Mixed Convection on Stagnation Point Flow of MHD Casson Fluid
Towards Stretched Sheet
M. Sreedhar Babub*, V. Venkata Ramanac,
b,cDepartment of Applied Mathematics, Yogi Vemana University, Vemana Puram,
Ganganapuram, Kadapa, Andhra Pradesh-516005, India
Email: [email protected], [email protected]
Abstract
In this article we investigated non-Newtonian fluid and its MHD stagnation point flow
with the thermal and mass buoyancy effects. A non-Newtonian fluid model obtained from the
Casson fluid model. By adopting similarity metamorphosis, the governing equations are
transferred as nonlinear ordinary differential equations. Those equations are solved
numerically by applying the R K Gill method forward with shooting technique. The nature of
Local Nusselt number, skin friction coefficient, Sherwood number and the profiles of
velocity, temperature and the concentration exhibited pictorially and deliberated by taking
assisting and opposing flow constraints into consideration. The numerical results turned up
are distinguished with formerly published work concealed by limiting cases and found to be
in remarkable agreement.
Keywords: MHD, stagnation point flow, heat and mass transfer, Casson fluid model,
assisting and opposing flows.
*Corresponding Author E-mail: [email protected]
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1. Introduction
Issues identified with convective boundary layer flow assume a key job in engineering
and industrial exercises. These flows are utilized to oversee thermal impacts in numerous
modern outlets, for example, motor cooling frameworks, gadgets, and PC power. Sakiadis
first concentrated the boundary layer flow on a ceaseless strong surface moving at a
consistent speed [1]. On a piece, the boundary layer flow is totally not quite the same as the
Blasius flow as far as diversion condition fluids. Tsou et al, They checked the hypothetical
forecasts of Newtonian liquids proposed by Sarkidis. [2] Viscous fluid stream in past
extended movies was an exemplary issue in fluid elements. Crane initially examined the
convective boundary layer stream on a stretchable sheet [3]. Guptha and others examined. [4]
On the extended sheet, suction or passing up heat and mass exchange. They likewise got
temperature and fixation profiles by considering isothermal moving plates. Chen and Char [5]
examined the stream and heat move of a layered limit layer, which is performed by suction or
blow trim of a ceaseless and stretchable direct sheet under a recommended divider
temperature and heat stream. Into, Gangadhar et al. [6] examined the progression of
nanofluids in an unsteady limit layer outside the extended surface. Wang [7] thought about
the state of the sanctioned divider temperature and concentrated the progression of stale
stream to the therapist film. Ishak et al. [8-9] contemplated the mixed convection of vertical
and persistent extended tablets, the progression of MHD stagnation focuses to extended
tablets, and the progression of slacking focuses to vertical and straight extended tablets.
While thinking about a recommended divider temperature or heat stream, numerous analysts
proceed with this kind of model utilizing viscoelastic fluids or micropolar and viscous
liquids. As of late, Khan et al. contemplated the progression of stagnation focuses on stretch
sheets. [10] Hayat et al. They considered the progression of a second-degree liquid on an
extended surface under Newtonian heating. [11]
In fluid mechanics, non-Newtonian fluid stream issues assume a significant job in
different fields, for example, therapeutic innovation and natural fluid mechanics. What's
more, the heat move of non-Newtonian fluids has a wide scope of uses in polymer
arrangements, compound designing, power building examined by Ibrahim and Gaddisa,
obstruction decrease, sweat cooling, and polymer smell compression. Powell limit layer
streams, for example, plastic preparing Non-Newtonian Eyring-based liquids containing
nanoparticles contain nanoparticles Furthermore, Hirschhorn et al. They analyzed MHD
stream under slip limit conditions. [13] On the tablet. As of late, Bear et al. They considered
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viscoelastic fluids on unbending bodies. [14] Li et al. [15] numerically inspected non-
Newtonian liquids in micro channels. Additionally, Reddy et al, They examined the impacts
of the entropy heat of non-Newtonian liquids [Sixteen]. Casson [17] presented another liquid
called Casson viscoplastic fluid, which is a non-Newtonian liquid. Casson presented the
investigation while concentrating the stream condition of printing ink shade suspensions.
Milk, gelatin, soup, ketchup, human blood, concentrated natural product juice, nectar are
entering the Casson fluid classification, and this fluid is more appropriate for rheological
information than common place viscoplastic models numerous materials. A Casson fluid
shear conveyance fluid was acquired. The fluid has unbounded consistency and zero shear
rate, and no stream happens beneath as far as possible and zero thickness endless shear rate
[18]. A few specialists [19, 20] have contemplated the issue of Casson fluids by utilizing
fitting changes. Abbas et al, They dissected comparative issues with the progression of heat
move fluid over the surface/stretch sheet for each situation. [21] and Mustafa et al. [22]
Mythili and Sivaraj assessed the physical impacts of Casson fluids in various geometries
under the requirements of transient stream [23], while Das et al. Created MHD consequences
for plates under a similar transient stream. [24] Nadeem et al. They let MHD's Casson fluid
move through a bit of paper. [25] Raju et al. Raju et al. They dissected the heat and mass
exchange of Casson fluids from porous extended hot surfaces. [26] Gangadhar et al.
contemplated the dissemination and heat impacts of digression hyperbolic fluids on extended
surfaces. [27] An ongoing report by Gangadhar et al. [28] The Cattaneo-Christov heat stream
model has been utilized to instigate thick dissemination impacts in non-Newtonian Maxwell
fluids.
Right now, impacts of heat and mass lightness on non-Newtonian fluids are consolidated
to concentrate on the investigation of heat and mass exchange. The Casson fluid model was
utilized to analyze the rheological properties of shear diminishing and shear thickening of
non-Newtonian fluids. This examination might be valuable for comparable investigations in
heat, mass and modern fields.
2. Mathematical Formulation
Consider a Cartesian facilitate framework and a limit layer of a Casson fluid. The fluid
can't go through a consistently moving extended surface situated on the y = 0 plane. The fluid
can't consolidate through laminar free convection close to the stagnation point and is
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incompressible Figure 1. The outer stream speed to the limit layer is xcue , and the direct
speed of the stretch sheet is xauw . At the point when two comparable and inverse powers
are presented, the sheet extends at a speed relative to the good ways from the starting point.
At the point when a consistent attractive field B0 is applied, the initiated attractive field is
disregarded on the grounds that its Reynolds attraction is little. It is expected that the x-pivot
is toward the extended surface and that the hub is symmetrical to the extended surface.
Rheological conditions speak to isomorphic progressions of Casson fluids (Eldabe and Salwa
[30])
cij
y
B
c
y
B
ij
ep
eijp
,2
2
,2
2
(1)
Here, ijijee,……, ije speaks to the
thji ),( part of the strain rate, py speaks to the fluid wet
blanket power, speaks to the result of the strain rate segment and itself, c speaks to the
basic worth, and the consistency of the item dependent on non-Newtonian model and B
speak to Dynamic plastic thickness of non-Newtonian liquids.
The conditions for the mass, force, energy and types of the limit layer are as per the
following.
0
y
v
x
u , (2)
),()()(
11
2
0
2
20
CCgTTguuB
y
u
dx
duu
y
uv
x
uu
CTe
e
e
(3)
,2
2
y
T
y
Tv
x
Tu
(4)
,2
2
y
CD
y
Cv
x
Cu
(5)
Where speed segments in x -and y -bearings spoke to by u and v separately. In eq. (3), the
assisting flow represented by ‘+’ symbol while the opposing flow represented by ‘_’ symbol
and also the parameter δ can be positive or negative which is in eq. (3), as assumed by Bejan
and Khair [30] and Khair and Bejan [31], the kinematic viscosity /0 , the viscosity of
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the fluid represented by0 , the non-Newtonian Casson parameter represented by
ycB p/2 , the electrical conductivity represented by σ, the coefficient of thermal
expansion represented by T , the coefficient of expansion with concentration represented by
C , fluid temperature represented byT , fluid concentration represented by C , fluid density,
heat capacity and thermal conductivity represented by , pc , k respectively and thermal
diffusivity represented by pck /
The boundary conditions in the present issue are
www CCTTvxaxuu ,,0,)( , at 0y (6)
,,, CCTTuu e As y (7)
In the equation given above, wT and wC are temperature and fixation at the divider individually.
The accompanying dimensionless amounts are utilized to change the administering
conditions.
CC
CC
TT
TTfxvay
v
a
ww
)(,)(,, (8)
In the above equations the stream function represented by
Equations (3) - (5) are altered as
01
13
32
2
2
2
22
f
a
cfM
a
cfff (9)
0Pr
12
2
ff (10)
01
2
2
ff
Sc (11)
Here the subsidiary as for notation was prime. The Prandtl number /Pr , where
denotes thermal diffusivity, the Hartmann number a
BM
2
02 , the local Reynolds number
)(Re
xux w
x , the thermal buoyancy parameter2Re x
Gr ,the Grashof number
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2
3)(
xTTgGr wT
, the solutal buoyancy parameter2Re x
Gc , the local Grashof number
2
3)(
xCCgGc wC
and the Schmidt number DSc / .
The boundary conditions which are represented by equations (6) and (7) are transformed as
,1)0(,1)0(,1)0(
,0)0(
ff (12)
,0)(,0)(,)(
a
cf (13)
The physical quantities skin friction coefficient, local Nusselt number and Sherwood number
are represented by Cf, Nu and Sh respectively and are defined as
,)(
,)(
,2
CCD
qxSh
TTk
qxNu
uC
w
m
w
w
w
w
f
(14)
Here w stands for surface shear stress, wq denotes surface heat flux and mq denotes surface
mass flux and are prearranged by
000
,,2
y
m
y
w
yc
y
Bwy
CDq
y
Tkq
y
up
(15)
In the above equations denotes dynamic viscosity, k and D denotes thermal and mass
conductivity’s.
By applying similarity transformations to equation (7), we get
,)0(
Re,
)0(
Re
,)0(1
1Re2
2
xx
fx
ShNu
fC
(16)
Where
2
Rexa
x denotes local Reynolds number.
3.1The Runge–Kutta–Gill method:
By utilizing R-K-Gill strategy along the shooting system and Newton-Raphson method, the
nonlinear coupled conditions (9)–(11) with the limit conditions (12)- (13) are fathomed
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numerically. To this technique, we characterize
765432
2
21 ,,,,,, YYYYYf
Yf
Yf
(17)
72
2
652
2
433
3
22
2
1 ,,,,,, FFFFFf
Ff
Ff
(18)
Substitute into conditions (9) - (11) are transforms into an arrangement of synchronous
conditions with the qualities which are characterize in conditions (17) and (18),
,21 YF (19)
,32 YF (20)
642
2
2
2
31
2
23
11 YY
a
c
a
cYMYYYF
(21)
,54 YF (22)
),Pr( 51425 YYYYF (23)
,76 YF (24)
),( 71627 YYYYScF (25)
The boundary conditions given in (12) and (13) are altered as
,1)0(,1)0(,1)0(,0)0( 6421 YYYY (26)
,0)(,0)(,/)( 642 YYacY (27)
Here, taken as 10 , based on physical parameters.
The initial conditions are noted as tYY )0(,)0( 53 and sY )0(7 . By the utilization of
Newton–Raphson method, we calculate the values of t, and s such that they satisfy
solutions of equations (19)-(25) and also fulfill the outer boundary conditions (26) – (27). We
start this with the initial estimated values ))0(),0(),0(( st by the shooting method.
Calculations of Newton–Raphson strategy are extended to consolidate the halfway
subsidiaries concerning measurement of every factor. Coming up next are the subsidiaries of
).,..,,( 521 FFFF with respect to t, and s are as follows:
),...,,(),...,,(),...,,( 2017161512111076 FFFFFFFFFFFF tt (28)
So, we need to find ,0,0,0 st FFF respectively. Following Cebeci and Keller [32],
yields a system of algebraic equations which satisfy the boundary conditions when ξ = 0.
0,0,0 ststfsftff ststst (29)
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The system of equations which are represented by equation (29) can be written in matrix
equation AX = B:
f
s
t
fff
st
st
st
(30)
The above system of equations was solved by using Cramer’s rule and the next
approximation of t, and s can be calculated by utilizing the below formulas:
)det(
),(det
,)det(
),(det,
)det(
),(det
)()(
)()()()(
A
JIAss
A
JIAtt
A
JIA
Boldnew
BoldnewBoldnew
(31)
By the values of t, and s which are evaluated by the above equations, we apply the fourth-
order R-K-Gill method to evaluate the solutions of first order ordinary differential equations
F1, F2, . . . , F20. Following Gill [33] and R-K formula is
,6
1
2
22
3
1
2
22
3
1
6
143211 hkhkhkhkYY ii
(32)
iYFk 1 (33)
12
2k
hYFk i (34)
213
2
2121
2
1kkYFk i (35)
323
2
21
2
2kkYFk i (36)
In the above conditions h speaks to step size. Right now, acquired numerical arrangements
are seen as good when step size h= 0.01. The relative contrast between two emphases is
greatest by the pre-allotted resilience 610 to assembly. The iterative plan is abrogated
when the distinction meets the assembly criteria.
3.2 Validation of the Numerical Procedure:
To state these ends, they are connected to ends previously recorded in the overview.
Connection is performed for explicit qualities of stream parameters. These related
outcomes are appeared in Table 1, and Shehzad et al. have distributed these outcomes.
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[34] Relevance is adequate and is built up in close participation with existing ends and
with subtle blunders.
Table 1 Examination of skin friction coefficient with the reachable endings in writing for
various estimations of β and M on account of non-Newtonian fluid
).0/Pr( caSc
β
M
−(1 +1
𝛽) 𝑓′′(0)
Shehzad et al. [34] Present study
0.5
0.8
1.4
2.0
0.8
0.5
0
0.5
1.0
1.5
1.93649
1.67705
1.46385
1.36931
1.50000
1.67705
2.12132
2.70416
1.936768
1.677122
1.463867
1.369314
1.500364
1.677122
2.121321
2.704163
4 Analysis and computations:
By utilizing the RK Gill numerical strategy technique, the change conditions existing
apart from everything else, energy, and material conditions spoke to by conditions (8), (9),
and (10) are spoken to by conditions (11) and (12). The count aftereffects of RK Gill strategy
were gotten by utilizing MATLAB. The heat and mass exchange stream qualities are
demonstrated graphically, and determined and showed as tables as different parameter
esteems, specifically a/c, β, M, λ, δ, Pr, and Sc. The information Pr = 0.7, β = 0.5, M = 0.5,
λ = 1, δ = 0.5, and Sc = 0.66 (except if in any case noted) are utilized in all figures spoke to
by Figure 2-19.
Fig. 2 shows the speed bends of various impact esteems of the speed proportion c/a,
where “a” speaks to the extending pace of the limit surface and “c” (c> 0) speaks to the
pressure speed c of the stagnation point stream. At the point when c/a> 0, the stream is made
out of a transformed limit layer structure with a similar thickness as c/a. At the point when c/a
<1, the speed of the stretch sheet c�̅� is littler than the outside speed of the surface a�̅�. On the
off chance that c/a <1 the thickness of the limit layer speeds up proportion c/an increments.
Figure 3 talks about the adjustments in the Casson parameter β in the assistant and convection
speed bends. Here, all qualities of β speak to Casson liquid, and endlessness for β speaks to
Newtonian liquid. It very well may be seen that for the contrary stream case, a diminishing in
the estimation of the Casson parameter brings about a decrease in the thickness and energy of
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the speed constraining layer, and it is seen that the contrary pattern is useful for the stream
case. It has likewise been seen that Casson liquids have lower transient speed esteems than
Newtonian liquids. For the helper stream and the contrary stream, the speed bends of various
qualities of M are appeared in FIG. 4. It very well may be seen that the speed circulation
diminishes on account of helper stream and increments on account of inverse stream. Here, M
= 0 implies that there is no attractive field. This is because of the way that the use of an
attractive field to an undermined conductive liquid makes an obstruction called the Lorentz
power; this power makes the liquid speed decline. The difference in the speed bend regarding
the parameters of heat lightness (λ) and mass (δ) is appeared in Figures 5 and 6, separately.
These charts show that the speed increment with expanding λ and δ, which helps the stream,
while the contrary pattern is watched for the contrary stream. Effect of the Prandtl Pr is
number appeared in Figure 7 on the assistant and inverse procedures. An expansion in the
Prandtl number diminishes the speed of the liquid and the thickness existing apart from
everything else limit layer, which helps the stream, and the contrary movement is seen in the
contrary stream. Right now, synthesis of hydrogen is considered as follows: Sc = 0.22, water
fume SC =0.66 and carbon dioxide Sc = 0.94. Figure 8 talks about the impact of these
synthetic concoctions on the speed profile. He additionally called attention to that within the
sight of heavier species (Schmidt Lager number), the speed of the liquid in the limit layer
diminishes.
Figure 9 delineates the connection between the temperature bend and β, assistant and
inverse streams. These temperature bends start at the point of confinement of θ and afterward
reduction to zero. To help with the stream, the temperature bend is diminished as the Casson
β parameter increments. What's more, the temperature profile of Casson liquid is higher than
that of Newtonian liquid. For Carson liquids, this change is more noteworthy than for
Newtonian liquids. It was likewise recognized that the temperature profile of Carson liquid
(β> 0) was fundamentally not the same as Newtonian liquid (β = ∞). On account of
countercurrent, the temperature appropriation and the thickness of the heat limit layer become
more slender as β increments. It tends to be seen from Fig. 10 that on account of helper
stream (λ> 0), the temperature and thickness of the heat limit layer decline with the expansion
of λ, in spite of the fact that the contrary pattern will be distinguished on account of inverse
stream (λ<0). In FIG. 11, it is recognized that the thickness of the heat limit layer diminishes
with the expansion of δ on account of the helper stream (δ> 0), and switches on account of
the countercurrent (δ <0). From Figure 12, the adjustment in Pr can be distinguished in the
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temperature bend to help and contradict stream. What's more, higher Pr gauges bring down
the heat conductivity and in this way the temperature of the two streaming fluids.
The impact of β on the fixation conveyance of helper and convection is appeared in Figure
13. It tends to be seen from this figure as the helper stream area and β in β increment, the
convergence of the liquid declines. A contrary pattern was seen in the contrary stream zone.
It very well may be seen from FIGS. 14 and 15 that the convergence of the limit layer
thickness diminishes with the expansion of the heat lightness and mass lightness parameters,
and the contrary example is seen on account of invert stream. FIG. 16 shows a fixation profile
of Schmidt Sc. With the advancement of Sc, the diffusivity of the substance diminishes, so
the focus circulation diminishes. Figures 17-19 show the obstruction conduct, heat move and
mass exchange rate at various qualities of the assistant stream and the Carson parameter β in
inverse ways. An expansion in the Carson parameter β brings about a decline in the
estimation of the skin contact coefficient, a diminishing in the pace of heat and mass
exchange to convection, and a convective conduct is recognized under the state of assistant
stream.
5. Conclusion:
The motivation behind this investigation was to read mixed convection MHD for heat and
mass exchange on extended surfaces. To this end, a non-Newtonian liquid was acquired from
the Casson liquid model. The finishes of this examination are as per the following.
1. The thickness and speed appropriation of the momentum boundary layer brings about
a decrease in Hartmann number, thermal buoyancy, mass buoyancy, Prandtl number,
or Schmidt number to help stream, and the other way around on account of inverse
stream. Furthermore, for helper and inverse stream limitations, the speed circulation
and the thickness of the beat restricting layer increment as the speed proportion
parameter increments.
2. As far as possible stream, where Casson parameters are utilized, decrease the
thickness of the limit layer and lessen the temperature of the liquid, and increment the
heat or mass lightness parameters. For the contrary stream confinement, the
temperature of the liquid will increment with the Casson parameter, while the
temperature of the liquid will increment with the heat or mass convection parameter.
For the assistant stream limit and the contrary stream limit, as the quantity of Prandtl
expands, the temperature of the liquid and the thickness of the heat limit layer decline.
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3. To help stream conditions, on account of Casson parameters, both the grouping of the
liquid and the thickness of the limit layer are expanded, while the heat lightness or
mass lightness parameters are expanded. For inverse stream conditions, the
temperature of the liquid increments as the Casson parameter, heat convection, or
mass convection parameter increments. For assistant stream conditions and inverse
stream conditions, liquid fixation and focus boundary layer thickness decline with
expanding Schmidt number.
4. At last, as far as possible stream, expanding Casson parameters builds surface shear
pressure and heat and mass exchange rates. For inverse stream limitations, these
outcomes are turned around.
Fig. 1: Physical model and facilitate framework
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Fig. 2: Velocity distribution )(f for dissimilar standards of a/c.
Fig. 3: Velocity distribution )(f for dissimilar standards of β.
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Fig. 4: Velocity distribution )(f for dissimilar standards of M.
Fig. 5: Velocity distribution )(f for dissimilar standards of λ.
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Fig. 6: Velocity distribution )(f for dissimilar standards of δ.
Fig. 7: Velocity distribution )(f for dissimilar standards of Pr.
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Fig. 8: Velocity distribution )(f for dissimilar standards of Sc.
Fig. 9: Temperature distribution )( for dissimilar standards of β.
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Fig. 10: Temperature distribution )( for dissimilar standards of λ.
Fig. 11: Temperature distribution )( for dissimilar standards of δ.
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Fig. 12: Temperature distribution )( for dissimilar standards of Pr.
Fig. 13: Attentiveness distribution )( for dissimilar standards of β.
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Fig. 14: Attentiveness distribution )( for dissimilar standards of λ.
Fig. 15: Attentiveness distribution )( for distinct values of δ.
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Fig. 16: Attentiveness distribution )( for dissimilar standards of Sc.
Fig. 17: Skin friction coefficient fx C2/1Re for dissimilar standards of β for both flows.
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Fig. 18: Nusselt number Nux
2/1Re for dissimilar standards of β for both flows.
Fig. 19: Sherwood number Shx
2/1Re for dissimilar standards of β for both flows.
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Table 1 Examination of skin friction coefficient with the reachable endings in writing for
various estimations of β and M on account of non-Newtonian fluid
).0/Pr( caSc
β
M
−(1 +1
𝛽) 𝑓′′(0)
Shehzad et al. [34] Present study
0.5
0.8
1.4
2.0
0.8
0.5
0
0.5
1.0
1.5
1.93649
1.67705
1.46385
1.36931
1.50000
1.67705
2.12132
2.70416
1.936768
1.677122
1.463867
1.369314
1.500364
1.677122
2.121321
2.704163
References:
[1] B.C. Sakiadis, Boundary-layer behavior on continuous solid surfaces: I. Boundary-
layer equations for two-dimensional and axisymmetric flow,AIChE J. 7(1961) pp.26-28.
https://doi.org/10.1002/aic.690070108
[2] F.K. Tsou,E.M. Sparrow, R.J. Goldstein,Flow and heat transfer in the boundary layer
on a continuous moving surface, Int. J. Heat Mass Transf. 10(1967) pp.219-235.
https://doi.org/10.1016/0017-9310(67)90100-7
[3] L. J., Crane, Flow past a stretching plate, Z. Angew. Math. Phys. 21 (1970)pp.645-647.
https://doi.org/10.1007/BF01587695
[4] P.S. Gupta, A.S. Gupta, Heat and mass transfer on a stretching sheet with suction or
blowing, Can. J. Chem. Eng. 55(1977) pp.744-746.https://doi.org/10.1002/cjce.5450550619
[5] C.K. Chen, M. Char, Heat transfer on a continuous, stretching surface with suction
and blowing, J. Math. Anal Appl. 135 (1988) pp 568-580. https://doi.org/10.1016/0022-
247X(88)90172-2
[6] K. Gangadhar, T. Kannan, G. Sakthivel, K. Dasaradha Ramaiah, Unsteady free
convective boundary layer flow of a nanofluid past a stretching surface using a spectral
relaxation method, International Journal of Ambient Energy.
(2018).https://doi.org/10.1080/01430750.2018.1472648
Journal of Xi'an University of Architecture & Technology
Volume XII, Issue III, 2020
Issn No : 1006-7930
Page No: 2725
[7] C.Y. Wang,Stagnation flow towards a shrinking sheet, Int. J. Non-Linear Mech.
43(2008) pp.377-382. https://doi.org/10.1016/j.ijnonlinmec.2007.12.021
[8] A. Ishak, R. Nazar, I. Pop, Mixed convection on the stagnation point flow toward a
vertical, continuously stretching sheet, J. Heat Transf. 129(2007)pp.1087-1090.
https://doi.org/10.1115/1.2737482
[9] A. Ishak, K. Jafar, R. Nazar, I. Pop, MHD stagnation point stream towards an extending
sheet,Phys.A,Stat.Mech.Appl.388(2009)pp.3377-3383.
https://doi.org/10.1016/j.physa.2009.05.026
[10] A. Q.Khan, A.Rasheed,Mixed Convection Magnetohydrodynamics Flow of a Nanofluid
with Heat Transfer: A Numerical Study, Mathematical Problems in
Engineering.2019(2019)8129564. https://doi.org/10.1155/2019/8129564
[11] T. Hayat, Z. Iqbal, M. Mustafa,Flow of a second grade liquid over an extending surface
with Newtonian warming, J. Mech. 28(2012) pp.209-216.
https://doi.org/10.1017/jmech.2012.21
[12] W. Ibrahim and G. Gadisa, Finite component Method arrangement of Boundary Layer
Flow of Powell – Eyring Nanofluid over a Nonlinear Stretching Surface, Journal of Applied
Mathematics, 2019(2019) 3472518. https://doi.org/10.1155/2019/3472518
Journal of Xi'an University of Architecture & Technology
Volume XII, Issue III, 2020
Issn No : 1006-7930
Page No: 2726