

study the movement of natural gas, oil and water through the oil reservoirs. The way in which

fluid flow through porous media is of great importance for the petroleum extraction

processes. Most of the problems have considerable practical significance besides contributing

more to the existing knowledge. Porous materials due to their inherent capability are widely

used in Ocean engineering, Mechanical engineering, Medical appliances and Modern

industrial fields. The practical applications in geophysics and engineering promote the study

of flow in a rotating porous channel. Besides that, its applications are extended to food

processing industry, the chemical processing industry, centrifugation filtration processes, and

rotating machinery. In addition to that, the hydrodynamic rotating flow of electrically

conducting viscous incompressible fluids has garnered adequate consideration due to its

variegated applications in engineering and physics. In the domain of geophysics, it can be

applied to measure and study position and velocities with respect to a fixed frame of

reference on the surface of the earth, which rotates with respect to an inertial frame in the

presence of its magnetic field. The area of geophysical dynamics, of late has become a

significant branch of fluid dynamics due to the enormous interest towards environment and

its concerns. As there is growing inquisitiveness to study astrophysics, the research on fluid

dynamics has become vital as through it stellar and solar structures, interplanetary and

interstellar matter, solar storms, etc. can be studied. In engineering, it finds its application in

MHD generator ion propulsion, MHD bearing, MHD pumps, MHD boundary layer control of

re-entry vehicles, etc. Several scholars, viz. Crammer and Pai [1], Ferraro and Plumpton

[2],and Shercliff [3]have studied such flows because of their varied importance and

applications.

The process of heat transfer is used in the cooling of nuclear reactors, providing heat

sink in turbine blades and aeronautics. There are countless, significant engineering and

geophysical applications of channel flows through a porous medium: for example, in the field

of agricultural engineering, for channel irrigation and for studying underground water

resources; and in petroleum technology, to study the environment of natural gas, oil and

water through the oil channels/reservoirs. The transient natural convection between two

vertical walls with porous material having variable porosity has been studied by Paul et al [4].



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In recent years, the effect of transversely applied magnetic field on the flow of an electrically

conducting viscous fluid has been studied extensively owing to its astrophysics, geophysical,

and engineering applications. Attia and Kotb [5] have studied MHD flow between the two

parallel plates with heat transfer. When the strength of magnetic field is strong, one cannot

overlook or negate the effect of Hall current. The rotating flow of an electrically conducting

fluid in the presence of a magnetic field is encountered in geophysical and comical fluid

dynamics. It is indispensable in solar physics involved in sun spot development. The Hall

effect on unsteady MHD free and forced convection flow in a porous rotating channel has

been investigated by several researchers, including Sivaprasad et al.[6], Singh and Kumar [7],

Singh and Pathak [8], and Ghosh et al [9]. Radiative convective flows have gained the

attention of many researchers in recent years. Radiation plays a crucial role in several

engineering, environment and industrial processes: for example, heating and cooling

chambers; fossil fuel combustion energy processes; astrophysical flows; and space vehicle re-

entry. Raptis[10]studied radiation free convective flow through a porous medium. Alagoa et

al. [11] have analysed the radiation effect on MHD flow through a porous medium between

infinite parallel plates in the presence of time-dependent suction. Singh and Kumar [12] have

studied the radiation effect on the exact solution of free convective oscillatory flow through a

porous medium in a rotating porous channel.

The behaviour of the fluid under extreme confinement is of great interest from both the

scientific and technological point of view. One of the great complexities and intricacies is to

discover and unravel the type of boundary condition that is appropriate for solving the

continuum fluid problems. Inspite of the widespread acceptance of the no-slip assumption,

there has existed for many years indirect experimental evidence, based on anomalous flow in

capillaries and other systems, that in some cases, a simple liquid can slip against the solid

when walls are sufficiently smooth, and the no-slip boundary condition is no longer valid.

The no-slip boundary condition holds good only when particles close to the surface fail to

move along with the flow, i.e., when adhesion is stronger than cohesion. However, this is

only true microscopically. Some other limitations of no-slip conditions are: they fail for large

contact angles; they do not hold at very low pressure; they do not work for polyethylene,

rubber compounds or suspensions; and they fail in hydrodynamics for hydrophobic surfaces

(Vinogradova [13]). Recently, the slip condition has become much more pronounced, and it

is now reasonably sure that viscous fluid can slip against solid surfaces in the event of the

surface being very smooth. The slip boundary condition has important application in

lubrication, extrusion, and the medical sciences, especially in polishing article heart valves,



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flows through porous media, micro and nanofluids, friction studies and biological fluids

(Blake [14]). On the other hand, chemical reactions have numerous applications, such as

ceramic manufacturing, food processing, and polymer production. Muthucumaraswamy [15]

has analyzed that the rate of diffusion is affected by chemical reaction. Many researchers

have shown interest in propulsion engines for aircraft technology. Recently, Reddy et al. [16]

have analysed the effect of radiation on an unsteady MHD convective flow through a non-

uniform horizontal channel. Chand et al. [21] investigated the Hall effect on an unsteady

MHD oscillatory convective heat transfer flow of a radiating and chemically reacting fluid

through a porous medium in a rotating vertical porous channel. Recently, Veera Krishna and

Swarnalathamma [22] discussed the peristaltic MHD flow of an incompressible and

electrically conducting Williamson fluid in a symmetric planar channel with heat and mass

transfer under the effect of an inclined magnetic field. Swarnalathamma and Veera Krishna

[23] discussed the theoretical and computational study of peristaltic hemodynamic flow of

couple stress fluids through a porous medium under the influence of a magnetic field with

wall slip condition. Veera Krishna and M.G.Reddy [24] discussed the MHD free convective

rotating flow of visco-elastic fluid past an infinite vertical oscillating plate. Veera Krishna

and G.S.Reddy [25] discussed the unsteady MHD convective flow of second grade fluid

through a porous medium in a rotating parallel plate channel with a temperature-dependent

source. Veera Krishna et al. [26] discussed heat and mass transfer on unsteady MHD

oscillatory flow of blood through porous arteriole.

The effects of radiation and Hall current on an unsteady MHD free convective flow in

a vertical channel filled with a porous medium have been studied by Veera Krishna et al.

[27]. The heat generation/absorption and thermo-diffusion on an unsteady free convective

MHD flow of radiating and chemically reactive second grade fluid near an infinite vertical

plate through a porous medium and taking the Hall current into account have been studied by

Veera Krishna and Chamkha [28]. Veera Krishna et al. [29] discussed the heat and mass

transfer on unsteady, MHD oscillatory flow of second-grade fluid through a porous medium

between two vertical plates under the influence of fluctuating heat source/sink, and chemical

reaction. Veera Krishna et al. [30] investigated the heat and mass transfer on MHD free

convective flow over an infinite non-conducting vertical flat porous plate. Veera Krishna and

Jyothi [31] discussed the effect of heat and mass transfer on free convective rotating flow of a

visco-elastic incompressible electrically conducting fluid past a vertical porous plate with

time dependent oscillatory permeability and suction in presence of a uniform transverse



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magnetic field and heat source. Veera Krishna and Subba Reddy [32] investigated the

transient MHD flow of a reactive second grade fluid through a porous medium between two

infinitely long horizontal parallel plates. Veera Krishna et al. [33] discussed heat and mass-

transfer effects on an unsteady flow of a chemically reacting micropolar fluid over an infinite

vertical porous plate in the presence of an inclined magnetic field, Hall current effect, and

thermal radiation taken into account. Veera Krishna et al.[34] discussed Hall effects on

steady hydromagnetic flow of a couple stress fluid through a composite medium in a rotating

parallel plate channel with porous bed on the lower half. Veera Krishna et al. [35] discussed

Hall effects on unsteady hydromagnetic natural convective rotating flow of second grade

fluid past an impulsively moving vertical plate entrenched in a fluid inundated porous

medium, while temperature of the plate has a temporarily ramped profile. Veera Krishna and

Chamkha [36] discussed the MHD squeezing flow of a water-based nanofluid through a

saturated porous medium between two parallel disks, taking the Hall current into account.

Veera Krishna et al. [37] discussed Hall effects on MHD peristaltic flow of Jeffrey fluid

through porous medium in a vertical stratum. The effects of heat and mass transfer on free

convective flow of micropolar fluid were studied over an infinite vertical porous plate in the

presence of an inclined magnetic field with a constant suction velocity and taking Hall

current into account have been discussed by Veera Krishna et al.[38]. Veera Krishna and

Chamkha [39] have discussed the systematic solution of time-dependent mean velocity on

MHD peristaltic rotating flow of an electrically conducting couple stress fluid in a uniform

elastic porous channel. Veera Krishna and Chamkha [40] investigated The diffusion-thermo,

radiation-absorption and Hall and ion slip effects on MHD free convective rotating flow of

nano-fluids (Ag and TiO2) past a semi-infinite permeable moving plate with constant heat

source. Veera Krishna et al.[41] discussed the Soret and Joule effects of MHD mixed

convective flow of an incompressible and electrically conducting viscous fluid past an

infinite vertical porous plate taking Hall effects into account. Hall and ion slip effects on

Unsteady MHD Convective Rotating flow of Nanofluids have been discussed by Veera

Krishna and Chamkha[42]. Veera Krishna [43] investigated the heat transport on steady

MHD flow of copper and alumina nanofluids past a stretching porous surface. More recently

Veera Krishna et al.[44] discussed investigated the Hall and ion slip effects on the unsteady

MHD free convective rotating flow through porous medium past an exponentially accelerated

inclined plate. The combined effects of Hall and ion slip on MHD rotating flow of ciliary

propulsion of microscopic organism through porous medium have been studied by Veera

Krishna et al.[45]. Veera Krishna and Chamkha [46] investigated the Hall and ion slip effects



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on the MHD convective flow of elastico-viscous fluid through porous medium between two

rigidly rotating parallel plates with time fluctuating sinusoidal pressure gradient. Veera

Krishna [47] reported that the Hall and ion slip effects on MHD free convective rotating flow

bounded by the semi-infinite vertical porous surface. Veera Krishna [47] discussed the MHD

laminar flow of an elastico-viscous electrically conducting Walter’s fluid through a circular

cylinder or a pipe.

In view of the above facts, in this paper, a probe or investigation is carried out of the

influence of Hall effects on the flow of an oscillatory convective MHD viscous

incompressible, radiating, and electrically conducting second grade fluid in a vertical porous

rotating channel in slip flow regime.

2. Formulation and Solution of the Problem

Consider the flow of a viscous, incompressible, and electrically conducting second

grade fluid through a porous medium bounded by two infinite vertical insulated plates at d

distance apart, under the influence of a uniform transverse magnetic field normal to the

channel, and taking Hall current into account. We introduce a Cartesian co-ordinate system

with the x-axis oriented vertically upward along the centreline of this channel, and the z-axis

taken perpendicular to the planes of the plates, which is the axis of the rotation, and the entire

system comprising of the channel and the fluid are rotating as a solid body about this axis

with constant angular velocity Ω. A constant injection velocity 0w is applied at the plate

/ 2z d and the same constant suction velocity, 0w , is applied at the plate / 2z d . The

schematic diagram of the physical problem is shown in Fig. 1.

Fig. 1. Physical Configuration of the Problem



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Since the plates of the channel occupying the planes / 2z d are of infinite extent,

all the physical quantities depend upon z and t only. Under the Boussinesq approximation, the

flow of the fluid through a porous medium in a rotating channel is governed by the following

equation:

0w

z

(1)

2 301

0 2 2

12

yB Ju u p u ui v w u g T

t z x z z t k

(2)

2 3

010 2 2

12 xB Jv v p v zi u w v

t z y z z t

(3)

2

10 1 02p

qT T TC w K Q T

t z z z

(4)

The boundary conditions are

1

1

2 f L u uu L

f z z

, 1

1

2, 0

f L v vv L T

f z z

at

2

dz (5)

00, cosu v T T t at2

dz (6)

where,

1/2

2L

p

is the mean free path which is constant for an incompressible fluid.

When the strength of the magnetic field is very large, the generalized Ohm’s law is

modified to include the Hall current, so that

0

1( ) ( )e e

e

e

wJ J B E V B p

B

(7)

In equation (7), the electron pressure gradient, the ion–slip effect, and the thermo-

electric effect are neglected. We also assume that the electric field E=0 under assumption

reduces to,

2

0x yJ m J B v (8)

2

0y xJ mJ B u (9)

Where, e em is the Hall parameter.

On solving the equations (8) and (9),



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we obtain,

0

2( )

1x

BJ v mu

m

(10)

0

2( )

1y

BJ mv u

m

(11)

Using the equations (10) and (11), the equations (2) and (3) reduce:

22 3

010 2 2 2

12 ( )

(1 )

Bu u p u ui v w mv u u g T

t z x z z t m k

(12)

22 3

010 2 2 2

12 ( )

(1 )

Bv v p v vi u w v mu v

t z y z z t m k

(13)

Following Cogley et al.[18], the last term in the energy equation stands for radiative heat

flux, which is given by

21 4q

Tz

(14)

Where, is the mean variation absorption coefficient.

Introducing non-dimensional variables,

* * * * * * * *0

2

0 0 0

, , , , , , , ,t wz x y u v T p d

z x y u v t pd d d d d T d w w

Making use of the non-dimensional variables, the equations (12), (13) and (4) reduce to (with

the asterisks being dropped)

2 3 2

2 2 2

1Re 2 Re ( ) Gr

1

u u p u u MiRv S mv u u

t z x z z t m K

(15)

2 3 2

2 2 2

1Re 2 Re ( )

1

v v p v v MiRu S mu v v

t z y z z t m K

(16)

22

2RePr Pr Pr N

t z z

(17)

The corresponding boundary conditions in non-dimensional form are

, , 0u v

u h v h Tz z

at

1

2z (18)



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00, cosu v T T t at1

2z (19)

where, 0Rew d

is the Reynolds number,

2dR

is the rotation parameter,

2

kK

d is the

permeability parameter,

2

0

0

Grg d T

w

is the thermal Grashof number,

1

PrpC

K

is the

Prandtl number, 1

2 dN

K

is the radiation parameter,

2

0

p

Q d

C

is the heat absorption

parameter, 2 2

2 0B dM

is the Hartmann number, and 1

2S

d

is the second grade fluid

parameter.

Combining equations (15) and (16), let andq u iv x iy , we obtain

2 3 2

2 2

1Re Re 2 Gr

1

q q p q q MS iR q

t z z z t im K

(20)

We assume the flow under the influence of pressure gradient varying periodically

with time,

cosp

A t

(21)

Upon substitution of equation (21) into equation (20), we obtain

2 3 2

2 2

1Re Re cos 2 Gr

1

q q q q MA t S iR q

t z z z t im K

(22)

The corresponding boundary conditions are

1, 0 at

2

qq h z

z

(23)

10, cos at

2q t z (24)

In order to solve the equations (22) & (17) with the boundary conditions (23) & (24),

following Choudary et al. [19], we assume a solution of the form



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0( , ) ( ) i tq z t q z e (25)

0( , ) ( ) i tz t z e (26)

Substituting the equations (25) and (26) in (22) and (17) respectively yields the

following equations:

2 2

0 00 02

1(1 ) Re (2 Re) Re Gr

1

d q dq MSi i R q A

dz dz im K

(27)

2

20 002

RePr Pr Pr RePr 0d d

N idz dz

(28)

The corresponding boundary conditions are

0 0

1, 0 at

2

qq h z

z

(29)

0 0

10, 1 at

2q z (30)

Solving the equations (27) and (28) with respect to the boundary conditions (29) and

(30), we obtain the following expressions for the velocity and the temperature.

( , )q z t

3 1 4 2

1

Recosh sinh

AB m z B m z

a

3 3 4 4

1 2

2 3 4 5

Gr Gr

2 2

m z m z m z m zi tB Be e e e

ea a a a

(31)

1 3 2 4cosh sinh i tB m z B m z e (32)

The validity and the correctness of the present solution are verified by taking

= Gr = = = 0 andS M R = h K .

i.e., for the horizontal channel in the absence of rotation, slip flow and the condition of the

ordinary medium so that

2

2

Re Re 4 Recosh

2( , ) 1

Re Re 4 Recosh

2

i t

iz

Aq z t e

i i

(33)

This solution is reported by Schlichting and Gersten [20] for periodic variation of

the pressure gradient along axis of the channel.



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We find the skin friction L at the left plate in terms of its amplitude and the phase angle as

1

2

cosL

z

qq t

z

(34)

1 23 1 4 2sinh cosh

2 2r i

m mq iq B m B m

3 3 4 4

2 2 2 21 3 2 4

2 3 4 5

Gr Gr

2 2

m m m m

B m B me e e e

a a a a

The amplitude and phase angleare given by

2 2

r iq q q and1tan i

r

q

q

The rate of heat transfer (NU), in terms of amplitude and the phase angle, can be obtained as

1

2

cosz

TNu H t

z

(35)

3 41 3 2 4sinh cosh

2 2r i

m mH iH B m B m

Where, the amplitude2 2

r iH H H and phase angle 1tan i

r

H

H

All the contents used above have been mentioned in the Appendix.

3. Results and Discussions

We have discussed the heat transfer on the flow of an oscillatory convective MHD

viscous incompressible, radiating, and electrically conducting second grade fluid in a vertical

porous rotating channel in slip flow regime, taking Hall current into account. The flow is

governed by the non-dimensional parameters, namely, Re the Reynolds number, A the

amplitude of pressure gradient, h the slip parameter, m the Hall parameter, R the rotation

parameter, K the permeability parameter, Gr the thermal Grashof number, Pr the Prandtl

number, N the Radiation parameter, 1 the Heat absorption parameter, M the Hartmann

number, S the second grade fluid parameter and the frequency of oscillation. The velocity



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profiles are depicted in Figs. (2-5), while the temperature profiles are depicted in Figs. (6).

The skin friction and the Nusselt number are tabulated in Tables 1-2. For computational

purposes, we are fixing the values Re = 2, = 0.1t for the velocity profiles.

The effect of the rotation number R, the Hall parameter m, and the second grade fluid

parameter S on the velocity profile is shown in Figs. 2. We notice that the magnitudes of

velocity components u and v enhance with increasing values of R, m and S throughout the

fluid region. A similar behaviour is observed for increasing thermal Grashof number Gr,

permeability parameter K and the amplitude of the pressure gradient A ((Figs. (3)). It is

observed from these Figs. (4-5) that the velocity profile is diminished with the increase of all

these parameters: the heat absorption parameter 1 , the slip parameter h, the frequency of

oscillation , the Hartmann number M, the Prandtl number Pr, and the radiation parameter N.

That is, it starts decreasing near the left plate of the channel.

The variations in the temperature profile are presented in Figs. (6). It is observed that

the temperature profiles decrease with the increasing Reynolds number Re and the Prandtl

number Pr. Likewise, the temperature profile is diminished with increasing the radiation

parameter N and the frequency of oscillation .

The skin friction and Nusselt number are evaluated analytically and tabulated in

Tables 1-2. The amplitude and phase angle of the skin friction are shown in Table 1. The

negative values in this table indicate that there is always a phase lag, and this phase goes on

increasing with increasing frequency of oscillation. We notice that the amplitude q and

phase angle of frictional force are enhanced with increasing the parameters S and A, and

diminished with increasing h, Pr and R. The amplitude of the frictional force increases and

the phase angle of frictional force reduces with increasing the parameters K, Gr, m and 1 .

The opposite behaviour is observed with increasing M, N and .

Likewise, the amplitude of the Nusselt number and the magnitude of the phase angle

increases with Re, Pr, N and 1 . The amplitude of the Nusselt number increases and the phase

angle decreases with increasing .



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Figs. 2 The velocity profiles for u and v against R, m and S with

1Gr = 3, = 0.5, Pr = 0.71, = 0.5, = 5, = 0.2, 0.1, 1, / 6M K A h N

Figs. 3 The velocity profiles for u and v against Gr, K and A with

1=1, = 0.5, Pr = 0.71, =1, =1, = 0.2, 0.1, 1, / 6R M m S h N

Figs. 4 The velocity profiles for u and vagainst 1 , h and with

Gr = 3, = 0.5, Pr = 0.71, = 0.5, = 1, = 1, 1, 1, 5M K S m R N A



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Figs. 5 The velocity profiles for u and v against M, Pr and N with

1Gr = 3, =1, =1, 1, = 0.5, = 5, = 0.2, 0.1, / 6S m R K A h

Figs. 6 The temperature profiles for against Pr, Re, N and



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Table 1: Amplitude and phase angle of skin friction for Re 0.5, 0.1t

M K R m S Gr Pr N A h 1 Amplitude

q

Phase angle

0.5 0.5 1 1 1 3 0.71 1 5 0.2 / 6

0.2 7.71825 -0.941407

1 7.06899 -0.961037

1.5 6.21020 -0.986480

1 8.41201 -0.755380

2 8.42697 -0.632667

1.5 6.68314 -0.863153

2 5.91425 -0.806275

2 7.80433 -0.932313

3 7.85395 -0.930859

2 8.77737 -0.995119

3 7.25606 -1.015140

4 8.56564 -0.916300

5 9.41745 -0.895723

3 6.35484 -0.852423

7 3.56824 -0.698480

2 4.76206 -0.957294

3 4.52711 -1.771980

8 10.8357 -0.977135

10 12.9186 -0.991364

0.3 6.96514 -0.900558

0.4 6.38554 -0.868982

/ 4 7.65388 -0.958290

/ 3 7.16499 -0.991776

0.5 7.76577 -0.928239

1 7.90141 -0.899951

Table 2: Amplitude and phase angle of Nusselt number for 0.1t

Re Pr N 1 Amplitude

r

Phase angle

2 0.71 1 0.2 / 6 1.80834 1.40967

3 2.15392 1.41469

5 2.87840 1.42482

3 4.13127 1.45389

7 8.07952 1.48488

2 2.45473 1.46107

3 3.26711 1.48566

0.5 1.86154 1.41643

1 1.94703 1.42590

/ 4 1.96826 1.35577

/ 3 1.99643 1.28804



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4. Conclusions

The flow and heat transfer of an oscillatory convective MHD viscous incompressible,

radiating and electrically conducting second grade fluid in a vertical porous rotating channel

in slip flow regime and taking Hall current into account has been discussed.

1. The resultant velocity enhances with increasing R, m, Gr, K, A and S throughout the

fluid region.

2. The velocity profile is diminished with the increase of 1 , h, , M, Pr and N.

3. The temperature profiles decrease with the increasing Re, N, Prand .

4. The amplitude and phase angle of frictional force are enhanced with increasing S and A,

and diminished with increasing h, Pr and R.

5. The amplitude of the frictional force increases and the phase angle of frictional force

reduces with increasing the parameters K, Gr, m and 1 . The opposite behaviour is

observed with increasing M, N and .

6. The amplitude of the Nusselt number and the magnitude of the phase angle increase

with Re, Pr, N and 1 .

7. The amplitude of the Nusselt number increases and the phase angle decreases with

increasing .

Appendix:

2

1

1(2 Re)

1

Ma i R

im K

2

1 Pr RePrb N i

2

1

1

Re Re 4 (1 )

2

a Sim

2

1

2

Re Re 4 (1 )

2

a Sim

2

1

3

Pr Re Pr Re 4

2

bm

2

1

4

Pr Re Pr Re 4

2

bm



Issn No : 1006-7930

Page No: 4306

1

3

1

2cosh2

Bm

, 2

4

1

2sinh2

Bm

2 21 2 2 2

3

1 2 1 2 1 22 1

sinh cosh2 2

2cosh sinh cosh cosh sinh sinh2 2 2 2 2 2

m mA A A h m

Bm m m m m m

h m m

1 12 1 2 1

4

1 2 1 2 1 22 1

cosh sinh2 2

2cosh sinh cosh cosh sinh sinh2 2 2 2 2 2

m mA A A h m

Bm m m m m m

h m m

1

1

Re AA

a

3 3

2 21

2 3

Gr

2

m m

B e e

a a

4 4

2 22

4 5

Gr

2

m m

B e e

a a

3 3

2 21 3

2 3

Gr

2

m m

B m e eh

a a

4 4

2 22 4

4 5

Gr

2

m m

B m e e

a a

2

1

Re AA

a

3 3

2 21

2 3

Gr

2

m m

B e e

a a

4 4

2 22

4 5

Gr

2

m m

B e e

a a

2

2 3 3 1(1 ) Re (1 )a Si m m a Si

2

3 3 3 1(1 ) Re (1 )a Si m m a Si

2

4 4 4 1(1 ) Re (1 )a Si m m a Si ,

2

5 4 4 1(1 ) Re (1 )a Si m m a Si

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