Hall effects on Unsteady MHD Rotating flow of Second grade fluid … · 2021. 3. 8. · transient...
Transcript of Hall effects on Unsteady MHD Rotating flow of Second grade fluid … · 2021. 3. 8. · transient...
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
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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
Journal of Xi'an University of Architecture & Technology
Volume XII, Issue III, 2020
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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