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Marangoni convection radiative flow of dusty nanoliquid with exponential space dependent heat source
Accepted Manuscript Marangoni convection radiative flow of dusty nanoliquid with exponential space dependent heat source B. Mahanthesh, B.J. Gireesha, B.C. Prasannakumara, N.S. Shashikumar PII:
S1738-5733(17)30357-1
DOI:
10.1016/j.net.2017.08.015
Reference:
NET 427
To appear in:
Nuclear Engineering and Technology
Received Date: 13 June 2017 Revised Date:
1 August 2017
Accepted Date: 16 August 2017
Please cite this article as: B Mahanthesh, B.J Gireesha, B.C Prasannakumara, N.S Shashikumar, Marangoni convection radiative flow of dusty nanoliquid with exponential space dependent heat source, Nuclear Engineering and Technology (2017), doi: 10.1016/j.net.2017.08.015. 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.
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Marangoni convection radiative flow of dusty nanoliquid with exponential space dependent heat source
1
Department of Mathematics, Christ University, Bangalore-560029, INDIA.
2
Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta 577451, Shimoga, Karnataka, INDIA.
Government First Grade College, Koppa, Chikkamagaluru-577126, Karnataka, INDIA.
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3
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B.Mahanthesh1,2,a, B.J.Gireesha2,b, B.C.Prasannakumara3, N.S.Shashikumar2
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Emails: [email protected], [email protected],[email protected]
Abstract:The flow of liquids submerged with nanoparticles is of significance to industrial applications, specifically in nuclear reactors and the cooling of nuclear systems to improve energy efficiency. The application of nanofluids in water-cooled nuclear systems can result in a
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significant improvement of their economic performance and/or safety margins. Therefore, in this paper, Marangoni thermal convective boundary layer dusty nanoliquid flow acrossa flat surface in the presence of solar radiation is studied. A two phase dusty liquid model is considered.
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Unlike classical temperature dependent heat source effects, an exponential space dependent heat source aspect is considered. Stretching variables are utilized to transform the prevailing partial
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differential system into a nonlinear ordinary differential system, which is then solved numerically via Runge-Kutta-Fehlberg approach coupled with shooting technique. The roles of physical parameters are focused
in momentum and heat transport distributions. Graphical
illustrations are also used to consider local and average Nusselt numbers. We examined the results under both linear and quadratic variation of the surface temperature. Our simulations established that the impact of Marangoni flow is useful for an enhancement of the heat transfer rate.
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Keywords: Marangoni convection; Dusty fluid; Nano fluid; Thermal radiation; exponential space dependent heat source; Average Nusselt number. ∞
average Nusselt number
(x, y)
cartesian coordinates [m]
ambient carrier fluid temperature [K] Greek symbols ∆
, , constants
dimensionless Nusselt number
exponential index
heat source parameter
∗
heat source coefficient [kgs-3]
Marangoni number
mean absorption coefficient [ ]
%
radiation parameter
Prandtl number
&
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radius of dust particles
(
specific heat [m2s-2K-1]
thermal relaxation time of dust particles
ν
kinematic viscosity [m2s-1]
mass of dust particles
momentum relaxation time of dust
positive fluid property
!∗
', Ψ !
particles
specific heat parameter
Stefan–Boltzmann constant
[" # $ ]
stream functions surface tension
thermal conductivity [kgms-3K-1]
Subscripts
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(, 4)
momentum dust parameter
surface length [m]
surface carrier fluid temperature [K]
(., /)
∗
dynamic viscosity [m2s-1]
*
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)
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dust particles mass concentration
+
density [kgm-3]
∗ constant exponent of the temperature
constant characteristic temperature [K]
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velocities of dust phases along x and y direction [ms-1]
velocities of nanofluid phases along
x and y direction [ms-1]
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Nomenclature
thermal dust parameter
,
volume fraction of nanoparticles
3
base fluid
5
∞
conditions at the free stream conditions at the surface
p
dust phase
6
nanofluid
3
nanoparticles
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1. Introduction The deviation of liquid surface tension with the temperature (thermocapillarity) or with
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the concentration (solutocapillarity) is recognized as the Marangoni effect. The discrepancy of surface tension leads to convective transport of the liquid, or Marangoni convection. This phenomenon emerges in various practical applications in fields such as aerospace, material science and chemical engineering, silicon melts, thin liquid films, chemical reactions and
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polymer physics. Christopher and Wang [1] reported a similarity solution for Marangoni convective flow towards a planar surface by accounting for boundary layer development along
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the surface. Zheng et al. [2] proposed a theoretical study of thermal Marangoni convection flow induced by temperature gradient on a free surface. They solved the governing expression via the Adomian decomposition method combined with the Padeapproximant procedure. Marangoni convection driven by thermo and solute gradient on flow along a permeable surface is addressed
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by Mudhaf and Chamkha [3]. They found that the wall velocity, and the heat and mass transport rates increased with the thermo-solutal surface tension ratio parameter. Dandapat et al. [4,5] illustrated the impact of changeable liquid properties and thermocapillarity on time dependent
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flow in a thin liquid film on a deformable horizontal sheet. Noor and Hashim [6] extended the study of Dandapat et al. [4] by including magnetohydrodynamic effects. Arifin et al. [7] utilized
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nanoliquids to study boundary layer flow driven by the Marangoni effect. By considering MHD, the work of Arifin et al. [7] was extended by Sastry et al. [8]. Marangoni radiative flow of pseudo-plastic nano liquids due to temperature gradient with changeable thermal conductivity was proposed by Lin et al. [9]. Recently, several papers have focused on Marangoni convection flow under distinct aspects [see 10-18 and references there in].
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Modern evaluation in engineering has led to enhanced performance of advanced thermal systems. Improvementcan be achieved by using nanoliquids in such systems because the characteristic feature of nanofluids is thermal conductivity enhancement. Particular applications
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of nanofluids are engine cooling, solar water heating, cooling of electronics, cooling of transformer oil, cooling of heat exchanging devices, and improving of heat transfer efficiency of chillers, domestic refrigerator-freezers, etc. Nanoliquids also find their relevance in nuclear
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reactor and nuclear system cooling. Buongiorno and Hu [19] carried out a project in a nuclear reactor to examine nanofluids and the mechanisms of enhanced heat transfer. They pointed out
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that nanofluids are best used in water-cooled nuclear systems and could result in a significant improvement of economic performance and/or safety margins. Mousavizadeh [20] scrutinized the impact of nanofluids on heat transfer characteristics in a VVER-1000 nuclear reactor via computational fluid dynamics (CFD) simulations. Their simulations established that heat transfer
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coefficient is augmented, and usage of the nanoliquids reduced in an MDNBR of a VVER-1000 nuclear reactor. Several studies (see [21]-[31]) have focused on nanoliquid flow and nuclear applications.
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During the past decade,the topic of dusty liquid material flow has grabbed the attention of engineers and scholars due to the two-phase nature of this flow. Relevance of dusty liquid flow
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can be visualized in various applications, for instance nuclear reactor cooling, powder technology, cement and steel production, atmospheric fallout, paint spray, dust collection, rain erosion, acoustics, performance of solid fuel rock nozzles, sedimentation and guided missiles. Thus, modeling, simulating and analys is of dustyliquid flow can be very interesting. Singleton [32] studied the two-phase motion of dusty liquid under boundary layer assumptions. Ezzat et al. [33] examined the impact of porosity on free convective flow of a dusty liquid. Jha and Apere
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[34] obtained exact solutions for unsteady two-phase dusty liquid flow in an annulus by employing the Laplace transformation technique. Numerical simulations were presented by Siddique et al. [35] for two-phase particulate suspension flow on planar semi-infinite vertical
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surface. Stagnation point three dimensional flow utilizing dusty-liquid material was considered by Mohaghegh and Rahimi [36]. An extensive advancement of the topic of boundary layer dusty liquid in various aspects has been done by Gireesha and co-workers ([37]-[40]). Impacts of small
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discrepancies in surface temperature and free stream velocity in dusty liquid flow were examined by Hossain et al. [41]. Mustafa [42] presented analytic solutions for two-phase dusty liquid flow
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and a heat model over deforming isothermal surfaces. Besides these studies, the impacts of nanoparticles on dusty liquid flowwith various physical aspects over deformable surfaces have been scrutinized [43-45].
Clearly, studies involving Marangoni convection and dusty nanofluids have not been
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undertaken so far. Thus, we focus on two-phase Marangoni boundary layer radiated flow of dusty nanofluid driven by thermal gradient. Unlike the traditional temperature dependent heat source effect, an exponential space dependent heat source aspect (see [46], [47]) is taken into
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account. The impact of active parameters inflowing into the problem on hydrothermal behavior has been examined via graphical illustrations.
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2. Problem Statement
Marangoni convective radiated flow driven by thermal gradient above a planar interface
is considered. A copper-water nanoliquid containing dust particles is utilized. The expression ) (7 )/9 indicates surface/ambient carrier fluid temperature. The surface temperature is supposed to be power law variation. A rectangular coordinate system (7, :) is considered, in
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Fig. 1). The surface tension !is supposed to fluctuate with temperature in linear form. That is, ! = !< [1 − ( − < )],
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here !< and < are the surface tension and the temperature characteristics, and is the positive
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fluid property;we assume that < = ∞.
Fig. 1: Schematic diagram of the problem. With the above-mentioned assumptions, governing equations for both nanofluid and dust phase are (see [35], [46]);
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+ 4A = 0,
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. + /A = 0,
CDE
+ 4A = F AA + F DE
.. + /.A = −
HI
FG
DE HI
( − .),
(( )JK L + 4A M = NJK + .W + /WA = − H ( − W),
V
(2) (3) (. − ),
(4) (5)
OP ∗ ∞ Q RS ∗
T AA +
FG UG HV
(W − ) + ∗ X7Y N− 7
IZ[ Q
:T,
(6) (7)
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Boundary conditions are (see [7], [8]) JK A =
\P \
\ \
, 4 = 0, = < + 7 ] ,at: = 0,
→ 0, . → 0, / → 4, → ∞ , W → ∞ as: → ∞.
(8)
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in which (, 4) and (., /) denote the velocities of the nanofluid and the dust phases, is the
constant exponent of the temperature,-density, & -particle density, -dynamic viscosity, ν∗ kinematic viscosity, -thermal conductivity, = 7
&
, = ∗ 7
a(IZ[) Q
-thermal relaxation time of dust particles ∗ =
are the mass and radius of the dust particles, ( and (& are the specific
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& (& /4c & , & and
&
-momentum relaxation time of dust
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∗ = & /6c particles
a(IZ[) Q
heat of the nanofluid and the dust particles, ∗ = < 7
eIf[ a
-heat source coefficient, < is a
positive constant, ! ∗ -Stefan–Boltzmann constant, -mean absorption coefficient and -
exponential index. The subscripts (x, y) denote partial derivatives and (3, 3, 6) symbolize the
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(nanofluid, base fluid, nanoparticles).
The effective properties of the nanofluid are: ` JK / K = (1 − ,) .i , SDE SE
=
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JK = (1 − ,)K + ,j , (Sk ] SE ) l(SE Sk ) (Sk ] SE )]l(SE Sk )
,
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(( )JK = (1 − ,)(()K + ,(( )j .
(9)
here ,-volume fraction of nano particles. Now, consider the following stretching transformation, as in (see [7],[8]), ' = 7
Ψ = 7
N
afI T Q
N
afI T Q
3(m ), n(m ) =
∞
o If[
, m = 7
∞ p (m ), Θ(m ) = o If[ ,
*
IZ[ Q
:, (10)
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qr qA
,. =
qΨ qA
and 4 = − q , / = − q . qr
qΨ
Further, , , and are constants with , and given by ([7], [8]). ∆
If[
, = s!< K / K , = sK /!< K , Q
Q
with+- surface length and ∆-constant characteristic temperature. Substituting (10) into(2)-(8), we have
R
t
wx l]l (u|)k (u|)E R
n ′′ +
]
R
33 ′′ + (p′ − 3′) −
] R
p ′ = 0,
] R
3 ′ = 0,
Θ′ p − ( + 1)p ′ Θ + (n − Θ) = 0,
(13)
(14)
(15)
3′′(0) = −2(1 − ,) .i , 3 (0) = 0, n (0) = 1,
3 ′ (∞) = 0, p ′ (∞) = 0, p (∞) = 3 (∞), n(∞) = 0, Θ(∞) = 0.
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where prime signifies the derivative with respect to concentration, = H∗ UE
UG
=F
I [ a
-specific heat parameter, =
E UE o[ a
(16) m, =
-momentum dust parameter, * = H∗
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=
(12)
n ′3 + (Θ − n ) − ( + 1)3 ′ n + exp(−m ) = 0,
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with,
R
pp ′′ + (3 ′ − p′) −
yDE z{yDE ] v yE QyE
]
]
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]
3 ′′′ +
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u ( l)a.e t l]l k v uE
(11)
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=
LCUG M SE
E
V [ a
-Prandtl number, % =
FG
FE
-dust particlemass
-thermal dust parameter,
Q OP∗ ∞
RSE S ∗
-radiation parameter and
heat source parameter.
Thelocal Nusselt number is = S
SDE N
V T
E [(,<) (,∞)]
,
(17)
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Dimensionless Nusselt number is (see [7], [8]); = −
SDE SE
N1 + %T 7 $ R
If[ Q
n′(0).
(18)
surface and the ambient temperature is given by (see [7], [8]) = −
R]O SDE $]i SE
[ Q
[ Q
n′(0),
(19)
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here = (K + ] /K K is the Marangoni number.
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The average Nusselt number , based on the average temperature difference between the
3. Solution Procedure:
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Nonlinear two-point boundary value problems referred to in Eqns. (12)-(16) are
integrated via the Runge-Kutta-Fehlberg technique. Now, we use the following set of variables: 3 = 7 , 3 = 7 , 3 = 7R , p = 7$ , p = 7i ,
n = 7O , n = 7 , Θ = 7 .
(20)
In view of (20), equations (12)-(15) take following form:
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7 = 7 , 7 = 7R ,
7R = − N (7i − 7 ) +
]
R
7 7R −
]
R
R
] R
7
T (1 − ,) .i 1 − , + ,
7i N(] ) T , 7O = 7 , R
t l]l
Fk
FE
,
z
7 7 + (7 − 7O ) − ( + 1)7 7O + exp(−m )
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7 = −
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7$ = 7i , 7i = − (7i − 7 ) −
]
Lu|GM k vwx Lu|G M E
yDE z t vN ] T yE Q
, 7 = −[ (7O − 7 ) − ( + 1)7i 7 ] N(] ) T. R
z
(21)
Respective boundary conditions are:
7 (0) = 0, 7 (0) = , 7R (0) = −2(1 − ,) .i , 7$ (0) = , 7i (0) = R , 7O (0) = 1, 7 (0) = $ , 7 (0) = i .
(22)
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[– n′ (0)] reported earlier by Grubka and Bobba [48] and Chen [49] for various values of Prandtl number in the absence of nanoparticle volume fraction and mass concentration of particles parameter ϕ=l; these values are presented in Table 1. It can be seen from the table that an
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excellent agreement exists between the results. This favorable comparison lends confidence to the numerical results that will be reported in the next section. % = n) = = 0 and = 1.
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Table 1: Comparison results for surface temperature gradient [– n (0)] in the case of , = 0, Grubka et al. [48]
Chen [49]
Present study
0.72
1.0885
1.0885
1.0884
1.0 3.0 10.0
1.3333 2.5097 4.7969
1.3333 2.5097 4.7968
1.3333 2.5096 4.7968
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4. Results and discussion
Physical interpretation of pertinent variables for distinct flow fields is the main purpose of this section. For this reason, we have prepared Figs. (2)-(12), in which green solid lines
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correspond to linear fluctuation of the surface temperature ( = 0 ) and red dotted lines
correspond to quadratic variation of the surface temperature ( = 1 ). Fig. 2 presents the behavior of the average Nusselt number for the Marangoni number, , and the dust particle
mass concentration parameter, . Here the heat transfer rate increases with increases of and .
Thus,the Marangoni effect is constructive for cooling processes. Fig. 3 shows the effects of %
and , on the Nusselt number, . The Nusselt number improves with %, while it decreases for
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of = 1. Fig. 4 depicts the impacts of and on . Here,the exponential space dependent
heat source aspect significantly enhances the heat transfer rate. However, this trend is reversed
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for larger .
Fig. 5 shows the impact of on the fluid and dust phase temperatures, n(m)and Θ(m).
We see that both n(m) and Θ(m) and their corresponding layers are significantly enhanced for
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larger . Physically more and more heat is supplied into the fluid when is enhanced, due to which both n(m) and Θ(m) are enhanced. Fig. 6 shows the responses of n(m) and Θ(m)for %. It is
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anticipated that both n(m) and Θ(m) increased when we boosted the thermal radiation strength because the radiation phenomena act as sources of energy to the fluid system.
The thermal conductivity increases for higher ,, and consequently a bulk quantity of heat
is transferred into the liquid from the plate. Thus, both n(m) and Θ(m) improve with , (see Fig.
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7). The effect of , on 3′(m) and p′(m) is portrayed in Fig. 8. Here, the results for the dusty liquid
can be recovered for , = 0. We see that the momentum layer is lower for the dusty nanoliquid
than for the dustyfluid. It can be seen in Fig. 7 that both 3′(m) and p′(m)decayed for larger ,
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because higher , increases the strength of the frictional force within the liquid motion, which
results in drops of 3′(m)andp′(m). The behavior of the velocity and temperature with variation in
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is illustrated in Figs. 9 and 10. The results for a nanoliquid can be recovered for = 0. Here,
both velocities and temperatures decrease when we increase . This is because of the relative drag force and heat exchange between the nanofluid and the dust phases. Thus,we can conclude that the use of dust particle suspensions in working liquids is a smart technique to control the flow fields. Fig. 11 reveals the effect of on n(m)and (m). Here,n(m)decreases and (m)
increases when we increase . This phenomenon is qualitatively the same in the case of the
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5. Concluding Remarks
An exponential space dependent heat source and Marangoni effects on dusty nanofluid
the surface temperature. Key remarks are:
•
The Marangoni effect is constructive for heat transfer rate.
Temperature is higher due to larger , while larger values of decay decrease both temperatures and velocities.
•
Results for (dusty fluid, nano fluid, viscous fluid) can be recovered for (, = 0, = 0,
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•
Heat transfer rate is improved for and %, whereas it decreases for and ,.
, = = 0).
Impact of linear variation of surface temperature dominates nonlinear variation of surface temperature.
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•
Thermal layers are enlarged via , while velocity layers decrease via ,.
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•
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•
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boundary layer flow toward a flat surface are examined by utilizing linear/quadratic variation of
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Fig. 2: Impact of and on .
Fig. 3: Impact of and on .
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Fig. 4: Impact of and on .
Fig. 5: Impact of on ¡(¢)and £(¢).
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Fig. 6: Impact of and ¡(¢)and £(¢).
Fig. 7: Impact of and ¡(¢)and £(¢).
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Fig. 8: Impact of and ¤′(¢)and ¥′(¢).
Fig. 9: Impact of and ¡(¢)and £(¢).
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Fig. 10: Impact of and ¤′(¢)and ¥′(¢).
Fig. 11: Impact of and ¡(¢)and £(¢).
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Fig. 12: Impact of and ¡(¢)and £(¢).
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Acknowledgment: The authors are thankful to referees and editor for their constructive suggestions. And one oftheauthors Mahanthesh B would like to express sincere gratitude to the
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Management of Christ University for their kind support.
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considering Darcy law. The European Physical Journal Plus, 132(1), 55. 27. Sheikholeslami, M. (2017). Magnetic field influence on nanofluid thermal radiation in a cavity with tilted elliptic inner cylinder. Journal of Molecular Liquids, 229, 137-147.
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28. Sheikholeslami, M., & Shehzad, S. A. (2017). Thermal radiation of ferrofluid in existence of Lorentz forces considering variable viscosity. International Journal of Heat
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and Mass Transfer, 109, 82-92.
29. Sheikholeslami, M., & Shehzad, S. A. (2017). Magnetohydrodynamic nanofluid convective flow in a porous enclosure by means of LBM. International Journal of Heat and Mass Transfer, 113, 796-805.
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30. Kumar, R., Sood, S., Sheikholeslami, M., & Shehzad, S. A. (2017). Nonlinear thermal radiation and cubic autocatalysis chemical reaction effects on the flow of stretched nanofluid under rotational oscillations. Journal of Colloid and Interface Science.
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34. Jha, B. K., &Apere, C. A. (2011). Unsteady MHD two-phase Couette flow of fluidparticle suspension in an annulus. AIP Advances, 1(4), 042121. 35. Siddiqa, S., Hossain, M. A., &Saha, S. C. (2015). Two-phase natural convection flow of
1542-1556.
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a dusty fluid. International Journal of Numerical Methods for Heat & Fluid Flow, 25(7),
36. Mohaghegh, M. R., & Rahimi, A. B. (2016). Three-Dimensional Stagnation-Point Flow and Heat Transfer of a Dusty Fluid Toward a Stretching Sheet. Journal of Heat Transfer,
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37. Gireesha, B. J., Mahanthesh, B., Manjunatha, P. T., & Gorla, R. S. R. (2015). Numerical
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38. Gireesha, B. J., Mahanthesh, B., Gorla, R. S. R., & Manjunatha, P. T. (2016). Thermal radiation and Hall effects on boundary layer flow past a non-isothermal stretching surface
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42. Turkyilmazoglu, M. (2017). Magnetohydrodynamic two-phase dusty fluid flow and heat
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44. Sandeep, N., Sulochana, C., & Kumar, B. R. (2016). Unsteady MHD radiative flow and heat transfer of a dusty nanofluid over an exponentially stretching surface. Engineering
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48. L.J. Grubka and K.M. Bobba, Heat transfer characteristics of a continuous, stretching surface with variable temperature, J. Heat Transf., 107(1) (1985) 248-250. 49. C.H. Chen, Laminar mixed convection adjacent to vertical continuously stretching sheets, J. Heat and Mass Trans., 33 (1998) 471-476.
S1738-5733(17)30357-1
DOI:
10.1016/j.net.2017.08.015
Reference:
NET 427
To appear in:
Nuclear Engineering and Technology
Received Date: 13 June 2017 Revised Date:
1 August 2017
Accepted Date: 16 August 2017
Please cite this article as: B Mahanthesh, B.J Gireesha, B.C Prasannakumara, N.S Shashikumar, Marangoni convection radiative flow of dusty nanoliquid with exponential space dependent heat source, Nuclear Engineering and Technology (2017), doi: 10.1016/j.net.2017.08.015. 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.
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Marangoni convection radiative flow of dusty nanoliquid with exponential space dependent heat source
1
Department of Mathematics, Christ University, Bangalore-560029, INDIA.
2
Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta 577451, Shimoga, Karnataka, INDIA.
Government First Grade College, Koppa, Chikkamagaluru-577126, Karnataka, INDIA.
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3
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B.Mahanthesh1,2,a, B.J.Gireesha2,b, B.C.Prasannakumara3, N.S.Shashikumar2
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Emails: [email protected], [email protected],[email protected]
Abstract:The flow of liquids submerged with nanoparticles is of significance to industrial applications, specifically in nuclear reactors and the cooling of nuclear systems to improve energy efficiency. The application of nanofluids in water-cooled nuclear systems can result in a
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significant improvement of their economic performance and/or safety margins. Therefore, in this paper, Marangoni thermal convective boundary layer dusty nanoliquid flow acrossa flat surface in the presence of solar radiation is studied. A two phase dusty liquid model is considered.
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Unlike classical temperature dependent heat source effects, an exponential space dependent heat source aspect is considered. Stretching variables are utilized to transform the prevailing partial
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differential system into a nonlinear ordinary differential system, which is then solved numerically via Runge-Kutta-Fehlberg approach coupled with shooting technique. The roles of physical parameters are focused
in momentum and heat transport distributions. Graphical
illustrations are also used to consider local and average Nusselt numbers. We examined the results under both linear and quadratic variation of the surface temperature. Our simulations established that the impact of Marangoni flow is useful for an enhancement of the heat transfer rate.
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Keywords: Marangoni convection; Dusty fluid; Nano fluid; Thermal radiation; exponential space dependent heat source; Average Nusselt number. ∞
average Nusselt number
(x, y)
cartesian coordinates [m]
ambient carrier fluid temperature [K] Greek symbols ∆
, , constants
dimensionless Nusselt number
exponential index
heat source parameter
∗
heat source coefficient [kgs-3]
Marangoni number
mean absorption coefficient [ ]
%
radiation parameter
Prandtl number
&
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radius of dust particles
(
specific heat [m2s-2K-1]
thermal relaxation time of dust particles
ν
kinematic viscosity [m2s-1]
mass of dust particles
momentum relaxation time of dust
positive fluid property
!∗
', Ψ !
particles
specific heat parameter
Stefan–Boltzmann constant
[" # $ ]
stream functions surface tension
thermal conductivity [kgms-3K-1]
Subscripts
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(, 4)
momentum dust parameter
surface length [m]
surface carrier fluid temperature [K]
(., /)
∗
dynamic viscosity [m2s-1]
*
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)
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dust particles mass concentration
+
density [kgm-3]
∗ constant exponent of the temperature
constant characteristic temperature [K]
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velocities of dust phases along x and y direction [ms-1]
velocities of nanofluid phases along
x and y direction [ms-1]
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Nomenclature
thermal dust parameter
,
volume fraction of nanoparticles
3
base fluid
5
∞
conditions at the free stream conditions at the surface
p
dust phase
6
nanofluid
3
nanoparticles
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1. Introduction The deviation of liquid surface tension with the temperature (thermocapillarity) or with
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the concentration (solutocapillarity) is recognized as the Marangoni effect. The discrepancy of surface tension leads to convective transport of the liquid, or Marangoni convection. This phenomenon emerges in various practical applications in fields such as aerospace, material science and chemical engineering, silicon melts, thin liquid films, chemical reactions and
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polymer physics. Christopher and Wang [1] reported a similarity solution for Marangoni convective flow towards a planar surface by accounting for boundary layer development along
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the surface. Zheng et al. [2] proposed a theoretical study of thermal Marangoni convection flow induced by temperature gradient on a free surface. They solved the governing expression via the Adomian decomposition method combined with the Padeapproximant procedure. Marangoni convection driven by thermo and solute gradient on flow along a permeable surface is addressed
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by Mudhaf and Chamkha [3]. They found that the wall velocity, and the heat and mass transport rates increased with the thermo-solutal surface tension ratio parameter. Dandapat et al. [4,5] illustrated the impact of changeable liquid properties and thermocapillarity on time dependent
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flow in a thin liquid film on a deformable horizontal sheet. Noor and Hashim [6] extended the study of Dandapat et al. [4] by including magnetohydrodynamic effects. Arifin et al. [7] utilized
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nanoliquids to study boundary layer flow driven by the Marangoni effect. By considering MHD, the work of Arifin et al. [7] was extended by Sastry et al. [8]. Marangoni radiative flow of pseudo-plastic nano liquids due to temperature gradient with changeable thermal conductivity was proposed by Lin et al. [9]. Recently, several papers have focused on Marangoni convection flow under distinct aspects [see 10-18 and references there in].
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Modern evaluation in engineering has led to enhanced performance of advanced thermal systems. Improvementcan be achieved by using nanoliquids in such systems because the characteristic feature of nanofluids is thermal conductivity enhancement. Particular applications
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of nanofluids are engine cooling, solar water heating, cooling of electronics, cooling of transformer oil, cooling of heat exchanging devices, and improving of heat transfer efficiency of chillers, domestic refrigerator-freezers, etc. Nanoliquids also find their relevance in nuclear
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reactor and nuclear system cooling. Buongiorno and Hu [19] carried out a project in a nuclear reactor to examine nanofluids and the mechanisms of enhanced heat transfer. They pointed out
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that nanofluids are best used in water-cooled nuclear systems and could result in a significant improvement of economic performance and/or safety margins. Mousavizadeh [20] scrutinized the impact of nanofluids on heat transfer characteristics in a VVER-1000 nuclear reactor via computational fluid dynamics (CFD) simulations. Their simulations established that heat transfer
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coefficient is augmented, and usage of the nanoliquids reduced in an MDNBR of a VVER-1000 nuclear reactor. Several studies (see [21]-[31]) have focused on nanoliquid flow and nuclear applications.
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During the past decade,the topic of dusty liquid material flow has grabbed the attention of engineers and scholars due to the two-phase nature of this flow. Relevance of dusty liquid flow
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can be visualized in various applications, for instance nuclear reactor cooling, powder technology, cement and steel production, atmospheric fallout, paint spray, dust collection, rain erosion, acoustics, performance of solid fuel rock nozzles, sedimentation and guided missiles. Thus, modeling, simulating and analys is of dustyliquid flow can be very interesting. Singleton [32] studied the two-phase motion of dusty liquid under boundary layer assumptions. Ezzat et al. [33] examined the impact of porosity on free convective flow of a dusty liquid. Jha and Apere
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[34] obtained exact solutions for unsteady two-phase dusty liquid flow in an annulus by employing the Laplace transformation technique. Numerical simulations were presented by Siddique et al. [35] for two-phase particulate suspension flow on planar semi-infinite vertical
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surface. Stagnation point three dimensional flow utilizing dusty-liquid material was considered by Mohaghegh and Rahimi [36]. An extensive advancement of the topic of boundary layer dusty liquid in various aspects has been done by Gireesha and co-workers ([37]-[40]). Impacts of small
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discrepancies in surface temperature and free stream velocity in dusty liquid flow were examined by Hossain et al. [41]. Mustafa [42] presented analytic solutions for two-phase dusty liquid flow
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and a heat model over deforming isothermal surfaces. Besides these studies, the impacts of nanoparticles on dusty liquid flowwith various physical aspects over deformable surfaces have been scrutinized [43-45].
Clearly, studies involving Marangoni convection and dusty nanofluids have not been
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undertaken so far. Thus, we focus on two-phase Marangoni boundary layer radiated flow of dusty nanofluid driven by thermal gradient. Unlike the traditional temperature dependent heat source effect, an exponential space dependent heat source aspect (see [46], [47]) is taken into
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account. The impact of active parameters inflowing into the problem on hydrothermal behavior has been examined via graphical illustrations.
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2. Problem Statement
Marangoni convective radiated flow driven by thermal gradient above a planar interface
is considered. A copper-water nanoliquid containing dust particles is utilized. The expression ) (7 )/9 indicates surface/ambient carrier fluid temperature. The surface temperature is supposed to be power law variation. A rectangular coordinate system (7, :) is considered, in
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Fig. 1). The surface tension !is supposed to fluctuate with temperature in linear form. That is, ! = !< [1 − ( − < )],
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here !< and < are the surface tension and the temperature characteristics, and is the positive
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fluid property;we assume that < = ∞.
Fig. 1: Schematic diagram of the problem. With the above-mentioned assumptions, governing equations for both nanofluid and dust phase are (see [35], [46]);
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+ 4A = 0,
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. + /A = 0,
CDE
+ 4A = F AA + F DE
.. + /.A = −
HI
FG
DE HI
( − .),
(( )JK L + 4A M = NJK + .W + /WA = − H ( − W),
V
(2) (3) (. − ),
(4) (5)
OP ∗ ∞ Q RS ∗
T AA +
FG UG HV
(W − ) + ∗ X7Y N− 7
IZ[ Q
:T,
(6) (7)
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Boundary conditions are (see [7], [8]) JK A =
\P \
\ \
, 4 = 0, = < + 7 ] ,at: = 0,
→ 0, . → 0, / → 4, → ∞ , W → ∞ as: → ∞.
(8)
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in which (, 4) and (., /) denote the velocities of the nanofluid and the dust phases, is the
constant exponent of the temperature,-density, & -particle density, -dynamic viscosity, ν∗ kinematic viscosity, -thermal conductivity, = 7
&
, = ∗ 7
a(IZ[) Q
-thermal relaxation time of dust particles ∗ =
are the mass and radius of the dust particles, ( and (& are the specific
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& (& /4c & , & and
&
-momentum relaxation time of dust
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∗ = & /6c particles
a(IZ[) Q
heat of the nanofluid and the dust particles, ∗ = < 7
eIf[ a
-heat source coefficient, < is a
positive constant, ! ∗ -Stefan–Boltzmann constant, -mean absorption coefficient and -
exponential index. The subscripts (x, y) denote partial derivatives and (3, 3, 6) symbolize the
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(nanofluid, base fluid, nanoparticles).
The effective properties of the nanofluid are: ` JK / K = (1 − ,) .i , SDE SE
=
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JK = (1 − ,)K + ,j , (Sk ] SE ) l(SE Sk ) (Sk ] SE )]l(SE Sk )
,
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(( )JK = (1 − ,)(()K + ,(( )j .
(9)
here ,-volume fraction of nano particles. Now, consider the following stretching transformation, as in (see [7],[8]), ' = 7
Ψ = 7
N
afI T Q
N
afI T Q
3(m ), n(m ) =
∞
o If[
, m = 7
∞ p (m ), Θ(m ) = o If[ ,
*
IZ[ Q
:, (10)
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qr qA
,. =
qΨ qA
and 4 = − q , / = − q . qr
qΨ
Further, , , and are constants with , and given by ([7], [8]). ∆
If[
, = s!< K / K , = sK /!< K , Q
Q
with+- surface length and ∆-constant characteristic temperature. Substituting (10) into(2)-(8), we have
R
t
wx l]l (u|)k (u|)E R
n ′′ +
]
R
33 ′′ + (p′ − 3′) −
] R
p ′ = 0,
] R
3 ′ = 0,
Θ′ p − ( + 1)p ′ Θ + (n − Θ) = 0,
(13)
(14)
(15)
3′′(0) = −2(1 − ,) .i , 3 (0) = 0, n (0) = 1,
3 ′ (∞) = 0, p ′ (∞) = 0, p (∞) = 3 (∞), n(∞) = 0, Θ(∞) = 0.
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where prime signifies the derivative with respect to concentration, = H∗ UE
UG
=F
I [ a
-specific heat parameter, =
E UE o[ a
(16) m, =
-momentum dust parameter, * = H∗
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=
(12)
n ′3 + (Θ − n ) − ( + 1)3 ′ n + exp(−m ) = 0,
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with,
R
pp ′′ + (3 ′ − p′) −
yDE z{yDE ] v yE QyE
]
]
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]
3 ′′′ +
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u ( l)a.e t l]l k v uE
(11)
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=
LCUG M SE
E
V [ a
-Prandtl number, % =
FG
FE
-dust particlemass
-thermal dust parameter,
Q OP∗ ∞
RSE S ∗
-radiation parameter and
heat source parameter.
Thelocal Nusselt number is = S
SDE N
V T
E [(,<) (,∞)]
,
(17)
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Dimensionless Nusselt number is (see [7], [8]); = −
SDE SE
N1 + %T 7 $ R
If[ Q
n′(0).
(18)
surface and the ambient temperature is given by (see [7], [8]) = −
R]O SDE $]i SE
[ Q
[ Q
n′(0),
(19)
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here = (K + ] /K K is the Marangoni number.
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The average Nusselt number , based on the average temperature difference between the
3. Solution Procedure:
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Nonlinear two-point boundary value problems referred to in Eqns. (12)-(16) are
integrated via the Runge-Kutta-Fehlberg technique. Now, we use the following set of variables: 3 = 7 , 3 = 7 , 3 = 7R , p = 7$ , p = 7i ,
n = 7O , n = 7 , Θ = 7 .
(20)
In view of (20), equations (12)-(15) take following form:
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7 = 7 , 7 = 7R ,
7R = − N (7i − 7 ) +
]
R
7 7R −
]
R
R
] R
7
T (1 − ,) .i 1 − , + ,
7i N(] ) T , 7O = 7 , R
t l]l
Fk
FE
,
z
7 7 + (7 − 7O ) − ( + 1)7 7O + exp(−m )
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7 = −
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7$ = 7i , 7i = − (7i − 7 ) −
]
Lu|GM k vwx Lu|G M E
yDE z t vN ] T yE Q
, 7 = −[ (7O − 7 ) − ( + 1)7i 7 ] N(] ) T. R
z
(21)
Respective boundary conditions are:
7 (0) = 0, 7 (0) = , 7R (0) = −2(1 − ,) .i , 7$ (0) = , 7i (0) = R , 7O (0) = 1, 7 (0) = $ , 7 (0) = i .
(22)
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[– n′ (0)] reported earlier by Grubka and Bobba [48] and Chen [49] for various values of Prandtl number in the absence of nanoparticle volume fraction and mass concentration of particles parameter ϕ=l; these values are presented in Table 1. It can be seen from the table that an
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excellent agreement exists between the results. This favorable comparison lends confidence to the numerical results that will be reported in the next section. % = n) = = 0 and = 1.
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Table 1: Comparison results for surface temperature gradient [– n (0)] in the case of , = 0, Grubka et al. [48]
Chen [49]
Present study
0.72
1.0885
1.0885
1.0884
1.0 3.0 10.0
1.3333 2.5097 4.7969
1.3333 2.5097 4.7968
1.3333 2.5096 4.7968
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4. Results and discussion
Physical interpretation of pertinent variables for distinct flow fields is the main purpose of this section. For this reason, we have prepared Figs. (2)-(12), in which green solid lines
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correspond to linear fluctuation of the surface temperature ( = 0 ) and red dotted lines
correspond to quadratic variation of the surface temperature ( = 1 ). Fig. 2 presents the behavior of the average Nusselt number for the Marangoni number, , and the dust particle
mass concentration parameter, . Here the heat transfer rate increases with increases of and .
Thus,the Marangoni effect is constructive for cooling processes. Fig. 3 shows the effects of %
and , on the Nusselt number, . The Nusselt number improves with %, while it decreases for
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of = 1. Fig. 4 depicts the impacts of and on . Here,the exponential space dependent
heat source aspect significantly enhances the heat transfer rate. However, this trend is reversed
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for larger .
Fig. 5 shows the impact of on the fluid and dust phase temperatures, n(m)and Θ(m).
We see that both n(m) and Θ(m) and their corresponding layers are significantly enhanced for
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larger . Physically more and more heat is supplied into the fluid when is enhanced, due to which both n(m) and Θ(m) are enhanced. Fig. 6 shows the responses of n(m) and Θ(m)for %. It is
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anticipated that both n(m) and Θ(m) increased when we boosted the thermal radiation strength because the radiation phenomena act as sources of energy to the fluid system.
The thermal conductivity increases for higher ,, and consequently a bulk quantity of heat
is transferred into the liquid from the plate. Thus, both n(m) and Θ(m) improve with , (see Fig.
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7). The effect of , on 3′(m) and p′(m) is portrayed in Fig. 8. Here, the results for the dusty liquid
can be recovered for , = 0. We see that the momentum layer is lower for the dusty nanoliquid
than for the dustyfluid. It can be seen in Fig. 7 that both 3′(m) and p′(m)decayed for larger ,
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because higher , increases the strength of the frictional force within the liquid motion, which
results in drops of 3′(m)andp′(m). The behavior of the velocity and temperature with variation in
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is illustrated in Figs. 9 and 10. The results for a nanoliquid can be recovered for = 0. Here,
both velocities and temperatures decrease when we increase . This is because of the relative drag force and heat exchange between the nanofluid and the dust phases. Thus,we can conclude that the use of dust particle suspensions in working liquids is a smart technique to control the flow fields. Fig. 11 reveals the effect of on n(m)and (m). Here,n(m)decreases and (m)
increases when we increase . This phenomenon is qualitatively the same in the case of the
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5. Concluding Remarks
An exponential space dependent heat source and Marangoni effects on dusty nanofluid
the surface temperature. Key remarks are:
•
The Marangoni effect is constructive for heat transfer rate.
Temperature is higher due to larger , while larger values of decay decrease both temperatures and velocities.
•
Results for (dusty fluid, nano fluid, viscous fluid) can be recovered for (, = 0, = 0,
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•
Heat transfer rate is improved for and %, whereas it decreases for and ,.
, = = 0).
Impact of linear variation of surface temperature dominates nonlinear variation of surface temperature.
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•
Thermal layers are enlarged via , while velocity layers decrease via ,.
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•
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•
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boundary layer flow toward a flat surface are examined by utilizing linear/quadratic variation of
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Fig. 2: Impact of and on .
Fig. 3: Impact of and on .
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Fig. 4: Impact of and on .
Fig. 5: Impact of on ¡(¢)and £(¢).
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Fig. 6: Impact of and ¡(¢)and £(¢).
Fig. 7: Impact of and ¡(¢)and £(¢).
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Fig. 8: Impact of and ¤′(¢)and ¥′(¢).
Fig. 9: Impact of and ¡(¢)and £(¢).
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Fig. 10: Impact of and ¤′(¢)and ¥′(¢).
Fig. 11: Impact of and ¡(¢)and £(¢).
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Fig. 12: Impact of and ¡(¢)and £(¢).
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Acknowledgment: The authors are thankful to referees and editor for their constructive suggestions. And one oftheauthors Mahanthesh B would like to express sincere gratitude to the
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Management of Christ University for their kind support.
6. References
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1. Christopher, D. M., & Wang, B. (2001). Prandtl number effects for Marangoni convection over a flat surface. International Journal of Thermal Sciences, 40(6), 564-
570.
2. Zheng, L., Zhang, X., &Gao, Y. (2008). Analytical solution for Marangoni convection over a liquid–vapor surface due to an imposed temperature gradient. Mathematical and
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