Effects of Hall Current on an Unsteady MHD Flow of Heat and Mass Transfer along a Porous Flat Plate with Chemical Reaction and Viscous Dissipation

International Journal of Engineering Inventions
ISSN: 2278-7461, ISBN: 2319-6491
Volume 2, Issue 1 (January 2013) PP: 22-30
www.ijeijournal.com P a g e | 22
Effects of Hall Current on an Unsteady MHD Flow of
Heat and Mass Transfer along a Porous Flat Plate with
Chemical Reaction and Viscous Dissipation
BalaSidduluMalga1
, Nagaraju.L2
, N.Kishan3
, Ramarao Kotikalapudi4
1
Chaitanya Bharathi Institute Technology, Gandipet, Hyderabad-500075, 2
Research Scholar, Osmania
University, Hyderabad-500007, 3
University College of Science, Osmania University, Hyderabad-500007,
4
Chaitanya Bharathi Institute Technology, Gandipet, Hyderabad-500075,
Abstract:- The influence of Hall current, chemical reaction and viscous dissipation on an unsteady MHD flow
and heat transfer along a porous flat plate with mass transfer is investigated. The transformed governing
equations in the present study were solved numerically by using Galerkin Finite element method. The effects of
Hall current, viscous dissipation, chemical reaction over primary velocity u, secondary velocity w, heat transfer
and mass transfer are analyzed. Effects of magnetic parameter M, Prandtl number Pr, Schmidt number Sc,
Modified Grashof number Gr for heat transfer and Modified Grashof number Gc for mass transfer on the
velocity components and temperature is also examined.
Keywords:- MHD, Hall current, viscous dissipation, chemical reaction, finite element method.
I. INTRODUCTON
The combined heat and mass transfer problems with chemical reaction are of importance in many
processes, and therefore have received considerable amount of attention in recent years. In processes such as
drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower and flow in a desert
cooler, the heat and mass transfer occurs simultaneously. Chemical reaction can be codified as either
homogeneous or heterogeneous processes. A homogeneous reaction is one that occurs uniformly through a
given phase; in contrast, a heterogeneous reaction takes place in a restricted region or within the boundary of a
phase. A reaction is said to be the first order reaction if the rate of reaction is directly proportional to the
concentration itself. In many chemical engineering processes, a chemical reaction between foreign mass and
fluid occur. Das et.al. [1] Studied the effect of mass transfer on flow past an impulsively started infinite
vertical plate with constant heat flux and chemical reaction. Muthukumarswamy and Ganesan,P [2,3] discussed,
flow past an impulsively started infinite plate with constant heat and mass flux and effect of chemical reaction
on moving isothermal surface with suction.
In the recent years theoretical study of MHD channel flows has been a subject of great interest due to
its widespread applications in designed cooling systems with liquid metals, petroleum industry, purification of
crude oil, polymer technology, centrifugal separation of matter from liquid, MHD generators, pumps,
accelerators and flow meters. In an ionized gas where the density is low and/or the magnetic field is very strong,
the conductivity normal to the magnetic field is reduced to the free spiraling of electrons and ions about the
magnetic lines of force before suffering collision; also, a current is induced in a direction normal to both the
electric and magnetic fields. The phenomenon, well known in the literature, is called the Hall Effect. The study
of magnetohydrodynamic flows with Hall currents has important engineering applications in problems of
magnetohydrodynamic generators and of Hall accelerators as well as in flight magnetohydrodynamics. Sato [4]
has studied the effect of Hall currents on the steady hydromagnetic flow between two parallel plates. Various
workers Sherman and Sutton [5], Hossain [6], Ram [7], Pop [8] studied the Hall effects. This effect can be taken
into account within the range of magnetohydrodynamical approximation. Katagiri [9] studied the steady
incompressible boundary layer flow past a semi-infinite flat plate in a transverse magnetic field at small
magnetic Reynolds number considering with the effect of Hall current. Agarwal et.al [10] gave the solution of
flow and heat transfer of a micro polar fluid over a stretching sheet using finite element technique. Sri Ramulu
et.al [11] studied the Effect of Hall Current on MHD Flow and Heat Transfer along a Porous Flat Plate with
Mass Transfer.
Hossain and Rashid [12] investigated the effect of Hall current on the unsteady free-convection flow of
a viscous incompressible fluid with mass transfer along a vertical porous plate subjected to a time dependent
transpiration velocity when the constant magnetic field is applied normal to the flow. And an analysis of heat
transfer taking dissipation function into account, in boundary layer flow of a hydromagnetic fluid over a

Effects of Hall Current on an Unsteady MHD Flow of Heat and Mass Transfer along a…
stretching sheet in the presence of uniform transverse magnetic field has been given by Lodha and Tak [13],
Gupta [14] studied the Hall current effects in the steady hydromagnetic flow of an electrically conducting fluid
past an infinite porous flat plate. Sattar and Maleque [15] studied the effect of viscous dissipation as well as
joule heating on two-dimensional steady MHD free-convection and mass transfer flow with Hall current of an
electrically conducting viscous incompressible fluid past an accelerated infinite vertical porous plate with wall
temperature and concentration. Recently, Bala Siddulu Malga et.al [16] studied the effects of viscous dissipation
on unsteady free convection and mass transfer boundary layer flow past an accelerated infinite vertical porous
flat plate with suction. In this paper, we investigated Hall current, viscous dissipation and chemical reaction of
an unsteady free-convection MHD flow of an electrical conducting viscous incompressible fluid with mass
transfer along an infinite vertical porous plate. We employed Galerkin finite element method to solve the
governing partial differential equations. The effects of various important physical parameters on the flow and
heat, mass transfer are analyzed.
II. MATHEMATICAL FORMULATION
An unsteady free-convection flow of an electrically conducting viscous incompressible fluid with mass
transfer along an infinite vertical porous plate has been considered. The flow is assumed to be in 𝑥′
direction
which is taken along the plate in upward direction. The 𝑦′
axis is taken to be normal to the direction of the plate.
At time𝑡 ≥ 0 , the temperature and the species concentration at the plate are raised to 𝑇𝑤 (≠ 𝑇∞) and 𝐶 𝑤 (≠ 𝐶∞)
and thereafter maintained uniform. The level of species concentration is assumed to be very low and hence
species thermal diffusion and diffusion thermal energy effects can be neglected. A magnetic field of uniform
strength is assumed to be applied transversely to the porous plate. The magnetic Reynolds number of the flow is
taken to be small enough so that the induced magnetic field can be neglected. The equation of conservation of
electric change ∇. 𝐽 = 0 gives 𝑗𝑦 =constant, where 𝐽 = ( 𝑗𝑥 , 𝑗𝑦 , 𝑗𝑧) .It is also assumed that the plate is non-
conducting. This implies 𝑗𝑦 = 0 at the plate and hence zero everywhere. When the strength of magnetic field is
very large the generalized Ohm’s law, in the absence of electric field takes the form
𝐽 +
𝑤 𝑒𝑇 𝑒
𝐵0
𝐽𝐵 = 𝜎(𝜇 𝑒 𝑉𝐵 +
1
𝑒𝑛 𝑒
∇𝑃𝑒) (1)
where V is the velocity vector; 𝜎, the electric conductivity; 𝜇 𝑒, the magnetic permeability; 𝜔𝑒 , the electron
frequency; 𝜏 𝑒, the electron collision time; e, the electron charge; 𝑛 𝑒, the number density of the electron; and
𝑃𝑒, the electron pressure. Under the assumption that the electron pressure (for weakly ionized gas), the thermo-
electric pressure and ion-ship are negligible, equation (1) becomes
jx =
𝜎 𝜇 𝑒 𝐵0
1+𝑚2 (mu − w)
jz =
𝜎 µ𝐵0
1+𝑚2 (u + mw)
(2)
Where u is the x component of V; w, the z component of V and m (=𝜔𝑒 𝜏 𝑒), the Hall parameter. Within
this frame work, the equations which govern the flow under the usual Boussinesq approximation are:
∂V′
y′ = 0 (3)
𝜕𝑢′
𝜕𝑡′
+ 𝑉′
𝜕𝑢′
𝜕𝑦′
= 𝑣
𝜕2 𝑢′
𝜕𝑦′
′ 2 −
σ𝜇 𝑒
2 𝐵0
2
𝜌 1+𝑚2 𝑢′
+ 𝑚𝑤′
+ g𝛽(𝑇 − 𝑇∞) + g𝛽∗
𝐶′
− 𝐶′
∞ (4)
𝜕𝑤′
𝜕𝑡′
+ 𝑉′
𝜕𝑤′
𝜕𝑦′
= 𝑣
𝜕2 𝑤′
𝜕𝑦′2 −
σ𝜇 𝑒
2 𝐵0
2
𝜌 1+𝑚2 𝑤′
− 𝑚𝑢′ (5)
𝜕𝑇
𝜕𝑡′
+ 𝑉′ 𝜕𝑇
𝜕𝑦′
=
𝑘
ρCp
𝜕2 𝑇
𝜕𝑦′ 2 +
μ
ρCp
∂u′
∂y′
2
(6)
𝜕𝐶′
𝜕𝑡′
+ 𝑉
′
𝜕 𝐶′
𝜕 𝑦′
= 𝐷
𝜕2 𝐶′
𝜕𝑦 ′ 2 − 𝐾1
′
c′
(7)
Where g is the acceleration due to gravity; 𝛽, the volumetric coefficient of thermal expansion; 𝛽∗
, the
coefficient of volume expansion with species concentration; T, the temperature of the fluid within the boundary
layer; 𝐶′
, the species concentration; 𝜌, 𝜇, 𝑣, 𝑘, 𝑐 𝑝 are respectively density, viscosity, kinematic viscosity, thermal
conductivity, specific heat at constant pressure; and D, the chemical molecular diffusivity, Chemical reaction
parameter𝐾1
′
. The initial and boundary conditions of the problem are
𝑡′
≤ 0: 𝑢′
= 0, 𝑤′
= 0, 𝑇 = 𝑇∞, 𝐶′
= 𝐶∞ 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑦′
𝑡′
> 0:
𝑢′
= 0, 𝑤 = 0, 𝑇 = 𝑇𝑤 , 𝐶′
= 𝐶 𝑤 𝑎𝑡 𝑦′
= 0
𝑢′
= 0, 𝑤′
= 0, 𝑇 = 𝑇∞, 𝐶′
= 𝐶∞, 𝑎𝑡 𝑦′
→ ∞
(8)
The non-dimensional quantities introduced in the equations (3) to (7) are

𝑡 =
𝑡′ 𝑈0
2
𝑣
, 𝑦 =
𝑦′ 𝑈0
𝑣
, 𝑢, 𝑉, 𝑤 =
(𝑢′, 𝑉′, 𝑤′)
𝑈0
𝜃 =
(𝑇−𝑇∞)
(𝑇 𝑤 −𝑇∞)
, 𝐶 =
(𝐶′−𝐶∞)
(𝐶 𝑤 −𝐶∞)
, 𝐾1 =
𝐾1
′ 𝑣
𝑉0
2
(9)
Where 𝑈0 is the reference velocity. Therefore, the governing equation in the dimensionless form are
obtained as
∂V
∂y
= 0 (10)
𝜕𝑢
𝜕𝑡
+ 𝑉
𝜕𝑢
𝜕𝑦
=
𝜕2 𝑢
∂y2 −
𝑀
1+𝑚2 u + mw + 𝐺𝑟 𝜃 + 𝐺𝑐 𝐶 (11)
𝜕𝑤
𝜕𝑡
+ 𝑉
𝜕𝑤
𝜕𝑦
=
𝜕2 𝑤
∂y2 −
𝑀
1+𝑚2 w − mu (12)
𝜕𝜃
𝜕𝑡
+ 𝑉
𝜕𝜃
𝜕𝑦
=
1
Pr
𝜕2 𝜃
∂y2 + Ec
∂u
∂y
2
(13)
𝜕𝐶
𝜕𝑡
+ 𝑉
𝜕𝐶
𝜕𝑦
=
1
Sc
𝜕2 𝐶
∂y2 − 𝐾1 𝑐 (14)
Where
𝑀 =
σ𝜇 𝑒
2 𝐵0
2 𝑣
𝜌 𝑈2 𝑖𝑠 Magnetic field parameter, 𝑃𝑟 =
μ Cp
k
is Prandtl number
Sc =
v
D
is Schmidt number, Ec =
U0
2
Cp(Tw −T∞)
is Eckert number
𝐺𝑟 =
vgβ(Tw −T∞)
U0
3 is Modified Grashof number for heat transfer
𝐺𝑐 =
vgβ∗
Cw −C∞
U0
3 is Modified Grashof number for mass transfer
And the boundary conditions (equation (8)) in the non-dimensional form are
𝑡 ≤ 0: 𝑢 = 0, 𝑤 = 0, 𝑇 = 0, 𝐶 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑦
𝑡 > 0:
u = 0, w = 0, θ = 1, C = 1 at y = 0
u = 0, w = 0, θ = 0, C = 0 as y → ∞
(15)
From equation (10), it is seen that 𝑣 is either constant or a function of time t. Similarly, solutions of equations
(11) to (14) with the boundary conditions equation (15) exist only if
𝑉 = −𝜆𝑡−
1
2 (16)
Where λ is a non-dimensional transpiration parameter. For suction 𝜆 > 0 and for blowing λ < 0. From equation
(16), it can be observed that the assumption is valid only for small values of time variable.
III. METHOD OF SOLUTION
By applying the Galerkin finite element method for equation (11) over a typical two-noded linear
element 𝑒 ( 𝑦𝑗 ≤ 𝑦 ≤ 𝑦 𝑘 ) is
𝑁 𝑇 𝜕2 𝑢 𝑒
𝜕𝑦2 −
𝜕𝑢 𝑒
𝜕𝑡
− 𝑉
𝜕𝑢 𝑒
𝜕𝑦
−
𝑀
1+𝑚2 𝑚𝜔 + 𝑢 𝑒
+ 𝐺𝑟 𝜃 + 𝐺𝑐 𝐶 𝑑𝑦
𝑦 𝑘
𝑦 𝑗
(17)
𝜕𝑁
𝜕𝑦
.
𝜕𝑢 𝑒
𝜕𝑦
− 𝑁 𝑇
−𝑉
𝜕𝑢 𝑒
𝜕𝑦
−
𝜕𝑢 𝑒
𝜕𝑡
−
𝑀
1+𝑚2 𝑢 𝑒
+ 𝑅∗
𝑑𝑦
𝑦 𝑘
𝑦 𝑗
=0 (18)
𝑤ℎ𝑒𝑟𝑒 𝑅∗
= −
𝑀𝑚
1+𝑚2 ω + 𝐺𝑟 𝜃 + 𝐺𝑐 𝐶 , 𝑁 = 𝑁𝑗 , 𝑁𝑘 , ∅ 𝑒
=
𝑢𝑗
𝑢 𝑘
,
𝑢 𝑒
= 𝑁. ∅ 𝑒
, 𝑁𝑗 =
𝑦 𝑘 −𝑦
𝑙(𝑒) , 𝑁𝑘 =
𝑦−𝑦 𝑗
𝑙(𝑒) , 𝑙(𝑒)
= 𝑦 𝑘 − 𝑦𝑗 = ℎ
The element equation given by
𝑁𝑗
′
𝑁𝑗
′
𝑁𝑗
′
𝑁𝑘
′
𝑁𝑘
′
𝑁𝑗
′
𝑁𝑘
′
𝑁𝑘
′
𝑢𝑗
𝑢 𝑘
𝑑𝑦 − 𝑉
𝑁𝑗 𝑁𝑗
′
𝑁𝑗 𝑁𝑘
′
𝑁𝑘 𝑁𝑗
′
𝑁𝑘 𝑁𝑘
′
𝑢𝑗
𝑢 𝑘
𝑑𝑦 +
𝑦 𝑘
𝑦 𝑗
𝑁𝑗 𝑁𝑗 𝑁𝑗 𝑁𝑘
𝑁𝑘 𝑁𝑗 𝑁𝑘 𝑁𝑘
𝑢𝑗
𝑢 𝑘
𝑑𝑦 +
𝑀
1+𝑚2
𝑢𝑗
𝑢 𝑘
𝑑𝑦 − 𝑅∗
𝑁𝑗
𝑁𝑘
𝑑𝑦 = 0 (19)
𝑆 𝑒
+ 𝐴 𝑒
+ 𝑅 𝑑𝑦 = 0
𝑦 𝑘
𝑦 𝑗
(20)
where 𝑆 𝑒
=
𝑁𝑗
′
𝑁𝑗
′
𝑁𝑗
′
𝑁𝑘
′
𝑁𝑘
′
𝑁𝑗
′
𝑁𝑘
′
𝑁𝑘
′
𝑢𝑗
𝑢 𝑘
− 𝑉
𝑁𝑗 𝑁𝑗
′
𝑁𝑗 𝑁𝑘
′
𝑁𝑘 𝑁𝑗
′
𝑁𝑘 𝑁𝑘
′
𝑢𝑗
𝑢 𝑘
+
𝑀
1+𝑚2
𝑢𝑗
𝑢 𝑘
,
𝐴 𝑒
=
𝑢𝑗
𝑢 𝑘
𝑎𝑛𝑑 𝑅 = 𝑅∗
𝑁𝑗
𝑁𝑘
Here the prime and dot denote differentiation with respect to y and t. we obtain

𝑆 𝑒
=
1
𝑙(𝑒)
1 −1
−1 1
𝑢𝑗
𝑢 𝑘
−
𝑉
2
−1 1
−1 1
𝑢𝑗
𝑢 𝑘
+
𝑀
1 + 𝑚2
𝑙(𝑒)
6
2 1
1 2
𝑢𝑗
𝑢 𝑘
𝐴 𝑒
=
𝑙(𝑒)
6
2 1
1 2
𝑢𝑗
𝑢 𝑘
𝑎𝑛𝑑 𝑅 = 𝑅∗
𝑙(𝑒)
2
1
1
We write the element equation for the elements 𝑦𝑖−1 ≤ 𝑦 ≤ 𝑦𝑖 and 𝑦𝑖 ≤ 𝑦 ≤ 𝑦𝑖+1. Assembling these element
equations, we get
1
𝑙 𝑒
1 −1 0
−1 2 −1
0 −1 1
𝑢𝑖−1
𝑢𝑖
𝑢𝑖+1
−
𝑉
2
−1 1 0
−1 0 1
0 −1 1
𝑢𝑖−1
𝑢𝑖
𝑢𝑖+1
+
𝑀
1+𝑚2
𝑙 𝑒
6
2 1 0
1 4 1
0 1 2
𝑢𝑖−1
𝑢𝑖
𝑢𝑖+1
+
𝑙 𝑒
6
2 1 0
1 4 1
0 1 2
𝑢𝑖+1
𝑢𝑖
𝑢𝑖+1
= 𝑅∗ 𝑙 𝑒
2
1
2
1
(21)
Now put row corresponding to the node i to zero, from equation (21) the difference schemes with 𝑙 𝑒
= ℎ is
ℎ
6
𝑢𝑖−1 + 4𝑢𝑖 + 𝑢𝑖+1 +
𝑀
1 + 𝑚2
ℎ
6
−
𝑉
2
−
1
ℎ
𝑢𝑖−1 +
2
ℎ
+
2𝑀
1 + 𝑚2
ℎ
3
𝑢𝑖 +
𝑀
1+𝑚2
ℎ
6
+
𝑉
2
−
1
ℎ
𝑢𝑖+1 = R and here R = 𝑅∗ 𝑙 𝑒
2
1
2
1
(22)
Using the Cranck-Nicolson method to the equation (22), we obtain:
𝐴1 𝑢𝑖−1
𝑗+1
+ 𝐴2 𝑢𝑖
𝑗+1
+ 𝐴3 𝑢𝑖+1
𝑗+1
+ 𝐴4 𝑢𝑖−1
𝑗
+ 𝐴5 𝑢𝑖
𝑗
+ 𝐴6 𝑢𝑖+1
𝑗
= 𝑅 (23)
Similarly, the equations (12), (13), (14) are becoming as follows:
𝐵1 𝑤𝑖−1
𝑗 +1
+ 𝐵2 𝑤𝑖
𝑗 +1
+ 𝐵3 𝑤𝑖+1
𝑗 +1
+ 𝐵4 𝑤𝑖−1
𝑗
+ 𝐵5 𝑤𝑖
𝑗
+ 𝐵6 𝑤𝑖+1
𝑗
= 𝑅 𝐴 (24)
𝐶1 𝜃𝑖−1
𝑗+1
+ 𝐶2 𝜃𝑖
𝑗 +1
+ 𝐶3 𝜃𝑖+1
𝑗+1
+ 𝐶4 𝜃𝑖−1
𝑗
+ 𝐶5 𝜃𝑖
𝑗
+ 𝐶6 𝜃𝑖+1
𝑗
= 𝑅 𝐵 (25)
𝐷1 𝑐𝑖−1
𝑗 +1
+ 𝐷2 𝑐𝑖
𝑗+1
+ 𝐷3 𝑐𝑖+1
𝑗 +1
+ 𝐷4 𝑐𝑖−1
𝑗
+ 𝐷5 𝑐𝑖
𝑗
+ 𝐷6 𝑐𝑖+1
𝑗
= 0 (26)
The initial and boundary conditions (15) reduce to
𝑢 𝑖, 0 = 0, 𝑤 𝑖, 0 = 0, 𝜃 𝑖, 0 = 0, 𝐶 𝑖, 0 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖
𝑢 0, 𝑗 = 0, 𝑤 0, 𝑗 = 0, 𝜃 0, 𝑗 = 1, 𝐶 0, 𝑗 = 1
𝑢 100, 𝑗 = 0, 𝑤 100, 𝑗 = 0, 𝜃 100, 𝑗 = 1, 𝐶 100, 𝑗 = 0
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗
(27)
𝐴1 =
1
6
− +
𝑘
6
𝑀
1+𝑚2 − 𝑟 +
𝑃𝑉
2
, 𝐴2 =
4
6
+ 2𝑟 +
2𝑘
3
𝑀
1+𝑚2 ,
𝐴3 =
1
6
+
𝑘
6
𝑀
1+𝑚2 −
𝑃𝑉
2
− 𝑟 , 𝐴4 = −
𝑘
6
𝑀
1+𝑚2 + 𝑟 −
𝑃𝑉
2
−
1
6
,
𝐴5 = −2𝑟 −
𝑀
1+𝑚2
2𝑘
3
−
4
6
, 𝐴6 = −
𝑘
6
𝑀
1+𝑚2 + 𝑟 −
𝑃𝑉
2
−
1
6
,
𝐵1 =
1
6
+
𝑘
6
𝑀
1+𝑚2 − 𝑟 +
𝑃𝑉
2
, 𝐵2 =
4
6
+ 2𝑟 +
2𝑘
3
𝑀
1+𝑚2 ,
𝐵3 =
1
6
+
𝑘
6
𝑀
1+𝑚2 −
𝑃𝑉
2
− 𝑟 , 𝐵4 = −
𝑘
6
𝑀
1+𝑚2 + 𝑟 −
𝑃𝑉
2
−
1
6
𝐵5 = −2𝑟 −
𝑀
1+𝑚2
2𝑘
3
−
4
6
, 𝐵6 = −
𝑘
6
𝑀
1+𝑚2 + 𝑟 −
𝑃𝑉
2
−
1
6
𝐶1 =
1
6
−
𝑟
𝑆𝑐
+
𝑃𝑉
2
, 𝐶2 =
4
6
+
2𝑟
𝑆𝑐
, 𝐶3 =
1
6
−
𝑟
𝑆𝑐
−
𝑃𝑉
2
, 𝐶4 =
1
6
+
𝑟
𝑆𝑐
−
𝑃𝑉
2
,
𝐶5 =
4
6
−
2𝑟
𝑆𝑐
, 𝐶6 =
1
6
+
𝑟
𝑆𝑐
+
𝑃𝑉
2
,
𝐷1 = 1 − 6𝑟
1
𝑆 𝑐
+ 3𝑝𝑉 + 𝐾1 𝑘 , 𝐷2 = 4 + 12𝑟
1
𝑆 𝑐
+ 4𝐾1 𝑘 ,
𝐷3 = 1 − 6𝑟
1
𝑆 𝑐
− 3𝑝𝑉 + 𝐾1 𝑘 , 𝐷4 = 1 + 6𝑟
1
𝑆 𝑐
− 3𝑝𝑉 − 𝐾1 𝑘 ,
𝐷5 = 4 − 12𝑟
1
𝑆 𝑐
− 4𝐾1 𝑘 , 𝐷6 = 1 + 6𝑟
1
𝑆 𝑐
+ 3𝑝𝑉 − 𝐾1 𝑘 ,
𝑅 𝐴 =
𝑀𝑚
1+𝑚2 𝑢, 𝑅 𝐵 = 𝐸𝑐
𝑈 𝑖+1−𝑈 𝑖−1
2ℎ
2
Here 𝑟 =
𝑘
ℎ2 where k, h is mesh sizes along y direction and time direction respectively. Index i refers to space
and j refers to time. The mesh system consists of h=0.1 for velocity profiles and concentration profiles and
k=0.05 has been considered for computations. In equation (23)-(26), taking i=1(1) n and using initial and
boundary conditions (15), the following system of equation are obtained.
𝐴𝑖 𝑋𝑖 = 𝐵𝑖 , 𝑖 = 1,2,3 …. (28)
Where 𝐴𝑖’s are matrices of order n and 𝑋𝑖 𝑎𝑛𝑑 𝐵𝑖’ 𝑠 are column matrices having n-components. The
solution of above system of equations are obtained using Thomas algorithm for velocity, temperature and
concentration. Also, numerical solutions for these equations are obtained by MATLAB-Program, Galerkin finite

element method. The same MATLAB-Program was run with slightly changed values of h and k and no
significant change was observed in the values of 𝑢, 𝑤, 𝜃, 𝑐.
IV. RESULTS AND DISCUSSION
To get the physical insight of the problem, the results are discussed through graphs. The numerical
results are obtained for the velocity profiles of main (primary) flow and cross (secondary) flow and the
temperature profiles, concentration profiles have been shown graphically for different flow parameters, such as
magnetic field parameter M, Hall current m, Eckert number Ec, Chemical reaction parameter 𝐾1 Modified
Grashof number Gr for heat transfer, Modified Grashof number Gc for mass transfer, Schmidt number Sc,
transpiration parameter 𝜆, and time t. The primary velocity profiles are shown in fig.1 (a), fig.2 (a) and fig.3 (a).
The Hall current effect is to increases the primary velocity component u, and this is because of the fact that Hall
Effect, in general reduces the level of Lorentz forces effect on the fluid, whose tendency is to suppress the flow.
The primary velocity flow is decreases with the increase of magnetic field parameter M and the transpiration
parameter λ. It is also observed that the primary velocity u increases with the time t. It can be seen that the effect
of Gr, and Gc is to increase the primary velocity profiles u, which is due to with an increase in buoyancy force
due to temperature differences, the flow is accelerated in the boundary layer. Here the positive of Gr (>0)
correspond to cooling of the plate by natural convection. The effect of Prandtl number Pr is reduces velocity
profiles, the reason is that the smaller values of Pr are equivalent to increasing the thermal conductivities, and
therefore heat is able to diffuse away from the heated surface more rapidly than for higher values of Pr. The
influence of Schmidt number Sc is to reduce the velocity profiles. It is also noticed that the influence of Eckert
number Ec, on the velocity profiles is increases the primary velocity profiles u, and designates the ratio of
kinetic energy of the flow into the boundary layer enthalpy differences. It embodies the conversion of kinetic
energy to internal energy by work done against the viscous fluid stresses. Hence, greater viscous dissipative heat
causes a rise in the velocity profiles.
The secondary velocity profiles w are shown graphically in fig.1 (b), fig.2 (b) and fig.3 (b). The
secondary velocity profiles w increase with time t, Hall parameter m and the magnetic field parameter M. But
decreases owing to increase of transpiration parameter λ, it is observed from figures. It also can be observed that
the secondary velocity profiles w increases as the Grashof number Gr (for heat transfer) and Gc (for mass
transfer) increases, but it decreases as Prandtl number Pr and Schmidt number Sc increases. It also seen that
effects of viscous dissipation increases the secondary velocity profiles w.
From the fig.5 it is evident that the temperature of the fluid increases with the increase of time t. But it decreases
with the increase of transpiration parameter λ and Prandtl number Pr. The viscous dissipation effect on
temperature profiles shown in fig.3(c). The viscous dissipation heat causes a rise in the temperature which is
shown from the figure.
Fig.6 is drawn for the concentration profiles. From the figure it can be seen that the concentrations
profiles are increases with the increase of time t, while it is decreases with the increase of transpiration
parameter λ and Schmidt number Sc.
The effect of chemical reaction parameter 𝐾1 on primary velocity and secondary velocity profiles,
temperature and concentration profiles are shown in fig.4 (a)-(d) respectively. From the fig.4 (a) & (b) it can be
observed that the effect of 𝐾1, leads to decreases the primary velocity profiles as well as secondary velocity
profiles. Fig4(c) reviles that an increase of chemical reaction parameter 𝐾1 leads to decrease the temperature
profiles. Fig.4 (d) shows that there is fall in concentration profiles due to the increasing the value of chemical
reaction parameter 𝐾1 is noticed.
Fig.1(a): The Primary velocity profiles

Fig.1 (b): The Secondary velocity profiles
Fig.2 (a): The Primary velocity profiles
Fig.2 (b): The Secondary velocity profiles
Fig.3 (a): The Primary velocity profiles for different values of Ec

Fig.3 (b): The Secondary velocity profiles for different values of Ec
Fig.3 (c): Temperature Profiles for different values of Ec
Fig.4 (a): The Primary Velocity Profile for different values of 𝐾1
Fig.4 (b): The Secondary Velocity Profile for different values of 𝐾1
t=1.0, λ=0.5, m=0.2, M=1

Fig.4(c): The Temperature Profile for different values of 𝐾1
Fig.4 (d): Concentration Profile for different values of 𝐾1
Fig.5: The Temperature Profiles
Fig.6: The Concentration Profiles

REFERENCES
1) Das U.N., Deka, R.K., and Soundalgekar, V.M, ‘Effects of mass transfer past an impulsively started
vertical infinite plate with constant heat flux and chemical reaction’. Forschung in ingenieurwesen
Engineering Research, Vol.60, pp. 284-287, 1994.
2) Muthukumarswamy R. and Ganesan, P, ‘Flow past an impulsively started infinite plate with constant
heat flux and mass transfer’, Comp. Method in applied Mechanics and Engg, Vol.187 (1-2):pp.79-90,
1997.
3) Muthukumarswamy, R, ‘Effect of a chemical reaction on moving isothermal vertical surface with
suction’, Acta Mech; Vol.155,pp. 65-70, 2002).
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Magnetic Field’. Journal of the Physical Society of Japan, Vol. 16(7), pp.1427-1433, 1961.
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infinite vertical porous Plate’ Journal of the Physical Society of Japan, Vol. 55(7), pp. 2183-2190,
1998.
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J.Math.Phys.Sci.Vol. 5, pp. 375-385, 1971.
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10) Agarwal, R. S, Bhargava, R., Balaji, A.V. S., Finite Element Solution of Flow and Heat Transfer of a
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Plate with Mass Transfer’. Journal of the Physical Society of Japan, Vol. 56(1), pp. 97-104, 1987.
13) Lodtha, A., Tak, S. S., Heat Transfer with Viscos Dissipation in Hydromagnetic Flow over a Stretching
Sheet, Ganita Sandesh, Vol 19, no 2, pp 209-215, 2005.
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pp. 281-287, 1975.
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16) Bala Siddulu Malga, Naikoti Kishan, ‘Viscous Dissipation Effects on Unsteady free convection and
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Effects of Hall Current on an Unsteady MHD Flow of Heat and Mass Transfer along a Porous Flat Plate with Chemical Reaction and Viscous Dissipation

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