1 s2.0 S1110016816302290 Main
1 s2.0 S1110016816302290 Main
1 s2.0 S1110016816302290 Main
H O S T E D BY
Alexandria University
ORIGINAL ARTICLE
Department of Mathematics, Statistics and Computer Science, G.B. Pant University of Agriculture and Technology,
Pantnagar, Uttarakhand 263145, India
KEYWORDS Abstract The purpose of present study is to identify the effects of viscous dissipation and suction/
Nanofluid; injection on MHD flow of a nanofluid past a wedge with convective surface in the appearance of
Viscous dissipation; slip flow and porous medium. The basic non-linear PDEs of flow and energy are altered into a
Wedge; set of non-linear ODEs using auxiliary similarity transformations. The system of equations together
Convective surface; with coupled boundary conditions have been solved numerically by applying Runge-Kutta-
Porous medium; Fehlberg procedure via shooting scheme. The influence of relevant parameters on non-
Suction/Injection dimensional velocity and temperature profiles are depicted graphically and investigated in detail.
The results elucidate that as enhance in the Eckert number, the skin friction coefficient increases,
while heat transfer rate decreases. The outcomes also specify that thermal boundary layer thickness
declines with an increase in suction parameter. Moreover, it is accelerated with augment in injection
parameter. The results are analogized with the study published earlier and it creates a fine concord.
Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an
open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Nomenclature
solid particles, which are suspended in a fluid. One of them is transfer and hydrodynamic properties of the flow [3,7–9].
two-phase model, this nanofluid model is applied by Sheik- Loganathan and Arasu [10] have introduced that heat and
holeslami and Abelman [2] for simulation of heat transfer mass transfer flow on non-Darcy magnetohydrodynamic flow
and nanofluid flow. Recently, Sheikholeslami and Ganji [3] with suction/injection, thermophoresis and porous medium.
have preferred two phase model, in which slip velocity between They depicted that concentration boundary layer is thinner
solid particles and fluid cannot be zero, for the simulation as increase in thermophoretic parameter. Yacob et al. [11] have
nanofluid flow and heat transfer performance of nanofluid. discussed steady flow of nanofluid over a static or moving
The second approach is single-phase model, in which both wedge by employing Tiwari and Das nanofluid model [6]. They
solid particles and fluid are in thermal equilibrium and solid found that skin friction coefficient with rate of heat transfer
particles are move with fluid with equal velocity. Various fac- boosts as raise in volume fraction of nanoparticles and wedge
tors may affect heat transfer enhancement, for example, Brow- angle. Moreover, Rahman et al. [12] have also implemented
nian diffusion, Brownian forces, sedimentation, gravity, Tiwari and Das approach to describe the influence of heat gen-
diffusion and friction between solid particles and fluid. In eration/absorption on two dimensional steady flow over a
the lack of any appropriate theoretical studies and experimen- wedge with convective surface for nanofluid. They found that
tal data in the literature to examine these aspects, the active Nusselt number enhances with raising the values of Biot num-
macroscopic two-phase model is not relevant for analyzing ber and slip parameter. The impact of mixed convection on
nanofluids [4–6]. If the primary significance is focused on heat boundary layer flow over an inclined plate in the presence of
transfer process, single-phase model, accounting for some of porous medium in nanofluid is intentional by Rana et al.
the above issues, is more expedient than two-phase model. [13]. Rashad et al. [14] inspected the parallel impact of mixed
Hence, the single-phase model is simpler and expedient in com- convection and porous medium on nanofluid boundary layer
putational steps. Furthermore, advanced characteristics of flow past a circular cylinder. They analyzed that both heat
nanofluid permit it to perform similar to a fluid that the con- and mass transfer coefficients are escalating function of either
ventional solid-fluid mixtures. Several studies have developed parameter of mixed convection or Biot number. Kuznetsov
the single phase model to find the numerical results of heat and Nield [15] have considered the classical problem of natural
Effect of viscous dissipation and suction/injection on MHD nanofluid flow 3117
convection of nanofluid flow past an erect plate in the existence magnetic field of steady strength is applied analogous to the
of porous medium. The effect of mixed convection on flow of direction of y or which acts toward the normal of flow direc-
boundary layer over a sphere in the occurrence of porous med- tion, and the external flow velocity UðxÞ ¼ axm , u0 is a con-
ium and nanofluid restraining gyrotactic microorganisms have stant and power-law parameter is m, which is also termed as
b
been proposed by Tham et al. [16]. The authors found that on Falkner-Skan parameter by 0 6 m 6 1, where, m ¼ 2b (see
escalating in biconvection parameter, heat and mass transfer Yacob et al. [11]). The total wedge angle is bp, and further
rate reduces. Kandasamy et al. [17] premeditated the impact we consider Tw is the temperature at wedge surface and T1 ,
of thermal radiation on an unsteady nanofluid MHD bound- is the ambient temperature. The wedge surface is heated by
ary layer flow over a porous wedge. They found that on convection from a hot fluid at temperature T0 , which gives a
enhancing the values of unsteady parameter temperature of coefficient of heat transfer h. The transparent porous medium
nanoparticle rises. Srinivasacharya et al. [18] have introduced is considered. The working fluid is a water based nanofluid
the effect of heat and mass transfer on magnetohydrodynamic including copper (Cu) as a spherical shape of solid nanoparti-
nanofluid flow over a wedge surface. They found that on grow- cle and presumed that regular fluid and nano-solid particle are
ing the numerical values of magnetic parameter both heat and in thermal equilibrium. The geometrical configuration of the
mass rate increase. Similarly, Grosßan et al. [19] have analyzed present model is shown in Fig. 1. Here the path of the surface
the consequence of natural convection and heat generation in a of wedge is x-axis and axis of y is normal to the surface of
square cavity packed with nanofluid in the presence of porous wedge. Under the boundary layer approximation and employ-
medium. Some exploration of heat transfer between two paral- ing Tiwari and Das nanofluid approach [6], the governing non-
lel plates utilizing nanofluid flow is indicated in Refs. [20,21]. linear PDEs of mass, momentum and heat transfer are
Sheikholeslami and Rashidi [22] have investigated the com- expressed as (see in Ref. [12]) follows:
bined effect of mixed convection and magnetic field flow in a
lid driven semi annuals containing Fe3O4-water nanofluid. @u @v
þ ¼0 ð1Þ
They found that as amplify in Richardson number heat trans- @x @y
fer rate enhances. In recent times many researchers have pro-
posed the influence of magnetic field on nanofluid flow due @u @u dU lnf @ 2 u g
u þv ¼U þ þ ðT T1 Þ
to a different geometries [23–37]. Further, the combined influ- @x @y dx qnf @y2 qnf
ences of MHD and FHD on an enclosure filled with Fe3O4- h i
bp
water nanofluid were studied by Sheikholeslami and Rashidi /ðqb Þsp þ ð1 /Þðqb Þbf sin
2
[38]. Recently, Sheikholeslami et al. [39] have utilized the lat-
tice Boltzmann model to simulate convective heat transfer
rB20 mnf
and nanofluid flow in a rectangular heated body. Presently, ðu UÞ ðu UÞ ð2Þ
qnf kqnf
control volume based finite element method (CVFEM) is
applied to simulate electric field effect on Fe3O4-ethylene gly- 2
col nanofluid due to a complex geometry were studied by @T @T @2T lnf @u Q0
u þv ¼ anf 2 þ þ ðT T 1 Þ
[40,41]. Uddin et al. [42] introduced the analytical simulation @x @y @y ðqCp Þnf @y ðqCp Þnf
of free convection flow of nanofluid over a vertical cone in ð3Þ
the presence of porous medium. They found that local Sher-
wood number enhances as increase in Brownian parameter. The related boundary conditions for the dimensionless
The impact of slip, mixed convection and joule heating flow velocity and temperature are as follows:
of nanofluid in a symmetric channel was deliberated by Hayat @u @T
et al. [43]. u¼l ; v ¼ vw ; knf ¼ qw ¼ hðTw TÞ at y ¼ 0
@y @y
The intent of the current model is to study the influence of
viscous dissipation, slip and suction/injection on two- u ¼ UðxÞ ¼ axm ; T ¼ T1 as y ! 1 ð4Þ
dimensional MHD boundary flow past a wedge with convec-
tive surface in the existence of porous medium utilized Cu- where vw the velocity of suction ð> 0Þ or injection ð< 0Þ; l the
water nanofluid. The impact of diverse governing parameters slip length, h the coefficient of heat transfer and ðu; vÞ compo-
on velocity and temperature are studied graphically and nents of velocity along with directions of ðx; yÞ respectively.
described in details. In the cognizance of the authors, no work
U (x ) = ax m
has been done in the past to simulate the effect of viscous dis-
sipation, slip and suction/injection on MHD nanofluid flow Tw
over a wedge surface with porous medium and convective v
u B0
boundary conditions. Hence our upshots are novel and y
authentic. The present study may have significant role in the vw g
fields of crude oil removal, heat exchangers, ground water pol- x
βπ 2
lution and storage of nuclear wastes, etc.
B0 Tw
Consider an incompressible steady convection flow of nano-
A porous wedge surface
fluid (Cu-water) varying over the wedge with uniform velocity
ðm1Þ
UðxÞ, and variable magnetic field BðxÞ ¼ B0 x 2 , where B0 is a Figure 1 Flow arrangement and coordinate system.
3118 A.K. Pandey, M. Kumar
The effective dynamic viscosity lnf ; the effective density qnf ; the the local Reynolds number of the base fluid, Q be the local
thermal diffusivity anf ; the heat capacitance ðqCp Þnf ; the ther- heat generation/absorption parameter, Bi be the Biot number,
mal expansion coefficient ðbnf Þ; and the thermal conductivity S be the mass flux parameter, for suction parameter ðS > 0Þ
knf of the nanofluid are defined as [12]: and for injection parameter ðS < 0Þ and A, B, C, D and E
9 are the constants respectively.
lbf
lnf ¼ ð1/Þ
knf
2:5 ; qnf ¼ ð1 /Þqbf þ /qsp ; anf ¼ ðqC Þ ;
>
>
p nf >
= mbf rB20 gbbf ðTw T1 Þx3 Grbf
Pr ¼ ;M ¼ ; Grf ¼ ; Nr ¼ 2 ;
k
ðqCp Þnf ¼ ð1 /ÞðqCp Þbf þ /ðqCp Þsp ; knfbf ¼
ksp þ2kbf 2/ðkbf ksp Þ
; abf qbf u0 v2 Rebf
ksp þ2kbf þ/ðkbf ksp Þ >
>
>
ðqb Þnf ¼ ð1 /Þqbf bbf þ /qsp bsp : ; lbf U2 hx 12
Ec ¼ ; Bi ¼ Re ;
ð5Þ ðTw T1 Þ jbf bf
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Moreover, the solid volume fraction is / and the subscripts Q0 x 2
bf and sp stand for the water and solid particles, respectively. Q¼ ; S ¼ vw ;
ðqCp Þbf U axm1 vbf ðm þ 1Þ
The thermophysical properties of water and Cu-nanoparticle
are specified in Table 1. qsp
A ¼ ð1 /Þ2:5 1 / þ / ;
The following non-dimensional similarity transformations qbf
are invoked: 2 3
!
1=2 1/ bsp /
1 m1 B¼4 þ 5;
u ¼ axm f 0 ; v ¼ ðm þ 1Þvbf axm1 fþ gf0 ; q bbf qbf
2 mþ1 1 / þ / qspbf ð1 /Þ qsp
þ/
1=2 1=2
2mbf Ux ðm þ 1ÞU T T1 C ¼ ð1 /Þ2:5 ;
w¼ fðgÞ; g ¼ y; hðgÞ ¼ ;
ðm þ 1Þ 2mbf x TW T1
" #
ð6Þ qsp ðCp Þsp ksp þ 2kbf þ /ðkbf ksp Þ
D¼ 1/þ/ ;E ¼ :
where wðx; yÞ is the stream function, which satisfied the mass qbf ðCp Þbf ksp þ 2kbf 2/ðkbf ksp Þ
Eq. (1), and is expressed as follows: ð11Þ
@w @w For the practical interest the physical quantities, the shear
u¼ and v ¼ : ð7Þ
@y @x stress rate and heat transfer rate are articulated as follows:
Now substituting Eqs. (5)–(7) into the Eqs. (2) and (3), respec- 2lnf @u knf x @T
tively, we acquire the subsequent system of coupled non-linear Cf ¼ ; Nux ¼ : ð12Þ
qU @y y¼0
2
kbf ðTw T1 Þ @y y¼0
ODEs:
Now, using Eqs. (6) and (7) in Eq. (12), the skin friction coef-
bp
f 000 þ Aff 00 þ Abð1 f 02 Þ þ ð2 bÞABNrh sin ficient (shear stress rate) together with the local Nusselt num-
2
ber (heat transfer rate) is converted into reduced skin friction
þ ð2 bÞðCM þ kAÞð1 f 0 Þ ¼ 0; ð8Þ coefficient and reduced Nusselt number, respectively:
Table 1 Thermophysical properties of water and Cu-nanoparticle (also see in Ref. [12]).
Physical properties q (kg/m3) Cp (J/kg K) j (W/m K) b 105 ð1=KÞ
Pure water 997.1 4179 0.613 21
Copper (Cu) 8933 385 400 1.67
Effect of viscous dissipation and suction/injection on MHD nanofluid flow 3119
0 1 0 1
y01 1
B y0 C B C
B 2C B y3 C
B 0C B B
C
C
B y3 C B y4 C
B C¼B ð14Þ
B y0 C B Ay2 y4 Abð1 y3 y3 Þ þ ðb 2ÞABNry5 sinðbpÞ þ ðb 2Þð1 y3 ÞðCM þ kAÞ C
C
B 4C B 2 C
B 0C B C
@ y5 A @ y6 A
y60 ðb 2ÞPrEQy5 PrDEy2 y6 Cknf Ecy4
1 2
and subsequent initial conditions are graphs satisfy the boundary conditions (4) and (10) asymptoti-
0 1 0 0
1 cally. Fig. 2 displays that velocity profile of Cu-water nanofluid
y1
increases with increase in Ec, corresponding to each value of
By C B C
B 2C B B
S C
C dimensionless variable g, in a particular range ð0 6 g 6 2:5Þ,
B C B pK ffiffiffiffiffiffiffiffiffi q1 C
B y3 C B C due to this reason velocity boundary layer thickness of the nano-
B C¼B ð2bÞ
C ð15Þ
By C B C fluid reduces. Fig. 3 reveals that temperature distribution profile
B 4C B q1 C
B C B
@ y5 A @ 1 þ pffiffiffiffiffiffi Cq2
A
of nanofluid ascends with enhancement in Eckert number, when
ð 2bÞBiE the value of horizontal axis ðg axisÞ lies between 0 6 g 6 1:5:
y6 q2 However, thickness of thermal boundary layer boosts, as
The system of first order ODEs (14) via initial conditions amplify in Eckert number.
(15) are solved using order of fourth-fifth RKF-integration Figs. 4 and 5 portray the consequence of porosity parame-
process and appropriate values of unknown initial conditions ter k, for Pr ¼ 6:2, M ¼ S ¼ 0:2; Ec ¼ Nr ¼ 5, Q ¼ 0:01,
q1 ; q2 are preferred and then numerical integration is applied. b ¼ 1, Bi ¼ K ¼ 1, on flow field and heat transfer process. It
is obvious from Fig. 4, as boosted in porosity parameter k,
Here we contrast the computed values of f 0 and h asg ¼ 1;
the velocity distribution graph escalates and velocity boundary
through the specified boundary condition f 0 ð1Þ ¼ 1 and
layer thickness reduces. Moreover, the rate of flow increases as
hð1Þ ¼ 0; and adjust the estimated values of q1 and q2 to gain
increase in porosity parameter k. Fig. 5 reveals that variation
a better approximation for result. The unfamiliar q1 and q2
of temperature with respect to porosity parameter. On observ-
have been approximated by Newton’s scheme such a way that
ing from this figure, as ascending in porosity parameter k, tem-
boundary conditions suited at highest numerical values of
perature profile graph regularly decreases. The graph also
g ¼ 1; with error less than 108 :
depicts that thermal boundary thickness becomes thinner as
enhancing in porosity parameter k.
4. Code validation The impact of slip parameter K, on flow and temperature
Profiles of nanofluid are illustrated in outlines 6 and 7 respec-
tively, for the fixed values of Pr ¼ 6:2;
In order to confirm the correctness of our code, we contrast our
M ¼ S ¼ k ¼ 0:2,Ec ¼ Nr ¼ 5; Q ¼ 0:01, b ¼ 1, Bi ¼ 1.
results of reduced Nusselt number ðh0 ð0ÞÞ, for the diverse val-
Fig. 6 shows that velocity of the nanoparticle sharply
ues of slip parameter K, with those obtained by Rahman et al.
increases with raise in slip parameter K; corresponding to every
[12] and found that results are in superior agreement, as pre-
value of non-dimensional variable g, in the range of
sented in Table 2. Hence employing of current code is authentic.
0 6 g 6 2:5, because velocity boundary layer thickness reduces
as ascending in values of slip parameter. As noticed from
5. Results and discussion Table 3, the skin friction coefficient reduces, on escalating in
slip parameter. Fig. 7 demonstrates the variation temperature
The effects of different objective parameters on f 0 ðgÞ and hðgÞ, profile graph with respect to slip parameter K: It is seen from
furthermore, on flow and rate of heat transfer are thoroughly this outline thermal boundary layer thickness continuously
determined with preset mesh size Dg ¼ 0:001; where 0 6 g 6 3: diminishes as magnify slip parameter. Furthermore, Table 3
It is examined that for the regular fluid ð/ ¼ 0Þ, the constant indicates that Nusselt number enhances for higher values of K:
parameters takes a unit value i.e. A ¼ B ¼ C ¼ D ¼ E ¼ 1: Figs. 8 and 9 present the velocity and temperature distribu-
The coefficient of skin friction and heat transfer rate is tion profiles of nanofluid for the various values of suction
revealed in Table 3 for diverse values of Eckert number Ec, por- parameter ðS > 0Þ. It is detected that from these graphs veloc-
ous medium parameter k, slip parameter K, and suction/injec- ity of Cu-water nanofluid escalates continuously with increase
tion parameter S. From the table it is detected that heat in suction parameter, for all values of g, in the range of
transfer rate rises with raise in the numerical values of both slip 0 6 g 6 2:5, while temperature profile graph decreases as aug-
and porosity parameter. The impacts of dissimilar governing menting in suction parameter, and consequently both the coef-
factor on the dimensionless velocity along with temperature ficient of skin friction and Nusselt number are escalating
are depicted in Figs. 2–11. In the present study we consider Cu functions of mass flux parameter, as mention in Table 3.
like a nanoparticle and water is equivalent to base fluid. The The velocity and temperature profile graphs of injection
influence of Eckert number Ec, on velocity and temperature pro- parameter ðS < 0Þ are shown in Figs. 10 and 11, respectively.
files for Pr ¼ 6:2, M ¼ S ¼ k ¼ 0:2; Nr ¼ 5, Q ¼ 0:01, b ¼ 1, From Fig. 10, as enhancing in injection parameter ðS < 0Þ,
Bi ¼ K ¼ 1, are depicted in Figs. 2 and 3, respectively. These the nanofluid velocity increases in the region of 0 6 g 6 2:5,
3120 A.K. Pandey, M. Kumar
Table 2 Comparison of reduced Nusselt number ðh0 ð0ÞÞ, for Pure water and Cu-water for various values of K when
Pr ¼ 6:2; / ¼ 0:1; Bi ¼ 1; Nr ¼ 5; M ¼ 0:2; Q ¼ 0:01; b ¼ 0:1; Ec ¼ S ¼ k ¼ 0.
Fluids Bi K h0 ð0Þ
Rahman et al. [12] Present results
Pure water 0.1 0 0.087810 0.08778988
0.2 0.089159 0.08915740
0.5 0.090340 0.09033630
1 0.091357 0.09135269
Cu-water 1 0 0.477845 0.47781933
0.2 0.509338 0.50929108
0.5 0.539131 0.53912255
1 0.566410 0.56636414
Table 3 Several values of skin friction coefficient ðf 00 ð0ÞÞ; and Nusselt number ðh0 ð0ÞÞ; for Cu-water when
Pr ¼ 6:2; / ¼ 0:1; Bi ¼ 1; Nr ¼ 5; M ¼ 0:2; Q ¼ 0:01; b ¼ 0:1.
Ec k K S f 00 ð0Þ h0 ð0Þ
0.2 0.2 1 0.2 0.752519 0.68451
30 0.7543109 0.677129
60 0.7560609 0.669726029
100 0.7584109 0.659826029
500 0.78082509 0.562086
5 0.4 1 0.2 0.788999 0.6863908
0.6 0.8179709 0.688708
1 0.8624259 0.6920408
2 0.9335519 0.697000408
5 0.2 2 0.2 0.499348 0.69734
2.5 0.426719 0.70084
3 0.372379 0.70334
6 0.21064 0.71023
20 0.069424 0.71568
5 0.2 1 0.1 0.7343309 0.536271
0.3 0.739719 0.405691
0.5 0.756961 0.265289
0 0.73730519 0.592304
0.5 0.7904905 0.77592
1 0.862459 0.86116
1 1.5 0.927501459 0.9060916
1.0 0.5
Pr = 6.2, M = S = λ = 0.2, Nr = 5, Q = 0.01, β = 0.1, Bi = K = 1
Ec=0.2
Ec=30 0.4
0.9 Ec=60
zoom
Ec=100
Ec=500
0.3
0.8
f ' (η ) θ (η ) Ec=0.2
Ec=30
0.2 Ec=60
0.7 zoom Ec=100
Ec=500
0.1
0.6
Pr = 6.2, M = S = λ = 0.2, Nr = 5, Q = 0.01, β = 0.1, Bi = K = 1
0.0
0.0 0.5 1.0 1.5 2.0 2.5
η 0.0 0.5 1.0 1.5
η
Figure 2 Velocity profiles for numerous values of Ec.
Figure 3 Temperature profiles for numerous values of Ec.
Effect of viscous dissipation and suction/injection on MHD nanofluid flow 3121
1.0 0.33
0.30 Pr = 6.2, M = S = λ = 0.2, Ec = Nr = 5, Q = 0.01, β = 0.1, Bi = 1
0.27
0.9 K=2
0.24
λ = 0.4 K=2.5
0.21 K=3
λ = 0.6
0.8 0.18 zoom K=6
λ =1
θ (η ) K=20
f ' (η ) 0.15
λ =2 0.12
0.7
0.09
0.06
0.6 0.03
Pr = 6.2, M = S = 0.2, Ec = Nr = 5, Q = 0.01, β = 0.1, Bi = K = 1 0.00
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5 2.0 2.5 η
η
Figure 7 Temperature profiles for numerous values of K.
Figure 4 Velocity profiles for numerous values of k.
1.0
0.35
Pr = 6.2, M = S = 0.2, Ec = Nr = 5, Q = 0.01, β = 0.1, Bi = K = 1 S=0
0.9
0.30
S=0.0
0.25 λ = 0.4 S=0.5
0.8
λ = 0.6 S=1
0.20 f '(η )
S=1.5
λ =1
θ (η ) zoom
0.7
0.15 λ =2
0.10
0.6
0.05 Pr = 6.2, M = λ = 0.2, Ec = Nr = 5, Q = 0.01, β = 0.1, Bi = K = 1
0.5
1.02 Pr = 6.2, M = λ = 0.2, Ec = Nr = 5, Q = 0.01, β = 0.1, Bi = K = 1
0.81
0.0
0.78 0.0 0.5 1.0 1.5
η
0.75
Pr = 6.2, M = S = λ = 0.2, Ec = Nr = 5, Q = 0.01, β = 0.1, Bi = 1
0.72 Figure 9 Temperature profiles for numerous values of suction
0.0 0.5 1.0 1.5 2.0 2.5 parameter (S 0).
η
0.9 S=-0.1
and due to this cause momentum boundary layer thickness S=-0.3
declines with an increase in injection parameter. Fig. 11 dis- S=-0.5
0.8
plays the temperature profile of Cu-water nanofluid between f '(η )
dimensionless temperature hðgÞ, versus dimensionless variable 0.7
g, for several values of injection parameter. It is examined that
on escalating the values of injection parameter, temperature 0.6
outline escalates corresponding to each value of g, in a definite
range ð0 6 g 6 2Þ. This graph also indicates that thermal Pr = 6.2, M = λ = 0.2, Ec = Nr = 5, Q = 0.01, β = 0.1, Bi = K = 1
0.5
boundary thickness increases with an increase in injection 0.0 0.5 1.0
η 1.5 2.0 2.5
parameter. Moreover, Table 3 indicates that the skin factor
escalates on growing in the numerical values of injection Figure 10 Velocity profiles for numerous values of injection
parameter (S 0).
3122 A.K. Pandey, M. Kumar
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