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Boundary layer analysis of an incessant moving needle in MHD radiative nanofluid with joule heating
Accepted Manuscript
Boundary layer analysis of an incessant moving needle in MHD radiative nanofluid with joule heating C. Sulochana , S.P. Samrat , N. Sandeep PII: DOI: Reference:
S0020-7403(17)30964-5 10.1016/j.ijmecsci.2017.05.006 MS 3681
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
16 April 2017 9 May 2017 12 May 2017
Please cite this article as: C. Sulochana , S.P. Samrat , N. Sandeep , Boundary layer analysis of an incessant moving needle in MHD radiative nanofluid with joule heating, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.05.006
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ACCEPTED MANUSCRIPT Highlights 2D magnetohydrodynamic flow of water and kerosene based nanofluids.
Flow past a persistent moving horizontal needle is evaluated.
Joule heating, non-uniform heat source/sink and dissipation effects are discussed.
Heat transfer rate of Ag-kerosene is high when equated to Ag-water nanofluid.
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Boundary layer analysis of an incessant moving needle in MHD radiative nanofluid with joule heating C. Sulochana*
S.P. Samrat
N. Sandeep
Department of Mathematics, Gulbarga University, Gulbarga-585106, India.
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E-mail: [email protected] Abstract
The boundary layer analysis of a 2D forced convection flow along an incessant moving horizontal needle in magnetohydrodynamic radiative nanofluid is investigated. The energy
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equation is incorporated with the joule heating, non-constant heat source/sink, and viscid dissipation effects. To check the variation in the boundary layer nature, we considered the two nanofluids namely, Ag-water and Ag-Kerosene. The reduced system of governing PDEs
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is solved by employing the Runge-Kutta Fehlberg integration scheme. Computational results of the local Nusselt number and friction factor are tabulated and discussed. Velocity and
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temperature fields are discussed with the help of graphical illustrations. Increasing the needle size significantly reduces the flow and energy boundary layers of both nanofluids. In
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particular, thermal and velocity fields of Ag-kerosene nanofluids are highly dissed when
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equated with the Ag-water nanofluid. Keywords: MHD, Thin needle, Nanofluid, Joule heating, viscous dissipation. Introduction
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1.
This study investigates the boundary layer analysis of a continuously moving thin needle. Here, the needle has the parabolic revolution about its axes direction, in addition to the variable thickness. The movement of thin needle distracts the free stream direction. This problem is highly attentive in experimental studies to measure the momentum and heat transfer behavior. It has a significant application in medicine, engineering and industries.
ACCEPTED MANUSCRIPT Such application encloses the blood flow problem, transportation, hot wire anemometer for measuring velocity of wind, fiber technology, coating of wires, geothermal power generation, aerospace, polymer ejection, metal spinning, and lubrication. The whole study of boundary layer thin needle has substantial practical importance. Inspiring by all these impacts, Lee [1] was first described the boundary layer analysis of a thin needle drenched in a viscous fluid
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and established the asymptotic characteristics of the motion of a fluid. Later on, Narain and Uberoi [2] deliberate the mixed convection flow past a tiny needle and determined the results for isothermal case and found that the thin needle has analogous nature among the movement and heat transmission. The natural, involuntary and diverse convection flows over a
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dissimilar geometrical aspects of cylinder have been studied by the researchers [3-6]. They discriminate the results numerically and analytically with the anomalous fluid flow by considering the cooled, wormed and heated surface of the cylinder. Ahmed et al [7]
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investigated free and involuntary convection over a thin needle occupied in viscous fluid. He conveyed the secondary and opposing flow cases by adopting the Keller-box method. Ishak et
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al. [8] analysed the flow on the surface of a thin needle, which moves incessantly and placed
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parallel to the free stream. They stated that the dual solution occurs when both the needle and free stream moves in reverse direction
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The nanofluid is used to enrich and diminish the thermal conductivity of a material by adding some metallic and non-metallic elements. It has pertinent applications in physical and
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chemical sciences. Specifically, in industrial process, bio-medicine, paper production, electronic device, etc. Copious researchers are acknowledged that the properties of a nanofluids are unique. These phenomena were numerically described by Nandy et al [9]. Siti et al. [10] illustrated the forced convection nanofluid flow on incessantly moving tinny needle. Mahian et al [11] presented the thermal transport of a nanofluid in a spinning cylinder with heat source. This report labels that the TiO2-water nanofluid is having effective heat
ACCEPTED MANUSCRIPT transfer performance when equated to Al2O3-EG nanofluid. Aba-Nada et al. [12] found that the heat flow among parallel annulus increases effectively by adopting the nanofluids. Sheikholeslami et al. [13] investigated the nanofluid flow between internal and external portion of the circular stretching cylinder by using magnetic field. Koriko et al. [14] examined the nanofluid flow under a parboiled of revolution with 29nm CuO nanoparticles.
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Trimbitas et al. [15] and Nazar et al. [16] deliberated the varied convection flow on the surface of a thin needle engrossed in a nanofluid. Ashorynejad et al [17] probed the heat flow of MHD viscid nanofluid past an elongating cylinder. Sandeep and Sulochana [18] distinguished the dusty nanofluid flow past a broadening surface and found that collision of
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nanoparticle in water advances the heat flow ratio. Sandeep [19] considered the film flow of nanofluid by immersing the grapheme nanoparticles in water. Recently, nanofluid flow over a dissimilar geometries like cylinder, vertical plate, and parabolic revolution has reported by
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the authors [20-22].
The knowledge of thermal radiation utilized in various fields of technologies, namely
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biomedicine, drilling process, space machinery, cancer treatment, high temperature methods,
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etc. Water Carbon nanofluid flow by a thin needle with heat flux was studied by Hayat et al. [23]. Hossain and Alim [24] argued the boundary layer flow past a vertical cylinder. They
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found the association between free convection and thermal radiation by employing Kellerbox method. Khan et al. [24] examined the viscous flow with dissipative, magnetic and joule
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heating effects over stretching cylinder under slip condition. Ramana Reddy et al [26] worked on the ferrofluid flow with thermal radiation past a heated boundary surface. Hayat et al. [27] presented a new model for nanofluid squeezing flow by considering the cross diffudion effects. Nisha et al. [28] considered the chemically reacting MHD nanofluid flow with dissipative effects past a flat cylinder. Very recently, the investigators [29-34] explored the
ACCEPTED MANUSCRIPT heat transfer of incessant moving objects in nanofluids by considering various flow geometries. All the above investigations limited to explore the boundary layer nature of nanofluid flows by considering the one or more physical aspects. But to the authors knowledge, no study has been reported the boundary layer nature and heat transfer of incessant moving
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horizontal needle in magnetohydrodynamic radiative nanofluid. To check the variation in the boundary layer nature, we considered the two nanofluids namely, Ag-water and AgKerosene. Dual solutions are obtained in both nanofluid cases. The reduced system of
2.
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governing PDEs are solved by employing the Runge-Kutta-Fehlberg integration scheme. Mathematical formulation
Let us assume the steady, forced convection MHD Newtonian nanofluid flow under the stagnation region of thin needle. Fig.1 shows the cylindrical coordinate ( x, r ) such that the x-
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axis is axial direction and r is normal to the x-axis and termed as radial direction and c is the
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r
T u
u
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magnitude of the needle. We examined the flow with the assistance of thermal stratification.
nanoparticle
Tw c
x
0 uw
Fig. 1 Schematic representation
ACCEPTED MANUSCRIPT We assume that the temperature of the cylindrical surface Tw is much greater than the ambient fluid T are constants where Tw T (heated needle). Also, U 0 uw u be composite velocity along x-axis, needle moves with constant velocity uw . The meanstream of constant velocity is u . The electromagnetic field B0 is imposed to the direction of flow and snubbing
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the induced magnetic field due to the trivial magnitude of Reynolds number. Joule heating, uneven heat source/sink and thermal radiation effects are taken into account. Under the presence of above discussed conditions and in the absence of pressure gradient, we imposed the governing equations proposed by Siti et al [10], written in the following form
nf uux vur nf
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(ru) x (rv)r 0
1 rur r B02u r
(2)
(3)
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1 16 *T3 2 ( C p )nf uTx vTr knf rTr q ''' B02u 2 f ur * r 3k r
(1)
The suitable boundary conditions are,
T T , u u ,
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u uw , T Tw v 0, at r R( x) as r
(4)
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where, u, v are velocities towards the direction of the axial and radial ( x, r ) respectively.
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( C p )nf , knf , nf , nf are the specific heat capacity, thermal conductivity, effective density and
viscosity and of the nanofluids, which are given as ( C p ) nf ( C p ) f ( C p ) f ( C p ) s ,
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k s 2k f 2 k f 2 k s knf k , k s 2k f 2 k f 2 k s f nf f f s , nf f (1 ) 2.5 ,
(5)
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(T T )
The non-uniform heat source/sink is stated as q ''' k f U 0 (Tw T ) xv f A* f ' B* . ( T T w )
To compute the basic eqns. (1) to (3) under the boundary restrictions (4), we assumed the relevant transformations as follows,
f xf ( ), U 0 r 2 f x , ( ) (T T ) (Tw T ),
(6)
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Here, is defined as a stream function which satisfy the continuity eqn. (1) identically and the velocity components are defined as u r 1 r , v r 1 x . Assume, c in Eqn. (6) predicts magnitude of the needle r R( x) f cx U 0 along the surface. Using eqn. (6) into
2(1 )2.5 ( f ''' f '') (1
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basic eqns. (2) to (4) reduces in the following nonlinear differential form,
s )( ff '' Mf ') 0 f
* * 2 2 Pr f ' ( A f ' B ) Ec(2 f '' Mf ' ) 0
(8)
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k ( C p )s 2 nf R ( '' ') 1 k ( C p ) f f
(7)
The suitable boundary conditions are,
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f (c) c / 2, f '(c) 2, (c) 1, f '( ) 1 2, ( ) ( 2) as
(9)
and 1 relates to a fixed needle in a moving fluid (Blassius flow) and moving
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0
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where uw U 0 defined as the ratio of velocity of needle and composite velocity. Here,
needle in a stationary fluid (Sakiadis flow), respectively. In this study 1 is restricted to all
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boundary conditions, i.e. free stream always moves in a positive direction. The physical parameter M ,Pr, R, A* , B* and Ec denotes the magnetic field, Prandtl number, thermal radiation, non-uniform heat source/sink respectively. They are described as M
C p B02 2U 0 2 16 *T3 , Pr ,R , Ec , 2 U 0 kf 3k *k f C p (Tw T )
The friction factor is given by,
(10)
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(11)
The Nusselt number is given by, Rex1 2 Nux 2c1 2 Knf K f '(c),
(12)
where Re x U 0 x f .
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3. Discussion of the results The nonlinear ODE Eqs. (7) and (8) with suitable restrictions Eq. (9) are unravelled by employing Runge-Kutta Fehlberg integration scheme. The current study probes the flow and energy transport of Ag-water and Ag-kerosene nanofluids. For computational purposes, the
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non-dimensional constants are used throughout the study as A* B* c 0.1; 0.01;
M 1; Ec 0.1; 1 . Table.1 directs the thermal and physical significance of the silver (Ag), water and kerosene. Here, we are presented and examined the Figs. 2-11 for velocity
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and temperature fields. Tables 2 and 3 directs the impact of pertinent parameters on Nusselt number and friction factor. Table 4 depicts the validation of the numerical technique by
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comparing with the other techniques.
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The profiles of velocity and temperature are examined with an impact of magnetic field measured in Figs. 2 and 3 respectively. As seen, the greater values of magnetic field
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diminish the fluid velocity and enlarge the temperature field. Physically, increasing the magnetic fields improves the resistive force (Lorentzs force) which plays opposite to the flow
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field. This Lorentz force capable to decline the flow and upsurge the thermal field. Figs. 4 and 5 depict the flow and thermal fields performance by means of volume fraction. It is clear that the ascending values of ϕ causes to decay in the flow velocity and enlarge the temperature profile. The interruption of nanoparticle in base fluid will improve the density and viscosity of nanofluid. These physics diminish the fluid velocity and supply the energy to enhance thermal field.
ACCEPTED MANUSCRIPT Figs. 6 and 7 describe the temperature field for consideration of A* and B* respectively. It holds that the temperature profile is cumulative for increasing values of A* and B* . The optimistic values of A* and B* shows the relation of heat source. So that the
rising values of A* and B* produces the heat source, which directs to grow the temperature field. Fig. 8 displays the nature of the temperature field with Eckert number. As seen, the
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temperature of the flow upgrades by emerging values of Eckert number. It happens due to increasing values of Eckert number criticizes the thermal boundary layer thickness which causes to rise temperature field. Therefore, temperature field upgrades by enlarging the
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dissipative parameter. The radiation influence on temperature profile is shown in Fig. 9. It is obvious that the ascending value of thermal radiation improves the temperature field. Radiation is basically a transmission of heat energy from one region to another. Thermal radiation yields peripheral heat energy to augment the temperature field.
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Figs. 10 and 11 respectively construct the effect of needle thickness over the velocity and temperature fields. It shows that the velocity and temperature boundary layers are highly
It is evident from the tables 2 and 3 that the variation in the physical
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flow profiles.
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influenced by the needle thickness. The higher values of needle thickness compresses both
parameters doesn’t show significant influence on the wall friction. But, rising values of the
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nanoparticle volume fraction and needle thickness boosts the heat transfer rate of both nanofluids. In particular, the Ag-water nanofluid has a low heat transfer rate when equated to
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Ag-kerosene nanofluid. This concludes that the Ag-water nanofluid is useful for cooling applications.
Physical properties C p ( J kgK )
(kg m3 ) K ( w mK )
Table 1. Thermophysical properties Water Kerosene 4179 2090 997.3 783 0.613 0.145
Silver (Ag) 240 10500 429
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Fig.2. Impact of M on flow field.
Fig.3. Impact of M on thermal field.
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Fig.4. Impact of on flow field.
Fig.5. Impact of on thermal field.
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Fig.6. Impact of A* on thermal field.
Fig.7. Impact of B* on thermal field.
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Fig.8. Impact of Ec on thermal field.
Fig.9. Impact of R on thermal field.
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Fig.10. Impact of c on flow field.
Fig.11. Impact of c on thermal field.
ACCEPTED MANUSCRIPT Table. 2 : Computational values of friction factor and Nusselt number for Ag-water nanofluid * c ''(c) f ''(c) M A B* Ec R 1 3 5 0.1 0.2 0.3
2.123383 2.059242 2.015638 1.024909 1.299485 1.653047 1.988383 1.847818 1.707253 1.928072 1.716431 1.467723 1.731927 1.367429 1.002931 2.017324 1.989627 1.977592 2.017324 3.685294 5.006804
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1 3 5
-3.536154 -4.696964 -5.565820 -1.322329 -1.826821 -2.515014 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -6.325205 -8.428255
1 2 3 0.5 1.0 1.5
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1 2 5
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0.1 0.3 0.5
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Table. 3: Computational values of friction factor and Nusselt number for Ag-Kerosene nanofluid * c ''(c) f ''(c) M A B* Ec R
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1 3 5
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0.1 0.2 0.3
1 3 5 1 2 3 0.5 1.0 1.5 1 2 5 0.1 0.3 0.5
-3.749723 -5.064285 -6.039849 -1.374336 -1.926008 -2.666025 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -6.738152 -9.018642
2.338522 2.232877 2.162826 1.106078 1.415015 1.812827 2.216355 2.086088 1.955822 2.165253 1.983296 1.776377 1.840069 1.328999 0.817931 2.226375 2.122672 2.076027 2.226375 4.225294 5.937770
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Table 4: Validation of the numerical technique for Nu x
RKF
M 1 3 5 7
RKS
BVP4C
BVP5C
2.3385226322 2.2328777610 2.1628260323 2.1024321230
2.3385226321 2.2328777611 2.1628260323 2.1024321230
2.3385226321 2.2328777611 2.1628260322 2.1024321231
Conclusions
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4.
2.338522 2.232877 2.162826 2.102432
RKN 2.3385226321 2.2328777611 2.1628260323 2.1024321230
Heating and cooling processes are very essential in industrial needs and holding the stability to control heat transfer capacity by using the nanofluid is ideal in future industrial needs. The prominence of nanofluid in heat transfer era is demonstrated in this work. The outcomes of
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present study are as follows:
The heat transfer rate of Ag-water nanofluid is higher than the Ag-Kerosene nanofluid.
Existences of A* and B* has a capacity to regulate the heat transfer rate.
Magnetic field has proclivity to criticize the Nusselt number and flow rate.
Increasing the needle thickness criticize the friction factor and improves Nusselt
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number.
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Thermal radiation produces peripheral heat energy to amplify the temperature
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fields.
Nanoparticle volume fraction makes a crucial role in the nature of the heat flow
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rate.
For cooling processes Ag-water nanofluids are very useful.
Acknowledgement The authors acknowledge the UGC for financial support under the UGC Dr. D. S. Kothari Postdoctoral Fellowship Scheme (No.F.4-2/2006 (BSR)/MA/13-14/0026).
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ACCEPTED MANUSCRIPT Graphical Abstract: The steady, two dimensional MHD Newtonian nanofluid flow under the stagnation region of tinny needle is considered. Figure shows the cylindrical coordinate ( x, r ) such that the x-axis is axial direction and r is normal to the x-axis and termed as radial direction and c is the magnitude of the needle. We examined the flow with the assistance of thermal stratification.
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We assume that the temperature of the cylindrical surface is much greater than the ambient fluid. Also, U 0 uw u be composite velocity along x-axis, where uw is the velocity of a needle and u is opposite of mainstream velocity.
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Boundary layer analysis of an incessant moving needle in MHD radiative nanofluid with joule heating C. Sulochana , S.P. Samrat , N. Sandeep PII: DOI: Reference:
S0020-7403(17)30964-5 10.1016/j.ijmecsci.2017.05.006 MS 3681
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
16 April 2017 9 May 2017 12 May 2017
Please cite this article as: C. Sulochana , S.P. Samrat , N. Sandeep , Boundary layer analysis of an incessant moving needle in MHD radiative nanofluid with joule heating, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.05.006
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ACCEPTED MANUSCRIPT Highlights 2D magnetohydrodynamic flow of water and kerosene based nanofluids.
Flow past a persistent moving horizontal needle is evaluated.
Joule heating, non-uniform heat source/sink and dissipation effects are discussed.
Heat transfer rate of Ag-kerosene is high when equated to Ag-water nanofluid.
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Boundary layer analysis of an incessant moving needle in MHD radiative nanofluid with joule heating C. Sulochana*
S.P. Samrat
N. Sandeep
Department of Mathematics, Gulbarga University, Gulbarga-585106, India.
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E-mail: [email protected] Abstract
The boundary layer analysis of a 2D forced convection flow along an incessant moving horizontal needle in magnetohydrodynamic radiative nanofluid is investigated. The energy
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equation is incorporated with the joule heating, non-constant heat source/sink, and viscid dissipation effects. To check the variation in the boundary layer nature, we considered the two nanofluids namely, Ag-water and Ag-Kerosene. The reduced system of governing PDEs
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is solved by employing the Runge-Kutta Fehlberg integration scheme. Computational results of the local Nusselt number and friction factor are tabulated and discussed. Velocity and
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temperature fields are discussed with the help of graphical illustrations. Increasing the needle size significantly reduces the flow and energy boundary layers of both nanofluids. In
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particular, thermal and velocity fields of Ag-kerosene nanofluids are highly dissed when
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equated with the Ag-water nanofluid. Keywords: MHD, Thin needle, Nanofluid, Joule heating, viscous dissipation. Introduction
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1.
This study investigates the boundary layer analysis of a continuously moving thin needle. Here, the needle has the parabolic revolution about its axes direction, in addition to the variable thickness. The movement of thin needle distracts the free stream direction. This problem is highly attentive in experimental studies to measure the momentum and heat transfer behavior. It has a significant application in medicine, engineering and industries.
ACCEPTED MANUSCRIPT Such application encloses the blood flow problem, transportation, hot wire anemometer for measuring velocity of wind, fiber technology, coating of wires, geothermal power generation, aerospace, polymer ejection, metal spinning, and lubrication. The whole study of boundary layer thin needle has substantial practical importance. Inspiring by all these impacts, Lee [1] was first described the boundary layer analysis of a thin needle drenched in a viscous fluid
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and established the asymptotic characteristics of the motion of a fluid. Later on, Narain and Uberoi [2] deliberate the mixed convection flow past a tiny needle and determined the results for isothermal case and found that the thin needle has analogous nature among the movement and heat transmission. The natural, involuntary and diverse convection flows over a
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dissimilar geometrical aspects of cylinder have been studied by the researchers [3-6]. They discriminate the results numerically and analytically with the anomalous fluid flow by considering the cooled, wormed and heated surface of the cylinder. Ahmed et al [7]
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investigated free and involuntary convection over a thin needle occupied in viscous fluid. He conveyed the secondary and opposing flow cases by adopting the Keller-box method. Ishak et
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al. [8] analysed the flow on the surface of a thin needle, which moves incessantly and placed
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parallel to the free stream. They stated that the dual solution occurs when both the needle and free stream moves in reverse direction
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The nanofluid is used to enrich and diminish the thermal conductivity of a material by adding some metallic and non-metallic elements. It has pertinent applications in physical and
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chemical sciences. Specifically, in industrial process, bio-medicine, paper production, electronic device, etc. Copious researchers are acknowledged that the properties of a nanofluids are unique. These phenomena were numerically described by Nandy et al [9]. Siti et al. [10] illustrated the forced convection nanofluid flow on incessantly moving tinny needle. Mahian et al [11] presented the thermal transport of a nanofluid in a spinning cylinder with heat source. This report labels that the TiO2-water nanofluid is having effective heat
ACCEPTED MANUSCRIPT transfer performance when equated to Al2O3-EG nanofluid. Aba-Nada et al. [12] found that the heat flow among parallel annulus increases effectively by adopting the nanofluids. Sheikholeslami et al. [13] investigated the nanofluid flow between internal and external portion of the circular stretching cylinder by using magnetic field. Koriko et al. [14] examined the nanofluid flow under a parboiled of revolution with 29nm CuO nanoparticles.
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Trimbitas et al. [15] and Nazar et al. [16] deliberated the varied convection flow on the surface of a thin needle engrossed in a nanofluid. Ashorynejad et al [17] probed the heat flow of MHD viscid nanofluid past an elongating cylinder. Sandeep and Sulochana [18] distinguished the dusty nanofluid flow past a broadening surface and found that collision of
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nanoparticle in water advances the heat flow ratio. Sandeep [19] considered the film flow of nanofluid by immersing the grapheme nanoparticles in water. Recently, nanofluid flow over a dissimilar geometries like cylinder, vertical plate, and parabolic revolution has reported by
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the authors [20-22].
The knowledge of thermal radiation utilized in various fields of technologies, namely
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biomedicine, drilling process, space machinery, cancer treatment, high temperature methods,
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etc. Water Carbon nanofluid flow by a thin needle with heat flux was studied by Hayat et al. [23]. Hossain and Alim [24] argued the boundary layer flow past a vertical cylinder. They
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found the association between free convection and thermal radiation by employing Kellerbox method. Khan et al. [24] examined the viscous flow with dissipative, magnetic and joule
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heating effects over stretching cylinder under slip condition. Ramana Reddy et al [26] worked on the ferrofluid flow with thermal radiation past a heated boundary surface. Hayat et al. [27] presented a new model for nanofluid squeezing flow by considering the cross diffudion effects. Nisha et al. [28] considered the chemically reacting MHD nanofluid flow with dissipative effects past a flat cylinder. Very recently, the investigators [29-34] explored the
ACCEPTED MANUSCRIPT heat transfer of incessant moving objects in nanofluids by considering various flow geometries. All the above investigations limited to explore the boundary layer nature of nanofluid flows by considering the one or more physical aspects. But to the authors knowledge, no study has been reported the boundary layer nature and heat transfer of incessant moving
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horizontal needle in magnetohydrodynamic radiative nanofluid. To check the variation in the boundary layer nature, we considered the two nanofluids namely, Ag-water and AgKerosene. Dual solutions are obtained in both nanofluid cases. The reduced system of
2.
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governing PDEs are solved by employing the Runge-Kutta-Fehlberg integration scheme. Mathematical formulation
Let us assume the steady, forced convection MHD Newtonian nanofluid flow under the stagnation region of thin needle. Fig.1 shows the cylindrical coordinate ( x, r ) such that the x-
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axis is axial direction and r is normal to the x-axis and termed as radial direction and c is the
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r
T u
u
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magnitude of the needle. We examined the flow with the assistance of thermal stratification.
nanoparticle
Tw c
x
0 uw
Fig. 1 Schematic representation
ACCEPTED MANUSCRIPT We assume that the temperature of the cylindrical surface Tw is much greater than the ambient fluid T are constants where Tw T (heated needle). Also, U 0 uw u be composite velocity along x-axis, needle moves with constant velocity uw . The meanstream of constant velocity is u . The electromagnetic field B0 is imposed to the direction of flow and snubbing
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the induced magnetic field due to the trivial magnitude of Reynolds number. Joule heating, uneven heat source/sink and thermal radiation effects are taken into account. Under the presence of above discussed conditions and in the absence of pressure gradient, we imposed the governing equations proposed by Siti et al [10], written in the following form
nf uux vur nf
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(ru) x (rv)r 0
1 rur r B02u r
(2)
(3)
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1 16 *T3 2 ( C p )nf uTx vTr knf rTr q ''' B02u 2 f ur * r 3k r
(1)
The suitable boundary conditions are,
T T , u u ,
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u uw , T Tw v 0, at r R( x) as r
(4)
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where, u, v are velocities towards the direction of the axial and radial ( x, r ) respectively.
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( C p )nf , knf , nf , nf are the specific heat capacity, thermal conductivity, effective density and
viscosity and of the nanofluids, which are given as ( C p ) nf ( C p ) f ( C p ) f ( C p ) s ,
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k s 2k f 2 k f 2 k s knf k , k s 2k f 2 k f 2 k s f nf f f s , nf f (1 ) 2.5 ,
(5)
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(T T )
The non-uniform heat source/sink is stated as q ''' k f U 0 (Tw T ) xv f A* f ' B* . ( T T w )
To compute the basic eqns. (1) to (3) under the boundary restrictions (4), we assumed the relevant transformations as follows,
f xf ( ), U 0 r 2 f x , ( ) (T T ) (Tw T ),
(6)
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Here, is defined as a stream function which satisfy the continuity eqn. (1) identically and the velocity components are defined as u r 1 r , v r 1 x . Assume, c in Eqn. (6) predicts magnitude of the needle r R( x) f cx U 0 along the surface. Using eqn. (6) into
2(1 )2.5 ( f ''' f '') (1
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basic eqns. (2) to (4) reduces in the following nonlinear differential form,
s )( ff '' Mf ') 0 f
* * 2 2 Pr f ' ( A f ' B ) Ec(2 f '' Mf ' ) 0
(8)
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k ( C p )s 2 nf R ( '' ') 1 k ( C p ) f f
(7)
The suitable boundary conditions are,
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f (c) c / 2, f '(c) 2, (c) 1, f '( ) 1 2, ( ) ( 2) as
(9)
and 1 relates to a fixed needle in a moving fluid (Blassius flow) and moving
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0
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where uw U 0 defined as the ratio of velocity of needle and composite velocity. Here,
needle in a stationary fluid (Sakiadis flow), respectively. In this study 1 is restricted to all
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boundary conditions, i.e. free stream always moves in a positive direction. The physical parameter M ,Pr, R, A* , B* and Ec denotes the magnetic field, Prandtl number, thermal radiation, non-uniform heat source/sink respectively. They are described as M
C p B02 2U 0 2 16 *T3 , Pr ,R , Ec , 2 U 0 kf 3k *k f C p (Tw T )
The friction factor is given by,
(10)
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(11)
The Nusselt number is given by, Rex1 2 Nux 2c1 2 Knf K f '(c),
(12)
where Re x U 0 x f .
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3. Discussion of the results The nonlinear ODE Eqs. (7) and (8) with suitable restrictions Eq. (9) are unravelled by employing Runge-Kutta Fehlberg integration scheme. The current study probes the flow and energy transport of Ag-water and Ag-kerosene nanofluids. For computational purposes, the
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non-dimensional constants are used throughout the study as A* B* c 0.1; 0.01;
M 1; Ec 0.1; 1 . Table.1 directs the thermal and physical significance of the silver (Ag), water and kerosene. Here, we are presented and examined the Figs. 2-11 for velocity
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and temperature fields. Tables 2 and 3 directs the impact of pertinent parameters on Nusselt number and friction factor. Table 4 depicts the validation of the numerical technique by
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comparing with the other techniques.
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The profiles of velocity and temperature are examined with an impact of magnetic field measured in Figs. 2 and 3 respectively. As seen, the greater values of magnetic field
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diminish the fluid velocity and enlarge the temperature field. Physically, increasing the magnetic fields improves the resistive force (Lorentzs force) which plays opposite to the flow
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field. This Lorentz force capable to decline the flow and upsurge the thermal field. Figs. 4 and 5 depict the flow and thermal fields performance by means of volume fraction. It is clear that the ascending values of ϕ causes to decay in the flow velocity and enlarge the temperature profile. The interruption of nanoparticle in base fluid will improve the density and viscosity of nanofluid. These physics diminish the fluid velocity and supply the energy to enhance thermal field.
ACCEPTED MANUSCRIPT Figs. 6 and 7 describe the temperature field for consideration of A* and B* respectively. It holds that the temperature profile is cumulative for increasing values of A* and B* . The optimistic values of A* and B* shows the relation of heat source. So that the
rising values of A* and B* produces the heat source, which directs to grow the temperature field. Fig. 8 displays the nature of the temperature field with Eckert number. As seen, the
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temperature of the flow upgrades by emerging values of Eckert number. It happens due to increasing values of Eckert number criticizes the thermal boundary layer thickness which causes to rise temperature field. Therefore, temperature field upgrades by enlarging the
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dissipative parameter. The radiation influence on temperature profile is shown in Fig. 9. It is obvious that the ascending value of thermal radiation improves the temperature field. Radiation is basically a transmission of heat energy from one region to another. Thermal radiation yields peripheral heat energy to augment the temperature field.
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Figs. 10 and 11 respectively construct the effect of needle thickness over the velocity and temperature fields. It shows that the velocity and temperature boundary layers are highly
It is evident from the tables 2 and 3 that the variation in the physical
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flow profiles.
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influenced by the needle thickness. The higher values of needle thickness compresses both
parameters doesn’t show significant influence on the wall friction. But, rising values of the
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nanoparticle volume fraction and needle thickness boosts the heat transfer rate of both nanofluids. In particular, the Ag-water nanofluid has a low heat transfer rate when equated to
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Ag-kerosene nanofluid. This concludes that the Ag-water nanofluid is useful for cooling applications.
Physical properties C p ( J kgK )
(kg m3 ) K ( w mK )
Table 1. Thermophysical properties Water Kerosene 4179 2090 997.3 783 0.613 0.145
Silver (Ag) 240 10500 429
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Fig.2. Impact of M on flow field.
Fig.3. Impact of M on thermal field.
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Fig.4. Impact of on flow field.
Fig.5. Impact of on thermal field.
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Fig.6. Impact of A* on thermal field.
Fig.7. Impact of B* on thermal field.
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Fig.8. Impact of Ec on thermal field.
Fig.9. Impact of R on thermal field.
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Fig.10. Impact of c on flow field.
Fig.11. Impact of c on thermal field.
ACCEPTED MANUSCRIPT Table. 2 : Computational values of friction factor and Nusselt number for Ag-water nanofluid * c ''(c) f ''(c) M A B* Ec R 1 3 5 0.1 0.2 0.3
2.123383 2.059242 2.015638 1.024909 1.299485 1.653047 1.988383 1.847818 1.707253 1.928072 1.716431 1.467723 1.731927 1.367429 1.002931 2.017324 1.989627 1.977592 2.017324 3.685294 5.006804
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1 3 5
-3.536154 -4.696964 -5.565820 -1.322329 -1.826821 -2.515014 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -3.536154 -6.325205 -8.428255
1 2 3 0.5 1.0 1.5
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1 2 5
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0.1 0.3 0.5
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Table. 3: Computational values of friction factor and Nusselt number for Ag-Kerosene nanofluid * c ''(c) f ''(c) M A B* Ec R
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1 3 5
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0.1 0.2 0.3
1 3 5 1 2 3 0.5 1.0 1.5 1 2 5 0.1 0.3 0.5
-3.749723 -5.064285 -6.039849 -1.374336 -1.926008 -2.666025 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -3.749723 -6.738152 -9.018642
2.338522 2.232877 2.162826 1.106078 1.415015 1.812827 2.216355 2.086088 1.955822 2.165253 1.983296 1.776377 1.840069 1.328999 0.817931 2.226375 2.122672 2.076027 2.226375 4.225294 5.937770
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Table 4: Validation of the numerical technique for Nu x
RKF
M 1 3 5 7
RKS
BVP4C
BVP5C
2.3385226322 2.2328777610 2.1628260323 2.1024321230
2.3385226321 2.2328777611 2.1628260323 2.1024321230
2.3385226321 2.2328777611 2.1628260322 2.1024321231
Conclusions
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4.
2.338522 2.232877 2.162826 2.102432
RKN 2.3385226321 2.2328777611 2.1628260323 2.1024321230
Heating and cooling processes are very essential in industrial needs and holding the stability to control heat transfer capacity by using the nanofluid is ideal in future industrial needs. The prominence of nanofluid in heat transfer era is demonstrated in this work. The outcomes of
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present study are as follows:
The heat transfer rate of Ag-water nanofluid is higher than the Ag-Kerosene nanofluid.
Existences of A* and B* has a capacity to regulate the heat transfer rate.
Magnetic field has proclivity to criticize the Nusselt number and flow rate.
Increasing the needle thickness criticize the friction factor and improves Nusselt
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number.
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Thermal radiation produces peripheral heat energy to amplify the temperature
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fields.
Nanoparticle volume fraction makes a crucial role in the nature of the heat flow
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rate.
For cooling processes Ag-water nanofluids are very useful.
Acknowledgement The authors acknowledge the UGC for financial support under the UGC Dr. D. S. Kothari Postdoctoral Fellowship Scheme (No.F.4-2/2006 (BSR)/MA/13-14/0026).
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ACCEPTED MANUSCRIPT Graphical Abstract: The steady, two dimensional MHD Newtonian nanofluid flow under the stagnation region of tinny needle is considered. Figure shows the cylindrical coordinate ( x, r ) such that the x-axis is axial direction and r is normal to the x-axis and termed as radial direction and c is the magnitude of the needle. We examined the flow with the assistance of thermal stratification.
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We assume that the temperature of the cylindrical surface is much greater than the ambient fluid. Also, U 0 uw u be composite velocity along x-axis, where uw is the velocity of a needle and u is opposite of mainstream velocity.
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