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Effect of heat absorption in natural convection nanofluid flow along a vertical wavy surface
Effect of heat absorption in natural convection nanofluid flow along a vertical wavy surface Ahmer Mehmood, Muhammad Saleem Iqbal PII: DOI: Reference:
S0167-7322(16)32693-9 doi:10.1016/j.molliq.2016.10.122 MOLLIQ 6526
To appear in:
Journal of Molecular Liquids
Received date: Accepted date:
10 September 2016 26 October 2016
Please cite this article as: Ahmer Mehmood, Muhammad Saleem Iqbal, Effect of heat absorption in natural convection nanofluid flow along a vertical wavy surface, Journal of Molecular Liquids (2016), doi:10.1016/j.molliq.2016.10.122
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Effect of heat absorption in natural convection nanofluid flow along a vertical wavy surface
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Ahmer Mehmood, Muhammad Saleem Iqbal1 Department of Mathematics and Statistics, FBAS, International Islamic University,
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Islamabad 44000, Pakistan.
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Abstract:
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Effects of heat absorption and nanoparticle on natural convection heat transfer along vertical wavy surface have been investigated. Transport equations have been solved numerically by accurate implicit finite difference scheme. The skin friction and Nusselt number are plotted against variation of several
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parameters for two types of nanoparticles namely, alumina (
) and magnetite (
). The impact
of nanoparticle concentration on flow and heat transfer process in the problem under investigation has
water based nanofluid. The influence of heat absorption
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heat transfer rate in comparison to
-water nanofluid exhibits higher skin friction and
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been studied in detail. The results indicate that
parameter is to increase the heat transfer rate and decrease the skin friction coefficient. For heat
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absorption case percent change in the skin friction and Nusselt number for two nanoparticles is shown in tabular form where comparison to the flat plate (pure fluid) and wavy surface (pure fluid) cases have also been made. The present results have been validated by producing the results available in literature and a very good agreement is observed. Keywords: Natural convection, nanofluid, heat absorption, vertical wavy surface, Keller-Box method
1. Introduction
The real heat transfer processes associated with the change of some form of energy into thermal energy have the possibility of involving internal heat absorption. Heat transfer and fluid flow phenomena with heat absorption is associated with large temperature gradient which is frequently met in several engineering and thermal processes such as, in the combustion chamber, in thermal control of space ships, in casting and blading of gas turbines and in spent fuel storage [1], in post-accident heat removal [2], in engine cooling system and in insulation of buildings etc.
1
Corresponding Author’s Email: [email protected] Cell No. +923335195130
1
ACCEPTED MANUSCRIPT Natural convection flows happen in many practical phenomenon caused by sink. The importance of heat sink increases in those flows where the chemical reaction occurs. A change due to heat absorption occurs in temperature distribution, particle decomposition rate in semiconductor products and nuclear
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reactors. Therefore, heat absorption, convection and conduction phenomena would have to be determined simultaneously.
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Vajravelu and Hadjinicolaou [3] considered an internal heat source over a stretching surface. Molla et
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al. [4] studied magneto hydrodynamic free convection flow on a sphere under the influence of heat source. Alam et al. [5] investigated heat source effect on MHD free convection flow over a sphere.
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Mamun et al. [6] analyzed effect of heat source along a vertical plate. Mamun et al. [7] discussed impact of viscous dissipation and heat source on heat transfer over a vertical plate. Mansoor and
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Ahmed [8] explored natural convection in porous triangular enclosure using nanofluid and heat generation.
Particularly, nanofluid is a fluid which has nanoparticles with size 1 to 100 nm spread in the fluid like
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water, ethylene, engine oil, menthol and glycol. Nanofluid can play vital role in the creation of new
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technologies for industrial and engineering processes. It can flow smoothly because they are so small to act similar to liquid molecules. Thermal conductivity of nanofluid with dispersed non-metallic or
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metallic particles are expected to be considerably greater than common heat transfer liquids. Consequently, the dispersed nanoparticles can considerably modify the thermal and transport properties of the base fluid. The considerable augmentation of convective heat transfer by forced convection has widely been studied [9-15]. The research on nanofluid has been focused mostly as it finds applications in various fields such as nano-drug delivery and advanced nuclear systems, etc. Therefore, all the normal heat transfer mods can be used for nanofluid only if the properties of base fluids are interchanged by nanofluid. Nanofluid can play vital role in the innovation of new technologies for industrial and engineering processes. The natural convection flow process commonly takes place in nature and also in several physical phenomenon such as combustion modeling, fire engineering, nuclear reactor cooling, petroleum reservoir, heat exchangers and in many industrial applications. The basic flow phenomena arises due to the buoyancy effects, which is generated from temperature difference. In natural convection flow heat transfer phenomena are the integral part and is regulated by bouncy effects which is generated from diffusion of thermal energy [16]. Consequently, neglecting the other effects, buoyancy term is combined with the Navier-Stoke’s equation and energy equations. In most of the physical situations 2
ACCEPTED MANUSCRIPT heat source enhances the free convective flow and heat transfer characteristic. Molla et al. [17] presented natural convective flow through a vertical wavy sheet under heat generation. Hady et al. [18] studied MHD natural convective flow through a vertical wavy sheet by considering heat source. Kabir
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et al. [19] discussed heat source impact on MHD free convective flow along a heated vertical wavy plate. Alim et al. [20] analyzed natural convective flow through a vertical wavy sheet by taking into
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account viscosity depending upon temperature and heat source. Parveen and Alim [21-22] discussed
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MHD natural convective flow through a vertical wavy sheet with heat source / sink, joule heating, viscosity depending upon temperature and temperature dependent thermal conductivity. Some
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additional related articles to the topic of wavy surface and viscous fluid are cited in the refs. [23-31]. The goal of this work is to explore more detailed information about the influence of heat source / sink
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on heat transfer rate in natural convection flow along vertical wavy surface in nanofluid. The nonsimilar solution of the coupled non-linear partial differential equations of natural convection flow has been obtained numerically using Keller-Box technique. The behavior of skin-friction and Nusselt
) and magnetite (
). Also the values of friction factor and heat transfer rate for two
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alumina (
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number has been discussed for variation in the governing parameters for two nanoparticles, namely,
nanoparticles alumina and magnetite are compared with the pure fluid by calculating their percentages.
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The calculated Nusselt number and skin friction are then compared with the existing data in the literature and percent change has been reported.
2. Mathematical formulation 2.1. Flow geometry and boundary conditions Consider two-dimensional incompressible, steady, natural convective flow of nanofluid along vertically fixed wavy plate. The configuration of the wavy sheet is described by sinusoidal function described ; is the wave length and is the amplitude of wavy surface as shown in Figure 1. It is assumed that all material properties of the fluid are constant except the density which varies with temperature. The wall temperature is denoted by and is the ambient temperature such that , temperature dependent heat absorption is described as the amount absorbed per unit volume and is the heat absorption constant. Due to the presence of temperature gradient, the flow is driven by the bouncy force. In view of above flow assumptions the appropriate boundary conditions read as: o all , (1) o all , where
denote pressure and
denotes the ambient pressure,
along - and - directions respectively.
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and
denote the velocity components
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2.2.
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Figure 1: Physical model and flow diagram.
Nanofluid: thermo-physical properties
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The material properties such as density, effective dynamic viscosity, heat capacity, thermal expansion coefficient, effective thermal conductivity of pure fluid and nanoparticles are given in Table 1. In this paper we consider the alumina and magnetite nanoparticles. Alumina is the strongest and stiffest of the
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oxide ceramics and widely used material in engineering ceramics. Alumina has a relatively high
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thermal conductivity in ceramic. It has acceptable price therefore, its cost is less to fabricating alumina shapes. Magnetite is normal and one of the three common natural oxides of iron. It has great
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significance such as, magnetic recording media, high density digital recording disc, in drug delivery system, cancer therapy and medical diagnostic etc. Table 1: Material properties of Properties
Fluid(water)
,
and base fluid.
)
)
4179
997.1
0.613
21.0
670
5180
9.7
0.5
765
3970
40
0.85
Based on theoretical and experimental findings, there are several different expressions for the calculation of the thermos-physical properties of the nanofluid [32]. However the most commonly used are given below:
which have been utilized in the present analysis.
2.3.
Governing equations 4
ACCEPTED MANUSCRIPT According to flow assumptions discussed in Sec. 2.1 the flow is two-dimensional and steady in nature. The base fluid is assumed to be water at normal temperature which allows the utilization of boundary layer assumptions. The convective transport of the nanoparticles has been described by the famous
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Tiwari and Das model [15]. According to this model the mass, momentum and energy conservation laws read as
where
is the Laplacian operator,
density,
is the specific heat,
(2) (3)
,
,
(4)
represents components of the velocity vector, is thermal expansion coefficient,
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subscripts “ ” “
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-
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,
is the
is thermal diffusivity and
” and “ ” e e to fluid, nanofluid and nanoparticle respectively.
To transform the transport equations to the dimensionless form where the domain of interest is,
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, we introduce the following variables and parameters:
(5)
Accordingly, the system (3) – (4) transforms to the following non-linear partial differential equations (6) (7)
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ACCEPTED MANUSCRIPT where the eq. (2) is satisfied identically. In eq. (6) and (7) sink parameter,
is the wavy parameter and
is the Prandtl number,
, &
subsc ipt ’ ’ represents derivative with respect to
is the heat
are material parameters. The
and the ‘ ' ’ denotes differentiation with respect to
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. Boundary conditions in non-dimensional form read as
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(8)
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The surface shear stress in term of friction factor and the heat transfer rate in term of Nusselt number
(9)
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are respectively described as
3. Computational procedure and validation of results The governing equations (6)–(8) which are non-similar in nature have been integrated through a finite
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difference scheme [33, 34] which is commonly referred as Keller-Box scheme. For solving the system
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(6)–(8) the following order is followed: first the equations are converted in to a system of first order equations which are successively linearize by Newton linearization formula, written in matrix form and
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finally with the help of block-tridiagonal-elimination method linear system is solved. It is well established that this technique gives correct results for boundary layer equations. This algorithm is unconditionally stable and has second order convergence. The value of large for the convergence of the solution. A uniform grid size of
is taken sufficiently was utilized. Also,
for accuracy of the solution the difference between the current and previous iteration was fixed at in order to stop the iteration process.
For the validation of present solution tabularized data is compared with the already published results present in the open literature and displayed in Tables 2, 3 and 4. The calculated values of
,
,
are compared with those of Kabir et al. [19], Alim et al. [20] and Hossain et al. [35]. It is obvious from these Tables that the values of
,
,
are very well
matched with the already published results. Table 2: Comparison of Pr present
0.72
0.74643
and
values when
Alim et al.[20] 0.74641 6
Present 0.33683
Alim et al.[20] 0.33715
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Table 3: Comparison of values
present
Alim et al.[20] 096548 0.92773 0.90585 0.90512 0.91769
0.90814 0.59269 0.48733 0.41727 0.35559
when
Present
Hossain Alim et al. [35] [20]
Kabir et al. [19]
0.90813 0.59270 0.48732 0.41728 0.35558
0.4010 0.8268 1.0690 1.2896 1.5495
0.401 0.825 1.066 1.066 1.542
0.40102 0.82662 1.06848 1.28878 1.54828
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Alim et al. Kabir et [20] al. [19]
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4.
and
Alim et al.[20] -0.31624 -0.62799 -1.22414 -1.82501 -2.84207
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Table 4: Comparison of present Hossain et al. [34] 1 0.9082 0.908 10 0.5928 0.591 25 0.4876 0.485 50 0.4176 0.485 100 0.3559 0.352
Present -0.31579 -0.62468 -1.2180 -1.8157 -2.8271
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0.96720 0.92711 0.90460 0.90353 0.91563
0.43391 0.54159 0.61277 0.69810
when
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0.72 1.5 3.0 4.5 7.0
and
0.43354 0.54120 0.61241 0.69782
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0.6690 0.58220 0.53827 0.49260
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0.66054 0.58201 0.53822 0.49270
SC
1.5 3.0 4.5 7.0
0.40101 0.82663 1.06847 1.28879 1.54827
Results and discussion
In this paper we have analyzed the effects of heat sink ( nanoparticle concentration (
), waviness parameter ( ) and
on the free convection flow of nanofluid over a vertical wavy surface.
The numerical solution of the problem is obtained by employing the implicit finite-difference technique. (
The
physical
parameters
and usselt numbe
of
primary
-
interest
are
the
skin
friction
. These quantities are obtained and shown in
equation (9). The computations are carried for two kinds of nanoparticles assuming water as base fluid, range of nanoparticle centration is taken from 0 to 0.2. The nanoparticles used in this study are from alumina ( the case of friction (
) and magnetite ( -water and
). The distributions of the skin friction and Nusselt number in -water are plotted in Figs. 2-10. Numerical values of local skin
), and local Nusselt number (
absorption parameter (
). 7
-
) are tabulated in Table 5 for heat
ACCEPTED MANUSCRIPT The influence of heat sink ( and
-
), waviness parameter ( ), and nanoparticle concentration (
are presented in Figures 2 - 6 for two nanoparticles alumina and
. From these figures, it is observed that an increase in the value of heat
) leads to an decrease in the values of
-
. Because the increase in heat absorption ( -
heated surface which results in increase in and
) forms a layer of cold fluid near the
. It is noted that for the case of alumina more
-
are attained as compared to magnetite. The
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increment in the values of
and a increase in the value of
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absorption (
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-
), on the
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magnetite. The Figures 2 and 3 illustrate the behavior of heat sink parameter, ( and
on
Figure 4 illustrates the effects of wavy amplitude on
). It is
decreases with the increase of waviness behavior. Effects of nanoparticle and
concentration on
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observed that
at heat absorption (
-
and
) are shown in Figures 5 -
that as the nanoparticle
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and 6. It is noted from the graphs of
at heat absorption (
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concentration increases, the viscous force of the nanofluid increases. Rate of flow is decreased due to more resistive fluid and
is increased. Also, it is noted that the heat transfer rate rises with
in case of and
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the increment in the nanoparticles concentration at heat absorption ( -nanoparticle as compared to -
-
).
is higher
-nanoparticle. Dimensionless
as a function of wavy amplitude, nanoparticle concentration, and heat
absorbtion parameter for two nanoparticles alumina and magnetite water based nanofluid are plotted in Figs. 7-10 at fixed
-
. Figure 7 illustrates variation in the
as
function of waviness parameter for the said two nanoparticles. Increasing the values of waviness parameter
-
decreases whereas, both quantities are greater for alumina nanoparticle as -
compared to magnetite nanoparticle. Figure 8 predicts the variation in the of nanoparticle concentration for two nanoparticles. Here, we see that the 9 and 10 show the change in
and
-
parameter. From these Figures it is observed that
rises. Figures
as function of heat absorption ( decreases and
with increasing heat absorption (
. Also, these Figures reveal that
better enhancement as compared to
-nanoparticle.
8
-
as function
-
)
increases
-nanoparticle exhibit
ACCEPTED MANUSCRIPT and
In Table 5 percent change in
-
are calculated for heat absorption (
) at different positions, i.e.
on the wavy surface for two
nanoparticles alumina and magnetite by comparing the values with flat plate (pure fluid) and wavy
-
surface highest decrease in -
increase in
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is found in comparison to flat surface (pure fluid). In comparison to flat is
is
for magnetite nanoparticle at
for alumina nanoparticle at
is
and highest
. Also, in comparison to (pure
for alumina nanoparticle at
for alumina nanoparticle at
and
.
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-
is
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fluid) wavy surface highest increase in highest increase in
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increase in
and
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surface (pure fluid) results. Table 5 shows that the overall maximum decrease in
Fig. 2: Heat generation effects on skin friction.
Fig. 4: Behavior of skin friction aagainst
.
Fig. 3: Heat generation effects on Nusselt number.
Fig. 5: Graph of skin friction influenced by .
9
on Nusselt number.
Fig. 7: variation of Nusselt number against .
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MA
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Fig. 6: Impact of
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Fig. 8: Nusselt number variation against
.
Fig. 9: Skin friction verses H
Fig. 10: Nusselt number behavior against H
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ACCEPTED MANUSCRIPT Table 6: Percent change in
and
for different nanoparticle when
% increase in
Verses
Verses
Verses
Pure flat surface
Pure wavy surface
Pure flat surface
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Nano
% increase in
Pure wavy surface
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Particle
Verses
0.5
MA
NU
SC
Material
0.5
11.1
91.9
27.1
7.2
84.8
22.3
10.6
114.6
28.5
6.6
107.1
24.0
10.0
138.8
29.0
-11.9
5.9
130.5
24.5
-32.2
9.9
149.4
29.7
-34.7
5.6
141.0
25.3
-27.5 -30.2
2.0
-8.5
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1.5
TE
1.0
D
-3.0
7. Concluding remarks
In this manuscript the impact of heat absorption for two nanoparticles on natural convection boundary layer flow along vertical wavy surface has been investigated. For this problem we adopted the Tiwari and Das model which includes nanoparticle concentration, viscosity and thermal conductivity. Natural convective flow and heat transfer along the vertical wavy surface in nanofluid containing two types of nanoparticles, namely, alumina (
) and magnetite (
) have been studied. Influence of
emerging parameters such as, nanoparticle volume fraction ( ), heat absorption ( amplitude ( ) on Nusselt number (
-
11
) and skin friction (
) and wavy ) for both the
ACCEPTED MANUSCRIPT nanofluids have been examined. The important results of the current study can be summarized as follows: 1.
Skin friction coefficient is decreasing function of heat absorption parameter and the local
Heat transfer rate increases and skin friction decreases with increasing amplitude of wavy surface for fixed value of heat absorption parameter.
On increasing nanoparticle concentration the Nusselt number and skin friction coefficient
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3.
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2.
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Nusselt number is increasing function of absorption parameter.
increase for heat absorption parameter.
Skin friction coefficient and heat transfer rate are observed higher for Alumina in comparison
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4.
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to Magnetite nanoparticle.
References
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ACCEPTED MANUSCRIPT Highlights Water-based nanofluid the magnetite and alumina with their concentration ranging from 0% to 20%.
2
Non-similar flow due to the non-flat surface texture of the vertical wall.
3
Heat transfer phenomena is enhanced in the presence of heat absorption.
4
Flow and heat transfer properties are affected significantly by the nanofluid in comparison to the pure fluid.
5
The alumina ( nanoparticle.
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1
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) gives higher heat transfer rate as compared to the magnetite (
15
)
S0167-7322(16)32693-9 doi:10.1016/j.molliq.2016.10.122 MOLLIQ 6526
To appear in:
Journal of Molecular Liquids
Received date: Accepted date:
10 September 2016 26 October 2016
Please cite this article as: Ahmer Mehmood, Muhammad Saleem Iqbal, Effect of heat absorption in natural convection nanofluid flow along a vertical wavy surface, Journal of Molecular Liquids (2016), doi:10.1016/j.molliq.2016.10.122
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Effect of heat absorption in natural convection nanofluid flow along a vertical wavy surface
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Ahmer Mehmood, Muhammad Saleem Iqbal1 Department of Mathematics and Statistics, FBAS, International Islamic University,
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Islamabad 44000, Pakistan.
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Abstract:
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Effects of heat absorption and nanoparticle on natural convection heat transfer along vertical wavy surface have been investigated. Transport equations have been solved numerically by accurate implicit finite difference scheme. The skin friction and Nusselt number are plotted against variation of several
MA
parameters for two types of nanoparticles namely, alumina (
) and magnetite (
). The impact
of nanoparticle concentration on flow and heat transfer process in the problem under investigation has
water based nanofluid. The influence of heat absorption
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heat transfer rate in comparison to
-water nanofluid exhibits higher skin friction and
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been studied in detail. The results indicate that
parameter is to increase the heat transfer rate and decrease the skin friction coefficient. For heat
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absorption case percent change in the skin friction and Nusselt number for two nanoparticles is shown in tabular form where comparison to the flat plate (pure fluid) and wavy surface (pure fluid) cases have also been made. The present results have been validated by producing the results available in literature and a very good agreement is observed. Keywords: Natural convection, nanofluid, heat absorption, vertical wavy surface, Keller-Box method
1. Introduction
The real heat transfer processes associated with the change of some form of energy into thermal energy have the possibility of involving internal heat absorption. Heat transfer and fluid flow phenomena with heat absorption is associated with large temperature gradient which is frequently met in several engineering and thermal processes such as, in the combustion chamber, in thermal control of space ships, in casting and blading of gas turbines and in spent fuel storage [1], in post-accident heat removal [2], in engine cooling system and in insulation of buildings etc.
1
Corresponding Author’s Email: [email protected] Cell No. +923335195130
1
ACCEPTED MANUSCRIPT Natural convection flows happen in many practical phenomenon caused by sink. The importance of heat sink increases in those flows where the chemical reaction occurs. A change due to heat absorption occurs in temperature distribution, particle decomposition rate in semiconductor products and nuclear
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reactors. Therefore, heat absorption, convection and conduction phenomena would have to be determined simultaneously.
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Vajravelu and Hadjinicolaou [3] considered an internal heat source over a stretching surface. Molla et
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al. [4] studied magneto hydrodynamic free convection flow on a sphere under the influence of heat source. Alam et al. [5] investigated heat source effect on MHD free convection flow over a sphere.
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Mamun et al. [6] analyzed effect of heat source along a vertical plate. Mamun et al. [7] discussed impact of viscous dissipation and heat source on heat transfer over a vertical plate. Mansoor and
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Ahmed [8] explored natural convection in porous triangular enclosure using nanofluid and heat generation.
Particularly, nanofluid is a fluid which has nanoparticles with size 1 to 100 nm spread in the fluid like
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water, ethylene, engine oil, menthol and glycol. Nanofluid can play vital role in the creation of new
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technologies for industrial and engineering processes. It can flow smoothly because they are so small to act similar to liquid molecules. Thermal conductivity of nanofluid with dispersed non-metallic or
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metallic particles are expected to be considerably greater than common heat transfer liquids. Consequently, the dispersed nanoparticles can considerably modify the thermal and transport properties of the base fluid. The considerable augmentation of convective heat transfer by forced convection has widely been studied [9-15]. The research on nanofluid has been focused mostly as it finds applications in various fields such as nano-drug delivery and advanced nuclear systems, etc. Therefore, all the normal heat transfer mods can be used for nanofluid only if the properties of base fluids are interchanged by nanofluid. Nanofluid can play vital role in the innovation of new technologies for industrial and engineering processes. The natural convection flow process commonly takes place in nature and also in several physical phenomenon such as combustion modeling, fire engineering, nuclear reactor cooling, petroleum reservoir, heat exchangers and in many industrial applications. The basic flow phenomena arises due to the buoyancy effects, which is generated from temperature difference. In natural convection flow heat transfer phenomena are the integral part and is regulated by bouncy effects which is generated from diffusion of thermal energy [16]. Consequently, neglecting the other effects, buoyancy term is combined with the Navier-Stoke’s equation and energy equations. In most of the physical situations 2
ACCEPTED MANUSCRIPT heat source enhances the free convective flow and heat transfer characteristic. Molla et al. [17] presented natural convective flow through a vertical wavy sheet under heat generation. Hady et al. [18] studied MHD natural convective flow through a vertical wavy sheet by considering heat source. Kabir
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et al. [19] discussed heat source impact on MHD free convective flow along a heated vertical wavy plate. Alim et al. [20] analyzed natural convective flow through a vertical wavy sheet by taking into
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account viscosity depending upon temperature and heat source. Parveen and Alim [21-22] discussed
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MHD natural convective flow through a vertical wavy sheet with heat source / sink, joule heating, viscosity depending upon temperature and temperature dependent thermal conductivity. Some
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additional related articles to the topic of wavy surface and viscous fluid are cited in the refs. [23-31]. The goal of this work is to explore more detailed information about the influence of heat source / sink
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on heat transfer rate in natural convection flow along vertical wavy surface in nanofluid. The nonsimilar solution of the coupled non-linear partial differential equations of natural convection flow has been obtained numerically using Keller-Box technique. The behavior of skin-friction and Nusselt
) and magnetite (
). Also the values of friction factor and heat transfer rate for two
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alumina (
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number has been discussed for variation in the governing parameters for two nanoparticles, namely,
nanoparticles alumina and magnetite are compared with the pure fluid by calculating their percentages.
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The calculated Nusselt number and skin friction are then compared with the existing data in the literature and percent change has been reported.
2. Mathematical formulation 2.1. Flow geometry and boundary conditions Consider two-dimensional incompressible, steady, natural convective flow of nanofluid along vertically fixed wavy plate. The configuration of the wavy sheet is described by sinusoidal function described ; is the wave length and is the amplitude of wavy surface as shown in Figure 1. It is assumed that all material properties of the fluid are constant except the density which varies with temperature. The wall temperature is denoted by and is the ambient temperature such that , temperature dependent heat absorption is described as the amount absorbed per unit volume and is the heat absorption constant. Due to the presence of temperature gradient, the flow is driven by the bouncy force. In view of above flow assumptions the appropriate boundary conditions read as: o all , (1) o all , where
denote pressure and
denotes the ambient pressure,
along - and - directions respectively.
3
and
denote the velocity components
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ACCEPTED MANUSCRIPT
2.2.
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Figure 1: Physical model and flow diagram.
Nanofluid: thermo-physical properties
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The material properties such as density, effective dynamic viscosity, heat capacity, thermal expansion coefficient, effective thermal conductivity of pure fluid and nanoparticles are given in Table 1. In this paper we consider the alumina and magnetite nanoparticles. Alumina is the strongest and stiffest of the
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oxide ceramics and widely used material in engineering ceramics. Alumina has a relatively high
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thermal conductivity in ceramic. It has acceptable price therefore, its cost is less to fabricating alumina shapes. Magnetite is normal and one of the three common natural oxides of iron. It has great
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significance such as, magnetic recording media, high density digital recording disc, in drug delivery system, cancer therapy and medical diagnostic etc. Table 1: Material properties of Properties
Fluid(water)
,
and base fluid.
)
)
4179
997.1
0.613
21.0
670
5180
9.7
0.5
765
3970
40
0.85
Based on theoretical and experimental findings, there are several different expressions for the calculation of the thermos-physical properties of the nanofluid [32]. However the most commonly used are given below:
which have been utilized in the present analysis.
2.3.
Governing equations 4
ACCEPTED MANUSCRIPT According to flow assumptions discussed in Sec. 2.1 the flow is two-dimensional and steady in nature. The base fluid is assumed to be water at normal temperature which allows the utilization of boundary layer assumptions. The convective transport of the nanoparticles has been described by the famous
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Tiwari and Das model [15]. According to this model the mass, momentum and energy conservation laws read as
where
is the Laplacian operator,
density,
is the specific heat,
(2) (3)
,
,
(4)
represents components of the velocity vector, is thermal expansion coefficient,
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subscripts “ ” “
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-
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,
is the
is thermal diffusivity and
” and “ ” e e to fluid, nanofluid and nanoparticle respectively.
To transform the transport equations to the dimensionless form where the domain of interest is,
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, we introduce the following variables and parameters:
(5)
Accordingly, the system (3) – (4) transforms to the following non-linear partial differential equations (6) (7)
5
ACCEPTED MANUSCRIPT where the eq. (2) is satisfied identically. In eq. (6) and (7) sink parameter,
is the wavy parameter and
is the Prandtl number,
, &
subsc ipt ’ ’ represents derivative with respect to
is the heat
are material parameters. The
and the ‘ ' ’ denotes differentiation with respect to
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. Boundary conditions in non-dimensional form read as
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(8)
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The surface shear stress in term of friction factor and the heat transfer rate in term of Nusselt number
(9)
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are respectively described as
3. Computational procedure and validation of results The governing equations (6)–(8) which are non-similar in nature have been integrated through a finite
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difference scheme [33, 34] which is commonly referred as Keller-Box scheme. For solving the system
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(6)–(8) the following order is followed: first the equations are converted in to a system of first order equations which are successively linearize by Newton linearization formula, written in matrix form and
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finally with the help of block-tridiagonal-elimination method linear system is solved. It is well established that this technique gives correct results for boundary layer equations. This algorithm is unconditionally stable and has second order convergence. The value of large for the convergence of the solution. A uniform grid size of
is taken sufficiently was utilized. Also,
for accuracy of the solution the difference between the current and previous iteration was fixed at in order to stop the iteration process.
For the validation of present solution tabularized data is compared with the already published results present in the open literature and displayed in Tables 2, 3 and 4. The calculated values of
,
,
are compared with those of Kabir et al. [19], Alim et al. [20] and Hossain et al. [35]. It is obvious from these Tables that the values of
,
,
are very well
matched with the already published results. Table 2: Comparison of Pr present
0.72
0.74643
and
values when
Alim et al.[20] 0.74641 6
Present 0.33683
Alim et al.[20] 0.33715
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Table 3: Comparison of values
present
Alim et al.[20] 096548 0.92773 0.90585 0.90512 0.91769
0.90814 0.59269 0.48733 0.41727 0.35559
when
Present
Hossain Alim et al. [35] [20]
Kabir et al. [19]
0.90813 0.59270 0.48732 0.41728 0.35558
0.4010 0.8268 1.0690 1.2896 1.5495
0.401 0.825 1.066 1.066 1.542
0.40102 0.82662 1.06848 1.28878 1.54828
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Alim et al. Kabir et [20] al. [19]
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4.
and
Alim et al.[20] -0.31624 -0.62799 -1.22414 -1.82501 -2.84207
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Table 4: Comparison of present Hossain et al. [34] 1 0.9082 0.908 10 0.5928 0.591 25 0.4876 0.485 50 0.4176 0.485 100 0.3559 0.352
Present -0.31579 -0.62468 -1.2180 -1.8157 -2.8271
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0.96720 0.92711 0.90460 0.90353 0.91563
0.43391 0.54159 0.61277 0.69810
when
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0.72 1.5 3.0 4.5 7.0
and
0.43354 0.54120 0.61241 0.69782
PT
0.6690 0.58220 0.53827 0.49260
RI
0.66054 0.58201 0.53822 0.49270
SC
1.5 3.0 4.5 7.0
0.40101 0.82663 1.06847 1.28879 1.54827
Results and discussion
In this paper we have analyzed the effects of heat sink ( nanoparticle concentration (
), waviness parameter ( ) and
on the free convection flow of nanofluid over a vertical wavy surface.
The numerical solution of the problem is obtained by employing the implicit finite-difference technique. (
The
physical
parameters
and usselt numbe
of
primary
-
interest
are
the
skin
friction
. These quantities are obtained and shown in
equation (9). The computations are carried for two kinds of nanoparticles assuming water as base fluid, range of nanoparticle centration is taken from 0 to 0.2. The nanoparticles used in this study are from alumina ( the case of friction (
) and magnetite ( -water and
). The distributions of the skin friction and Nusselt number in -water are plotted in Figs. 2-10. Numerical values of local skin
), and local Nusselt number (
absorption parameter (
). 7
-
) are tabulated in Table 5 for heat
ACCEPTED MANUSCRIPT The influence of heat sink ( and
-
), waviness parameter ( ), and nanoparticle concentration (
are presented in Figures 2 - 6 for two nanoparticles alumina and
. From these figures, it is observed that an increase in the value of heat
) leads to an decrease in the values of
-
. Because the increase in heat absorption ( -
heated surface which results in increase in and
) forms a layer of cold fluid near the
. It is noted that for the case of alumina more
-
are attained as compared to magnetite. The
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increment in the values of
and a increase in the value of
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absorption (
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-
), on the
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magnetite. The Figures 2 and 3 illustrate the behavior of heat sink parameter, ( and
on
Figure 4 illustrates the effects of wavy amplitude on
). It is
decreases with the increase of waviness behavior. Effects of nanoparticle and
concentration on
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observed that
at heat absorption (
-
and
) are shown in Figures 5 -
that as the nanoparticle
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and 6. It is noted from the graphs of
at heat absorption (
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concentration increases, the viscous force of the nanofluid increases. Rate of flow is decreased due to more resistive fluid and
is increased. Also, it is noted that the heat transfer rate rises with
in case of and
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the increment in the nanoparticles concentration at heat absorption ( -nanoparticle as compared to -
-
).
is higher
-nanoparticle. Dimensionless
as a function of wavy amplitude, nanoparticle concentration, and heat
absorbtion parameter for two nanoparticles alumina and magnetite water based nanofluid are plotted in Figs. 7-10 at fixed
-
. Figure 7 illustrates variation in the
as
function of waviness parameter for the said two nanoparticles. Increasing the values of waviness parameter
-
decreases whereas, both quantities are greater for alumina nanoparticle as -
compared to magnetite nanoparticle. Figure 8 predicts the variation in the of nanoparticle concentration for two nanoparticles. Here, we see that the 9 and 10 show the change in
and
-
parameter. From these Figures it is observed that
rises. Figures
as function of heat absorption ( decreases and
with increasing heat absorption (
. Also, these Figures reveal that
better enhancement as compared to
-nanoparticle.
8
-
as function
-
)
increases
-nanoparticle exhibit
ACCEPTED MANUSCRIPT and
In Table 5 percent change in
-
are calculated for heat absorption (
) at different positions, i.e.
on the wavy surface for two
nanoparticles alumina and magnetite by comparing the values with flat plate (pure fluid) and wavy
-
surface highest decrease in -
increase in
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is found in comparison to flat surface (pure fluid). In comparison to flat is
is
for magnetite nanoparticle at
for alumina nanoparticle at
is
and highest
. Also, in comparison to (pure
for alumina nanoparticle at
for alumina nanoparticle at
and
.
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D
MA
-
is
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fluid) wavy surface highest increase in highest increase in
SC
increase in
and
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surface (pure fluid) results. Table 5 shows that the overall maximum decrease in
Fig. 2: Heat generation effects on skin friction.
Fig. 4: Behavior of skin friction aagainst
.
Fig. 3: Heat generation effects on Nusselt number.
Fig. 5: Graph of skin friction influenced by .
9
on Nusselt number.
Fig. 7: variation of Nusselt number against .
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MA
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Fig. 6: Impact of
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Fig. 8: Nusselt number variation against
.
Fig. 9: Skin friction verses H
Fig. 10: Nusselt number behavior against H
10
ACCEPTED MANUSCRIPT Table 6: Percent change in
and
for different nanoparticle when
% increase in
Verses
Verses
Verses
Pure flat surface
Pure wavy surface
Pure flat surface
PT
Nano
% increase in
Pure wavy surface
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Particle
Verses
0.5
MA
NU
SC
Material
0.5
11.1
91.9
27.1
7.2
84.8
22.3
10.6
114.6
28.5
6.6
107.1
24.0
10.0
138.8
29.0
-11.9
5.9
130.5
24.5
-32.2
9.9
149.4
29.7
-34.7
5.6
141.0
25.3
-27.5 -30.2
2.0
-8.5
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1.5
TE
1.0
D
-3.0
7. Concluding remarks
In this manuscript the impact of heat absorption for two nanoparticles on natural convection boundary layer flow along vertical wavy surface has been investigated. For this problem we adopted the Tiwari and Das model which includes nanoparticle concentration, viscosity and thermal conductivity. Natural convective flow and heat transfer along the vertical wavy surface in nanofluid containing two types of nanoparticles, namely, alumina (
) and magnetite (
) have been studied. Influence of
emerging parameters such as, nanoparticle volume fraction ( ), heat absorption ( amplitude ( ) on Nusselt number (
-
11
) and skin friction (
) and wavy ) for both the
ACCEPTED MANUSCRIPT nanofluids have been examined. The important results of the current study can be summarized as follows: 1.
Skin friction coefficient is decreasing function of heat absorption parameter and the local
Heat transfer rate increases and skin friction decreases with increasing amplitude of wavy surface for fixed value of heat absorption parameter.
On increasing nanoparticle concentration the Nusselt number and skin friction coefficient
SC
3.
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2.
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Nusselt number is increasing function of absorption parameter.
increase for heat absorption parameter.
Skin friction coefficient and heat transfer rate are observed higher for Alumina in comparison
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4.
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to Magnetite nanoparticle.
References
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ACCEPTED MANUSCRIPT Highlights Water-based nanofluid the magnetite and alumina with their concentration ranging from 0% to 20%.
2
Non-similar flow due to the non-flat surface texture of the vertical wall.
3
Heat transfer phenomena is enhanced in the presence of heat absorption.
4
Flow and heat transfer properties are affected significantly by the nanofluid in comparison to the pure fluid.
5
The alumina ( nanoparticle.
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1
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) gives higher heat transfer rate as compared to the magnetite (
15
)