Soret Effects on Free Convection Nanofluid Flows Due to a Stretching Sheet
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Copyright © 2023 by American Scientific Publishers Journal of Nanofluids
All rights reserved. Vol. 12, pp. 202–210, 2023
Printed in the United States of America (www.aspbs.com/jon)
Soret Effects on Free Convection Nanofluid Flows
Due to a Stretching Sheet
B. Shankar Goud1 , Pudhari Srilatha2 , and M. N. Raja Shekar1, ∗
1
Department of Mathematics, JNTUH University College of Engineering Hyderabad, Kukatpally, Hyderabad,
Telangana, 500085, India
2
Department of Mathematics, Institute of Aeronautical Engineering, Hyderabad, Telangana, 500043, India
This paper analyzes Nano liquids flow over a stretching sheet and the characteristics of heat and mass trans-
fer with uniform heat generation. Cu-water and Ag-water nano liquids are considered in the study. To obtain
a system of the non-linear ODEs similarity terms have been applied and an optimal homotopy asymptotic
method (OHAM) have been used to work out these equations. Various parametric influences on the velocity,
temperature, and mass transfer have been shown employing tables and graphs and the result we found is in
excellent agreement with the previous results in the works. The homotopy asymptotic approach is used to solve
these equations numerically, together with their boundary conditions and the Soret effect has been the primary
subject of this article. It was displayed that the Ag-water nano liquid displays higher thermal conductivity com-
pared to Cu-water nano liquid. In addition, the presence of a uniform heat source has a reducing effect on the
ARTICLE
concentration profile and an increasing effect on the temperature.
KEYWORDS: Nanoliquid Flow, OHAM, Stretching Sheet, Soret Number, Schmidt Number, Heat Generation.
IP: 5.10.31.151 On: Sat, 23 Sep 2023 12:38:29
Copyright: American Scientific Publishers
1. INTRODUCTION Delivered by Ingenta
Raju et al.11 presented their study on MHD Casson
A flow on a stretching surface has fascinated the con- fluid and the behavior of heat and mass transport through
centration of many researchers over the past few decades an exponentially stretching permeable surface in the inci-
because of its application in the field of engineering. dence of flow parameters. Also, the MHD flow of Jeffery
Some of its applications are like melt-spinning, etc. The fluid via a porous medium across a stretched sheet under
first person who studied about a two-dimensional steady the influence of chemical reactions and heat production
Newtonian fluid passion, in extrusion processes, wire has been studied by Jena et al.12 The porous matrix and
drawing, the manufacture of plastic, the performance of magnetic field improved the temperature distribution in
lubricants by an extending elastic sheet with a linearly the flow domain. Velocity slip parameter and heat source
fluctuating velocity a certain distance from a fixed point influences on MHD flow of a Jeffery fluid on a stretching
1
moving in its plane was Crane. Later on, this study was surface were analyzed by Gizachew and Shankar.13 Fur-
expanded by multiple writers to reveal different aspects of thermore, Gaffar et al.14 analyzed a steady boundary layer
flow and fluid heat transfer in an infinitely vast distance flow of an incompressible non-Newtonian Jeffery’s fluid
past a semi-infinite non-linear vertical plate. Ali et al.15
next to a stretched sheet. The following are amongst them.
studied a porous non-linear shrinking sheet with viscous
Awaludin et al.,2 Noor and Hashim,3 Abdel-Rahman,4
5 6 7 MHD flow. They discovered that dual solutions only exist
Bhattacharyya et al., Reza et al., Ullah et al., Ahmad
8 9 10 for positive values of the governing factors. Heat trans-
and Ishak, Shah et al. Kameswaran et al. investigated
mission via fluid and an MHD Boundary layer flow with
the convective transfer of mass and heat in a Nanofluid
changing viscosity in a porous medium approaching a
flow via a sheet stretched constraints to various impacts.
stretched sheet with an effect of viscous dissipation on
They obtained that the Ag-Water Nanofluid displays lower heat source/sink incidence were considered by Dessie and
rates of wall heat and mass transmission compared with Kishan.16
Cu-water Nanofluid. Thermophoresis and Brownian motion influences on a
nanofluid via a nonlinearly permeable stretching sheet
∗
have been analyzed by Falana et al.17 Ibrahim and
Author to whom correspondence should be addressed.
Email: mnrs@jntuh.ac.in Shankar18 numerically examined unsteady laminar viscous
Received: 27 February 2022 boundary layer flow and heat transport through a stretch-
Accepted: 16 March 2022 ing sheet. They explained that the stretching velocity of
202 J. Nanofluids 2023, Vol. 12, No. 1 2169-432X/2023/12/202/009 doi:10.1166/jon.2023.1946Goud et al. Soret Effects on Free Convection Nanofluid Flows Due to a Stretching Sheet
time-dependent and surface temperature causes the flow 2. MATHEMATICAL FORMULATION
and temperature unsteadiness. El-Aziz and Yahaya19 pre- An incompressible two-dimensional laminar steady nano
sented the combined impact of thermal and concentration liquid flow through a stretching sheet is reflected on. The
diffusions in unsteady MHD natural convection flow over a origin of the structure is situated at the slit where the sheet
moving plate embedded in a viscous fluid-saturated porous is formed. The stretching continuous surface is taken in
media and maintained at constant heat flux. A flow on a x-axis direction, and y-axis considered normal to the sur-
boundary layer and the influence of heat and mass trans- face of the sheet. The liquids are water-based nanofluid
mission in a viscoelastic electrically conducting fluid over with two kinds of nanoparticles: copper (Cu) and silver
a porous medium subject to a magnetic transverse field (Ag) nanoparticles. The base and the nanoparticles fluids
in the incidence of chemical reaction and heat source/sink are assumed to be in thermal equilibrium, with no-slip
has been deliberate by Nayak et al.20 In their examina- among them. The thermophysical parameters of nanoflu-
tion, they accomplished the field of temperature for both ids are listed in Table I. Flow geometry of the present
(PST) and (PHF) results. Swain et al.21 have made study- study as seen in Figure A. According to the aforemen-
ing heat and mass transfer in a boundary layer MHD flow tioned hypothesis, the governing boundary layer equations
of a viscous electrically conducting fluid across a stretched for nano liquid flow, heat, and concentration fields may be
exponential sheet. expressed in the dimensional form (Ref. [40])
There are a lot of studies that focus on boundary layer
flow and heat transmission in Newtonian fluids, but not u v
+ =0 (1)
all of them. Nevertheless, because of the growing sig- x y
nificance of nanofluids, a marvelous quantity of concern u v nf 2 u
has been specified to the analysis of nanofluids’ convec- u +v = (2)
x y nf y 2
tive transfer in current situations. The term ‘nanofluid’
ARTICLE
invented by Choi22 explains a liquid stretching sheet, it T T 2 T Q
u +v = nf 2 + T − T (3)
was formed by suspending nanoparticles into a base fluid. x y y cp nf
The nanoparticles that are used commonly are metals like
copper, aluminum, iron, and gold, andIP:the
5.10.31.151
base liquidsOn:
usu-Sat, 23 Sepu2023 C 12:38:29
C 2 C 2 T
+v = D 2 + D1 2 (4)
Copyright: American
ally employed are glycol, water, and methylbenzene or oil. Scientific x Publishersy y y
Later on, several authors studied Nanofluids. Ebaid Delivered
et al.23 byHere
Ingenta
u v are velocity elements in the x y axis and,
analyzed the impact of velocity slip and heat transmission denote the density of the fluid, T refers the tempera-
of Cu-Water and TiO2 -Water Nanofluids in the occurrence ture, C is the concentration,C is the concentration away
of a magnetic field. Though, the temperature of the Cu-
Water Nanofluid is greater the temperature of the TiO2 -
Table I. Thermophysical properties of water, copper, and silver nano-
Water Nanofluid. Ibrahim24 inspected Nanofluid stagnation
particles at 300 k (see Oztop and Abu-Nada42 ).
MHD flow and heat transport of a melting past a stretching
sheet. He found that an increase of melting heat transfer Properties
and magnetic parameter diminishes together with the skin Physical quantity ((kg/m )) kw/m k np x105 k−1 Cp j/kg k
3
friction factor and Sherwood number. Several researchers Water 997.1 0.613 21 4179
studied the different fluid flow concepts on various sheets Copper (Cu) 8933 401 1.67 385
(see Refs. [25–34, 39, 43–47]). The effects of Dufour and Copper (Cu) 10500 429 1.87 235
Soret are essential while density variations occur in the
flow system.
Ahmed35 explored the impact of Heat and mass transfer
through a stretched surface by suction or injection with
Dufour and Soret and established the factor of friction
and the local heat transfer factor raise as Dufour numbers
increase, but diminishes as Soret numbers increase. Alter-
natively, an increase of Dufour numbers enhances the local
Sherwood number, but it decreases with an increase of
Soret number. On observing that in almost above literature
the mass transfer effects for the considered nano-liquids
are ignored. Hence, we have investigated heat and mass
transfer characteristics of copper and silver nano-liquids
due to a stretching sheet in the presence of a uniform heat
source for which an OHAM is used. Fig. A. Flow geometry of the problem.
J. Nanofluids, 12, 202–210, 2023 203Soret Effects on Free Convection Nanofluid Flows Due to a Stretching Sheet Goud et al.
from the sheet of the fluid. Cp is the specific heat at pres- Equations (2)–(4) change into the following problems with
sure constant, D is the diffusivity of specifies, and D1 is two-point boundary values when Eqs. (6)–(10), and (12)
the coefficient that shows how temperature gradient affects are used.
mass flow.:
f − 1 f − ff = 0
2
Equations (1)–(4) are assumed to have the following (13)
boundary conditions.:
kf
x 2 ⎫ − Pr 3 2f
− f − = 0 (14)
knf
u = uw = bx v = 0 T = Tw = T + A ⎪
⎪
⎪
l ⎪
⎬
− Sc 2f − f + Sr = 0 (15)
x 2
C = Cw = C + Q at y = 0
⎪
⎪
l ⎪
⎪ The appropriate boundary constraints are:
⎭
u → 0 T → T C → C as y →
(5) f 0 = 0 f 0 = 1 f → 0 (16)
whereas A Q and b are fixed, l is the length of the 0 = 1 → 0 (17)
characteristic, T is the fluid temperature away from the
sheet. Brinkman36 defined the effective dynamic viscosity 0 = 1 → 0 (18)
of nanofluids as
The non-dimensional quantities are defined as:
f
nf = (6) vf cp vf
1 − 2 5 Pr =
f
Sc =
kf D
And other factors are given by:
D1 Tw − T Q
Sr = and =
ARTICLE
nf = 1 − f + s effective density (7) D Cw − C b cp nf
knf
nf = thermal diffusivity (8) Where
cp nf ⎧
IP: 5.10.31.151 On: Sat, 23 Sep 2023 ⎪
⎪
⎪
12:38:29 2 5
= 1 − 1 − +
s
cp nf = 1 − cp f + cp Copyright:
s
American Scientific Publishers
⎪
⎨ 1
Delivered by Ingenta f
(19)
the heat capacitance of the nanofluid (9) ⎪
⎪ cp s
⎪
⎩ 3 = 1 − + c
⎪
The Maxwell-Garnett model is used to determine the p f
thermal conductivity of nanofluids confined to spheri-
cal nanoparticles. (Ref: Max-well Garnett37 and Guerin 3. SKIN FRICTION, NUSSELT AND
et al.38 ). SHERWOOD NUMBER
If there is a specific relevance in free convection phenom-
ks + 2kf − 2 kf − ks ena these scientific factors define the surface drag, wall
knf = kf (10)
ks + 2kf + kf − ks heat and mass transfer rates.
Thus, At the wall, the shearing stress w is specified by
In Eqs. (7)–(10), the subscripts signify the nf : thermophys-
ical nf : thermophysical characteristics of the nano-fluid, u 1
w = −nf =− vf b 3 xf 0 (20)
25 f
f : base fluid, and: nano solid particles, respectively. y y=0 1 −
By adding a stream function x y and the Eq. (1) is
fulfilled such that where nf is viscosity coefficient. The skin friction factor
is determined as
2
u= v=− (11) Cf = 2w (21)
y x u w
and using Eq. (20) in Eq. (21) we achieve
Additionally, the following similarity factors are intro-
duced:
Cf 1 − 2 5 ∗ Rex = −2f 0 (22)
⎫
u = bxf v = − bvf f ⎪
⎪
⎪
⎪ At the wall, The Nusselt number at the surface flux is
⎬
T = T + Tw − T C = C + Cw − C specified by
⎪
⎪
b ⎪
⎪ x 2 b
= y = vf b xf ⎭ T
vf qw = −knf = −knf A 0 (23)
(12) y y=0 l vf
204 J. Nanofluids, 12, 202–210, 2023Goud et al. Soret Effects on Free Convection Nanofluid Flows Due to a Stretching Sheet
where knf is the Nanofluids thermal conductivity. The Nus- The substitution of the values of , and from
selt number is defined as: (31) into Eqs. (13)–(15), the residuals can be obtained as:
x qw R1 C11 C12 R2 C21 C22 and R3 C31 C32
N ux = (24) Afterward, achieve the Jacobians J1 J2 and J3 as in the
kf Tw − T
following ways:
Applying Eq. (23) in Eq. (24), the non-dimensional wall
b
heat transfer coefficient is estimated as follows: J1 C11 C12 = ∫ R1 2 C11 C12 d (32)
0
Nu kf
√ x = − 0 (25) b
Rex knf J2 C21 C22 = ∫ R2 2 C21 C22 d (33)
0
The mass flux at the wall surface is given by b
J3 C31 C32 = ∫ R3 2 C31 C32 d (34)
C x 2 b 0
Jw = −D = −D 0 (26)
y y=0 l vf With these defined constants, an estimated solution of the
problem (to order m) is found very fastly.
and the Sherwood number is specified as
x Jw
Shx = (27) 5. RESULTS AND DISCUSSION
D Cw − C
In the analytic solutions, the impacts of nanoparticle vol-
The non-dimensional wall rate of mass transfer is calcu- ume fraction, heat generation parameter, the influence of
lated using (26) in (27) as Schmidt and Soret numbers on the heat and mass transport
properties of nanofluids were examined. Two categories
Sh
ARTICLE
√ x = − 0 (28) of nanoparticles, specifically, Water is the base fluid, and
Rex copper and silver are added to it, and Prandtl number Pr =
Rex signifies the local Reynolds number specified as: 6 2 is fixed taken into account. The transformed nonlin-
Rex = xuw /vf ear ODEs (13)–(15) with the constraints (16)–(18) and the
IP: 5.10.31.151 On: Sat, 23 Sep 2023 12:38:29
analytically
Copyright: American Scientific solved by employing the optimal homotopy
Publishers
Delivered byasymptotic
Ingenta method (OHAM). The profiles of flow areas
4. NUMERICAL SOLUTION were acquired and practiced the outcomes were to calcu-
The solution of Eqs. (13)–(15) concerning the consistent late the coefficient of skin friction, heat, and mass transfer
boundary circumstances (16) to (18) can be explained by rate. Analytical results were looked at for different val-
optimal homotopy asymptotic method. For this intent, we ues of the factors on a graph and in a table. The analytic
take initial guesses for zeroth-order and first-order solution method was validated by comparing it with earlier pub-
with corresponding boundary conditions, respectively as: lished journals by Yirga and Tesfay40 and Hamad.41 As
⎫ seen in Table II, the outcomes are consistent. The coeffi-
f0 = 1 − e− ⎪ ⎪
⎬ cients of skin friction for distinct values of volume size of
0 = e − (29) the nanoparticles and when Pr = 6 2 Sc = 1 Sr =
⎪
⎪
− ⎭ 0 2 = 0 1 are given in Table II.
0 =e
⎫ For changed values of Prandtl number (Pr), the coeffi-
f1 = −C11 e− −1 + 1 −1 + e − ⎪
⎪ cients of heat transmission values are exposed in Table III.
⎪
⎪
⎪
⎬ It is obvious that the heat transfer factor rises as Pr grows.
1 = C21 e−2 kf Pr 3 − e kf Pr 3 + e knf The consequences are attained in an Excellent consistency
−e kf Pr 3 + e kf Pr 3 /knf ⎪
⎪ with prior outcomes by Kameswaran.10 An enhancement
⎪
⎪
−2
⎪
⎭
1 = −C31 e −Sc + e Sc − e + e Sc − e Sr
(30) Table II. Assessment of −f 0 for different values of Nanoparticle
Similarly, we can obtain second-order solutions too, for volume fraction , when magnetic parameter M = 0 and Prandtl num-
brevity, the solutions for f2 2 and 2 are not ber Pr = 6 2.
presented here. Present
Consequently, solutions for the equation of flow transfer Reference [39] Reference [40] results
Eqs. (up to second-order terms) are specified by M Cu-water Ag-water Cu-water Ag-water Cu-water Ag-water
f = f0 + f1 + f2 0 0 05 1.10892 1.13966 1.1089 1.1397 1.1089 1.1395
01 1.17475 1.22507 1.1747 1.2251 1.1745 1.2245
= 0 + 1 + 2 (31) 0 15 1.20886 1.27215 1.2089 1.2722 1.2084 1.2711
02 1.21804 1.28979 1.218 1.2898 1.2175 1.2885
= 0 + 1 + 2
J. Nanofluids, 12, 202–210, 2023 205Soret Effects on Free Convection Nanofluid Flows Due to a Stretching Sheet Goud et al.
Table III. Comparability of the values − 0 for diverse values of Pr
values.
Outcomes Pr = 0 72 Pr = 1 Pr = 3
Kameswaran10 1 08852 1 33333 2 50973
Present 1.0904 1.3333 2.4953
in the Pr values reveals that the thermal diffusivity is dom-
inated by momentum diffusivity. With each increase in
Prandtl number, the rate of heat transmission at the surface
enhances.
The volume size of nanoparticle influence on the
velocity, temperature, and concentration profiles for both
nanofluids is shown in Figures 1–3. These indicated that
as the values of raises, the Cu-water and Ag-water nano
liquid velocity reduces. Also found that the axial speed in
Fig. 2. Temperature profile as a function of when = 0 1 and Pr =
the instance of a Cu-water nano liquid is quite greater than 6 2.
an Ag-water nano liquid. As it is revealed in Figure 2, The
increase in the volume fraction of nanoparticles
enhances the heat transfer conductivity of the nano liq-
uid, which increases the thickness of the temperature field
as result. It is also detected that the distribution of tem-
ARTICLE
perature is higher in Ag-water nano liquid than in Cu-
water nano liquid. This is a predictable result because Ag
is a better heat conductor and electricity the nanoparticle
volume size increment made the thickness of the concen-
IP: 5.10.31.151 On: Sat, 23 Sep 2023 12:38:29
tration curves enlarged for both typesCopyright:
of nanoliquids con- Scientific Publishers
American
sidered in Figure 3. Delivered by Ingenta
The coefficient of skin friction(−f 0) as a role of the
is demonstrated by Figure 4. In the absence of , the
coefficient of skin friction value is one, a Crane’s9 standard
outcome. An enlargement of enlarges the coefficient
of skin friction monotonically to a maximum value before
declining. The findings explained relating to the coefficient
of skin friction agree for the two nanoliquids. For Cu-water
nanoliquid the highest value of the skin friction is accom- Fig. 3. Concentration profile as a function of when = 0 1, Pr = 6 2,
plished with the minor value of in association with an Sc = 1 and Sr = 0 2.
Ag-water nano liquid. Supplementary, the Cu-water nano
liquid displays lower drag to the flow as compared to the
Ag-water nano liquid. The non-dimensional − 0 and
− 0 are detained as a function of the size of a volume
of the nanoparticle ( in Figures 5 and 6, respectively. It
is observed that − 0 referred as diminishing term of ,
whereas the reverse is accurate in the situation of − 0.
It turns out that a nanofluid made of copper and water has
greater heat and mass transfer gradients than a nanofluid
made of silver and water. The existence of nanoparticles
tends to diminish the wall temperature gradient and to rise
the wall mass gradient.
Figures 7 and 8 elucidate the influence of heat genera-
tion factor ( on the temperature curves and concentration
profile in the event of Cu-water and Ag-water nanoliq-
uids. We saw that the temperature field enhances for both
Fig. 1. Velocity profile as a function of . situations of nanoliquids by enhancing the values of the
206 J. Nanofluids, 12, 202–210, 2023Goud et al. Soret Effects on Free Convection Nanofluid Flows Due to a Stretching Sheet
Fig. 4. Coefficient of skin friction as a function of when Sc = 1, Fig. 7. Temprature profile with variation of for Sc = 1 = 0 3, Sr =
Sr = 0 2, = 0 1 and Pr = 6 2. 0 2, and Pr = 6 2.
ARTICLE
IP: 5.10.31.151 On: Sat, 23 Sep 2023 12:38:29
Copyright: American Scientific Publishers
Delivered by Ingenta
Fig. 8. Concentration profile with variation of for Sc = 1, = 0 3,
Fig. 5. Heat transfer coefficient as a function of when Sc = 1, Sr = Sr = 0 2, and Pr = 6 2.
0 2, = 0 1 and Pr = 6 2.
Fig. 6. Mass transfer coefficient as a function of when Sc = 1, Sr =
0 2, = 0 1 and Pr = 6 2. Fig. 9. Temperature profile with variation of Pr for Sr = 0 2, Sc = 1,
= 0 3 and = 0 1.
J. Nanofluids, 12, 202–210, 2023 207Soret Effects on Free Convection Nanofluid Flows Due to a Stretching Sheet Goud et al.
Fig. 10. Concentration profile with variation of Pr for, Sr = 0 2, Sc = 1, Fig. 12. The effect of Sc on the concentration profile when Sr = 0 2,
= 0 3 and = 0 1. = 0 3, = 0 1 and Pr = 6 2.
heat generation factor (. This figure give the tempera- the diffusion coefficient is seen in Figure 11 in the case
ture profile of Ag-water is larger than Cu-water nanofluids. of Cu-water and Ag-water nanoliquids. As the Sr rises,
Enhancing the values of upsurges nanofluid the thermal the concentration curves of both nanoliquids increase. This
ARTICLE
conductivity width and increased temperature distribution. shows that the concentration increase of Ag-water nano
Also looked that the concentration profile decreases for liquid is greater than the concentration increment of Cu-
some cases of nanoliquids with enhancing the values of water nanoliquid.
the .
IP: 5.10.31.151 On: Sat, 23 It noted
Sep that12:38:29
2023 Eqs. (14) and (15) show that the functions
Figure 9 Shows that the temperature profile in both and have
Copyright: American Scientific been partly dropped. This indicates that Sc and
Publishers
cases of the nanoliquids declines as the PrandtlDelivered number bySr
Ingenta
do not influence heat transfer. Figure 12 displays the
upsurges. It is establishing that Ag-water had a higher tem- Schmidt number Sc impact on the diffusion field in the
perature profile than Cu-water. The concentration profile incident of Cu-water and Ag-water nanoliquids. The values
for both cases of nanoliquids increases as the Prandtl num- of Sc enlarge the concentration curves of both nanoliquids
ber increases as depicted in Figure 10. This shows that lessens. The concentration increase of Ag-water was found
the increase of the Prandtl number rises the thickening to be greater than that of Cu–water.
of the boundary layer of the concentration profile. And
the concentration profile of Ag-water is higher than that
of Cu-water. The influence of the Soret number Sr on 6. CONCLUSIONS
It is shown in this study that the problem of heat and mass
transfer in the flow of nanoliquids is discussed because
of a stretching sheet in the occurrence of heat generation.
The governing nonlinear PDEs were turned into ODEs by
employing the similarity variables & solved analytically
aid of OHAM. Two kinds of nanoliquids were deliber-
ated, Cu-water and Ag-water, and Among other factors,
the results of the current study revealed that.
• Cu-water nanoliquids have a thicker velocity boundary
than Ag-water nanoliquids. Meanwhile, the velocity field
decreases as the amount of volume size of nanoparticles
increases Meanwhile velocity field reduces with an incre-
ment of the volume size of nanoparticles.
• Cu-water nanoliquid thickness of the thermal boundary
layer is less than Ag-water nanoliquid. An increase of
enlarges the thermal boundary layer width.
• The width of the boundary layer of concentration for
Fig. 11. The effect of Sr on the concentration profile when Sc = 1, Ag-water nanoliquid is slightly more than Cu-water nano-
= 0 3, = 0 1 and Pr = 6 2. liquid.
208 J. Nanofluids, 12, 202–210, 2023Goud et al. Soret Effects on Free Convection Nanofluid Flows Due to a Stretching Sheet
• Increases in Pr increase the breadth of the concentration uw Temperature at the sheet surface
boundary layer, while increases in Sc diminish it. The heat generation parameter
• The coefficient of skin friction rises as the volume frac- knf Nanofluid’s thermal conductivity (Wm−1 K−1 )
tion of nanoparticles increases. Ag-water nanoliquid dis- cp nf Heat capacity of the nanofluid (Jkg2 m3 K−1 )
plays larger skin friction than Cu-water nanoliquid. N ux Nusselt number
• Nusselt at the plate surface declines with rising the vol- Rex Local Reynolds number
ume size of the nanoparticle. The Cu-water nanoliquid has Shx Sherwood number.
a maximum Nusselt number than Ag-water nanoliquid.
• Sherwood number at the plate surface raises with a rise
of the nanoparticle volume fraction, The Ag-water nano-
References and Notes
1. L. Crane, Zeitschrift fur Angew. Mathe-Matik und Phys. ZAMP 21,
liquid has a higher rate of mass transfer than the Cu-water 645 (1970).
nanoliquid. 2. I. S. Awaludin, A. Ishak, and I. Pop, Sci. Rep. 8, 13622 (2018).
• We have compared the values of skin friction and heat 3. N. F. M. Noor and I. Hashim, Am. Inst. Phys. 1750, 030019 (2016).
transfer rate for some particular parameters, it is observed 4. R. G. Abdel-Rahman, Math. Probl. Eng. 2013, 1 (2013).
5. K. Bhattacharyyaa, T. Hayatb, and A. Alsaedic, Chin. Phys. B 22,
that our results are in very good agreement with earlier
024702 (2013).
results. These values are tabulated in the Tables II and III. 6. M. Reza, R. Chahal, and N. Sharma, Int. J. Chem. Mol. Eng. 10,
585 (2016).
7. I. Ullah, K. Bhattacharyya, S. Shafie, and I. Khan, PLoS One 11,
e0165348 (2016).
NOMENCLATURE 8. K. Ahmad and A. Ishak, Propuls. Power Res. 6, 269 (2017).
9. Z. Shah, E. Bonyah, S. Islam, W. Khan, and M. Ishaq, Heliyon 4,
A Constant e00825 (2018).
Q Constant 10. P. Kameswaran, M. Narayana, P. Sibanda, and P. V. S. Murthy, Int.
b Constant J. Heat Mass Transf. 55, 7587 (2012).
ARTICLE
u v Velocity component in the (x y) direction 11. C. S. Raju, N. Sandeep, M. J. Babu, and V. Sugunamma, Alexandria
Eng. J. 55, 151 (2016).
(ms−1 )
−3 12. S. Jena, S. R. Mishra, and G. C. Dash, Int. J. Appl. Comput. Math
f Density of the base fluid (kgm ) 2016, 1225 (2017).
T The fluid temperature (K)IP: 5.10.31.151 On: Sat, 23 13. Sep 2023 12:38:29
A. Gizachew and B. Shankar, Adv. Appl. Sci. 3, 34 (2018).
C The nanoparticle concentration (kgm−3 ) American Scientific
Copyright: Publishers
14. S. A. Gaffar, V. R. Prasad, and E. K. Reddy, Ain Shams Eng. J. 8,
C The nanoparticle concentration far from Delivered
the by Ingenta
277 (2017).
15. F. M. Ali, R. Nazar, N. M. Arifin, and I. Pop, Bound. Value Probl.
sheet 32 (2013).
−1
cp The specific heat capacity (Jkg K) 16. H. Dessie and N. Kishan, Ain Shams Eng. J. 5, 967 (2014).
D Specifies diffusivity (ms−1 ) 17. A. Falana, O. A. Ojewale, and T. B. Adeboje, Adv. Nanoparticles 5,
D1 The coefficient indicating the contribution of a 123 (2016).
temperature gradient to mass flux (m2 s−1 ) 18. W. Ibrahim and B. Shanker, World J. Mech. 1, 288 (2011).
19. M. A. EL-AZIZ and A. S. YAHYA, J. Theor. Appl. Mech. Sofia 47,
l Is the characteristic distance 25 (2017).
T The fluid temperature extreme from the sheet 20. M. K. Nayak, G. C. Dash, and L. P. Singh, Propuls. Power Res. 5,
The solid volume portion of nanoparticles 70 (2016).
f Base fluid viscosity (kgm−1K−1 ) 21. I. Swain, S. R. Mishra, and H. B. Pattanayak, Hindawi Publ. Corp.
nf Nanofluid viscosity (kgm−1 K−1 ) J. Eng. 2015, Article ID 452592 (2015).
−3 22. S. U. S. Choi, ASME FED 66, 99 (1995).
nf Nanofluid density (kgm ) 23. A. Ebaid, F. Al Mutairi, and S. M. Khaled, Adv. Math. Phys. 2014,
nf Thermal diffusivity (m2 s−1 ) Article ID 538950 (2014).
Tw Temperature at the sheet surface 24. W. Ibrahim, Propuls. Power Res. 6, 214 (2017).
c Base fluid heat capacity (Jkg2 m3 K−1 ) 25. M. Veera Krishna, K. Jyothi, and A. J. Chamkha, Journal of Porous
p f 2 3 −1 Media 23, 751 (2020).
cp s Nanofluid’s effective heat capacity (Jkg m K ) 26. M. Veera Krishna, B. V. Swarnalathamma, and A. J. Chamkha, Spe-
The stream function cial Topics & Reviews in Porous Media: An International Journal
The similarity variable 9, 347 (2018).
f The dimensionless stream function 27. M. Veera Krishna, P. V. S. Anand, and A. J. Chamkha, Special Top-
The dimensionless temperature function ics & Reviews in Porous Media: An International Journal 10, 203
(2019).
The dimensionless concentration function 28. M. Veera Krishna and A. J. Chamkha, International Communications
Pr Prandtl number in Heat and Mass Transfer 113, 104494 (2020).
Sc Schmidt number 29. M. Veera Krishna, N. Ameer Ahamad, and A. J. Chamkha, Alexan-
Sr Soret number dria Engineering Journal 59, 565 (2020).
cf Skin friction coefficient 30. M. Veera Krishna, N. Ameer Ahamad, and A. J. Chamkha, Ain
Shams Engineering Journal 12, 2099 (2021).
Cw Concentration at the sheet surface 31. T. Tayebi, A. Sattar Dogonchi, N. Karimi, HuGe-JiLe, A. J.
kf Thermal conductivity of the base fluid Chamkha, Y. Elmasry, Sustainable Energy Technologies and Assess-
(Wm−1 K−1 ) ments. 46, 101274 (2021).
J. Nanofluids, 12, 202–210, 2023 209Soret Effects on Free Convection Nanofluid Flows Due to a Stretching Sheet Goud et al.
32. A. J. Chamkha, A. S. Dogonchi, and D. D. Ganji, Appl. Sci. 8, 2396 40. Y. Yirga and D. Tesfay, World Acad. Sci. Eng. Technol. Int. J. Mech.
(2018). Mechatronics Eng. 9, 709 (2015).
33. S. M. Seyyedi, A. S. Dogonchi, M. Hashemi-Tilehnoee, D. D. Ganji, 41. M. Hamad, Int. Commun. Heat Masstrans. 38, 487
and A. J. Chamkha, International Journal of Numerical Methods for (2011).
Heat & Fluid Flow 30, 4811 (2020). 42. H. F. Oztop and E. Abu-Nada, Int. J. Heat Fluid Flow 29, 1326
34. A. S. Dogonchi, M. Waqas, S. R. Afshar, S. M. Seyyedi, M. (2008).
Hashemi-Tilehnoee, A. J. Chamkha, and D. D. Ganji, International 43. B. Shankar Goud, International Journal of Thermofluids 7, 100044,
Journal of Numerical Methods for Heat & Fluid Flow 30, 659 (2020).
(2020). 44. B. Shankar Goud, International Journal of Thermofluid Science and
35. A. A. Ahmed, Commun. Nonlinear Sci. Numer. Simul 14, 2202 Technology 7, 070201 (2020).
(2009). 45. B. Shankar Goud and M. N. Raja Shekar, International Journal of
36. H. C. Brinkman, J. Chem. Phys. 20, 571 (1952). Computational and Applied Mathematics 12, 53 (2017).
37. J. C. M. Garnett, Philos. Trans. R. Soc. Lond. A203, 385 (1904). 46. B. Shankar Goud and P. Srilatha, International Journal of
38. C. -A. Guérin, P. Mallet, and A. Sentenac, J. Opt. Soc. Am. A23, Innovative Technology and Exploring Engineering 8, 263
349 (2006). (2019).
39. M. M. Nandeppanavar, K. Vajravelu, and P. S. Datti, Heat-trans. 47. D. R. Yanala, B. Shankar Goud, and A. J. Chamkha, Heat Transfer
Asian Res. 45, 15 (2016). 51, 1 (2022).
ARTICLE
IP: 5.10.31.151 On: Sat, 23 Sep 2023 12:38:29
Copyright: American Scientific Publishers
Delivered by Ingenta
210 J. Nanofluids, 12, 202–210, 2023You can also read
