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Journal of Applied Mathematics and Physics, 2019, 7, 968-982
http://www.scirp.org/journal/jamp
ISSN Online: 2327-4379
ISSN Print: 2327-4352
DOI:
10.4236/jamp.2019.74065 Apr. 29, 2019 968 Journal of Applied Mathematics and Physics
MHD Effects on Mixed Convective Nanofluid
Flow with Viscous Dissipation in Surrounding
Porous Medium
Md. Nasir Uddin1*, Md. Abdul Alim2, Md. Mustafizur Rahman2
1Department of Mathematics, Bangladesh Army University of Engineering & Technology, Natore, Bangladesh
2Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh
Abstract
In existence of concerning magnetic field, heat together with mass transfer
features on mixed convective copper-water
nanofluid flow through inclined
plate is investigated in surrounding porous medium together with viscous
dissipation. A proper set of usef
ul similarity transforms is considered as to
transform the desired governing equations into a system as ordinary differen-
tial equations which are nonlinear. The transformed equations for nanofluid
flow include interrelated boundary conditions which are res
olved numerically
applying Runge-Kutta integration process of sixth-order together with Nach-
tsheim and Swigert technique. The numerical consequences are compared
together with literature which was published previously and acceptable com-
parisons are found. The influence of significant parameters like as magnetic
parameter, angle for inclination, Eckert number, fluid suction parameter,
nanoparticles volume fraction, Schmidt number and permeability parameter
on concerning velocity, temperature along with concentration boundary lay-
ers remains examined and calculated. Numerical consequences are presented
graphically. Moreover, the impact regarding these physical parameters for
engineering significance in expressions of local skin friction coefficient in ad-
dition to local Nusselt together with Sherwood numbers is
correspondingly
examined.
Keywords
MHD, Mixed Convection, Nanofluid, Porous Medium, Viscous Dissipation
1. Introduction
Magnetic fluids which are types of particular nanofluids took advantage of mag-
How to cite this paper:
Uddin, Md.N.
,
Alim
, Md.A. and Rahman, Md.M. (2019
)
MHD Effects on Mixed Convecti
ve Nanof-
luid Flow with Viscous Dissipation in Su
r-
rounding Porous Medium
.
Journal of A
p-
plied Mathematics and Physics
,
7
, 968-982.
https:
//doi.org/10.4236/jamp.2019.74065
Received:
March 11, 2019
Accepted:
April 26, 2019
Published:
April 29, 2019
Copyright © 201
9 by author(s) and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution
International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
Md. N. Uddin et al.
DOI:
10.4236/jamp.2019.74065 969 Journal of Applied Mathematics and Physics
netic material goods of nanoparticles inside as for example liquid rotary seals
functioning with no maintenance along with tremendously low leakage in very
extensive variety of applications. Magnetohydrodynamics (MHD) nanofluid
flows have broadly uses of MHD generators, tunable optical fiber, optical grat-
ing, optical modulators and switches, polymer, petroleum technologies and then
metallurgical industries.
Heat transfer enhancement concerning boundary layer fluid flow for different
nanofluid flow passes through a vertical plate that has been studied with steady
case via Rana and Bhargava [1] including with the effect such as temperature
dependent heat source/sink. For different nanofluids, average of Nusselt number
was found to diminish by them. The influence regarding viscous dissipation,
chemical reaction together with Soret in existence of magnetic field into nanof-
luids flow has been considered pass through porous media through Yohannes
and Shankar [2]. The Keller box process was used to solve governing equation of
the concerning fluid flow, and numerical outcomes were presented for various
parameters of convective heat together with mass transfer properties.
Magnetic field including thermal radiation effects for nanofluid flow has been
analyzed along stretching surface through Khan
et al.
[3]. The flow field was
discussed by them with dissimilar time steps and reported that average shear
stress reductions with the development of magnetic field are observed. MHD
boundary layer nanofluid flow regarding heat with mass transfer has been stated
through porous media by Haile and Shankar [4] with considering thermal radia-
tion including viscous dissipation with chemical reaction effects. They were con-
sidered copper (Cu)-water and Al2O3-water nanofluids and noted out that veloc-
ity field decreases with increase of magnetic field.
For considering the steady case, MHD mixed convective nanofluid flow
through porous medium which has been deliberated past along a stretching
sheet by Ferdows
et al.
[5]. They concluded that velocity together with tempera-
ture increases while concentration decreases gradually with increase of Eckert
number. Heat transfer physical characteristics of flow field with three dissimilar
categories of nanofluid pass through permeable stretching/shrinking surface has
been considered and observed through porous medium via Pal
et al.
[6]. They
found with the increment of suction/injection parameters as local Nusselt num-
ber rises for stretching sheet while decreasing for shrinking sheet.
Through inclined porous plate, magnetohydrodynamic mixed convective flow
including Joule heating together with viscous dissipation on the field has been
studied via Das
et al.
[7]. The velocity along with temperature of flow fields rise
due to rise of particular magnetic field which was obtained by them. Within a
porous medium, MHD mixed convection happening on fluid flow has been in-
spected and analyzed toward a vertical plate through Hari
et al.
[8] with the rad-
iation, including heat generation with presence of chemical reaction effect. On
increasing particular magnetic field, velocity profile overshoots adjacent the
plate surface and then convergence to the boundary which was found by them.
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On boundary layer flow, MHD effects toward exponentially shrinking sheet have
been analyzed by Jain and Choudhary [9]. The numerical outcomes were ob-
served graphically and then explored through them. For MHD Williamson na-
nofluid flow, effects regarding chemical reaction, melting and radiation through
porous medium have been examined and discussed with specific physical para-
meters by Krishnamurthya
et al.
[10]. Considering Brinkman nature nanofluid
in unsteady magnetohydrodynamic flow towards on vertical plate has been dis-
cussed into porous medium via Ali
et al.
[11]. Fluid velocity diminishes with the
enhancement of such nanoparticle volume fraction which is noticed by them.
Through porous media, MHD flow regarding nanofluid with characteristic of
heat including mass transfer has been reported along stretching sheet via Reddy
and Chamkha [12]. The velocity reduces while temperature together with con-
centration enhances as per magnetic parameter increases which were found by
them. For unsteady case, MHD along with free convective nanofluid flow pass
through a flat plate has been stated with radiation absorption by Prasad
et al.
[13] and influence of regarding several physical factors on flow field was studied
by them.
Double diffusive magnetohydrodynamic nanofluids flow together with effects
of thermal radiation including viscous-Ohmic dissipation has been discussed
along nonlinear stretching/shrinking sheet via Pal and Mandal [14]. In existence
of magnetohydrodynamics, unsteady natural convective regarding nanofluid
flow through above a porous vertical plate has been studied for three dissimilar
categories of nanofluids by Geetha
et al.
[15]. The numerical outcomes were
analyzed and then presented for numerous physical factors which are concerned
by them. A mathematical relation for two dimensional flow of magnetic Maxwell
nanofluid subject to heat sink/source influenced by a stretched cylinder has been
attained by Irfan
et al.
[16]. As stated aforementioned, it remains more practical
to include MHD effects to study the impact regarding momentum including
heat with mass transfer flow, and to the best of author’s knowledge; no investi-
gation is considered as MHD effects on mixed convective through nanofluid
flow along viscous force in surrounding porous medium.
Therefore, in the light of above literatures, the purpose of the current work is
to observe MHD effects on mixed convective nanofluid flow along with viscous
dissipation in surrounding porous medium. The governing equations for consi-
dered nanofluid flow are converted into a combination as nonlinear ordinary
differential equations via introducing similarity variables and solved numerical-
ly. Moreover, for cogency of numerical results, a comparison is prepared with
the literature which is published and comparatively satisfactory comparison is
achieved. The terms of engineering interests such as wall shear stress including
rate of heat transfer together with rate of mass transfer are shown into tabular
form. The influence for various physical features as magnetic parameter, angle of
inclination, Eckert number and fluid suction parameter are presented on the
field of flow and analyzed thereafter.
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2. Mathematical Model
The two dimensional incompressible nanofluid flows which is steady and visc-
ous is considered for analysis. The nanofluid flow is passing inclined porous
plate in surrounding porous medium. In addition to consider magnetic field
B
0,
this is introduced normally to direction of nanofluid flow. For two dimensional
coordinate system which is shown as below in Figure 1, (
x
,
y
) is (direction along
of flow, direction normal of flow) and (
u
,
v
) is (velocity components of
x
, veloc-
ity components of
y
). It is supposed that
U∞
denotes free stream velocity of flow
field;
g
denotes the gravity by virtue of acceleration and
α
denotes angle to ver-
tical porous plate. Moreover, the temperature
Tw
at the wall is larger than am-
bient temperature
T∞
while concentration
Cw
at the wall is larger than ambient
concentration
C∞
.
Consequently under the above flow field consideration, conservation law for
mass is obeyed automatically as given below:
0
uv
xy
∂∂
+=
∂∂
(1)
The common fluid as water is considered such as base fluid while copper (Cu)
is considered such as nanoparticles for flow field together with both are locally
thermal equilibrium. The thermophysical properties which are used for nanof-
luid are specified into Table 1.
Figure 1. The diagram of flow configuration.
Table 1. Thermo physical properties of base fluid and nanoparticle.
Particles
Thermo Physical Properties
ρ
(kg/m3)
Cp
(J/kg K)
k
(W/m K)
β
× 10-5 (1/K)
σ
(S/m)
Water (H2O)
997.1
4179
0.613
21
5.5 × 10−6
Copper (Cu)
8933
385
401
1.67
59.6 × 10−6
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Using the present physical features, the modified dimensional boundary layer
equations of Pal and Mandal [17] as well as Bachok
et al.
[18] for describing na-
nofluid flow field in expressions of conservation equations such as momentum,
energy together with concentration equations can be expressed such as
( )
( )
( )
( )
{ }
( )
( )
2
2
2
0
cos
nf t c
nf nf
nf
nf nf
nf pp
u u ug
u v TT CC
xy y
BuU
k
ν ρβ ρβ α
ρ
σν
ρ
∞∞
∞
∂∂ ∂
+ = + −+ −
∂∂ ∂
−+−
(2)
( )
( )
2
22 2
2
0
22
1
nf nf nf
pnf
T T T uu
u v u BuU
xy y
yy
C
αµ σ
ρ
∞
∂∂ ∂ ∂ ∂
+= + + + −
∂∂ ∂
∂∂
(3)
2
2
CC C
uvD
xy y
∂∂ ∂
+=
∂∂ ∂
(4)
where for the nanofluid,
vnf
is the kinematic viscosity, (
βt
)
nf
and (
βc
)
nf
are the
coefficient for thermal and concentration expansions,
g
is the acceleration owing
to gravity,
α
is the angle for inclination,
σnf
is the electrical thermal conductivity,
ρnf
is the density, nf
α
is the thermal diffusivity and
( )
pnf
C
ρ
is the heat capa-
citance. Furthermore,
kpp
is permeability regarding porous medium whereas
D
is
mass diffusivity.
The operational dynamic viscosity for nanofluid was given via Brinkman [19]
while relation between physical quantities regarding nanoparticle and base fluid
which were familiarized by Abu-Nada [20] as:
( ) ()
()
()
() ( )
( )
()
()( )
( )
()
( ) ( )
( ) ( )
( ) ( ) ( )
2.5 ,1 ,
1
1
1,
1
22
, ,,
2
31
1
2
bf
nf nf bf np
t tt
nf bf np
c cc
nf bf np
p pp
nf bf np
np bf bf np
nf nf nf
nf nf
nf bf np bf bf np p nf
np
bf
nf
bf np
bf
C CC
K K KK
KK
KK K KK C
µ
µ ρ φ ρ φρ
φ
ρβ φ ρβ φ ρβ
ρβ φ ρβ φ ρβ
ρ φ ρ φρ
φ
µ
να
ρφρ
σφ
σ
σ
σσ
σ
= =−+
−
=−+
=−+
=−+
+− −
= = =
++−
−
= + +
1
np
bf
σφ
σ
−−
(5)
where, bf
µ
is considered for dynamic viscosity while
ϕ
is considered for nano-
particle solid volume fraction. The subscripts in aforementioned equations
bf
and
np
symbolize namely base fluid and nanoparticle respectively.
The accompanying boundary conditions for the existing nanofluid flow field
are as follows:
( )
0, , and at 0
ww w
u v vxT T CC y==±= ==
(6)
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, and atuUT T CC y
∞∞ ∞
= = = →∞
(7)
where permeability of porous plate is as
vw
(
x
) which is for suction (<0) or blow-
ing (>0) whereas the subscripts in aforementioned equations
w
and
∞
are de-
noted wall and boundary layer edges respectively.
The well-known stream function is convenient to consider as equation of con-
tinuity is satisfied identically through following relations:
anduv
yx
ψψ
∂∂
= = −
∂∂
(8)
In the above,
ψ
is the stream function. To reduce complexity of nanofluid flow
field, the succeeding dimensionless transformations which were introduced via
Cebeci
et al.
[21] are used:
( ) ( )
( )
,,
and
bf
bf w
w
U TT
y xU f
x TT
CC
sCC
η ψ ν η θη
ν
η
∞∞
∞
∞
∞
∞
−
= = = −
−
=−
(9)
In the above,
η
is the similarity transform. The components for velocity of
Equation (8) using Equation (9) can be re-written such as
( ) ( ) ( )
1
and 2
bf
U
u Uf v f f
x
ν
η ηη η
∞
∞
′′
= = −
(10)
wherever the prime of aforementioned equation indicates as differentiation with
respect to
η
.
Using similarity transformations in Equations (2) and (4) including the ac-
companying boundary conditions Equations (6) and (7), the transformed equa-
tions in which momentum, energy together with concentration equations are as
follows:
( )
( )
( )
( )
1 2 3 4 15
1cos 1 0
2
tc
f ff Ri Ri s M K f
φ φ φ θ φ α φφ
′′′ ′′ ′
+ + + + + −=
(11)
(
)
( )
2
2
76
61
1 Pr 2 2 10
2f Ec f f f MEc f
θ φθ φ
φφ
′′ ′ ′ ′′′ ′′ ′
+ + + + −=
(12)
10
2
s Sc f s
′′ ′
+=
(13)
and corresponding boundary conditions are supposed as below:
, 0, 1, 1, at 0
w
fff s
θη
′
= = = = =
(14)
1, 0 and 0 atfs
θη
′→ → → →∞
(15)
In the aforementioned Equation (14), the coefficient for wall mass transfer is
( )
ww
bf
x
f vx U
ν
∞
= −
(16)
such as
0
w
f>
for suction and
0
w
f<
for injection or blowing while physical
parameters are defined as under:
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( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )
( )
33
22
2
0
22
22.5
12
34
, ,Re ,
, , , ,Pr ,
Re Re
, ,1 ,1 ,
1 ,1
tw c w
bf bf
tc x
bf
bf bf
bf p
bf bf bf
tc
tc
bf pp bf
xx
bf np
bf
pw
bf
tc
np np
tbf
g T Tx g C Cx xU
Gr Gr
C
Bx x
Gr Gr
Ri Ri M K
U kU K
U
Ec Sc D
C TT
ββ
ν
νν
νρ
σν
ρ
νρ
φ φ φ φφ
ρ
ρβ ρβ
φ φφ φ φφ
ρβ
∞∞
∞
∞∞
∞
∞
−−
= = =
= = = = =
= = =− =−+
−
=−+ =−+
( )
( )
( )
( )
56
7
,,
and 1
nf nf
c bf bf
bf
pnp
pbf
K
K
C
C
σ
φφ
ρβ σ
ρ
φ φφ
ρ
= =
=−+
(17)
In the aforementioned Equations (11)-(13),
Grt
is the local thermal Grashof
number,
Grc
is the local mass Grashof number, Re
x
the local Reynolds number,
Rit
is represented as the local Richardson number of thermal,
Ric
is denoted as
the local Richardson number of mass,
K
is indicated as the parameter of per-
meability,
M
is designated as magnetic parameter,
Ec
is denoted as Eckert num-
ber,
Sc
is indicated as the Schmidt number whereas
( )
1, 2, , 7
i
i
φ
=
are con-
stants.
The parameters which are the engineering interest are skin-friction coeffi-
cient, local Nusselt together with Sherwood numbers. The nondimensional form
of local skin-friction coefficient
f
C
is
( )
( )
1
2
1
2Re 0
fx
Cf
φ
−
′′
=
(18)
Furthermore, using the thermophysical property of nanofluid, the local Nus-
selt number is converted through the resulting form
( )
( )
1
2
6Re 0
xx
Nu
φθ
′
= −
(19)
However, local Sherwood number is transformed into the succeeding form as:
( )
( )
1
2
Re 0
x
Sh s′
= −
(20)
3. Procedures of Numerical Solutions Using Nachtsheim and
Swigert Technique
The system regarding nonlinear boundary value problem which was represented
by Equations (11)-(13), together with corresponding boundary conditions Equa-
tions (14) and (15) is solved using Nachtsheim and Swigert [22] technique which
is used to find unspecific initial conditions. The transformed boundary value
system is re-transformed to initial value system and solved numerically employ-
ing sixth order Runge-Kutta initial value solver. The useful numerical methods
were described by Nachtsheim and Swigert [22] and sixth order Runge-Kutta in-
itial value solver by Al-Shimmary [23].
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4. Comparison of Numerical Results
For the accurateness of numerical results, comparisons are prepared considering
the special effects regarding velocity ratio parameter
λ
on velocity and tempera-
ture profiles. The results for Bachok
et al.
[18] which was published is compared
with verify of Bachok
et al.
[18] work by authors for copper (Cu)-water nanof-
luid flow along with Prnadtl number as Pr = 6.2, nanoparticle volume fraction as
ϕ
= 0.1 and velocity ratio parameter as
λ
= −0.5 as shown into Figure 2.
It is detected that, verified corresponding numerical results of first solution
are found excellent agreement. This favorable acceptable comparison indication
Figure 2. Comparison of first solution of (a) velocity distribution and (b) temperature
distribution for copper (Cu)-water nanofluid flow while Pr = 6.2,
ϕ
= 0.1,
λ
= −0.5 and
M
= 0.
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is to improve numerical approach of Nachtsheim and Swigert [22] for present
work. As well, the obtained numerical results for present work will report
graphically into next subsequent section and analyze thereafter.
5. Results and Discussion of Significant Parameters
The comprehensive numerical results are computed for water (H2O)-copper
(Cu) nanofluid flow including dissimilar values of nondimensional parameters
that described flow characteristics of the nanofluid. The influences of namely
magnetic parameter
M
, angle for inclination
α
, Eckert number
Ec
, fluid suction
parameter
fw
, nanoparticles volume fraction
ϕ
, Schmidt number
Sc
and permea-
bility parameter
K
are analyzed to describe flow characteristics. The values for
nondimensional parameters
Rit =
1,
Ric =
1,
M
= 0.5,
K
= 0.5,
α =
30˚, Pr = 6.2,
Ec
= 0.5,
Sc =
0.60,
fw
= 1.5,
ϕ
= 0.03 and
U∞
/
ν
= 1.0 are considered except oth-
erwise specified. The selective results are shown graphically with velocity, tem-
perature together with concentration flow fields. Moreover, the interest of engi-
neering terms as local skin friction coefficient
Cf
and local Nusselt number
Nux
together with local Sherwood number
Sh
are presented graphically.
The impacts for magnetic parameter namely
M
(
M
= 0, 1 and 2) on the copper
(Cu) and water (H2O) nanofluid flow fields are shown in Figure 3(a) and Figure
3(b) and Figure 4(a) and Figure 4(b). The fluid velocity for nanofluid flow rises
because of rise of concerning magnetic field as given away in Figure 3(a) and
alike result was established through Das
et al.
[7]. Because, as magnetic field
strength increases; the Lorentz force regarding magnetic field creates boundary
layer for nanofluid flow as thinner. On free stream velocity, magnetic lines of
forces move pass through the plate. The fluid which is decelerated by the viscous
force, receives an impulsion from magnetic field which counteracts a viscous ef-
fects. As results, velocity of considered nanofluid flow rises and converges to the
boundary together with rising magnetic field parameter. Additionally as detected
in Figure 3(b), the thickness of thermal boundary layer of considered nanofluid
flow rises owing to rise magnetic field because functional magnetic field which
has a tendency to heat the fluid owing to electromagnetic work reduces heat
transfer to the wall. Furthermore, skin friction coefficient namely
Cf
is estab-
lished to increases with increase magnetic field strength as given away in Figure
4(a). The reason for this is that, functioning magnetic field regarding nanofluid
flow tends to improve flow motion and thus to improve surface friction force. In
existence of growing magnetic field strength, the temperature gradient decreases
at the wall which in turn leads to a reduction in rate of heat transfer. Subse-
quently,
Nux
namely local Nusselt number which is schemed in Figure 4(b) de-
creases together with rise magnetic field strength.
The influence for
α
(
α
= 0˚, 30˚ and 60˚) which is angle for inclination for the
copper (Cu) and water nanofluid flow fields is revealed in Figure 5(a) and Fig-
ure 5(b). The angle regarding inclination is only in momentum Equation (12)
by cos
α
. For
α
= 0˚, plate assumes a vertical position whereas plate is horizontal
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Figure 3. Variation of (a) velocity and (b) temperature for effect of magnetic parameter
M.
Figure 4. Effects of magnetic parameter
M
on (a) local skin friction coefficient and (b) local Nusselt number against the stream-
wise distance.
Figure 5. Effects of angle of inclination
α
on (a) velocity and (b) local skin friction coefficient against the streamwise distance.
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for
α
= 90˚. The gravitational effects is maximum at a vertical position whereas
minimum at horizontal position. Consequently, momentum boundary layer
nanofluid flow decreases to increase of
α
. As a consequence this is observed into
Figure 5(a), maximum velocity of nanofluid flow is found at vertical position
while minimum velocity is found at horizontal position. Moreover, when, the
angle regarding inclination
α
increases, velocity of nanofluid flow decreases and
so its shear stress decreases. As per significance, local skin friction coefficient
namely
Cf
of nanofluid declines which is detected in Figure 5(b).
The variation of velocity, temperature together with concentration of wa-
ter-copper nanofluid flow for various values regarding fluid suction parameter
fw
(
fw
= 1.5, 3.0 and 4.5) are given away in Figure 6(a) and Figure 6(b). Simulta-
neously, as fluid suction of nanofluid increases then much number of nanofluid
passes through plate and subsequently concentration regarding nanofluid flow
field decreases at all points whereas local Sherwood number
Sh
increases which
are finding out in Figure 6(a) and Figure 6(b). The effects of nanoparticles vo-
lume fraction
ϕ
(
ϕ
= 0, 0.15 and 0.30) for water-Cu nanofluid flow are analyzed
in Figure 7(a) and Figure 7(b). With the increasing of nanoparticles volume
fraction
ϕ
, the thermal boundary layer breadth of nanofluid flow decreases as
seen in Figure 7(a). It is also observed from Figure 7(b) that with increase of
ϕ
,
local Nusselt number
Nux
increases for as much the temperature gradient at the
wall increased and as a result
Nux
increases.
On the other hand, the effect of the permeability parameter
K
(
K
= 0, 3 and 6)
on the momentum boundary layer as well as local skin friction coefficient
Cf
are
observed in Figure 8(a) and Figure 8(b). On the basis of the velocity variation
in Figure 8(a), it is seen that the velocity of the flow field within the velocity
boundary layer increases near the plate while decreases in the next and finally
converges to the boundary condition as the permeability of the porous medium
increases. However, Figure 8(b) shows the influence of permeability of the por-
ous medium on the local skin friction coefficient against the streamwise distance
x
. In Figure 8(b), it is found that the entire local skin friction coefficient in-
creases with the increase of permeability of the porous medium parameter.
Figure 6. Effects of fluid suction
fw
on (a) concentration and (b) local Sherwood number
Sh
against the streamwise distance.
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Figure 7. Effects of nanoparticles volume fraction
φ
on (a) temperature and (b) local Nusselt number
Nux
against the streamwise
distance.
Figure 8. Effects of permeability parameter
K
on (a) velocity and (b) skin friction coefficient
Cf
against the streamwise distance.
6. Conclusions
The effect of magnetohydrodynamic on mixed convective nanofluid flow with
viscous dissipation has been examined in surrounding porous medium. Conse-
quently, the concerning governing equations regarding nanofluid flow are con-
verted into nonlinear boundary layer equations by proper similarity transforma-
tions and solved numerically through sixth-order Runge-Kutta method with
Nachtsheim and Swigert [22] approaches. For validity of numerical results,
comparisons of numerical results remain prepared with Bachok
et al.
[18] and a
comparatively acceptable comparison is reached. The concerning numerical re-
sults mainly calculated for influence of magnetic parameter
M
, angle for inclina-
tion
α
, Eckert number
Ec
, fluid suction parameter
fw
, nanoparticles volume frac-
tion
ϕ
, Schmidt number
Sc
and permeability parameter
K
on flow field.
The following results on velocity, temperature together with concentration of
nanofluid flow fields including local skin friction coefficient
Cf
, local Nusselt
Md. N. Uddin et al.
DOI:
10.4236/jamp.2019.74065 980 Journal of Applied Mathematics and Physics
number
Nux
together with local Sherwood number
Sh
is deduced from the
present analysis:
The skin friction coefficient increases but local Nusselt number decreases due
to increasing values for magnetic parameter.
The local skin friction coefficient shows diminishing behavior for rising val-
ues angle of inclination.
The local Sherwood number of nanofluid flow displays raising behavior for
raising values of fluid suction parameter.
The local Nusselt number of nanofluid flow shows raising behavior for rais-
ing values of nanoparticles volume fraction.
The skin friction coefficient of nanofluid flow increases for increasing value
of permeability parameter.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this pa-
per.
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