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Active and zero flux of nanoparticles between a squeezing channel with thermal radiation effects
Active and zero flux of nanoparticles between a squeezing channel with thermal radiation effects M. Atlas, Rizwan Ul Haq, T. Mekkaoui PII: DOI: Reference:
S0167-7322(16)31733-0 doi: 10.1016/j.molliq.2016.08.032 MOLLIQ 6192
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
28 June 2016 4 August 2016 9 August 2016
Please cite this article as: M. Atlas, Rizwan Ul Haq, T. Mekkaoui, Active and zero flux of nanoparticles between a squeezing channel with thermal radiation effects, Journal of Molecular Liquids (2016), doi: 10.1016/j.molliq.2016.08.032
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ACCEPTED MANUSCRIPT Active and zero flux of nanoparticles between a squeezing channel with thermal radiation effects M. Atlas 1 , Rizwan Ul Haq 2,∗ , T. Mekkaoui 3 Department of Mathematics, Capital University of Science and Technology, Islamabad
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1
2
Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan
Department of Mathematics, Faculty of Science and Technology, Errachidia, Morocco
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3
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44000, Pakistan.
Abstract
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Present study deals the effect of thermal radiation on unsteady nanofluid flows between two parallel plates. Due to gravitational settling nanoparticles accumulate at the base of
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the channel and effect the state of homogeneous mixture of nanofluid so the present model describe the analysis of both active and passive control of nanoparticles at the lower surface of the channel. Similarity transformations are used to reduce the basic partial differential
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equation to nonlinear coupled ordinary differential equations. The equations along with the prescribed boundary conditions were solved numerically using the finite difference method
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in MATLAB for various values of emerging parameters. Numerical results are obtained for velocity, temperature and concentration for different parameter such as squeeze number,
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radiation parameter, Lewis number, and Eckert number. Flow behavior and temperature are also described through stream lines and isotherms profiles. Throughout the whole analysis, it is found that for zero normal flux there is low temperature profile and high concentration profile is obtained as compare to the active control of nanoparticles. Isotherms plots depict the significant influence of radiation effects to enhance temperature in the restricted domain. Keywords: Nanofluid, Brownian motion, thermophoresis, zero flux, active control.
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Introduction
Heat and mass transfer in a channel have great interest in engineering applications and many branches of science, a lot of study on heat and mass transfer have been done in recent years. Some practical examples related to channel flow are polymer processing, compression, and injection molding. Initially, Stefan [1] advertised a traditional paper on squeezing flow b y 2
* Mailing address: ideal [email protected], [email protected] (R. U. Haq)
Contact: +92 333 5371853
1
ACCEPTED MANUSCRIPT utilization of lubrication conjecture. After that, Reynolds [2] attains a solution for elliptic plates, and Archibald [3] further modifies this problem for rectangular plates. The theoretical and experimental studies of squeezing flows have been regulated by many researchers.
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Siddiqui et al. [4] examined the hydro magnetic squeezing flow of a viscous fluid between
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two parallel plates and used the homotopy perturbation method (HPM) to find out the solutions. Recently, Sheikholeslami and Ganji [5] analyzed the characteristic of heat and mass transfer for unsteady nanofluid in a squeezing channel in the presence of viscous dissipation
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and radiation effects. Operating on heat and mass transfer along with chemical reaction effect plays a crucial part in designs of chemical processing equipment, development and
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dispersion of fog, destruction of crops and cooling towers.
Most of the study on channel flow complications incorporates base fluid which has low
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thermal conductivity. Due to poor thermal conductivity the resulting production is slow. To boost up the thermal conductivity of the fluid, nanoparticles can incorporate with the base fluid. In this field a lot of work done in recent years. Nanofluids are more useful at industrial
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level because of their heat transfer properties; they intensify the thermal conductivity and convective properties over the properties of the base fluid. Initially Choi [6] is the foremost
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person who inaugurated the concept of nanofluids and employed the nanofluid in an extensive variation of industries, energy manufacturing and electronics production, textiles and paper
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construction. Masuda et al [7] initiate the concept of thermal conductivity magnification. Buongiorno and Hu [8] used the Nanofluids in nuclear systems. Kleinstreuer et al. [9] present the approach of the Nanofluids flow in the transportation of Nano-drugs. Buongiorno [10] present a new model construct on the nanoparticles mechanics. Nield and Kuznetsov [11] calculated the matter of boundary layer natural convection past a vertical plate and utilize the Buongiorno nanofluid model with the outcome of Brownian motion and thermophoresis of nanoparticles. Following that they reconsider a paper and enlarge it to the problem in which the boundary is passively controlled rather than actively controlled. Khanafer et al. [12] studied the two dimensional rectangular flow by using copper-water nanofluid. They originate that the heat transfer ratio grows by increasing the ratio of the eliminated particles. The identical problem resolved experimentally by Nnanna et al. [13] for ethylene glycol by utilizing Cu nanoparticles and Nnanna and Routhu [14] for Alumina-water nanofluids. Different experimental work done by Putra et al [15] on natural convection of Al2 O3 - and CuO-Water nanofluids. They have summarized that the natural convection coefficient was 2
ACCEPTED MANUSCRIPT lower than that of pure water. In recent years study on channel flow by utilizing nanfluid has considerable attraction. Abu-Nada et al. [16] originated natural convection heat transfer improvement in horizontal
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parallel annuli range by nanofluid. They initiate that nanoparticle with higher thermal con-
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ductivity at low Rayleigh numbers yields more improvements in heat transfer. Ellahi [17] observed that flow of non-Newtonian nanofluid along with magneto hydrodynamic (MHD) in a pipe and examined that the MHD parameter decreases the fluid motion in the existence
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of variable viscosities temperature graphs is slower than that of velocity graphs. Sheikholeslami et al. [18] measured the effect of magnetic field on free convection heat transfer in
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a cold square and heated elliptic cylinder. They initiate that increase in Hartmann number enhance the heat transfer but increase in Rayleigh number it will decrease. Current many
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authors present different ideas for nanofluid and heat transfer through various geometries that accomplish the wide attention at industrial level [19-30]. Thermal radiation also plays an important role in controlling heat transfer process in
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polymer processing industry. As a result of little convective heat transfer coefficient the thermal radiation has an important role on the surface. In space technology and high
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temperature the consequence of thermal radiation are meaningful. Bakier [31] reviewed the reaction of thermal radiation on mixed convection. He operated the equations by using the
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fourth order RK method. Effect of radioactive vertical isothermal on MHD flow with mixed convection was numerically solved by Damseh [32]. Hossain and Takhar [33] studied the effect of radiation with Roseland diffusion approximation along a vertical plate on free convection boundary layer flow. Zahmatkesh [34] concluded that the impact of thermal radiation on the growth of free convection flow inside an enclosure filled saturated porous media. Hayat et al. [35] used the Homotopy Analysis Method to study the impact of radiation on the MHD mixed convection stagnation point. Pal and Mondal [36] present the complete study by incorporate convective heat transfer in a porous medium by assuming heat generation and radiation impacts. In our work we have discussed the nanofluid flow model between two parallel plates and calculate the impact of thermal radiation on heat and mass transfer. Moreover, we have considered the active and passive control of nanoparticles near the lower surface of the channel. Mathematical model is constructed in the form of partial differential equations. Resulting equations are solved numerically with FDM. Important physical parameters: radiation pa3
ACCEPTED MANUSCRIPT rameter, squeezing number, Eckert number, Brownian motion parameter, thermophoresis parameter, Lewis number are described through graphs. Flow behavior is also discussed through stream lines. For both active and passive control of nanoparticles results are also
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constructed for local Nusselt number and Sherwood number.
Mathematical Model
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Consider the unsteady squeezing flow of two dimensional incompressible flow of viscous fluid between two infinite parallel plates. The two plates are placed apart from each other at the
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distance h(t) = l(1 − βt)1/2 . For β > 0 the two plates are squeezed until they touch t =
1 β
and for β < 0 the two plates are separated (see Fig. 1). Further, we have investigated both
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active and passive control of the nanoparticles at the lower surface of the channel. Radiation effects are also taken into the account. The governing equations for mass, momentum and
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energy in unsteady two dimensional flow of nanofluid are [5]
(1)
∂u ∂ 2u ∂ 2u 1 ∂p ∂u ∂u +u +v =− + ν nf ( 2 + 2 ), ∂t ∂x ∂y ρnf ∂x ∂x ∂y
(2)
∂v ∂v 1 ∂p ∂v ∂2v ∂2v +u +v =− + ν nf ( 2 + 2 ), ∂t ∂x ∂y ρnf ∂y ∂x ∂y
(3)
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∂u ∂v + = 0, ∂x ∂y
2
2
+ u ∂T + v ∂T = αnf ( ∂∂xT2 + ∂∂yT2 ) + (ρcµp )f (( ∂u )2 ) ∂x ∂y ∂y ! DB ∂C + . ∂T + ∂C . ∂T ∂x ∂x ∂y ∂y (ρcp )p ' − 1 ∂qr , $ " " # # +S (ρc 2 (ρcp )f ∂y & % 2 p )f DT ∂T ∂T + Tc ∂x ∂y ∂T ∂t
∂C ∂C ∂C +u +v = DB ∂t ∂x ∂y
+
∂ 2C ∂ 2C + ∂x2 ∂y 2
,
+
+
DT Tc
,$
∂ 2T ∂ 2T + ∂x2 ∂y 2
(4)
'
.
(5)
Where, the radiation heat flux consider according to the Rosseland approximation such that 4σ e qr = − 3β
R
∂T 4 , ∂y
where σ e , β R are the Stefan-Boltzman constant and the mean absorption
constant respectively. The fluid phase temperature difference within the flow is assumed to be sufficiently small so that T 4 may be expressed as a linear function of temperature. This is done by expanding T 4 in a Taylor series about the temperature Tc and neglecting higher order terms to yield T 4 = 4Tc3 T − Tc3 . In above equations u and v are the velocities in the x and y directions respectively. T is the temperature. C is the concentration, ν nf is viscosity 4
ACCEPTED MANUSCRIPT of nanofluid, P is the pressure, ρnf is the density of nanofluid, µnf is the dynamic viscosity of nanofluid, knf is the thermal conductivity of nanofluid, cp is the specific heat of nanofluid, DB is the diffusion coefficient of the diffusing species and DT is the thermophoretic diffusion
v=
∂u ∂y
dh ,T dt
=T =C=0y=0
= Th , C = Ch (Active control of nanoparticles) at y = h (t)
= T = DB ∂C + ∂y
DT ∂T Tc ∂y
= 0 (Passive control of nanoparticles) at y = 0
We introduce these parameters y
βl 1
2(1−βt) 2
,
u=
′ βx f 2(1−βt)
f (η) , θ =
T , Th
(η)
φ=
(7) C Ch
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v=−
1 l(1−βt) 2
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η=
(6)
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u = 0, v = vw =
∂u ∂y
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v=
T
coefficient. The relevant boundary conditions are
Substituting the above variables into (2) and (3) and then eliminating pressure gradient from the resulting equations gives
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, ! 4 ′′ ′ ′ ′ ′ ′ 2 ′′ 2 1 + Rd θ + Pr S (f θ − ηθ ) + N bθ φ + N t(θ ) + Ec (f ) = 0 3
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+
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# " ′′′ ′′ ′ ′′ ′′′ f iv − S ηf + 3f + f f − f f =0
φ′′ + Sc (Sf φ′ − ηSφ) +
N t ′′ θ =0 Nb
(8) (9) (10)
in the above expressions ”′” denote the derivative with respect to η. The corresponding boundary conditions are defined as, f (0) = 0, f (0) = 0, θ(0) = 0, φ(0) = 0 at η = 0 Active control of nanoparticles at η = 1, f (1) = 1, f ′ (1) = 0, θ(1) = 1, φ(1) = 1 (11) ′ ′ ′ f (0) = 0, f (0) = 0, θ(0) = 0, N bφ (0) + N tθ (0) = 0 at η = 0 Passive control of nanoparticles f (1) = 1, f ′ (1) = 0, θ(1) = 1, φ(1) = 1 at η = 1. (12) ′
Where, S is the squeeze number, Pr is the Prandtl number, Ec is the Eckert number, Sc is the Lewis number, Rd is the Radiation parameter, N b is the Brownian motion parameter and N t is the thermophoretic parameter. The governing parameters are defined by " #2 βl2 ρ βx 1 Ec = Cp 2(1−βt) , pr = ρ µα , S = 2µ f , Sc = DαB , f
Rd =
4σ e Tc3 , βRk
Nt =
(ρc)p DT (Th −Tc ) , (ρc)f αTc
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Nb =
(ρc)p DB (φh −φc ) (ρc)f α
ACCEPTED MANUSCRIPT Physical quantities of interest are the Nusselt number and sherwood number, which are defined as: Nu =
) −lk( ∂T ∂y y=0 kf TH
4 (1 + Rd), 3
Sh =
) −lk( ∂C ∂y y=0 DCh
Results and discussion
0
′
1 − βtSh = −φ (0).
(14)
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3
4 ′ 1 − βtN u = −(1 + Rd)θ (0), 3
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0
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In the view of Eq. (7), we have
(13)
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The above mentioned coupled differential equations (8)-(10) along with the boundary conditions defined in Eqs. (11) and (12) are solve through a numerically technique. Since the
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present mathematical model contains the two point boundary value problem (BVP) so we apply finite difference method (FDM) method to solve this system. The step size is taken as and the procedure for FDM method is repeated until we get the asymptotically conver-
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gent results within a tolerance level of 10 −7 . All these working schemes are assimilated in computational software Matlab.
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To validate present study we have made a comparison with existing literature in table 1. One can see that results produce by Sheikholeslami and Ganji [5] are in good agreement
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with the current study in the absence of zero flux of nanoparticles. Although we have plotted all the results for local Nusselt and Sherwood number to analyze the heat transfer influence and concentration of nanoparticle volume fraction via figures. 8 − 11 but in the tables 2 − 7 one can easily compute to validate our results numerically.
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Results and discussion
In order to evaluate the solutions of given system of equations, results are obtained for ′
velocity f (η), f (η) temperature θ(η) and concentration profile φ(η) for different values of Brownian motion parameter N b, Thermophoretic parameter N t, Squeezing parameter S, Eckert number Ec, and Radiation parameter Rd. The effect of governing parameters on velocity, temperature, concentration is shown with the help of graphs. Figure 2 exhibits ′
the result of squeeze number S on velocity profile f (η) and f (η). For increasing values of squeeze number S, the velocity profile of f increases near the lower surface and opposite 6
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behavior shown near the upper surface of the channel. Velocity profile f (η) increases near the upper and lower surface and decreases at the mean position of the channel. Figure 3 depicts the temperature and concentration profile for different values of Squeeze
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number S. It is found that for both active and passive control, temperature profile get-
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ting decrease with an increase of squeezing parameter S. Physically, it is determine that when plates moves away from each other particles of nanofluid mixture will disperse. Consequently, there is less collision of particles with in the channel and it will be less influence of
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inertial process. Similarly, it is formed that increase of squeezing number provides increase in concentration profile. It is found that from Fig. 2(b), for passive control of nanoparticles
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particles are dispersed at the lower surface of the channel. Figure 4 exhibits the behavior of temperature and concentration on the different values of Radiation Parameter Rd. For
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increasing values of Rd, temperature profile increases for both active and passive control of nanoparticles. However, concentration profile increases in both active and passive control. Figure 5 shows the result of temperature and concentration on the different values of Eckert
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number Ec. The increasing values of Ec the temperature profile increases for both active and passive control of nanoparticles. The concentration profile exhibits that with the in-
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crease in Eckert number the boundary layer thickness decreases both in active and passive control. Figure 6 shows the variation of temperature and concentration on different values
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of Brownian motion parameter N b. Since, Brownian motion is random motion of particles within the base fluid. Because of random motion, particles collide with each other and inertial forces transfer more rapidly, therefore this phenomena enhance the heat transfer. Consequently increases in the values of N b the temperature profile increases in active and passive control. On the other hand the concentrations graph increases in active control and decreases in passive control of nanoparticles. Figure 7 exhibits the graph of thermophoretic parameter N t on temperature and concentration. For increasing values of N t, temperature graph is decreases in active and passive control and concentration graph decreases on active control and increases on passive control of nanoparticles. Figure 8 demonstrate the variation of Nusselt and Sherwood number. There is no significant effect of Brownian motion parameter N b on the surface of the Nusselt number in the case of passive control and in the case of active control of nanoparticles the graph is increases. Fig. 8 also shows that with an increase of Eckert number Ec the graph of Nusselt number increases in both active and passive control of nanoparticles. Brownian motion 7
ACCEPTED MANUSCRIPT parameter N b shows the little effect of active control as compared to passive control. For increasing Eckert number Ec the Sherwood number increase in both active and passive control of nanoparticles. Figure 9 demonstrate the variation of Nusselt and Sherwood number.
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There is no effect of Brownian motion parameter N b on the surface of the Nusselt number
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in the case of passive control and in the case of active control of nanoparticles the graph is increases. Fig. 9 also shows that with an increase of Radiation parameter Rd the graph of Nusselt number increases in both active and passive control of nanoparticles. Brownian
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motion parameter N b shows the little effect of active control as compared to passive control. For increasing Radiation parameter Rd the Sherwood number decreases in both active and
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passive control of nanoparticles. Figure 10 shows that the Brownian motion parameter N b has no effect on the surface of Nusselt number in the passive control of nanoparticles and
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active control of nanoparticles is increases. As Eckert number Ec increases the Nusselt number increases in both active and passive control of nanoparticles. The graph of Sherwood number shows that Brownian motion parameter N b has greater effect of passive control as
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compared to active control and as a result of Eckert number increasing the active and passive control both increases. Figure 9 shows that both Nusselt and Sherwood number increases as
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we increase the thermophoretic parameter N t. Fig. 11, exhibits the behavior of isotherms for various values of radiation parameter Rd. It can be determined that for increasing values
5
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of Rd, behavior of isotherms shows the rapidly variation in the temperature.
Conclusion
In this paper, the effect of thermal radiation on unsteady nanofluid flow between two parallel plates is studied. By using the similarity transformation for velocity, temperature and concentration the basic equations are reduced to a set of ordinary differential equations. The set of ordinary differential equations along with the prescribed boundary conditions have solved numerically by using finite difference method. Effects of different parameters such as squeeze number, Radiation parameter Schmidt number, Brownian motion parameter and Eckert number on temperature, velocity and concentration profile are examined. The nanofluid flow has controlled actively and passively. For different values of Rd the temperature and concentration have opposite behavior. Temperature and concentration have same behavior for various values of both Brownian motion parameter N b and Thermophoretic pa8
ACCEPTED MANUSCRIPT rameter N t for active and passive control of nanoparticles. The result of the study indicate the different behavior of Brownian motion parameter N b on Nusselt and Sherwood number for different values of radiation parameter Rd, Eckert number Ec and Thermophoretic
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parameter N t.
Acknowledgment
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This work is supported by The World Academy of Science (TWAS) for research and advanced
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training with FR number: 3240288442.
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isothermal surface embedded in a saturated porous media. J Appl Mech 73:54–59 [33] Hossain MA, Takhar HS (1996) Radiation effect on mixed convection along a vertical plate with uniform surface temperature. Heat Mass Transf 31:243–248 [34] Zahmatkesh I (2007) Influence of thermal radiation on free convection inside a porous enclosure. Emir J Eng Res 12(2):47–52. [35] Hayat T, Abbas Z, Pop I, Asghar S (2010) Effects of radiation and magnetic field on the mixed convection stagnation-point flow over a vertical stretching sheet in a porous medium. Int J Heat Mass Transf 53:466–474. [36] Pal D, Mondal H (2009) Radiation effects on combined convection over a vertical flat plate embedded in a porous medium of variable porosity. Meccanica 44:133–144. Table 1: Comparison table for various values of radiation parameter and Schmidt number on Nusselt number when S = 0.5, Ec = 0.01, N b = 0.1, N t = 0.2, P r = 10 and in the absence of zero flux o nanoparticles. 12
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Fig. 1: Schematic illustration of the channel-ow con guration.
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1 .4
S = 0, 1 , 2
1 .2
1
0 .8
f ' (η)
0 .6
0 .4
f ( η) 0 .2
0 0
0.25
0 .5
0.75
1
η 0
Fig 2 : Variation of f () and f () for various values of S = 0; 1 and 2:
1
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P
co is ve as
o ntr
f lo
n na
op
es icl t r a
1.5 1.2
S=2 S=1 S=0 S=2 S=1 S=0
0
0.25
0.5
0.75
η
0.6 0.3 0
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(a)
S=2 S=1 S=0 S=2 S=1 S=0
0.9
0.2
0
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n
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θ(η)
0.4
of
Passive control of nanoparticles
1.8
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A
0.6
o ec v i ct
ol ntr
p
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0.8
o an
φ(η)
1
2.1
es icl art
1
Active control of nanoparticles
0
(b)
0.25
0.5
0.75
1
η
ED
Fig 3.Variation of temperature and concentration for di¤erent values of S = 0; 1 and 2:
PT
1
θ(η)
0.6
0.4
(a)
0.25
0.5
Rd=2 Rd=1 Rd=0 Rd=2 Rd=1 Rd=0
0.9 0.6
Rd=2 Rd=1 Rd=0 Rd=2 Rd=1 Rd=0
{ {
Passive control of nanoparticles 0
1.5 1.2
Active control of nanoparticles
0.2
Passive control of nanoparticles
1.8
φ(η)
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0.8
0
2.1
0.75
0.3 0 1
η
(b)
Active control of nanoparticles 0
0.25
0.5
0.75
η
Fig 4.Variation of temeperature and concentration for various values of Radiation parameter Rd = 0; 1 and 2:
2
1
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2.4
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Active control of nanoparticles 0
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η
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η
0.4
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Passive control of nanoparticles
0
(a)
Ec =0.04 Ec =0.02 Ec =0.00 Ec =0.04 Ec =0.02 Ec =0.00
{ {
Active control of nanoparticles
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0.8
0.2
Ec =0.04 Ec =0.02 Ec =0.00 Ec =0.04 Ec =0.02 Ec =0.00
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φ(η)
θ(η)
1.6
Fig 5.Variation of temperature and concentration for various values of Eckert number
1
0.5
0.6
0.45
0.4
0.425
0.45
4.5
3.5
0.475
2.5
θ(η )
Passive control of nanoparticles
(a)
0
0.25
0.5
0.75
ntro lo
2
f na
nop a
rticl es
1.5
Nb=0.7 Nb=0.5 Nb=0.3 Nb=0.7 Nb=0.5 Nb=0.3
{ {
Active control of nanoparticles
0
P as sive co
3
0.4
0.2
Nb=0.7 Nb=0.5 Nb=0.3 Nb=0.7 Nb=0.5 Nb=0.3
4
φ(η)
0.8
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Ec = 0:00; 0:02 and 0:04:
1
particles l of nano o tr n o c Active
0.5 0
1
η
(b)
0
0.25
0.5
0.75
η
Fig 6.Variation of temperature and concentration for various values of Brownian motion parameter N b = 0:3; 0:5 and 0:7: 3
1
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1
4
3
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0.6
P ass ive c o
2.5
0.5
0.4775
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θ(η )
φ(η)
0 .475
1
0
0.25
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η
nano
parti cles
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0.5 0
-0.5
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Passive control of nanoparticles
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(a)
Nt=0.7 Nt=0.5 Nt=0.3 Nt=0.7 Nt=0.5 Nt=0.3
{ {
Active control of nanoparticles
0.2
ntro l of
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1.5
0.4
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3.5
0.505
0.8
Nt=0.7 Nt=0.5 Nt=0.3 Nt=0.7 Nt=0.5 Nt=0.3
1
(b)
Active control of nanoparticles
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0.25
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0.75
1
η
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Fig 7.Variation of velocity and concentration for various values of Thermophoretic
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parameter N t = 0:3; 0:5 and 0:7:
4.8
3.6 3.2 2.8
7 6
Ec=0.00 Ec=0.01 Ec=0.02 Ec=0.00 Ec=0.01 Ec=0.02
-φ'(0)
4
l of ntrcoles o c e ti Acntiavnopar
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(1+4Rd/3)θ'(0)
4.4
8
Ec=0.00 Ec=0.01 Ec=0.02 Ec=0.00 Ec=0.01 Ec=0.02
{ {
Active control of nanoparticles Passive control of nanoparticles
5 4 3
Passive control of nanoparticles
2 1
2.4
0
(a)
0.2
0.4
Nb
0.6
0.8
(b)
0.2
0.4
Nb
0.6
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Fig 8.E¤ects of Brownian motion parameter N b on (a) Nusselt and (b) Sherwood number for various values of Eckert number Ec = 0:00; 0:01 and 0:02:
4
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(1+4Rd/3)θ'(0)
10 8
-φ'(0)
6.4 5.6 4.8
Rd=0 Rd=1 Rd=2 Rd=0 Rd=1 Rd=2
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3.2
2
2.4
Passive control of nanoparticles 0.2
0.4
0.6
Nb
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(a)
6 4
4
1.6
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7.2
{ {
Active control of nanoparticles Passive control of nanoparticles
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8
12
f rol os t n o e c icle Actaivnopart n
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Rd=0 Rd=1 Rd=2 Rd=0 Rd=1 Rd=2
8.8
0.8
0
0.2
(b)
0.4
Nb
0.6
0.8
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Fig 9:.E¤ects of Brownian motion parameter N b on (a) Nusselt and (b) Sherwood
Nt=0.3 Nt=0.5 Nt=0.7 Nt=0.3 Nt=0.5 Nt=0.7
4.8 4.4
of trolles n o ve c rtic Acntianopa
4
0.2
0.4
Nb
{ {
Active control of nanoparticles Passive control of nanoparticles
Nt=0.3 Nt=0.5 Nt=0.7 Nt=0.3 Nt=0.5 Nt=0.7
6
4
2
Passive control of nanoparticles
3.6
(a)
8
-φ'(0)
(1+4Rd/3)θ'(0)
5.2
10
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5.6
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number for various values of Eckert number Rd = 0; 1 and 2:
0.6
0
0.8
(b)
0.2
0.4
Nb
0.6
0.8
Fig 10.E¤ects of Brownian motion parameter N b on (a) Nusselt and (b) Sherwood number for various values of Eckert number N t = 0:3; 0:5 and 0:7:
5
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Fig. 11: Variation of isotherms for various
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values of Rd.
6
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!" = 20 0.270602 0.296133 0.298608 0.299544
Present Study !" = 1 !" = 10 0.270420 0.270521 0.296130 0.296131 0.298612 0.298677 0.299598 0.299512
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Sheikholeslami and Ganji [5] !" = 0.5 !" = 1 !" = 10 0.27041 0.270421 0.270507 0.296127 0.296127 0.296130 0.298606 0.298607 0.298607 0.299543 0.299543 0.299543
!" = 0.5 0.270420 0.296131 0.298610 0.299548
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↓ 0 3 6 12
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Table 1: Comparison table for various values of radiation parameter and Schmidt number on Nusselt number when = 0.5, = 0.01, = 0.1, = 0.2, ! = 10 and in the absence of zero flux o nanoparticles. !" = 20 0.270612 0.296132 0.298654 0.299538
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Table 2: Variation of Nusselt number (1 + 4/3)$′(0), when &' = 2, ! = 0.5, *" = 0.02, !" = 0.5 and = 1.
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-: ↓ 0.3 0.6 0.9
Active control of nanoparticles -; = 0.3 -; = 0.5 -; = 0.7 3.215093 3.457153 3.710134 3.581995 3.840261 4.109018 3.972927 4.246715 4.530456
Passive control of nanoparticles -; = 0.3 -; = 0.5 -; = 0.7 2.555036 2.554189 2.553343 2.555036 2.554189 2.553343 2.555036 2.554189 2.553343
Active control of nanoparticles = 1 = 2 = 3 2.973548 4.293644 5.620765 3.329252 4.629376 5.947042 3.70932 4.981083 6.28515
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-: ↓ 0.3 0.6 0.9
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Table 3: Variation of Nusselt number (1 + 4/3)$′(0), when &' = 2, ! = 0.5, -; = 0.2, *" = 0.01 and Sc = 0.5. Passive control of nanoparticles = 1 = 2 = 3 2.435778 3.769229 5.102617 2.435778 3.769229 5.102617 2.435778 3.769229 5.102617
Table 4: Variation of Nusselt number (1 + 4/3)$′(0), when &' = 2, ! = 0.5, -; = 0.8, = 1 and Sc = 0.5.
-: ↓ 0.3 0.6 0.9
Active control of nanoparticles *" = 0 *" = 0.01 *" = 0.02 3.579414 3.710026 3.840639 3.980239 4.113731 4.247225 4.403302 4.539618 4.675935
Passive control of nanoparticles *" = 0 *" = 0.01 *" = 0.02 2.313648 2.433284 2.55292 2.313648 2.433284 2.55292 2.313648 2.433284 2.55292
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Table 5: Variation of Sherwood number <′(0), when &' = 2, ! = 0.5, *" = 0.02, !" = 0.5 and = 1.
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-: ↓ 0.3 0.6 0.9
Passive control of nanoparticles -; = 0.3 -; = 0.5 -; = 0.7 -1.09502 -1.82442 -2.55334 -0.54751 -0.91221 -1.27667 -0.36501 -0.60814 -0.85111
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Active control of nanoparticles -; = 0.3 -; = 0.5 -; = 0.7 0.61922 0.195792 -0.37642 0.729197 0.459452 0.110896 0.762421 0.541774 0.265804
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Table 6: Variation of Sherwood number <′(0), when &' = 2, ! = 0.5, -; = 0.2, *" = 0.01 and Sc = 0.5. Passive control of nanoparticles = 1 = 2 = 3 -0.69594 -0.68531 -0.68035 -0.34797 -0.34266 -0.34017 -0.23198 -0.22844 -0.22678
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0.3 0.6 0.9
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-: ↓
Active control of nanoparticles = 1 = 2 = 3 0.813614 0.882304 0.913422 0.854114 0.908689 0.932988 0.865293 0.916516 0.938984
-: ↓
Active control of nanoparticles *" = 0 *" = 0.01 *" = 0.02 -0.42307 -0.57211 -0.72115 0.058053 -0.01812 -0.09428 0.209939 0.158084 0.106229
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0.3 0.6 0.9
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Table 7: Variation of Sherwood number<′(0), when &' = 2, ! = 0.5, -; = 0.8, = 1 and Sc = 0.5. Passive control of nanoparticles *" = 0 *" = 0.01 *" = 0.02 -2.64417 -2.7809 -2.91762 -1.32208 -1.39045 -1.45881 -0.88139 -0.92697 -0.97254
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Graphical Abstract:
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4
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(a)
Passive control of nanoparticles
Nt=0.3 Nt=0.5 Nt=0.7 Nt=0.3 Nt=0.5 Nt=0.7
4
2
Passive control of nanoparticles
3.6
0.2
Active control of nanoparticles
6
l of tro conticles e tiv ar Ac anop n
4.4
8
-f'(0)
4.8
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(1+4Rd/3)q'(0)
5.2
{ {
10
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Nt=0.3 Nt=0.5 Nt=0.7 Nt=0.3 Nt=0.5 Nt=0.7
5.6
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To avoid the gravitational settling, both active and passive control of nanoparticles is performed for nanofluid between squeezing channel. Isotherms plots for squeezing flow are plotted for radiation parameter.
0
0.4
0.6
Nb
0.8
(b)
0.2
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ACCEPTED MANUSCRIPT Highlights • Analysis is performed for squeezing channel between two parallel plates.
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• Results are analyzed for both active and zero flux of nanoparticles at the surface.
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• Radiation and viscous dissipation effects are also incorporated within the nanofluid.
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• Isotherms behavior is also described for radiation parameter.
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S0167-7322(16)31733-0 doi: 10.1016/j.molliq.2016.08.032 MOLLIQ 6192
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
28 June 2016 4 August 2016 9 August 2016
Please cite this article as: M. Atlas, Rizwan Ul Haq, T. Mekkaoui, Active and zero flux of nanoparticles between a squeezing channel with thermal radiation effects, Journal of Molecular Liquids (2016), doi: 10.1016/j.molliq.2016.08.032
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Active and zero flux of nanoparticles between a squeezing channel with thermal radiation effects M. Atlas 1 , Rizwan Ul Haq 2,∗ , T. Mekkaoui 3 Department of Mathematics, Capital University of Science and Technology, Islamabad
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1
2
Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan
Department of Mathematics, Faculty of Science and Technology, Errachidia, Morocco
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3
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44000, Pakistan.
Abstract
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Present study deals the effect of thermal radiation on unsteady nanofluid flows between two parallel plates. Due to gravitational settling nanoparticles accumulate at the base of
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the channel and effect the state of homogeneous mixture of nanofluid so the present model describe the analysis of both active and passive control of nanoparticles at the lower surface of the channel. Similarity transformations are used to reduce the basic partial differential
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equation to nonlinear coupled ordinary differential equations. The equations along with the prescribed boundary conditions were solved numerically using the finite difference method
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in MATLAB for various values of emerging parameters. Numerical results are obtained for velocity, temperature and concentration for different parameter such as squeeze number,
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radiation parameter, Lewis number, and Eckert number. Flow behavior and temperature are also described through stream lines and isotherms profiles. Throughout the whole analysis, it is found that for zero normal flux there is low temperature profile and high concentration profile is obtained as compare to the active control of nanoparticles. Isotherms plots depict the significant influence of radiation effects to enhance temperature in the restricted domain. Keywords: Nanofluid, Brownian motion, thermophoresis, zero flux, active control.
1
Introduction
Heat and mass transfer in a channel have great interest in engineering applications and many branches of science, a lot of study on heat and mass transfer have been done in recent years. Some practical examples related to channel flow are polymer processing, compression, and injection molding. Initially, Stefan [1] advertised a traditional paper on squeezing flow b y 2
* Mailing address: ideal [email protected], [email protected] (R. U. Haq)
Contact: +92 333 5371853
1
ACCEPTED MANUSCRIPT utilization of lubrication conjecture. After that, Reynolds [2] attains a solution for elliptic plates, and Archibald [3] further modifies this problem for rectangular plates. The theoretical and experimental studies of squeezing flows have been regulated by many researchers.
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Siddiqui et al. [4] examined the hydro magnetic squeezing flow of a viscous fluid between
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two parallel plates and used the homotopy perturbation method (HPM) to find out the solutions. Recently, Sheikholeslami and Ganji [5] analyzed the characteristic of heat and mass transfer for unsteady nanofluid in a squeezing channel in the presence of viscous dissipation
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and radiation effects. Operating on heat and mass transfer along with chemical reaction effect plays a crucial part in designs of chemical processing equipment, development and
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dispersion of fog, destruction of crops and cooling towers.
Most of the study on channel flow complications incorporates base fluid which has low
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thermal conductivity. Due to poor thermal conductivity the resulting production is slow. To boost up the thermal conductivity of the fluid, nanoparticles can incorporate with the base fluid. In this field a lot of work done in recent years. Nanofluids are more useful at industrial
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level because of their heat transfer properties; they intensify the thermal conductivity and convective properties over the properties of the base fluid. Initially Choi [6] is the foremost
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person who inaugurated the concept of nanofluids and employed the nanofluid in an extensive variation of industries, energy manufacturing and electronics production, textiles and paper
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construction. Masuda et al [7] initiate the concept of thermal conductivity magnification. Buongiorno and Hu [8] used the Nanofluids in nuclear systems. Kleinstreuer et al. [9] present the approach of the Nanofluids flow in the transportation of Nano-drugs. Buongiorno [10] present a new model construct on the nanoparticles mechanics. Nield and Kuznetsov [11] calculated the matter of boundary layer natural convection past a vertical plate and utilize the Buongiorno nanofluid model with the outcome of Brownian motion and thermophoresis of nanoparticles. Following that they reconsider a paper and enlarge it to the problem in which the boundary is passively controlled rather than actively controlled. Khanafer et al. [12] studied the two dimensional rectangular flow by using copper-water nanofluid. They originate that the heat transfer ratio grows by increasing the ratio of the eliminated particles. The identical problem resolved experimentally by Nnanna et al. [13] for ethylene glycol by utilizing Cu nanoparticles and Nnanna and Routhu [14] for Alumina-water nanofluids. Different experimental work done by Putra et al [15] on natural convection of Al2 O3 - and CuO-Water nanofluids. They have summarized that the natural convection coefficient was 2
ACCEPTED MANUSCRIPT lower than that of pure water. In recent years study on channel flow by utilizing nanfluid has considerable attraction. Abu-Nada et al. [16] originated natural convection heat transfer improvement in horizontal
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parallel annuli range by nanofluid. They initiate that nanoparticle with higher thermal con-
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ductivity at low Rayleigh numbers yields more improvements in heat transfer. Ellahi [17] observed that flow of non-Newtonian nanofluid along with magneto hydrodynamic (MHD) in a pipe and examined that the MHD parameter decreases the fluid motion in the existence
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of variable viscosities temperature graphs is slower than that of velocity graphs. Sheikholeslami et al. [18] measured the effect of magnetic field on free convection heat transfer in
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a cold square and heated elliptic cylinder. They initiate that increase in Hartmann number enhance the heat transfer but increase in Rayleigh number it will decrease. Current many
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authors present different ideas for nanofluid and heat transfer through various geometries that accomplish the wide attention at industrial level [19-30]. Thermal radiation also plays an important role in controlling heat transfer process in
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polymer processing industry. As a result of little convective heat transfer coefficient the thermal radiation has an important role on the surface. In space technology and high
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temperature the consequence of thermal radiation are meaningful. Bakier [31] reviewed the reaction of thermal radiation on mixed convection. He operated the equations by using the
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fourth order RK method. Effect of radioactive vertical isothermal on MHD flow with mixed convection was numerically solved by Damseh [32]. Hossain and Takhar [33] studied the effect of radiation with Roseland diffusion approximation along a vertical plate on free convection boundary layer flow. Zahmatkesh [34] concluded that the impact of thermal radiation on the growth of free convection flow inside an enclosure filled saturated porous media. Hayat et al. [35] used the Homotopy Analysis Method to study the impact of radiation on the MHD mixed convection stagnation point. Pal and Mondal [36] present the complete study by incorporate convective heat transfer in a porous medium by assuming heat generation and radiation impacts. In our work we have discussed the nanofluid flow model between two parallel plates and calculate the impact of thermal radiation on heat and mass transfer. Moreover, we have considered the active and passive control of nanoparticles near the lower surface of the channel. Mathematical model is constructed in the form of partial differential equations. Resulting equations are solved numerically with FDM. Important physical parameters: radiation pa3
ACCEPTED MANUSCRIPT rameter, squeezing number, Eckert number, Brownian motion parameter, thermophoresis parameter, Lewis number are described through graphs. Flow behavior is also discussed through stream lines. For both active and passive control of nanoparticles results are also
2
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constructed for local Nusselt number and Sherwood number.
Mathematical Model
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Consider the unsteady squeezing flow of two dimensional incompressible flow of viscous fluid between two infinite parallel plates. The two plates are placed apart from each other at the
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distance h(t) = l(1 − βt)1/2 . For β > 0 the two plates are squeezed until they touch t =
1 β
and for β < 0 the two plates are separated (see Fig. 1). Further, we have investigated both
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active and passive control of the nanoparticles at the lower surface of the channel. Radiation effects are also taken into the account. The governing equations for mass, momentum and
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energy in unsteady two dimensional flow of nanofluid are [5]
(1)
∂u ∂ 2u ∂ 2u 1 ∂p ∂u ∂u +u +v =− + ν nf ( 2 + 2 ), ∂t ∂x ∂y ρnf ∂x ∂x ∂y
(2)
∂v ∂v 1 ∂p ∂v ∂2v ∂2v +u +v =− + ν nf ( 2 + 2 ), ∂t ∂x ∂y ρnf ∂y ∂x ∂y
(3)
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∂u ∂v + = 0, ∂x ∂y
2
2
+ u ∂T + v ∂T = αnf ( ∂∂xT2 + ∂∂yT2 ) + (ρcµp )f (( ∂u )2 ) ∂x ∂y ∂y ! DB ∂C + . ∂T + ∂C . ∂T ∂x ∂x ∂y ∂y (ρcp )p ' − 1 ∂qr , $ " " # # +S (ρc 2 (ρcp )f ∂y & % 2 p )f DT ∂T ∂T + Tc ∂x ∂y ∂T ∂t
∂C ∂C ∂C +u +v = DB ∂t ∂x ∂y
+
∂ 2C ∂ 2C + ∂x2 ∂y 2
,
+
+
DT Tc
,$
∂ 2T ∂ 2T + ∂x2 ∂y 2
(4)
'
.
(5)
Where, the radiation heat flux consider according to the Rosseland approximation such that 4σ e qr = − 3β
R
∂T 4 , ∂y
where σ e , β R are the Stefan-Boltzman constant and the mean absorption
constant respectively. The fluid phase temperature difference within the flow is assumed to be sufficiently small so that T 4 may be expressed as a linear function of temperature. This is done by expanding T 4 in a Taylor series about the temperature Tc and neglecting higher order terms to yield T 4 = 4Tc3 T − Tc3 . In above equations u and v are the velocities in the x and y directions respectively. T is the temperature. C is the concentration, ν nf is viscosity 4
ACCEPTED MANUSCRIPT of nanofluid, P is the pressure, ρnf is the density of nanofluid, µnf is the dynamic viscosity of nanofluid, knf is the thermal conductivity of nanofluid, cp is the specific heat of nanofluid, DB is the diffusion coefficient of the diffusing species and DT is the thermophoretic diffusion
v=
∂u ∂y
dh ,T dt
=T =C=0y=0
= Th , C = Ch (Active control of nanoparticles) at y = h (t)
= T = DB ∂C + ∂y
DT ∂T Tc ∂y
= 0 (Passive control of nanoparticles) at y = 0
We introduce these parameters y
βl 1
2(1−βt) 2
,
u=
′ βx f 2(1−βt)
f (η) , θ =
T , Th
(η)
φ=
(7) C Ch
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v=−
1 l(1−βt) 2
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η=
(6)
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u = 0, v = vw =
∂u ∂y
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v=
T
coefficient. The relevant boundary conditions are
Substituting the above variables into (2) and (3) and then eliminating pressure gradient from the resulting equations gives
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, ! 4 ′′ ′ ′ ′ ′ ′ 2 ′′ 2 1 + Rd θ + Pr S (f θ − ηθ ) + N bθ φ + N t(θ ) + Ec (f ) = 0 3
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+
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# " ′′′ ′′ ′ ′′ ′′′ f iv − S ηf + 3f + f f − f f =0
φ′′ + Sc (Sf φ′ − ηSφ) +
N t ′′ θ =0 Nb
(8) (9) (10)
in the above expressions ”′” denote the derivative with respect to η. The corresponding boundary conditions are defined as, f (0) = 0, f (0) = 0, θ(0) = 0, φ(0) = 0 at η = 0 Active control of nanoparticles at η = 1, f (1) = 1, f ′ (1) = 0, θ(1) = 1, φ(1) = 1 (11) ′ ′ ′ f (0) = 0, f (0) = 0, θ(0) = 0, N bφ (0) + N tθ (0) = 0 at η = 0 Passive control of nanoparticles f (1) = 1, f ′ (1) = 0, θ(1) = 1, φ(1) = 1 at η = 1. (12) ′
Where, S is the squeeze number, Pr is the Prandtl number, Ec is the Eckert number, Sc is the Lewis number, Rd is the Radiation parameter, N b is the Brownian motion parameter and N t is the thermophoretic parameter. The governing parameters are defined by " #2 βl2 ρ βx 1 Ec = Cp 2(1−βt) , pr = ρ µα , S = 2µ f , Sc = DαB , f
Rd =
4σ e Tc3 , βRk
Nt =
(ρc)p DT (Th −Tc ) , (ρc)f αTc
5
Nb =
(ρc)p DB (φh −φc ) (ρc)f α
ACCEPTED MANUSCRIPT Physical quantities of interest are the Nusselt number and sherwood number, which are defined as: Nu =
) −lk( ∂T ∂y y=0 kf TH
4 (1 + Rd), 3
Sh =
) −lk( ∂C ∂y y=0 DCh
Results and discussion
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′
1 − βtSh = −φ (0).
(14)
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3
4 ′ 1 − βtN u = −(1 + Rd)θ (0), 3
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In the view of Eq. (7), we have
(13)
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The above mentioned coupled differential equations (8)-(10) along with the boundary conditions defined in Eqs. (11) and (12) are solve through a numerically technique. Since the
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present mathematical model contains the two point boundary value problem (BVP) so we apply finite difference method (FDM) method to solve this system. The step size is taken as and the procedure for FDM method is repeated until we get the asymptotically conver-
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gent results within a tolerance level of 10 −7 . All these working schemes are assimilated in computational software Matlab.
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To validate present study we have made a comparison with existing literature in table 1. One can see that results produce by Sheikholeslami and Ganji [5] are in good agreement
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with the current study in the absence of zero flux of nanoparticles. Although we have plotted all the results for local Nusselt and Sherwood number to analyze the heat transfer influence and concentration of nanoparticle volume fraction via figures. 8 − 11 but in the tables 2 − 7 one can easily compute to validate our results numerically.
4
Results and discussion
In order to evaluate the solutions of given system of equations, results are obtained for ′
velocity f (η), f (η) temperature θ(η) and concentration profile φ(η) for different values of Brownian motion parameter N b, Thermophoretic parameter N t, Squeezing parameter S, Eckert number Ec, and Radiation parameter Rd. The effect of governing parameters on velocity, temperature, concentration is shown with the help of graphs. Figure 2 exhibits ′
the result of squeeze number S on velocity profile f (η) and f (η). For increasing values of squeeze number S, the velocity profile of f increases near the lower surface and opposite 6
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behavior shown near the upper surface of the channel. Velocity profile f (η) increases near the upper and lower surface and decreases at the mean position of the channel. Figure 3 depicts the temperature and concentration profile for different values of Squeeze
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number S. It is found that for both active and passive control, temperature profile get-
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ting decrease with an increase of squeezing parameter S. Physically, it is determine that when plates moves away from each other particles of nanofluid mixture will disperse. Consequently, there is less collision of particles with in the channel and it will be less influence of
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inertial process. Similarly, it is formed that increase of squeezing number provides increase in concentration profile. It is found that from Fig. 2(b), for passive control of nanoparticles
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particles are dispersed at the lower surface of the channel. Figure 4 exhibits the behavior of temperature and concentration on the different values of Radiation Parameter Rd. For
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increasing values of Rd, temperature profile increases for both active and passive control of nanoparticles. However, concentration profile increases in both active and passive control. Figure 5 shows the result of temperature and concentration on the different values of Eckert
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number Ec. The increasing values of Ec the temperature profile increases for both active and passive control of nanoparticles. The concentration profile exhibits that with the in-
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crease in Eckert number the boundary layer thickness decreases both in active and passive control. Figure 6 shows the variation of temperature and concentration on different values
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of Brownian motion parameter N b. Since, Brownian motion is random motion of particles within the base fluid. Because of random motion, particles collide with each other and inertial forces transfer more rapidly, therefore this phenomena enhance the heat transfer. Consequently increases in the values of N b the temperature profile increases in active and passive control. On the other hand the concentrations graph increases in active control and decreases in passive control of nanoparticles. Figure 7 exhibits the graph of thermophoretic parameter N t on temperature and concentration. For increasing values of N t, temperature graph is decreases in active and passive control and concentration graph decreases on active control and increases on passive control of nanoparticles. Figure 8 demonstrate the variation of Nusselt and Sherwood number. There is no significant effect of Brownian motion parameter N b on the surface of the Nusselt number in the case of passive control and in the case of active control of nanoparticles the graph is increases. Fig. 8 also shows that with an increase of Eckert number Ec the graph of Nusselt number increases in both active and passive control of nanoparticles. Brownian motion 7
ACCEPTED MANUSCRIPT parameter N b shows the little effect of active control as compared to passive control. For increasing Eckert number Ec the Sherwood number increase in both active and passive control of nanoparticles. Figure 9 demonstrate the variation of Nusselt and Sherwood number.
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There is no effect of Brownian motion parameter N b on the surface of the Nusselt number
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in the case of passive control and in the case of active control of nanoparticles the graph is increases. Fig. 9 also shows that with an increase of Radiation parameter Rd the graph of Nusselt number increases in both active and passive control of nanoparticles. Brownian
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motion parameter N b shows the little effect of active control as compared to passive control. For increasing Radiation parameter Rd the Sherwood number decreases in both active and
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passive control of nanoparticles. Figure 10 shows that the Brownian motion parameter N b has no effect on the surface of Nusselt number in the passive control of nanoparticles and
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active control of nanoparticles is increases. As Eckert number Ec increases the Nusselt number increases in both active and passive control of nanoparticles. The graph of Sherwood number shows that Brownian motion parameter N b has greater effect of passive control as
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compared to active control and as a result of Eckert number increasing the active and passive control both increases. Figure 9 shows that both Nusselt and Sherwood number increases as
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we increase the thermophoretic parameter N t. Fig. 11, exhibits the behavior of isotherms for various values of radiation parameter Rd. It can be determined that for increasing values
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of Rd, behavior of isotherms shows the rapidly variation in the temperature.
Conclusion
In this paper, the effect of thermal radiation on unsteady nanofluid flow between two parallel plates is studied. By using the similarity transformation for velocity, temperature and concentration the basic equations are reduced to a set of ordinary differential equations. The set of ordinary differential equations along with the prescribed boundary conditions have solved numerically by using finite difference method. Effects of different parameters such as squeeze number, Radiation parameter Schmidt number, Brownian motion parameter and Eckert number on temperature, velocity and concentration profile are examined. The nanofluid flow has controlled actively and passively. For different values of Rd the temperature and concentration have opposite behavior. Temperature and concentration have same behavior for various values of both Brownian motion parameter N b and Thermophoretic pa8
ACCEPTED MANUSCRIPT rameter N t for active and passive control of nanoparticles. The result of the study indicate the different behavior of Brownian motion parameter N b on Nusselt and Sherwood number for different values of radiation parameter Rd, Eckert number Ec and Thermophoretic
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parameter N t.
Acknowledgment
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This work is supported by The World Academy of Science (TWAS) for research and advanced
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training with FR number: 3240288442.
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References
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[15] N. Putra,W. Roetzel, S.K. Das, Natural convection of nanofluids, Heat and Mass Transfer 39 (8–9) (2003) 775–784. [16] Abu-Nada E, Chamkha A J. Mixed convection flow in a lid-driven inclined square enclosure filled with a nanofluid transfer in nanofluids. European Journal of Mechanics B/Fluids 2010; 29: 472-482. [17] Ellahi R (2013) The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: analytical solutions. Appl Math Model 37(3):1451– 1467 10
ACCEPTED MANUSCRIPT [18] Sheikholeslami M, Gorji-Bandpay M, Ganji DD (2012) Magnetic field effects on natural convection around a horizontal circular cylinder inside a square enclosure filled with nanofluid. Int Commun Heat Mass Transf 39:978–986
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[19] Z. Liu, Y. Liu, Y. Chang, H. R. Seyf, A. Henry, A. L. Mattheyses, K. Yehl, Y. Zhang, Z.
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Huang, K. Salaita, Nanoscale optomechanical actuators for controlling mechanotransduction in living cells, Nature methods, 13(2):143-6, (2016).
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[20] B Rajabifa, H Seyf, Y Zhang, S.K. Khanna. Flow and Heat Transfer in Micro Pin Fin
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Heat Sinks with Nano-Encapsulated Phase Change Materials. ASME. J. Heat Transfer.
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effects on the thermal behavior and cooling performance of microchannel heat sinks, International Journal of Thermal Sciences, 58, 36-44 (2012).
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[22] Seyed Moein Rassoulinejad-Mousavi, Hamid Reza Seyf And S Abbasbandy, heat transfer through a porous saturated channel with permeable walls using two-equation energy
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[23] H. R. Seyf, S. M. Rassoulinejad-Mousavi, An Analytical Study for Fluid Flow in Porous Media Imbedded Inside a Channel With Moving or Stationary Walls Subjected to Injection/Suction, ASME. J. Fluids Eng. 133(9), 091203 (2011). [24] S. K. Mohammadian, Y. Zhang, Thermal management improvement of an air-cooled high-power lithium-ion battery by embedding metal foam, Journal of Power Sources 296, 305-313(2016). [25] S. M. Rassoulinejad-Mousavi, Saied Abbasbandy, Analysis of Forced Convection in a Circular Tube Filled With a Darcy–Brinkman–Forchheimer Porous Medium Using Spectral Homotopy Analysis Method. ASME. J. Fluids Eng. (2011), 133(10),101207. [26] S. M. Rassoulinejad-Mousavi, Saied Abbasbandy, H. H. Alsulami, Analytical flow study of a conducting Maxwell fluid through a porous saturated channel at various wall boundary conditions, Eur. Phys. J. Plus, 129: 181, 1-10, (2014).
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ACCEPTED MANUSCRIPT [27] S. M. Rassoulinejad-Mousavi, H. Yaghoobi, Effect of Non-linear Drag Term on Viscous Dissipation in a Fluid Saturated Porous Medium Channel with Various Boundary Conditions at Walls, Arab. J. Sci. Eng. 39 (2), 1231–1240, (2014).
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[28] S. T. Hussain, Rizwan Ul Haq, Z. H. Khan, S. Nadeem, Water Driven Flow of Carbon
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nanofluid nanotubes in a rotating channel, Journal of Molecular Liquid, 214, (2016) 136–144.
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[29] M. Sheikholeslami, R. Ellahi, M. Hassan, and S. Soleimani, A study of natural convection heat transfer in a nanofluid filled enclosure with elliptic inner cylinder, International
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Journal for Numerical Methods for Heat and Fluid Flow, 24(8) (2014) 1906-1927. [30] Rizwan Ul Haq, S.T. Hussain, Z.H. Khan, Z. Hammouch, Flow and Heat transfer anal-
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ysis of Water and Ethylene Glycol based Cu nanoparticles between two parallel disks with Suction/Injection effects, Journal of Molecular Liquids, 221 (2016) 298–304.
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[31] Bakier AY (2001) Thermal radiation effect on mixed convection from vertical surface in saturated porous media. Int Commun Heat Mass Transf 28(1):119–126
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[32] Damseh RA (2006) Magnetohydrodynamics-mixed convection from radiate vertical
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isothermal surface embedded in a saturated porous media. J Appl Mech 73:54–59 [33] Hossain MA, Takhar HS (1996) Radiation effect on mixed convection along a vertical plate with uniform surface temperature. Heat Mass Transf 31:243–248 [34] Zahmatkesh I (2007) Influence of thermal radiation on free convection inside a porous enclosure. Emir J Eng Res 12(2):47–52. [35] Hayat T, Abbas Z, Pop I, Asghar S (2010) Effects of radiation and magnetic field on the mixed convection stagnation-point flow over a vertical stretching sheet in a porous medium. Int J Heat Mass Transf 53:466–474. [36] Pal D, Mondal H (2009) Radiation effects on combined convection over a vertical flat plate embedded in a porous medium of variable porosity. Meccanica 44:133–144. Table 1: Comparison table for various values of radiation parameter and Schmidt number on Nusselt number when S = 0.5, Ec = 0.01, N b = 0.1, N t = 0.2, P r = 10 and in the absence of zero flux o nanoparticles. 12
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Fig. 1: Schematic illustration of the channel-ow con guration.
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co is ve as
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Fig 3.Variation of temperature and concentration for di¤erent values of S = 0; 1 and 2:
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θ(η)
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Rd=2 Rd=1 Rd=0 Rd=2 Rd=1 Rd=0
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Fig 4.Variation of temeperature and concentration for various values of Radiation parameter Rd = 0; 1 and 2:
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Ec =0.04 Ec =0.02 Ec =0.00 Ec =0.04 Ec =0.02 Ec =0.00
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φ(η)
θ(η)
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Fig 5.Variation of temperature and concentration for various values of Eckert number
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0.5
0.6
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particles l of nano o tr n o c Active
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Fig 6.Variation of temperature and concentration for various values of Brownian motion parameter N b = 0:3; 0:5 and 0:7: 3
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Fig 7.Variation of velocity and concentration for various values of Thermophoretic
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parameter N t = 0:3; 0:5 and 0:7:
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(1+4Rd/3)θ'(0)
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Fig 8.E¤ects of Brownian motion parameter N b on (a) Nusselt and (b) Sherwood number for various values of Eckert number Ec = 0:00; 0:01 and 0:02:
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f rol os t n o e c icle Actaivnopart n
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Fig 9:.E¤ects of Brownian motion parameter N b on (a) Nusselt and (b) Sherwood
Nt=0.3 Nt=0.5 Nt=0.7 Nt=0.3 Nt=0.5 Nt=0.7
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number for various values of Eckert number Rd = 0; 1 and 2:
0.6
0
0.8
(b)
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Fig 10.E¤ects of Brownian motion parameter N b on (a) Nusselt and (b) Sherwood number for various values of Eckert number N t = 0:3; 0:5 and 0:7:
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Fig. 11: Variation of isotherms for various
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!" = 20 0.270602 0.296133 0.298608 0.299544
Present Study !" = 1 !" = 10 0.270420 0.270521 0.296130 0.296131 0.298612 0.298677 0.299598 0.299512
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Sheikholeslami and Ganji [5] !" = 0.5 !" = 1 !" = 10 0.27041 0.270421 0.270507 0.296127 0.296127 0.296130 0.298606 0.298607 0.298607 0.299543 0.299543 0.299543
!" = 0.5 0.270420 0.296131 0.298610 0.299548
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↓ 0 3 6 12
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Table 1: Comparison table for various values of radiation parameter and Schmidt number on Nusselt number when = 0.5, = 0.01, = 0.1, = 0.2, ! = 10 and in the absence of zero flux o nanoparticles. !" = 20 0.270612 0.296132 0.298654 0.299538
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Table 2: Variation of Nusselt number (1 + 4/3)$′(0), when &' = 2, ! = 0.5, *" = 0.02, !" = 0.5 and = 1.
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-: ↓ 0.3 0.6 0.9
Active control of nanoparticles -; = 0.3 -; = 0.5 -; = 0.7 3.215093 3.457153 3.710134 3.581995 3.840261 4.109018 3.972927 4.246715 4.530456
Passive control of nanoparticles -; = 0.3 -; = 0.5 -; = 0.7 2.555036 2.554189 2.553343 2.555036 2.554189 2.553343 2.555036 2.554189 2.553343
Active control of nanoparticles = 1 = 2 = 3 2.973548 4.293644 5.620765 3.329252 4.629376 5.947042 3.70932 4.981083 6.28515
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-: ↓ 0.3 0.6 0.9
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Table 3: Variation of Nusselt number (1 + 4/3)$′(0), when &' = 2, ! = 0.5, -; = 0.2, *" = 0.01 and Sc = 0.5. Passive control of nanoparticles = 1 = 2 = 3 2.435778 3.769229 5.102617 2.435778 3.769229 5.102617 2.435778 3.769229 5.102617
Table 4: Variation of Nusselt number (1 + 4/3)$′(0), when &' = 2, ! = 0.5, -; = 0.8, = 1 and Sc = 0.5.
-: ↓ 0.3 0.6 0.9
Active control of nanoparticles *" = 0 *" = 0.01 *" = 0.02 3.579414 3.710026 3.840639 3.980239 4.113731 4.247225 4.403302 4.539618 4.675935
Passive control of nanoparticles *" = 0 *" = 0.01 *" = 0.02 2.313648 2.433284 2.55292 2.313648 2.433284 2.55292 2.313648 2.433284 2.55292
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Table 5: Variation of Sherwood number <′(0), when &' = 2, ! = 0.5, *" = 0.02, !" = 0.5 and = 1.
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-: ↓ 0.3 0.6 0.9
Passive control of nanoparticles -; = 0.3 -; = 0.5 -; = 0.7 -1.09502 -1.82442 -2.55334 -0.54751 -0.91221 -1.27667 -0.36501 -0.60814 -0.85111
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Active control of nanoparticles -; = 0.3 -; = 0.5 -; = 0.7 0.61922 0.195792 -0.37642 0.729197 0.459452 0.110896 0.762421 0.541774 0.265804
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Table 6: Variation of Sherwood number <′(0), when &' = 2, ! = 0.5, -; = 0.2, *" = 0.01 and Sc = 0.5. Passive control of nanoparticles = 1 = 2 = 3 -0.69594 -0.68531 -0.68035 -0.34797 -0.34266 -0.34017 -0.23198 -0.22844 -0.22678
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0.3 0.6 0.9
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-: ↓
Active control of nanoparticles = 1 = 2 = 3 0.813614 0.882304 0.913422 0.854114 0.908689 0.932988 0.865293 0.916516 0.938984
-: ↓
Active control of nanoparticles *" = 0 *" = 0.01 *" = 0.02 -0.42307 -0.57211 -0.72115 0.058053 -0.01812 -0.09428 0.209939 0.158084 0.106229
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0.3 0.6 0.9
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Table 7: Variation of Sherwood number<′(0), when &' = 2, ! = 0.5, -; = 0.8, = 1 and Sc = 0.5. Passive control of nanoparticles *" = 0 *" = 0.01 *" = 0.02 -2.64417 -2.7809 -2.91762 -1.32208 -1.39045 -1.45881 -0.88139 -0.92697 -0.97254
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Graphical Abstract:
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To avoid the gravitational settling, both active and passive control of nanoparticles is performed for nanofluid between squeezing channel. Isotherms plots for squeezing flow are plotted for radiation parameter.
0
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• Results are analyzed for both active and zero flux of nanoparticles at the surface.
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• Radiation and viscous dissipation effects are also incorporated within the nanofluid.
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• Isotherms behavior is also described for radiation parameter.
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