The Water-Based Nanofluid for the Influence of Dissipative Heat Energy

1. ARTICLE
Copyright © 2022 by American Scientific Publishers
All rights reserved.
Printed in the United States of America
Journal of Nanofluids
Vol. 11, pp. 1–9, 2022
(www.aspbs.com/jon)
Analytical Approach on the Water-Based Nanofluid
for the Influence of Dissipative Heat Energy
Priya Mathur1∗
and J. C. Misra2
1
Department of Mathematics, Poornima Institute of Engineering Technology, Jaipur, Rajasthan 302022, India
2
Department of Mathematics, IIT, Kharagpur, India
The current study reveals the influence of both dissipative heat in water-based conducting nanoliquid through
an impassioned semi-infinite vertical plate past embedding with a permeable medium. The effect of particle
concentration in conjunction with transverse magnetic field and heat source/sink is presented by introducing
the flow and heat transfer phenomena. However, various heat transfer assets of the considered nanofluid are
exaggerated by assuming the slip condition. The reason is that, for the diminution in the heat transfer, the
role of temperature slip is vital. For the transformation of governing PDEs into nonlinear coupled ODEs, it
is necessary to assume suitable self-similar transformations. Moreover, a semi-analytical technique such as
Homotopy Perturbation Method (HPM) is employed to solve these transformed equations. The novelty of the
study is to get a comparison of the result with the results obtained using the MATLAB in-build code bvp5c. The
numerical results for the comparison between both the methods have been presented in tabular form and the
control of various parameters for the flow phenomena is demonstrated via graphs.
KEYWORDS: MHD, Nanofluid, Joule Dissipation, Heat Source/Sink, Homotopy Perturbation Method.
1. INTRODUCTION
Today, industrial processes such as the preparation of com-
puter chips, in micro-level, car engines, in macro-level
both need the higher rate of cooling needs the quite ever,
the cooling rate for the better designing and shape of the
material. The heat generation rates in the system play an
important role to carry out the process. From the earlier
investigations, it is seen that the traditional liquids used for
the preparation of the materials i.e., water, kerosene, gly-
col, etc. have a low thermal conductivity that is not useful
for the production. In earlier decades, for the augmenta-
tion in the heat transfer phenomena, solid metal or oxide
particles are added in the conventional liquid of poor ther-
mal conductivity. A theoretical model was pioneered by
the scientist James Clerk Maxwell exhibiting the behav-
ior through electrical conductivity of heterogeneous solid
particles. Further, an investigation is going on using the
classical Maxwell model conductivity of the liquid mix-
tures with solid particles. It is observed that the micropar-
ticles settle very rapidly in liquids that causes a major
problem. This causes extra pressure drops within the sys-
tem, clogging of the solid particles, and abrasion. Further-
more, to get appreciable improvements in the conductivity
∗
Author to whom correspondence should be addressed.
Email:
Received: xx Xxxx xxxx
Accepted: xx Xxxx xxxx
properties high particle concentrations are required. The
formation of nanoparticles of iron and oxides of size, 1–
100 nm, is mixed in standard liquid sorts of nanofluids.
Choi1
began his work with nanofluid within the basic sense
where he searched for liquids like water, ethylene glycol,
etc. The effective properties of nanofluid are above that
of traditional liquids, for an equivalent reason the prop-
erties of heat transfer also improve. Now, scientists were
curious about studying the nanofluid that has various bio-
logical properties. The presence of solid particles results
in attractive features within the basic elements of ther-
mophysical nanofluids. They provided their research on
thermo-physical properties and their stability. In numer-
ous systems, the density of nanofluid acting a crucial role.
Such applications have a decrease in pressure and pump-
ing capacity where the dimensions feature a significant
behavior. Further, Dogonchi et al.23
studied the distribu-
tion of Cu particles in water via a triangular shaft of the
fluid magnetization. The conditions for warmth transfer
of the warmth/sinking effect of MHD flow were thrash-
ing out by Clarify et al.4
For the radiative heat impact, Li
et al.5
explored on the flow of nanofluid into a hole. More-
over, an analytical performance is to seek out an answer
for measurement control. Additionally, the behavior of
low-temperature humidity using Casson fluid; a compar-
ative analysis is presented by Souayeh et al.6
Makinde7
proposed the outcome of viscous dissipation along with
Newtonian heating conditions on Sakiadis flow. For the
J. Nanofluids 2022, Vol. 11, No. xx 2169-432X/2022/11/001/009 doi:10.1166/jon.2022.1844 1

2. Analytical Approach on the Water-Based Nanofluid for the Influence of Dissipative Heat Energy Mathur and Misra
ARTICLE
consequences of radiation and warmth dissipation
Motsumi and Makinde8
analyzed their flow over a moving
plate. Moreover, in9
it is characterize that, non-Newtonian
nanofluid deposits beyond the expansive space on the
impact of warmth transfer sites. Lie group analysis is pro-
posed by Das10
for the flow stagnation point nanofluid sus-
pension. Again, Das11
analyzed the flow through a skinny
surface with a divine slide. The work of Nadeem et al.12
analyzes the water-based nanofluid supported an advisory
sheet. A review work of Kuntsov and Neild13
carry out
the flow of natural boundaries that show nanofluid pass-
ing a straight plate. Additionally, a study of the rela-
tive boundary conditions of 3D Casson nanofluid over a
series of consecutive advisory sheets was accomplished
by Nadeem et al.14
An investigation for free flow of
MHD fluid to the visible fluid by varying the temper-
ature generation/absorption by industrial sources is car-
ried out by Pattnaik and Biswal.15
Recently, Mishra and
colleagues16–23
studied the occurrence of nanofluids in
their various body structures watching different nanopar-
ticles. For the optimal exercise of heat phenomenon,
Bahiraei et al.24
analyzed the utilization of nanoliquid
in tubes embedding with graphene-platinum. Additionally,
Bahiraei et al.25–27
worked on different thermal compo-
nents employing a hybrid nanofluid in several geometries.
Mishra and Mathur28
discussed the significance of melt-
ing heat transfer conditions on the Williamson nanofluid
flow past a porous medium. The author solves the set
of differential equations using the method of variation
parameter. The manipulation of various properties is ana-
lyzed and validated by previous results. Chamkha and
his co-researchers have worked conscionable to investi-
gated the different aspects of MHD nanofluid flow. To find
important facts about nanofluid properties and the influ-
ence of different parameters.29–35
The efficiency of elec-
tronic equipment is directly affected by its heat dissipation.
They need such coating or cooling systems so that their
life may increase. Nowadays nanofluids are in demand
for the cooling of computers and microchips which are
used in it. In automobile industries, nanofluids are using
in lubricants to improve their tribological properties like
load carrying, friction reduction within the moving parts.
Krishna et.al.36–41
had done brilliant work on Hall and
ion slip effects on MHD with different rotation of flows.
Authors4243
investigated hall effects on nanofluids on
puros and ramped surfaces. Krishna and his colleagues44–47
studied the heat and mass transfer in micropolar fluids also
in various mediums and discuss the effect of parameters
on the velocity. Dongonchi with his fellow researchers48–51
worked on entropy generation in nanofluid in recent years.
Their significant contribution explained heat generation in
different shapes of nanoparticles suspended in base fluids.
Many researchers52–58
have studied and developed mod-
els for natural convective flow. They also explained the
effect of various parameters on heat transfer and entropy
generation.
With a quick overview of the aforementioned literature,
it aims to debate the method of conducting nanofluid over
a vast straight plate to strengthen heat transfer structures.
Nanoparticles like Cu, Al2O3, and TiO2 are suspended in
the base fluid water to extend the thermophysical prop-
erties of conventional liquids. Additionally, the effect of
magnetic flux is considered. However, the characteris-
tic equations are solved using both numerical as well
as analytical techniques and presented both graphically
and tabular form for particular values of the contributing
parameters.
2. THE FORMALISM OF THE PROBLEM
The present study has been done for the nanofluids which
are flowing in a two-dimensional semi-infinite plate placed
vertically and embedding with permeable medium and
it is electrically conducting. More commonly, various
applications through porous media are useful like hydrol-
ogy, geophysics inkjet printers, nuclear waste disposal,
and biomechanics, etc. Further, the flow characteristic is
enhanced by incorporating the dissipative heat energy due
to the magnetic field and external heat source in the tem-
perature profile that affects the flow phenomena. Water
is considered as the base liquid by introducing both met-
als such as Copper (Cu) and metallic oxide (Al2O3) and
Titanium oxide (TiO2) for the preparation of nanofluid.
The applied magnetic field is forced to propose along the
transverse direction of the flow of nanofluid (Fig. 1). How-
ever, it is wise to neglect the impact of induced magne-
tization along with and Hall effects. Here, Tf is the plate
temperature and hf is assumed as the heat transfer coeffi-
cient. Assuming such conditions, the equations that govern
the flow phenomena with appropriate boundaries are as
follows:
ux +vy = 0 (1)
Fig. 1. Flow model.
2 J. Nanofluids, 11, 1–9, 2022

3. Mathur and Misra Analytical Approach on the Water-Based Nanofluid for the Influence of Dissipative Heat Energy
ARTICLE
nf uux +vuy = nf uyy +nf nf gT −T
−
nf 2
0 +
nf
k∗
p
u (2)
cpnf uTx +vTy = knf Tyy +nf u2
x +nf 2
0u2
+QT −T
(3)
ux0=U0vx0=0−knf Tyx0=hf Tw −T x0
ux=0T x=T
(4)
The effective properties i.e., density nf , heat capacity
cpnf , the viscosity nf are explained as;28
nf = 1− f + s nf =
f
1 − 2 5
Cpnf = 1− Cpf + Cps (5)
Following Maxwell–Garnetts model the effective con-
ductivity is defined as:28
knf = kf
ks +2kf −2 kf −ks
ks +2kf + kf −ks
and as deployed in29
the effective electrical conductivity is
followed as:
nf
f
= 1+
3r −1
r +2−r −1
r =
s
f
The following similarity variables and rules for the
renovations are worned to reconstruct the governing
equations.23
=
a
f
y =
a f xf =
T −T
Tw −T
Impose of above transformations that satisfy Eq. (1)
automatically.
The Eqs. (2)–(4) as well as Eq. (5) are reconstructed as
f
+ A1A2ff
−f 2
−MA1A3f
+A1A2A4Gr
−A1A2
1
Kp
f
= 0 (6)
+ A5A6 Pr f
+
A5
A1
Ec Pr f 2
+A3A5MEc Pr f 2
+A5 Pr S = 0 (7)
f 0 = 0f
0 = 1
0 = Bi0−1
f
→ 0 → 0 (8)
Table I. Values of the properties of base fluid and nanoparticles and
the.
(Kg/m3
) Cp(J/KgK) k(W/mK) ×105
K−1
(S/m)
Cu 8933 385 401 1 67 5 96×106
Al2O3 3970 765 40 0 85 35×106
TiO2
A1 = 1− 2 5
A2 = 1− +
s
f
A3 = 1+
3r −1
r +2−r −1
A4 = 1− +
s
f
A5 =
kf
knf
A6 = 1− +
cps
cpf
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
Using, the Lewis number, Le = /DB Eckert num-
ber, Ec = f ax2
/cpf
Tw −T, the heat source/sink
parameter, S = Q/cpf
, magnetic parameter, M =
f B2
0/af , Prandtl number, Pr = f /.
3. METHODOLOGY
The “Homotopy Analysis Method” (HAM) is reported an
approximate-analytical procedure to find the resulting out-
put of ordinary/partial differential equations but “Homo-
topy Perturbation Method,” we use perturbation method as
well as homotopy method both. The basic concept of this
method for solving a nonlinear differential equation is as
follows:
Du−F s = 0s ∈ (9)
and the designed boundaries as
B u
u
n
= 0s ∈ (10)
where D, the operator used for differentiation, B, operator
relating to boundaries, F s, termed as source, , bound-
ary domain and u
n
, normal to .
Now, the operator D can be deployed as
Du = Lu+Nu (11)
Now, Eq. (9) is written as
Lu+Nu−F s = 0 (12)
Defining the homotopy for the perturbation
vsp ×01 → R
Then
Hvp = 1−pLv−Lu0+pDv−F s = 0
p ∈ 01s ∈ (13)
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4. Analytical Approach on the Water-Based Nanofluid for the Influence of Dissipative Heat Energy Mathur and Misra
ARTICLE
Involvement of Taylor’s series of Eq. (13) leads to
v =
m=0
pm
vm (14)
u = lim
p→1
v =
m=0
vm (15)
Applying HPM to Eqs. (6)–(8) we have
1−p f
−f0
+p f
+A1A2ff
−f 2
−MA1A3f
+A1A2A4Gr −A1A2
1
Kp
f
= 0 (16)
1−p
−0
+p
+A5A6 Pr f
+
A5
A1
Ec Pr f 2
+A3A5MEc Pr f 2
+A5 Pr S = 0 (17)
f and are considered as
f =
m=0
fm =
m=0
m (18)
Arranging the like powers of pm
− terms with m = 0 in
Eqs. (19, 20) and m = 12··· in Eqs. (21)–(23),
f0
= 00
= 0 (19)
f00 = 0 f0
0 = 1 f0
= 0
00
= Bi00−10 = 0 (20)
f
m +
m−1
k=0
A1A2fm−k−1f
k −f
m−k−1f
k−MA1A3f
m−1
+A1A2A4Grm−1 −A1A2
1
Kp
f
m−1
= 0 (21)
m +
m−1
k=0
A5A6 Pr fm−k−1
k +
A5
A1
Ec Pr f
m−k−1f
k
+A3A5MEc Pr f
m−k−1fk
+A5 Pr Sm−1
= 0 (22)
with modified boundary condition:
fm0 = 0f
m0 = 0f
m = 0
m0 = 0m = 0
(23)
In particular, for the following five values of the param-
eters taking initial, the first order solutions for the flow are
computed and presented below:
For = 0 1M=0 1Kp = 1Gr = 0 1Pr = 6 2Ec =
0 1S = 0 1Bi = 0 1
f0=−0 252
f1=−0 4815552
+0 3498693
−0 1066714
+0 0167835
−0 0010496
f =−0 78262
+0 3447293
−0 1416094
+0 0531595
−0 0280966
+0 0144577
−0 0054478
+0 0014779
−0 00033110
−0 00011511
−0 00007612
+0 0000413
−0 00002714
+0 00001315
−0 00000516
−0 00000217
−0 000000518
+
0 = 0 152785−0 076393
1=0 034989−0 1549352
+0 0948913
−0 0108984
=0 471202−0 528803−0 5492422
+0 5131433
−0 3370304
+0 1419665
−0 0369066
+0 0165997
−0 0096498
+0 0042999
−0 00311310
−0 00295611
−0 00224312
+0 00133113
−0 00063914
+0 00025115
−0 00008116
−0 00002117
−0 000000418
+0 000000819
+
4. NUMERICAL COMPUTATIONS AND
VALIDATIONS
The influence of dissipative heat energy with inclusion
of viscous diffusion on the water-based nanofluid is ana-
lyzed in the present investigation. The flow phenomena
are considered over an infinite plate considering the effects
of the magnetization vis-a-vis the heat source/sink. The
water-based nanofluid is prepared with the addition of
TiO2 nanoparticles and the distorted equations are tack-
led numerically by utilizing the in-build code bvp5c, then
results compared with an approximate analytical method
i.e., HPM. The computations for both skin friction and
Nusselt number are presented via Table II. It is seen
from the table that the “resistive force” offered by the
appearance of the magnetic property and the porosity
the shear rate increases in magnitude. In comparison to
both, the methodology employed here are very much
significant to two or three decimal accuracy i.e., Cf =
−1 20278bvp5c and Cf = −1 20278HPM whereas
Nu = 0 67032bvp5c and HPM. The augmentation in
the thermal buoyancy parameter, as well as the volume
concentration of the nanoparticle, reduces the shear rate
but the significant increase is marked for the rate of heat
4 J. Nanofluids, 11, 1–9, 2022

5. Mathur and Misra Analytical Approach on the Water-Based Nanofluid for the Influence of Dissipative Heat Energy
ARTICLE
Table II. Validation as well as the numerical computations for the rate coefficients.
Cf Nu
M Kp Gr Pr S Ec Bi bvp5c HPM bvp5c HPM
1 1 0.1 0.1 6.2 0.1 0.1 1 −1.20278 −1.20234 0.67032 0.67032
2 −1.43906 −1.43873 0.63124 0.63124
1 2 −1.43812 −1.43806 0.64043 0.64043
3 −1.63705 −1.63673 0.61215 0.61215
1 0.2 −1.19228 −1.19210 0.67138 0.67138
0.3 −1.18185 −1.18135 0.67243 0.67243
0.1 0.2 −0.87107 −0.87007 0.76667 0.76667
0.3 −0.58709 −0.58615 0.86705 0.86705
0.1 2 −1.19193 −1.19185 0.43515 0.43515
6.2 −1.20278 −1.20233 0.67032 0.67032
1 0.2 −1.20096 −1.20034 0.60643 0.60643
0.3 −1.19809 −1.19751 0.50915 0.50915
0.1 0.2 −1.18786 −1.18626 0.09062 0.09062
0.3 −1.18014 −1.18001 −0.20925 −0.20925
0.1 2 −1.19922 −1.19630 0.92725 0.92725
3 −1.19735 −1.19303 1.06320 1.06320
transfer. It rises because of the lower in thermal diffusiv-
ity as presented that the Prandtl number increases how-
ever, significant reduction is employed due to the increase
of heat source. The coupling parameter i.e., the Eckert
number that occurs for the inclusion of viscous dissipa-
tion reduces the rate coefficient whereas drastic increase
of rate is obtained due to the increase in Biot number.
The comparative study is taken care of due to the varia-
tion in several nanoparticles i.e., Cu, TiO2, and Al2O3 on
the flow profiles. Figure 2 describes the manipulation of
nanoparticles on the fluid momentum. It is clear to see that
the profile diminishes greatly in the case of Cu nanoparti-
cles in comparison to the remaining. The fact is, the par-
ticles obstruction near the sheet region is more since the
metal like Cu has higher density of the. Further, Figure 3
rendered that the fluid temperature shows its maximum
Fig. 2. Comparison of nanoparticles on velocity profiles.
Fig. 3. Comparison of nanoparticles on temperature profiles.
magnitude in the case of Cu nanoparticles. The order of
retardation is Cu, Al2O3, and TiO2 respectively.
5. RESULTS AND DISCUSSION
The velocity and the temperature profiles are presented
in Figures 4–12. The significant performance of the reli-
able parameters appeared in the phenomena. Here, the
important characteristics of several parameters i.e., volume
concentration (), magnetization (M), Porosity, thermal
buoyancy (Gr), coupling parameter (Ec), Prantl number
(Pr), heat source (S) and Biot number (Bi) on the veloc-
ity and temperature distributions. However, the compu-
tations are obtained for TiO2-water nanofluid. In each
figure, the variation of a particular parameter on each pro-
file is presented keeping others fixed. Figure 4 illustrates
the variation of nanoparticle volume concentration on the
stream function. It is evident that fluid concentration is the
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6. Analytical Approach on the Water-Based Nanofluid for the Influence of Dissipative Heat Energy Mathur and Misra
ARTICLE
Fig. 4. Validation and influence of on stream function.
Fig. 5. Validation of on velocity profile.
Fig. 6. Characteristics of M and Kp on the velocity.
Fig. 7. Characteristics of Gr on velocity.
Fig. 8. Validation of on fluid temperature.
Fig. 9. Impact of M and Kp on the fluid temperature.
6 J. Nanofluids, 11, 1–9, 2022

7. Mathur and Misra Analytical Approach on the Water-Based Nanofluid for the Influence of Dissipative Heat Energy
ARTICLE
Fig. 10. Control of Ec on temperature.
particles present in a consistent volume of fluid. Increasing
volume fraction enhances the transverse velocity distribu-
tion so that the thickness decreases. The result signifies
that the clogging of nanoparticles is more near the sheet
region. In comparison to the methodologies employed
here, it can be concluded that the bvp5c numerical tech-
nique and the approximate analytical technique, HPM cor-
relates equivalently. The validation of the methods as well
as the variation of on the flow profiles are displayed
in Figure 5. The augmentation in the fluid velocity is
marked due to the enhanced concentration. A significant
fall in the velocity profile is observed within the region
1appox and further, it became smooth to meet
the necessary conditions. Figures 6(a) and (b) exhibits the
influence of the resistive forces offered by the occurrence
of a magnetic parameter as well as the porosity parame-
ter. The inclusion of magnetic parameters in the velocity
profiles causes the activity of the resistive force named
Fig. 11. Control of S on temperature.
Fig. 12. Control of Bi on temperature.
“Lorentz force” resisting the fluid motion in conjunction
with the particle concentration. Therefore, an increase in
magnetic parameters retards the fluid motion. As similar to
magnetic, porosity have retards the fluid motion. There is
a temperature change is measured by the buoyancy param-
eter that ventilates a building by allowing denser, cold air-
lift warm air up. The amount varies due to the difference
temperature conditions of the air. Therefore, Figure 7 indi-
cates the variation of thermal buoyancy parameter on the
velocity distribution. Here, Gr 0 signifies the cooling of
the sheet. As sheet is cooled the fluid velocity boost up
since the hotter fluids are less dense than that of colder
fluid. Hence, the fluid particle present in the sheet region
moves up with increasing thermal buoyancy. Figure 8 por-
trays the validation between the numerical and the approx-
imate analytical solution profiles for nanofluid tempera-
ture for the appearance of volume concentration. However,
the variation of volume fraction is also presented in the
same figure. Increasing concentration gives rise to enhance
the fluid temperature significantly. The solution profiles
obtained by using both the methodologies correlate signif-
icantly. Figure 9 deliberates the influence of the resistive
forces such as magnetic and the porosity parameter on the
fluid temperature. The amount of stored energy boosts up
the fluid temperature with an increase in both parameters.
Figure 10 displays the influence of coupling parameter on
the nanofluid temperature. Eckert number is inversely pro-
portional to the temperature difference between the sheets
and fluid. A amplify in Ec suggests a major decrease in
the temperature difference. Therefore, increasing Eckert
number enhances the fluid temperature significantly. The
amount of Ec is due to the appearance of viscous dissi-
pation that couples the velocity and temperature profiles.
As described earlier, the variation of heat source on the
temperature is demonstrated in Figure 11. An increase in
heat source enriches the fluid temperature significantly.
The control of Biot number on the fluid temperature is
J. Nanofluids, 11, 1–9, 2022 7

8. Analytical Approach on the Water-Based Nanofluid for the Influence of Dissipative Heat Energy Mathur and Misra
ARTICLE
exhibited in Figure 12. Biot number is due to the consider-
ation of heat flux conditions. Biot number is also directly
proportional to mean temperature difference at any layer
of the fluid. Therefore, the profile behaves similar to that
of the profiles of coupling parameters. Hence, it is con-
cluded that increasing Bi favors enhancing the nanofluid
temperature significantly.
6. CONCLUSIVE REMARKS
The flow of TiO2-water-based nanofluid over an infinite
vertical plate for the influence of dissipative heat energy is
analyzed in the present investigation. Further, the applied
magnetic field along with the exterior heat source affect-
ing the flow phenomena as well. The consideration of
heat flux conditions enriches the study. The novelty is
the comparison between the earlier numerical solution and
the present analytical technique coincides with each other
significantly. The manipulation of a few of the important
parameters is described as;
• Nanoparticle concentration favors to attenuation in the
sheet thickness that is supposed to enhance the transverse
velocity profiles.
• Clogging of the nanoparticles near the sheet region
causes an augmentation in the fluid temperature.
• Thermal buoyancy rises up the velocity due to cooling
of the plate surface.
• Both the coupling parameter and the Biot number cause
a significant increase in the nanofluid temperature through-
out the domain.
• The more amount of volume concentration retards the
shear rate whereas the rate of heat transfer increases.
• The excellent agreement between the numerical and
approximate analytical technique makes energize to pro-
ceed in further investigation.
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