MHD Mixed Convection Flow from a Vertical Plate Embedded in a Saturated Porous Medium with Melting and Heat Source or Sink

M.V.D.N.S.Madhavi et al. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 7, Issue 3, ( Part -4) March 2017, pp.47-58
www.ijera.com DOI: 10.9790/9622- 0703044758 47 | P a g e
MHD Mixed Convection Flow from a Vertical Plate Embedded in a
Saturated Porous Medium with Melting and Heat Source or Sink
*M.V.D.N.S.Madhavi1
, J.Siva Ram Prasad2
K.Hemalatha2
, ,
1
Scholar, Krishna University, Dept. of Mathematics, V.R.Siddhartha Engineering College, Vijayawada, India,
2
Dept. of Mathematics, V.R.Siddhartha Engineering College, Vijayawada, India
ABSTRACT
We analysed in this paper the problem of MHD mixed convection flow from a vertical plate embedded in a
saturated porous medium in the presence of melting, thermal dispersion, radiation and heat absorption or
generation effects for aiding and opposing external flows. Similarity solution for the governing equations is
obtained for the flow equations in steady state. The equations are numerically solved by Runge-Kutta fourth
order method coupled with shooting technique. The effect of melting and heat absorption or generation under
different parametric conditions on velocity, temperature and heat transfer was analyzed for both aiding and
opposing flows.
Keywords: Heat absorption or generation, Melting, MHD, Porous medium, Radiation, Thermal dispersion.
I. INTRODUCTION
Over the last few years the study of heat
transfer with melting effect in porous media has
been increased due to wide variety of applications
in industry such as magmasolidification, melting of
the permafrost, preparation of semiconductor
material. Roberts [1] was the first to study the
shielding effect to describe the melting phenomena
of ice placed in ahot stream of air in the steady.
Transport in porous media has received continuing
interest in the past five decades. This interest stems
from their importance in many industrial and
clinical applications. Because of these applications,
several investigators have turned their attention to
the study of fundamental and applied problems
related to heat transfer in porous media. Also the
problem of combined free and forced convection
(mixed convection) in a porous medium has many
important applications in geothermal reservoirs
where pressure gradients may be generated either
by artificial with drawl or injection of fluids or by
natural recharge or discharge of meteoric water. M
Kazmierczaketal [2] and P Cheng[3]studied
melting from a flat plate embedded in a porous
medium in the presence of natural convection,
combined free and forced boundary layer flows
about inclined surfaces. Merkin [4] has considered
mixed convection boundary-layer flow in porous
media adjacent to a vertical uniform heat flux
surface.
Due to the important and interesting
applications in geothermal energy extraction,
nuclear waste disposal industry, underground heat
exchangers for energy storage and recovery,
temperature controlled reactors, packed beds and
the utilization of porous layers for transpiration
cooling by water for fire fighting, in the storage of
food grains, etc., the study of convective heat
transfer in a non-Darcy porous medium has been
gaining the attention of several researchers.
Chamkha [5] presented a numerical study for non-
Darcy hydro magnetic free convection flow of an
electrically-conducting and heat-generating fluid
over a vertical cone and a wedge adjacent to a
porous medium. Murthy and Singh [6] analyzed
thermal dispersion effects on non-Darcy convection
over a cone. An analysis is performed by Alin v.
Roscaetal [7]to study the heat and mass transfer
characteristics of mixed convection flow along a
vertical plate embedded in a fluid saturated porous
medium under the combined buoyancy effects of
thermal and mass diffusion. They showed that dual
solutions exist for a certain range of parameters
in the problem. Very interesting analytical
solutions have also been included in the paper.A
note on the effect of surface melting on the steady
mixed convection boundary layer flow over
avertical flat surface embedded in afluid saturated
porous medium is studied further by J.H.Merkin
etal[8] which is previously studied by Ahmad and
Pop[9]. The main conclusion is that solutions are
possible only if M<1 with the limit as m  1 being
discussed. The critical values, identified in Ahmad
and Pop[9] are examined in more detail and the
free convection limit derived.
When dealing problems in porous media,
the effects of melting, radiation and heat absorption
or generation become important. The problem of
unsteady mixed convection boundary layer flow
near the stagnation point on a heated vertical plate
embedded in a fluid saturated porous medium with
thermal radiation and variable viscosity was
investigated by Hassanien and Al-arabi [10].
Murthy et al. [11] considered mixed convection
RESEARCH ARTICLE OPEN ACCESS

flow of an absorbing fluid up a uniform non–Darcy
porous medium supported by a semi-infinite ideally
transparent vertical flat plate due to solar radiation.
Chamkha et al. [12] presented a numerical study of
coupled heat and mass transfer by boundary-layer
free convection over a vertical flat plate embedded
in a fluid-saturated porous medium in the presence
of thermophoretic particle deposition and heat
generation or absorption effects. The effects of
Non-Darcy mixed convection with thermal
dispersion-radiation in a saturated porous medium
was studied by Prasad and Hemalatha [13]. They
observed that temperature decreases with
increasing melting parameter.Chamkha [14]
discussed heat and mass transfer for a non-
Newtonian fluid flow along a surface embedded in
a porous medium with uniform wall heat and mass
fluxes and heat generation or absorption.
AlsoChamkha et al [15] analyzed melting and
radiation effects on mixed convective flow from a
vertical surface embedded in a non-Newtonian
fluid saturated non-Darcy porous medium for
aiding and opposing external flows. They obtained
representative flow and heat transfer results for
various combinations of physical parameters.
Motivated by the works mentioned above, the
present paper aims at analysing the effect of
melting, thermal dispersion-radiation and heat
absorption and generation on mixed convection
from a vertical plate embedded ina saturated porous
medium for aiding and opposing external flows
.
II. MATHEMATICAL FORMULATION
Fig. 1 Schematic diagram of the problem.
We considered a steady mixed convection
boundary layer flow past a vertical melting surface
embedded in a fluid saturated porous medium. The
flow model and geometry are shown in the figure1.
Further we consider a Cartesian coordinate system
(x, y), where x and y are coordinates measured
along the plate and normal to it, respectively. In
this coordinate system it appears as if the frozen
porous medium moves towards the stationary
melting/solid interface with constant velocity equal
to the melting velocity (The melting front is
modelled as a vertical plate. This plate constitutes
the inter phase between the liquid phase and the
solid phase during melting inside the porous
matrix. The temperature of the solid region is
considered less than the melting point, i.e., T0< Tm.
On the right hand side of melting front, the liquid is
super- heated, i.e., Tm< T∞. A vertical boundary
layer flow, on the liquid side smoothes the
transition from Tm to T∞. The assisting external
flow velocity is taken as u.
When taking into consideration the effect of
thermal dispersion and thermal radiation, the
governing equations for steady non-Darcy flow in
aporous medium can be written as follows.
The continuity equation is = 0 (1)
The momentum Equation is
(2)
The Energy Equation is
(T-Tm)
(3)
Here, u and v are the velocities along x and y
directions respectively, T is temperature in the
thermal boundary layer, K is Permeability, k is
thermal conductivity, B0 Magnetic field strength,C
is Forchheimer empirical constant, β is coefficient
of thermal expansion, υ is kinematics viscosity, ρ is
density, Viscosity, electrical conductivity, Cp is
specific heat at constant pressure, g is acceleration
due to gravity, and thermal diffusivity α = αm+ αd ,
where αm is the molecular diffusivity and αd is the
dispersion thermal diffusivity due to mechanical
dispersion, Q0 is the volumetric heat generation or
absorption parameter. As in the linear model
proposed by Plumb [16] the dispersion thermal
diffusivity αd is proportional to the velocity
component i.e. αd = γud, where γ is the dispersion

coefficient and d is the mean particle diameter. The
radiative heat flux term q is written using the
Rosseland approximation (Sparrow and Cess [17],
Raptis [18] as
q = (4)
Where is the Stefan–Boltzmann constant and ‘a’
is the mean absorption coefficient.
The physical boundary conditions for the present
problem are
y =0, T=Tm, k = ρ[hsf+ Cs (Tm - T0)]v (5)
and y→ ∞, T→ T∞, u=u∞ (6)
Where hsf and Cs are latent heat of the solid and
specific heat of the solid phases respectively and u∞
is the assisting external flow velocity, k = αρCp is
the effective thermal conductivity of the porous
medium. The boundary condition Eq. 5 means that
the temperature on the plate is constant and thermal
flux of heat conduction to the melting surface is
equal to the sum of the heat of melting and the heat
required for raising the temperature of solid to its
melting temperature Tm.
Introducing the stream function ψ with u = , and
v = .
The continuity Eq. 1 will be satisfied and the Eq.
2 and Eq. 3 transform to
(7)
(T-Tm) (8)
Introducing the similarity variables as
Ψ = f(η)(αm u∞ x)1/2
, η = , θ(η) =
, the momentum equation Eq. 7 and energy
equation Eq. 8 are reduced to
= 0 (9)
(1+D ) + ( f + D ) +
R + (1 + D) r θ = 0
(10)
Where the prime symbol denotes the differentiation
with respect to the similarity variable η and
Rax/Pexis the mixed convection parameter,Rax =
is the local Rayleigh number, Pex=
is the local Peclet number, F = is the non-
Darcian parameter, D = is the dispersion
parameter, Cr = is the temperature ratio, MH
= is magnetic parameter, R = is
the radiation parameter, and r = is the
dimensionless heat generation or absorption
parameter (r<0 corresponds to heat absorption and
r>0 corresponds to heat generation).
Taking into consideration, the thermal dispersion
effect together with melting, the boundary
conditions Eq. 5 and Eq. 6 take the form
η=0,θ=0,f(0)+{1+Df1
(0)}2Mθ1
(0)=0. (11)
and η→∞, θ=1, f1
=1. (12)
where M = is the melting parameter.
The local heat transfer rate from the surface of the
plane is given by qw = -k
The Nusselt number is Nu = = , Where h
is the local heat transfer coefficient and k is the
effective thermal conductivity of the porous
medium, which is the sum of the molecular thermal
conductivity km and the dispersion thermal
conductivity kd .
The modified Nusselt number is obtained as
= [1+ R + D f1
(0)]θ1
(0) (13)
III. SOLUTION PROCEDURE
The dimensionless equations Eq. 9 and
Eq. 10 together with the boundary conditions Eq.11
and Eq.12 are solved numerically by means of the
fourth order Runge-Kutta method coupled with
double shooting technique. The solution thus
obtained is matched with the given values of f1
(∞)
and θ (0). In addition, the boundary condition η→∞
is approximatedby = 8 which is found
sufficiently large for the velocity and temperature
to approach the relevant free stream
properties.Numerical computations are carried out
for F = 1; D = 0, 0.5, 1; Ra/Pe=1, -1; M = 0, 0.8, 2;
R = 0.5; Cr = 0.1,0.5,1; r = - 0.1, 0, 0.1.

IV. RESULTS AND DISCUSSION
0 1 2 3 4 5 6 7 8
0.7
0.8
0.9
1
1.1
1.2
1.3
Dimensionless Distance, 
f1
M = 0, Ra/Pe = 1
M = 0.8, Ra/Pe = 1
M = 2, Ra/Pe = 1
M = 0, Ra/Pe = -1
M = 0.8, Ra/Pe = -1
M = 2, Ra/Pe = -1
F=1,R=0.5,D=0.5,MH=1,Cr=0.1,r = - 0.1
0 1 2 3 4 5 6 7 8
0.7
0.8
0.9
1
1.1
1.2
1.3
f'
M = 0, Ra/Pe = 1
M = 0.8, Ra/Pe = 1
M = 2, Ra/Pe = 1
M = 0, Ra/Pe = -1
M = 0.8, Ra/Pe = -1
M = 2, Ra/Pe = -1
F=1,R=0.5,D=0.5,MH=1,Cr=0.1,r = 0.1
Fig. 2. Velocity profiles for different values of melting parameter with heat absorption (r= -0.1) and generation
(r = 0.1).
The velocity profiles for aiding and
opposing flows are shown in Fig. 2 for different
melting parameter values in the presence of heat
absorption and heat generation coefficient ‘r’. In
aiding flow the increase in melting parameter leads
to increase in velocity of the fluid both in the
presence of heat absorption and heat generation
coefficient. This effect is found opposite for the
opposing flow.
0 1 2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature,
M = 0, r = 0.1
M = 0.8, r = 0.1
M = 2, r = 0.1
M = 0, r = -0.1
M = 0.8, r = -0.1
M = 2, r = -0.1
F=1,R=0.5,D=0.5,MH=1,Cr=0.1,Ra/Pe=1
0 1 2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature,
M = 0, r = 0.1
M = 0.8, r = 0.1
M = 2, r = 0.1
M = 0, r = -0.1
M = 0.8, r = -0.1
M = 2, r = -0.1
F=1,R=0.5,D=0.5,MH=1,Cr=0.1,Ra/Pe = -1
Fig. 3 Temperature profiles for different values of melting parameter in the presence of heat absorption(r= -0.1)
and heat generation(r= 0.1) in aiding and opposing flows.
Figure 3 shows the effect of melting
parameter on temperature profiles in the presence
of heat absorption and heat generation coefficient
‘r’ in aiding and opposing flows. In both flow
cases, the same effect is found. The temperature
decreases with the increase in the melting
parameter both in the presence of heat absorption
and heat generation coefficient. It is also noted that
as the ‘r’ value increases, the temperature of the
fluid decreases at a fixed melting parameter
value.The velocity profiles for aiding and opposing
flows are shown in Fig. 4 for different thermal
dispersion parameter values in the presence of heat
absorption and heat generation coefficient ‘r’. In
aiding flow the increase in thermal dispersion
parameter leads to increase in velocity of the fluid
both in the presence of heat absorption and heat
generation coefficient. This effect is found opposite
for the opposing flow.
Figure 5 shows the effect of thermal
dispersion parameter on temperature profiles in the
coefficient ‘r’ in aiding and opposing flows. In both
flow cases, the same effect is found. The
temperature decreases with the increase in the
melting parameter both in the presence of heat

absorption and heat generation coefficient. It is also
noted that as the ‘r’ value increases, the
temperature of the fluid decreases at a fixed
melting parameter value.
0 1 2 3 4 5 6 7 8
0.7
0.8
0.9
1
1.1
1.2
1.3
f'
D = 0, Ra/Pe = 1
D = 0.5, Ra/Pe = 1
D = 1, Ra/Pe = 1
D = 0, Ra/Pe = -1
D = 0.5, Ra/Pe = -1
D = 1, Ra/Pe = -1
F=1,M=2,R=0.5,MH=1,Cr=0.1,r = -0.1
0 1 2 3 4 5 6 7 8
0.7
0.8
0.9
1
1.1
1.2
1.3
f'
D = 0, Ra/Pe = 1
D = 0.5, Ra/Pe = 1
D = 1, Ra/Pe = 1
D = 0, Ra/Pe = -1
D = 0.5, Ra/Pe = -1
D = 1, Ra/Pe = -1
F=1,M=2,R=0.5,MH=1,Cr=0.1,r = 0.1
Fig. 4 Velocity profiles for different values of thermal dispersion parameter with heat absorption (r= -0.1) and
generation (r = 0.1).
0 1 2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature,
D = 0, r = 0.1
D = 0.5, r = 0.1
D = 1, r = 0.1
D = 0, r = -0.1
D = 0.5, r = -0.1
D = 1, r = -0.1
F=1,R=0.5,M=2,MH=1,Cr=0.1,Ra/Pe=1
0 1 2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature,
D = 0, r = 0.1
D = 0.5, r = 0.1
D = 1, r = 0.1
D = 0, r = -0.1
D = 0.5, r = -0.1
D = 1, r = -0.1
F=1,R=0.5,M=2,MH=1,Cr=0.5,Ra/Pe= -1
Fig. 5 Temperature profiles for different values of thermal dispersion parameter in the presence of heat
absorption(r= -0.1) and heat generation(r= 0.1) in aiding and opposing flows.
Figure 6 depict the effects of the heat
generation or absorption coefficient ‘r’ on the
velocity profiles in aiding and opposing flows
without and with thermal dispersion, in the case of
melting parameter M=2, radiation parameter R=0.5,
flow inertia parameter F=1. The presence of a heat
generation source and heat absorption sink in the
flow are represented by positive value r = 0.1 and
negative value r = - 0.1 respectively. In aiding flow,
It is noted that the velocity profile increases with
the increase of ‘r’ value in both the cases of
absence and presence of thermal dispersion effect.
But the effect is found opposite in opposing flow in
the presence and absence of thermal dispersion.
Figure 7 illustrate the influence of the heat
generation or absorption coefficient ‘r’ on the
temperature profiles in aiding and opposing flows
without and with thermal dispersion, in the case of
melting parameter M=2, radiation parameter R=0.5,
flow inertia parameter F=1. It is noted that the
temperature profile decreases with the increase of
‘r’ value in the absence as well as the presence of
thermal dispersion effect both in aiding and
opposing flow cases. The velocity profiles for
aiding and opposing flows are shown in Fig. 8 for
different temperature ratio parameter values in the
coefficient ‘r’. In aiding flow the increase in
temperature ratio parameter leads to increase in
velocity of the fluid both in the presence of heat
absorption and heat generation coefficient. This
effect is found opposite for the opposing flow.

0 1 2 3 4 5 6 7 8
0.7
0.8
0.9
1
1.1
1.2
1.3
f'
r = -0.1, Ra/Pe = 1
r = 0, Ra/Pe = 1
r = 0.1, Ra/Pe = 1
r = -0.1, Ra/Pe = -1
r = 0, Ra/Pe = -1
r = 0.1, Ra/Pe = -1
F=1,M=2,R=0.5,D=0,MH=1,Cr=0.1
0 1 2 3 4 5 6 7 8
0.7
0.8
0.9
1
1.1
1.2
1.3
f'
r = -0.1, Ra/Pe = 1
r = 0, Ra/Pe = 1
r = 0.1, Ra/Pe = 1
r = -0.1, Ra/Pe = -1
r = 0, Ra/Pe =-1
r = 0.1, Ra/Pe = -1
F=1,M=2,R=0.5,D=0.5,MH=1,Cr=0.1
Fig. 6 Velocity profiles for different values of ‘r’ without and with dispersion in aiding and opposing flows
0 1 2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature,
r = -0.1, Ra/Pe = 1
r = 0, Ra/Pe =1
r = 0.1, Ra/Pe = 1
r = -0.1, Ra/Pe = -1
r = 0, Ra/Pe = -1
r = 0.1, Ra/Pe = -1
F=1M=2,R=0.5,D=0,MH=1,Cr=0.1
0 1 2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature,
r = -0.1, Ra/Pe = 1
r = 0, Ra/Pe = 1
r = 0.1, Ra/Pe = 1
r = -0.1, Ra/Pe = -1
r = 0, Ra/Pe = -1
r = 0.1, Ra/Pe = -1
F=1,M=2,R=0.5,D=0.5,MH=1,Cr=0.1,Ra/Pe=1
Fig. 7 Temperature profiles for different values of ‘r’ in the absence and presence of thermal dispersion in
aiding and opposing flows.
0 1 2 3 4 5 6 7 8
0.7
0.8
0.9
1
1.1
1.2
1.3
f'
Cr = 0.1, Ra/Pe = 1
Cr = 0.5, Ra/Pe = 1
Cr = 1, Ra/Pe = 1
Cr = 0.1, Ra/Pe = -1
Cr = 0.5, Ra/Pe = -1
Cr = 1, Ra/Pe = -1
F=1,M=2,R=0.5,D=0.5,MH=1,r = -0.1
0 1 2 3 4 5 6 7 8
0.7
0.8
0.9
1
1.1
1.2
1.3
f'
Cr = 0.1, Ra/Pe = 1
Cr = 0.5, Ra/Pe = 1
Cr = 1, Ra/Pe = 1
Cr = 0.1, Ra/Pe = -1
Cr = 0.5, Ra/Pe = -1
Cr = 1, Ra/Pe = -1
F=1,M=2,R=0.5,D=0.5,MH=1,r = 0.1
Fig. 8 Velocity profiles for different values of temperature ratio parameter with heat absorption (r= -0.1) and
generation (r = 0.1).
Figure 9 show the effect of temperature
ratio parameter on temperature profiles in the
coefficient ‘r’ in aiding and opposing flows. In both
flow cases, the same effect is found. The
temperature decreases with the increase in the
temperature ratio parameter both in the presence of
heat absorption and heat generation coefficient. It is
also noted that as the ‘r’ value increases, the
temperature of the fluid decreases at a fixed
temperature ratio parameter value.

0 1 2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature,
Cr = 0.1, r = 0.1
Cr = 0.5, r = 0.1
Cr = 1, r = 0.1
Cr = 0.1, r = -0.1
Cr = 0.5, r = -0.1
Cr = 1, r = -0.1
F=1,M=2,R=0.5,D=0.5,MH=1,Ra/Pe=1
0 1 2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature,
Cr = 0.1, r = 0.1
Cr = 0.5, r = 0.1
Cr = 1, r = 0.1
Cr = 0.1, r = -0.1
Cr = 0.5, r = -0.1
Cr = 1, r = -0.1
F=1,M=2,R=0.5,D=0.5,MH=1,Ra/Pe= -1
Fig. 9 Temperature profiles for different values of temperature ratio parameter in the presence of heat
absorption(r= -0.1) and heat generation(r= 0.1) in aiding and opposing flows.
Fig. 10. Variation of local Nusselt number with the melting parameter for different values of heat absorption or
generation coefficient in aiding flow.
Fig. 11 Variation of local Nusselt number with the melting parameter for different values of heat absorption or
generation coefficient in opposing flow.
Figures10 and 11 shows the effect of heat
absorption or generation parameter on the Nusselt
number given in Eq. (13) for different values of
melting parameter for aiding and opposing flows
respectively. The same effect is found in both flow
cases. The increase in the value of absorption or
generation parameter causes the decrease in Nusselt
number, whereas increase in the melting parameter
results in the decrease in Nusselt number.

Figures12 and 13 shows the effect of
temperature ratio parameter in the presence of heat
absorption on the Nusselt number for different
values of melting parameter for aiding and
opposing flows respectively. The increase in the
value of temperature ratio parameter causes the
increase in Nusselt number, whereas increase in the
melting parameter results in the decrease in Nusselt
number in both flow cases. Figures14 and 15 shows
the effect of temperature ratio parameter in the
presence of heat generation on the Nusselt number
for different values of melting parameter for aiding
and opposing flows respectively. The increase in
the value of temperature ratio parameter causes the
increase in Nusselt number, whereas increase in the
melting parameter results in the decrease in Nusselt
number in both flow cases. Figures16 and 17 shows
the effect of thermal dispersion parameter in the
presence of heat absorption on the Nusselt number
for different values of melting parameter for aiding
and opposing flows respectively. The increase in
the value of thermal dispersion parameter causes
the increase in Nusselt number, whereas increase in
the melting parameter results in the decrease in
Nusselt number in both flow cases. Figures18 and
19 shows the effect of thermal dispersion parameter
in the presence of heat generation on the Nusselt
number for different values of melting parameter
for aiding and opposing flows respectively. The
increase in the value of thermal
dispersionparameter causes the increase in Nusselt
number, whereas increase in the melting parameter
results in the decrease in Nusselt number in both
flow cases.
Fig. 12.Variation of local Nusselt number with the melting parameter for different values of temperature ratio
parameter in the presence of heat absorptionin aiding flow.
parameter in the presence of heat absorption in opposing flow.

parameter in the presence of heat generation in aiding flow.
Fig. 15. Variation of local Nusselt number with the melting parameter for different values of temperature ratio
parameter in the presence of heat generation in opposing flow.
Fig. 16.Variation of local Nusselt number with the melting parameter for different values of thermal dispersion
parameter in the presence of heat absorptionin aiding flow.

parameter in the presence of heat absorptionin opposing flow.
parameter in the presence of heat generation in aiding flow.
parameter in the presence of heat generation in opposing flow.

V. CONCLUSION
The melting phenomenon has been
analyzed in the presence of heat absorption and
heat generation coefficient with mixed convection
flow and heat transfer in a saturated non-Darcy
porous medium considering the effects of thermal
dispersion, thermal radiation and applied magnetic
field by taking Forcheimer extension in the flow
equations. Numerical results for the velocity and
temperature profiles as well as the heat transfer rate
as a function of Nusselt number are obtained for
aiding and opposing flows and the same are
presented graphically. Same results are obtained in
the presence of heat absorption and heat generation
coefficient. This study shows that the increase in
melting, thermal dispersion, heat absorption or
generation and temperature ratio parameters tend to
increase / decrease the velocity within the boundary
in aiding / opposing flow. Further, it is noticed that
the temperature decreases with the increase in
melting, thermal dispersion, heat absorption or
generation and temperature ratio parameters in both
flow cases. Moreover it is found that the rate of
heat transfer decreases with the increase in melting
parameter while it increases with the increase in
thermal dispersion and temperature ratio parameter
values in both aiding and opposing flows. It is also
noticed that the Nusselt number decreases with the
increase in heat absorption or generation parameter
value in both flow cases.
REFERRENCES
[1] Leonard Roberts, “On the Melting of a
Semi – Infinite Body of Ice placed in a
Hot Stream of Air, J.Fluid Mech.4,
pp.505-528, 1958.
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MHD Mixed Convection Flow from a Vertical Plate Embedded in a Saturated Porous Medium with Melting and Heat Source or Sink

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