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Thermally radiative convective flow of magnetic nanomaterial: A revised model

A. Sohail, S.I.A. Shah, W.A. Khan, M. Khan

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S2211-3797(17)31000-8 http://dx.doi.org/10.1016/j-rinp.2017.07.011 RINP 784

Results in Physics

Please cite this article as: Sohail, A., Shah, S.I.A., Khan, W.A., Khan, M., Thermally radiative convective flow of magnetic nanomaterial: A revised model, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp. 2017.07.011

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Thermally radiative convective flow of magnetic nanomaterial: A

revised model

A. Sohail", S.I.A. Shah", W.A. Khan6'1 and M. Khan6

"Faculty of Numerical Sciences, Islamia College University, Peshawar, Pakistan ^Department of Mathematics, Quaid-i-Azam University, Islamabad 44000,

Pakistan

cDepartment of Sciences and Humanities, National University of Computer and Emerging

Sciences Islamabad 44000, Pakistan.

Abstract: The present paper endeavors to scrutinize the unsteady magnetohydrodynamic (MHD) second grade nanofluid over a porous stretching sheet in the presence of convective boundary and nanoparticles flux conditions. The influence of the thermal radiation is scrutinized by utilizing nonlinear Rosseland approximation. The self-similarity transformation is used to transfer the governing partial differential equations into the ordinary differential equations. The resulting problem under consideration is solved analytically by using the homotopy analysis method (HAM). The effect of non-dimensional parameters on the temperature, concentration and local Nusselt is discussed by using graphs and tables. It is perceived that the temperature of the second grade nanoliquid declines as unsteadiness Parameter enhances. Moreover, it is estimated from the plots that the concentration of the second grade nanoliquid drops as the Brownian motion parameter increases while the reverse trend is detected for the thermophoresis parameter.

Keywords: Second grade nanofluid; Nonlinear thermal radiation; Convective boundary conditions; Nanoparticle mass flux condition.

1 Introduction

During the most recent decades, a great deal of interest has been devoted to study the nanofluids owing to their many industrial importance, especially, in nanotechnology. Nanofluid is a suspension of solid nano particles or fibers of diameter 1-100 nm in a basic fluids such as water, oil and ethylene glycol. Nanofluids are generally used in industry because of the growing use of these smart fluids due to their enhanced thermal conductivity

1 Corresponding author: Electronic mail: waqar_qau85@yahoo.com

and convective heat transfer coefficient of the base fluid, which grows with increasing the volumetric fraction of nanoparticles. The addition of small particles of solids materials, such as (Cu, Ag, Au, Fe, Hg, Ti etc. metals and non metallic Al2O3, CuO, TiO2, SiO2) possessing high thermal conductivities to fluid that can enhance the thermal conductivity of the base fluid (such as oil, water and ethylene glycol). Moreover, nanofluids exhibit poor heat transfer rates as the thermal conductivities of such fluids are important in calculating the heat transfer coefficient. Due to better performance of heat exchange, nanofluids can be utilized in several industrial applications such as in transportation, chemical production, automotive, power generation in power plant and in nuclear system etc. (Choi et al. [1]). Owing to this, many researchers are persuaded and revolutionary work has been made by them. Rana and Bhargava [2] scrutinized numerically the flow induced by a flat plate stretching with the non-linear velocity in a nanofuid by using variational finite element method. The analysis of the slip effects on the boundary layer flow and heat transfer over a linearly stretching surface in the presence of nanoparticle fractions has been done by Noghrehabadi et al. [3]. Khan et al. [4] examined the three-dimensional flow of an Oldroyd-B nanofluid towards a stretching sheet with heat generation/absorption. Nield and Kuznetsov [5] scrutinized the Cheng-Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid. The steady flow of Burgers nanofluid over a stretching surface with heat generation/absorption was studied by Khan and Khan [6]. Hayat et al. [7] addressed the problem of mathematical study for three-dimensional boundary layer flow of Jeffrey nanofluid. Pal and Mondal [8] scrutinized the MHD convective stagnation-point flow of nanofluids over a non-isothermal stretching sheet with induced magnetic field. Mixed convection stagnation-point flow of nanofluids over a stretching/shrinking sheet in a porous medium with internal heat generation/absorption was studied by Pal and Mondal [9]. Khan and Khan [10] reported the forced convection analysis for generalized Burgers nanofluid flow over a stretching sheet. Some recent studies on nanoliquid may be found in the references

a stretc

impact magnetic field on the convective heat transfer is of much importance and plays a vital role in controlling heat transfer in manufacturing processes where the quality of the final product depends on heat control factors. Many equipments such as MHD generators, pumps, bearings and boundary layer control are affected by the interaction between the electrically conducting fluid and a magnetic field. Additionally, the behavior of the flow

strongly depends on the orientation and intensity of the applied magnetic field. The exerted magnetic field manipulates the suspended particles and rearranges their concentration in the fluid which strongly changes heat transfer characteristics of the flow. A magnetic nanofluid has both the liquid and magnetic characteristics. Such materials have fascinating applications like optical modulators, magneto-optical wavelength filters, nonlinear optical materials, optical switches, optical gratings etc. Magnetic particles have key role in the construction of loud speakers as sealing materials and in sink float separation. Magnetic nanofluids are useful to guide the particles up the blood stream to a tumor with magnets. This is due to the fact that the magnetic nanoparticles are regarded more adhesive to tumor cells than non-malignant cells. Such particles absorb more power than microparticles in alternating current magnetic fields tolerable in humans i.e. for cancer therapy. Numerous applications involving nanofluids include drug delivery, hyperthermia, contrast enhancement in magnetic resonance imaging and magnetic cell separation. Motivated by all the aforementioned facts, various scientists and engineers are engaged in the steady of MHD flow of nanofluids. Aziz et al. [29] studied MHD flow over an inclined radiating plate with temperature dependent thermal conductivity, variable reactive index and heat generation. Matin et al. [30] presented the MHD mixed convective flow of a nanofluid over a stretching sheet. Zeeshan et al. [31] examined the MHD flow of a third grade nanofluid between coaxial porous porous cylinders. Hayat et al. [32] examined the Cattaneo-Christov heat flux in MHD flow of Oldroyd-B fluid with homogeneous-heterogeneousreactions. Salahuddin et al. [33] studied MHD flow of Cattanneo-Christov heat flux model for Williamson fluid over a stretching sheet with variable thickness. Mahanthesh et al. [34] examined the impact of magneto nanoparticle on prescribed surface heat flux boundary over a bidirectional non-linear stretched surface. Waqas et al. [35] studied the MHD mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition. Raju and Sandeep [36] inspected the characteristics of Opens Heat and mass transfer on MHD non-Newtonian bio-convection flow over a rotating cone/plate with cross diffusion. Hayat et al. [37] scrutinized the features of Carreau fluid in presence of chemical processes

In view of all the above mentioned applications, the purpose of the present work is to investigate effects of MHD on unsteady second grade nanofluid with non-linear thermal radiation and new mass flux condition. The homotopy analysis method (HAM) [38-41] is used to obtain the analytical solutions of the problem. The effects of different emerging

parameters are demonstrated through their graphical representations and discussed in detail.

2 Problems formulation

Let us study the unsteady forced convective two-dimensional MHD flow of an incompressible second grade nanofluid past a stretching surface with velocity Uw(x,t) = , where a and c are constants and x the coordinate measured along the stretching surface. A transverse magnetic field of strength B(t) = ^j^ct is applied parallel to the y — axis, where B0 is the constant applied magnetic field. Heat transfer analysis is carried out in the presence of non-linear thermal radiation. The surface is heated by the convection from a hot fluid temperature Tf which has a linear variation with the x — axis and an inverse square law for its decrease with the time, while the temperature of the ambient cold fluid is T^. Moreover, it is also assumed that the ambient concentration is constant C^. Unsteady conservation of mass, momentum, thermal energy and nanoparticles equations for a nanofluid can be written in Cartesian coordinates as:

du dv o dx dy '

du du du d2u ai dt dx dy dy2 Pf

a u i u d u i au a u dtdy2 dxdy2 dx dy2

du d2i

I uud_v I v d-u dy dy2 dy3

aB2(t)u Pf

udT dT _ k dqr

dt dx dy (pc)f dy2 (pc) f dy

dödT Dt (dT

dy dy T^ y dy

dC dC dC _ d^ö DL d^L

dt dx dy B dy2 T^ dy2'

dT dC D dT

u _ Uw' v _ Vw' - k— _ h [Tf - T]' Db+ dT_ 0 at y _ 0,

u — 0, T — Too, C — Coo as y —y to,

dy T^ dy

where Vw is the mass transfer at surface with Vw < 0 for suction and Vw > 0 for injection and are expressed by

ACCEPTED MANUSCRIPT

Vï-ct '

where v0 denotes a uniform suction/injection velocity.

The radiative heat flux expression in Eq. (3) is given by the Rosseland approximation as

16aT3 dT 3x dy '

where a is the Stefan-Boltzmann constant and x the mean absorption coefficie: Consequently, the energy (3) becomes

dT dT dT k d2T 16kaT3 82

— + u— + v— =--+-

dt dx dy (pc) f dy2 3x (pc)

Introducing the dimensionless velocities, temperature, concentration and variable n as :

0 (n) :

In perspective of Eq. (10) problem:

v (1 - ct)'

— (6) are reduced to the following boundary value

f+ ff'' — f'2+a 2f'f— f"2 — fr+a ( 2/'" + 2fiv

(2f' + 2 f ) — M2f' = 0,

[{1 + Rd (1 + (0f — 1)0)3}0']' + Pr (f0 — f 0) — Pr A (20 + 20') + Pr (Nb4>'0' + Nt0'2) =0,

00' + Pr Le (f 0' — f » + N 0" — Pr LeA (20 + 2 0') = 0

f = S, f ' = 1, 9' (0) = -Y [1 - 9 (0)] , N^'(0) + Nte'(0) = 0 at n = 0, (14) f ^ 0, 9 ^ 0,0 ^ 0 as n (15)

where A (= is the unsteadiness parameter, S \ = -¡Vaj source the mass transfer parameter with S > 0 for suction and S < 0 for inject, a\ the second grade dimensionless

^ ^^^ ^^^^^^^ naramíilfir Ñ - I —

parameter, M2 apf) the magnetic parameter, 9f f J the temperature ratio parameter, Pr (= a) the Prandtl number, Rd the radiation parameter, Nb = (tDbcC'x' ) the Brownian motion parameter, Nt ( = tDt(Tf the thermophoresis parameter and Le

\ 1 ^a J

Dob) the Lewis number.

~ , , of flow is the lo

From physical point of view, the important characteristics of flow is the local Nusselt number Nux which can be characterized as

x fdT\ NUx = - (Tf~T^){ -d-y)

+, ■, (16)

y=0 k (Tf - T~) ,

The above dimensionless variables reduce the above quantities in the following form

Re-2 Nux = - [1 + Rd{1 + (9f - 1)9 (0)}3] 9' (0), (17)

in which Re = ax2/v is the local Reynolds number.

3 The analytic series solutions

The governing non-linear ordinary differential Eqs. (11) - (13) with the boundary conditions (14) and (15) are solved analytically by utilizing the homotopy analysis method (HAM). For such analytic solution corresponding to the heat and mass transfers, the appropriate initial guesses and linear operators are chosen as follows:

fo(n) = S +1 - e-n, 9o(n) = e-n, 0o(n) =--e-Nn, (18)

- +1 -+1

^ £f [f (n)] = f''' - f', £ [9(n)] = 9" - 9', £+ [0(n)] = 0' - 0. (19)

4 Graphical results and discussion

The analytically calculated results for the temperature and concentration fields are displayed graphically, in order to get definite perception of the problem, for different values

of the unsteadiness parameter A, Biot number 7, Prandtl number Pr, radiation parameter Rd, Brownian motion parameter Nb, thermophoresis parameter Nt and Lewis number Le appearing in the above problem.

Figures 1(a,b) portray the effect of the unsteadiness parameter A and Biot number 7 on the temperature profile. These figures highlight the fact that the temperature and the associated thermal boundary layer thickness diminishe for enlarging values of the unsteadiness parameter while the opposite behavior is observed for the Biot number. Physically, the Biot number represents the ratio of internal thermal resistance of a solid to boundary layer thermal resistance.

The influence of the Prandtl number Pr and radiation parameter Rd on the temperature distribution is elucidated through figures 2(a,b). It is demonstrated through figure 2(a) that the temperature and the associated thermal boundary layer thickness is decreasing functions of the Prandtl number. This happens because of the way that the fluids with higher Prandtl number have low thermal conductivity which reduces the conduction and hence the thermal boundary layer thickness. Moreover, an observation of figure 2(b) makes it clear that the temperature distribution and the associated thermal boundary layer thickness enhances for large values of the radiation parameter. This is due to fact that the surface heat flux increases under the impact of thermal radiation which results in increasing temperature inside the boundary layer region.

Figures 3(a,b) delineate the variation of the temperature in response to a change in the values of the Brownian motion parameter Nb and the thermophoresis parameter Nt, respectively. It is worth noting from figure 3(a) that the temperature profile and the associated thermal boundary layer thickness enhance for the enlarging values of the Brownian motion parameter. Physically, by enlarging the values of the Brownian motion parameter Nb the random motion of the particles increases which results in an enhancement in the temperature profile and associated thermal boundary layer thickness. Moreover, it is also observed from figure 3(b) that the temperature profile and associated thermal boundary layer thickness enhance for the enlarging values of the thermophoresis parameter. Physically, with an increase of the thermophoresis parameter, difference between the wall temperature and the reference temperature enhances which increases the temperature profileand the associated thermal boundary layer thickness. It is further observed from these figures that the effects of the thermophoresis parameter on the temperature profiles are more dominant when compared

with the Brownian motion parameter

TED MANUSCRIPT

Figures 4(a,b) indicate the impact of the Biot number 7 and Prandtl number Pr on the concentration field. It is revealed through these figures that the magnitude of the nondi-mensional concentration profile diminishes with the increase of the Biot number and Prandtl number while the associated concentration boundary layer thickness also enhances.

Figures 5(a,b) portray the influence of the Brownian motion parameter Nb and ther-mophoresis parameter Nt on the nondimensional concentration field. It is illustrated through figure 5(a) that the concentration profile and the associated boundary layer thickness enhance with the increase of the Brownian motion parameter. Physically, the higher values of the Brownian motion parameter stifle the diffusion of nanoparticles into the fluid regime away from the surface which as a result enhances the nanoparticle concentration in the boundary layer. Furthermore, it is also noticed that the concentration profile and associated boundary layer thickness diminishe with the increase of the thermophoresis parameter. Physically, the thermophoresis force increments with the increase in the thermophoresis parameter which tends to move nanoparticles from hot to cold areas and hence diminishes the magnitude of nanoparticle volume fraction profile.

Table 1 is presented for the numerical values of the local Nusselt number for different values of M, Pr, Nb, Le and Nt. From table it is noticed that magnitude of local Nusselt number enhances for for large val ues of M and decreases for Le.

Table 1: Numerical values of the reduced Nusselt number for different values of physical

parameters.

My Pr Nb Nt Le Re-2 Nux

0.5 1.0 1.5

0.2 0.4 0.6

0.9 1.3 1.7

1.305714 1.594711 2.01419 2.520191 2.520191 2.520191 2.520189 2.520189 2.520

0.1 0.6 0.9

520180 2.52019 2.520

191 .520191 .520191 2.520191 0.9 2.52019

0.9 2.52019 1.3 2.520189 1.7 2.520189

5 Summary and conclusions

The effects of thermal radiation on unsteady MHD second grade nanofluid in the presence of convective and new mass boundary conditions were explored analytically. In the energy equation the radiative heat flux term was introduced by using the nonlinear Rosseland approximation. It is worth mentioning that rather than the linear Rosseland diffusion approximation, when the nonlinear Rosseland diffusion approximation was utilized the problem was also governed by the temperature ratio parameter by which the temperature and heat transfer characteristics were highly affected. The important findings of our analysis are listed below:

The temperature distribution and associated thermal boundary layer thickness were enhanced for the higher values of the Brownian motion parameter Nb and thermophoresis parameter Nt.

An increase in the radiation parameter showed an enhancement in the fluid temperature.

The temperature and associated thermal boundary layer thickness were reduced for the higher values of the unsteadiness parameter A.

• The concentration distribution and assciated concentration boundary layer thickness were increasing functions of the Biot number 7.

• It was anticipated that the concentration distribution was decrease for the ther-mophoretic parameter.

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Figures caption

Figures 1: Variation of temperature field 9 (n) with n for several values of the unsteadiness parameter A (panel-a) and Biot number y (panel-b).

Figures 2: Variation of temperature field 9 (n) with n for several values of the Prandtl number for Pr (panel-a) and radiation parameter Rd (panel-b).

Figures 3: Variation of temperature field 9 (n) with rq for several values of the Brownian motion parameter Nb (panel-a) and thermophoresis parameter Nt (panel-b).

Figures 4: Variation of concentration field 0 (n) with rq for several values of the Biot number y (panel-a) and Prandtl number Pr (panel-b).

Figures 5: Variation of concentration field 0 (n) with rq for several values of the Brownian motion parameter Nb (panel-a) and thermophoresis parameter Nt (panel-b"1