Electrical MHD Carreau nanofluid over porous oscillatory stretching surface with variable thermal conductivity: Applications of thermal extrusion system

Journal Pre-proof Electrical MHD Carreau nanofluid over porous oscillatory stretching surface with variable thermal conductivity: Applications of thermal extrusion system Sami Ullah Khan, Sabir Ali Shehzad

PII: DOI: Reference:

S0378-4371(20)30001-7 https://doi.org/10.1016/j.physa.2020.124132 PHYSA 124132

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Physica A

Received date : 17 May 2019 Revised date : 6 December 2019 Please cite this article as: S.U. Khan and S.A. Shehzad, Electrical MHD Carreau nanofluid over porous oscillatory stretching surface with variable thermal conductivity: Applications of thermal extrusion system, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2020.124132. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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.

© 2020 Published by Elsevier B.V.

Journal Pre-proof Highlights (for review)

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The problem is composed by utilizing the electrical MHD and radiation aspects. An incompressible flow of Carreau nanofluid is considered. Flow is induced from periodically oscillation of the moving surface. The thermophoretic force and Brownian movement are also accounted. The results governed by present study can be more effective for enhancement of thermal system.

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   

Journal Pre-proof *Manuscript Click here to view linked References

Electrical MHD Carreau nanofluid over porous oscillatory stretching surface with variable thermal conductivity: Applications of thermal extrusion system

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Sami Ullah Khana,* and Sabir Ali Shehzada Department of Mathematics, COMSATS University Islamabad, Sahiwal 57000, Pakistan *

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Corresponding author: [email protected]

Abstract: Present contribution is the theoretical application of the thermal extrusion system associated with the various manufacturing processes. The problem is composed by utilizing the

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electrical MHD and radiation aspects on flow of Carreau nanofluid. Keeping industrial and mechanical applications in mind, the thermophoretic force and Brownian movement are also accounted. Here, flow is induced from periodically oscillation of the moving surface. The thermal system governed the partial differential equations of momentum, temperature and concentration distributions. These partial differential equations are non-dimensionalized first and

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then treat analytically. Accuracy of solution is carefully confirmed. The involved flow

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constraints are constructed as function of material constraints, electrical parameter, combined parameter, thermophoretic constraint, convection parameters, effective Prandtl and Schmidt numbers, thermophoresis parameter and Brownian parameter. The results governed by present study can be more effective for enhancement of thermal system. We believed that the present

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flow model in presence of nanoparticles can be useful to improve the efficiency of thermal energy extrusion system.

Keywords: Carreau nanofluid; electric effects; thermal radiation; variable thermal conductivity; oscillatory stretching surface

1. Introduction 1

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Nowadays, the study of non-Newtonian fluids has dragged a remarkable attention of the scholars due to their highly complex and interdisciplinary nature. The frequent use of these fluids in

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chemical, mechanical and food industries, petroleum drilling, paper production, wire coating etc., many investigations are re-vamped to examine the diverse rheological behavior of such

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materials. The non-Newtonian fluids are classified by introducing the different constitutive models due to their diverse nature. Among these, Carreau fluid is one which successfully pretends the shear thinning and thickening aspects. For example, the numerical investigation on radiative flow employing Carreau fluid has been illustrated by Reddy and Sandeep [1]. Hayat et

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al. [2] thrash out the rheological behavior of Carreau fluid flow induced by the stretched sheet. Irfan et al. [3] employed shooting technique to simulate the heat transfer phenomenon in steadystate flow of Carreau fluid. In their analysis, generalized Fourier’s law and variable thermal conductivity have been also executed. Khan et al. [4] performed numerical computations based on Keller Box method to analyze the rheology of Carreau fluid model. They observed that the

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presence of Weissenberg number deliberated the fluid motion while power law index causes

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amplification of particles velocity. Hsiao [5] conducted an interesting analysis of Carreaunanofluid flow under activation energy and pointed out that the obtained simulations are useful to enhance the efficiency of various manufacturing systems. The homogenous and heterogeneous reactions in stretching flow of Carreau fluid are reported by Khan et al. [6]. The numerical

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imitation of Carreau fluid flow has been analyzed by Khan and Hashim [7]. Kumar et al. [8] implemented Maxwell-Cattaneo model to investigate the transportation of heat in melting surface carrying Carreau fluid. The slip flow of Carreau fluid through porous medium is examined by Shah et al. [9]. The dynamic of two famous non-Newtonian models namely, Casson-Carreau fluids appeared over moving surface is addressed by Raju et al. [10]. 2

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In modern world, the low thermal conductivity materials are not proffered in distinct industrial and technological systems due to low convective heat transportation. However, the involution in

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nanotechnology introduces the various metallic or nonmetallic particles. As variety of working fluids can be considered in mechanical processes, the interaction of nanofluids becomes

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promising solution to enhance the heat transfer rates with changing the desired geometry and structure of the system. The diameter of nanofluids lies between 1 to 100nm. The augmentation of heat transport is based on the interaction of nanofluids may effectively contributed in imperative cooling problems and thermal energy storage systems. The primary idea of nanofluid

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was suggested by Choi [11] in 1995. This fascinating idea leads the investigators to present abnormal characteristics of nanofluids. Buongiorno [12] explained the two important mechanisms known as thermophoretic force and Brownian movement which are major factors in the improvement of heat transportation ability of ordinary fluids. Zhu et al. [13] performed a theatrical investigation on flow of Cu- Al2O3-water based nanofluid. Babu and Sandeep [14]

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employed Runge-Kutta method to explore the slip flow of Cu and CuO nanoparticles immersed

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in base fluid. The non-uniform heat absorption and generation features in nanofluid flow are elaborated by Raju et al. [15]. The flow problem involving finite thin film phenomenon under Brownian movement has been numerically investigated by Jing et al. [16]. The enhancement of heat transfer in slip flow of Casson fluid under condition of Newtonian heating has been

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visualized by Ullah et al. [17]. The characteristics of water- Fe3O4 nanofluid through Lorentz field is described by Sheikholeslami et al. [18]. The bio-convected nanofluid behavior is numerically addressed by Shen et al. [19]. Abbasi et al. [20] described the features of peristaltic nanofluid flow under Ohmic dissipation. Some more recent works on the flow of nanofluid with different configurations have been executed in the references [21-28]. 3

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There are numerous applications of fluid flow through saturated porous media in the area of petroleum engineering, food industries, hydrology and biological systems. These effects are

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further encountered in oil production, bio films, biological membranes, fossil fuels, thin glass bead packs etc. Referred to this subject, the initial work on this topic was suggested Darcy [29]

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and later on many interesting continuations have been made in this direction [30-33]. In all above aforementioned attempts, the investigators have contributed by considering the stretching surfaces. However, in some circumstances, the flow is not induced by the surface that not only stretched but also oscillates periodically. This interesting phenomenon is the time

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dependent flow for which the various features of fluid motion can be studied at various times. This idea was elaborated by Wang [34] and later on some attempts have been made by scholars. The phenomenal work of Wang is extended by Abbas et al. [35] by using second grade fluid. Both analytical and numerical solutions are provided and compared with each other. The flow induced by periodically oscillating and moving surface for viscous fluid in existence of thermo-

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diffusion effects is elucidated by Zheng et al. [36]. On the same systemic geometry, Ali et al.

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[37] reported the heat transportation in flow of Jeffrey fluid. A numerical based continuation for flow of second grade fluid induced by convective heated oscillatory surface has been declared by Khan et al. [38]. Abdullah et al. [39] discovered the thermal radiation role in Eyring-Powell fluid flow induced by the oscillation of moving surface.

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As a fundamental contribution towards the extrusion of thermal system manufacturing process and heat transportation augmentation, we considered the Carreau nanofluid flow in presence of combined electrical and MHD field effects. In fact, current work is the extension of Abbas et al. [35] in three directions. Firstly, by considering Carreau fluid, secondly by including heat and mass transportation with nanoparticles and lastly by taking variable thermal conductivity and 4

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thermal radiation features. This proposed model is quite useful to enhance the efficiency of manufacturing system. The problem is formulated in form of PDE’s with various flow features

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under boundary layer assumptions. After truncated the dimensionless variables, we utilized the homotopic algorithm to compute the solutions. The impact of control parameters is studied

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graphically.

2. Physical model and novelty

A schematic representation of unsteady mixed convection flow of Carreau nanofluid confined by

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periodically moving thermal forming sheet is presented in Fig. 1. The applications of such thermal extrusion surface may be attributed in many mechanical industries. The metal sheet, plastic sheet, food sheet are the most fascinating examples of this phenomenon. The present structure aims to improve and enhance the efficiency of thermal energy transportation systems. The extrusion sheet is extensively used in various industrial products, as extrusion process commonly contains extruded objects like various polymers, foodstuffs, ceramics etc. However,

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in the extrusion process, the extrusion sheet is assumed to be hot thin surface which need to be

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cool down by using traditional fluid. In order to achieve maximum efficiency and enhance the production process in present attempt, we have exploited the non-Newtonian fluid with some more useful factors. Based on these flow parametric assumptions and parameters, the competence of thermal system can be progressed.

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3. Flow equations

The flow problem consists of periodically stretching surface under the influence of two parallel and equal forces. The velocity of surface is represented as u  bx sin t ,  being frequency. In the coordinate system,  u, v  being the velocity components along  x , y  direction, respectively. 5

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The considered Carreau nanofluid is incompressible and electrically conducting. Some physical effects like mixed convection, electric field, radiation, porous medium and variable conductivity

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have been also included. The surface temperature of convectively heated surface is represented by Tw while Cw is surface concentration. Following to these assumptions, the governing

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mathematical model is represented as [5]:

u v   0, x y

(1)

u u u  2u 3  n  1 2  2u  u    B02   B0 E0   u  v   2   2   u u   u  g 1 T  T   g  2  C  C  , (2) t x y y 2 y  y  f f k

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2

 C T D  T 2   T  T T T 1  1 qr u v  K T    T ,   1  DB   t x y   c  f y y y T  y     c  f y   y 

(3)

C C C  2C DT  2T u v  DB 2  . t x y y T y 2

(4)

assumed as:

u  0,

v  0, T  Tw , C  Cw ; y  0, t  0,

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u  u  bx sin t ,

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Following the geometry of the problem, boundary conditions for the present flow domain are

T  T , C  C ;

y  .

(5) (6)

where  is time constant,   indicates the electrical conductivity,  f

represents the fluid

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density, E0 electric field, k  denotes the permeability parameter, g gravity, 1 and  2 denotes coefficient of thermal and mass diffusion coefficients, respectively,  is porous medium, T is temperature, K T  stands for thermal conductivity, qr is the radiative heat flux,   c  f is fluid density, C is concentration, DB denotes the Brownian diffusion coefficients,  1    c  p /   c  f 6

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is the ratio of heat capacitance of nanoparticles to heat capacitance of fluid and DT represents the thermophoretic diffusion coefficient. The energy equation (3) for thermal conductivity is

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modified as T  T   K T   K  1   , T  

(7)

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where K  ambient liquid conductivity while  is the parameter for thermal dependence conductivity. In present study, we consider flowing appropriate variables to reduce the

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independent variables in the governing equations [26-30]:

u  bxf y  y,  , v    b f  y,  , y 

 ( y , ) 

T  T C  C ,  ( y , )  . Tw  T C w  C

b



y ,   t,

(8)

(9)

The interaction of above quantities in (1-5), following dimensionless form are obtained 3  n  1We f yy f yy2   f y  Gr  Gc  Ea  0, 2

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f yyy  Sf y  f y2  ff yy 

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2 2  1  Rd        1    yy    y     S  f  y  Nb y y  Nt  y    0,  Pr 

 yy  S ( Sc)  Scf  y 

Nt  yy  0. Nb

(10)

(11)

(12)

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The governing boundary conditions are

f y  0,   sin  , f  0,   0, (0, )  1, (0, )  1,

(13)

f y  ,   0,  (, )  0,  (, )  0,

(14)

7

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where We   2b 2 x 2 / is the material parameter, S   / b the ratio of oscillating frequency to stretched rate,     B02 /  f b  / k b the combined Hartmann number and porosity parameter

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(combined parameter), Ee    B0 E0 /  f b electric parameter, Gr  g 1 Tw  T  / b2 x and

Gc  g 2  Cw  C  / b2 x the mixed convection parameters, Pr   /  f the Prandtl number,

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Nt  1DT Tw  T  / T the thermophoresis constraint, Sc   / DB the Schmidt number and Nb   1 DB  C f  C  / the Brownian motion parameter.

1    yy    y 

2

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By using one parametric approach for linearized thermal radiation, Eq. (11) can be written as 2  Preff  S  f  y  Nb y y  Nt  y    0,  

(15)

where Preff  1  Rd  / Pr combined parameter of Prandtl number and thermal radiation and termed as effective Prandtl number.

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Key physical quantities to examine the heat and mass transportation rates are named as local Nusselt and Sherwood numbers, respectively, can be justified in following mathematical forms

xqs xjs , Shx  , k Tw  T  DB  Cw  C 

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Nu x 

(16)

 T   C  qs   k   , js   DB   .  y  y 0  y  y 0

After utilization of Eqs. (8-9) in (16), flowing transmuted equations are obtained

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Nux Shx   y  0,  ,   y  0,  , Re x Re x

(17)

where Nu x and Shx signify the effective local Nusselt and Sherwood numbers, respectively.

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4. Solution methodology The system of dimensionless flow Eqs. (10), (12) and (15) with the boundary conditions (13) and

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(14) are tackled analytically by homotopic algorithms by using MATHEMTICA software. The homotopy analysis method is one analytical method which can be used to compute the series

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solution of many nonlinear problems with excellent convergence. This analytical technique is free of any small or large parameter constraints [40-47]. The convergence rate of solutions achieved by homotopic algorithm is strictly depend on suitable selection of auxiliary constraints

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namely h f , h and h . For this purpose, h  curves for quantities of interest are presented in Fig. 2 for specified values of emerging parameters. It is observed that suitable values for preeminent solution can be picked from 1.4  h f  0, 1.8  h  0.2, 1.9  h  0.1. 4.1. Validation of results

Before analyzing the results, we verified our solution by comparing it with already reported data.

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The obtained results are compared as a limiting case, with exact solution, suggested by Turkyilmazoglu [40] in Table 1. Table 1 reports that our results have good accuracy with the

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results of Turkyilmazoglu [48].

Table 1: Numerical data of f ''  0,  for case of linear stretching when We  Ee  H a  0 and



Turkyilmazoglu [48]

Present results

0

-1.000000

-1.000000

0.5

-1.22474487

-1.2247470

1

-1.41421356

-1.4142172

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   / 2.

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1.5

-1.58113883

-1.581147

2.0

-1.73205081

-1.7320575

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5. Analysis of results This section quantify the significance of various flow parameters like material constraint We ,

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electric parameter Ee , mixed convected parameters Gc , Gr , effective parameter Preff , thermophoresis constraint Nt , variable thermal conductivity  , Brownian movement Nb and Schmidt number Sc on velocity f y , temperature  and nano-particles volume-fraction  . In

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order to vary the associated parameter, all other parameters accomplished some constant values like We  0.5, Ee  0.5, H a  0.5, Gc  0.5,   0.2, Preff  0.7, Gr  0.5, Nt  0.5, Nb  0.5 and Sc  0.5.

The flow of various parameters like material parameter We , combined parameter H a , electric

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parameter Ea and mixed convection parameter Gr is illustrated in term of velocity profile f y with time  Fig. 3(a-d). The response of We against velocity distribution f y is scrutinized in

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Fig. 3(a). The periodic motion reached at maximum range as We varies. In fact, the fluid near the surface oscillates due to no slip boundary conditions. The effects of combined parameter H a on f y are referred in Fig. 3(b). A rapidly decay in velocity is attributed by increasing H a .

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Physically combined parameter H a is associated with both magnetic field and porosity effects. The interaction of magnetic force results a Lorentz force which resists the movement of fluid particles in the whole flow domain. Similarly, presence of porous medium also decays the velocity distribution due to permeability of porous medium. To enumerate the velocity for distinct values of electric parameter Ea , Fig. 3(c) is dispatched. An approximately in phase flow 10

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is observed by increasing distance from the surface. Whilst a rapid increasing trend is noted by varying Ea . Fig. 3(d) portrayed the impact of convected parameters Gr on velocity. The

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amplitude of the velocity periodically oscillates and increases gradually as Gr varies. The physical consequence of such trend is attributed as Gr entails buoyancy forces which respond as

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an increment of velocity distribution.

Fig. 4(a-d) is prepared to analyze the variation of velocity f y with y at fixed time instants

  8.5 for specific values of material parameter We, combined parameter H a , electric

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parameter Ea , convected parameter Gr and power law index n. The implication of We on f y at

  8.5 is analyzed in Fig. 4(a). An increment in We results low viscosity of fluid and as result the velocity enhanced. The graphical explanation of for distinct H a is explained in Fig. 4(b). Being combination of both magnetic force and porosity parameter, the velocity profile decreases

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as H a increases. The consistent behavior of convected parameter Gr on f y is concealed in Fig. 4(c). The distribution of velocity altered due to buoyancy forces and gradually increases by

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varying Gr . The impact of power law index n on f y is illustrated in Fig. 4(d). The velocity profile enhanced by as n  0,1, 2,3. It is remarked that n  1 represents the viscous fluid while fluid behaves non-Newtonian behavior for rising values of n which leads to enhancement of

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velocity. A relatively thinner boundary layer is truncated for shear thickening fluid. The behavior of temperature profile  and concentration profile  for variation of We is utilizes in Fig. 5(a-b). A decrement in temperature and concentration profiles is accompanied by varying We . A thinner trend of thermal and concentration boundary layer is noted. The evaluation of

temperature and concentration distributions with increasing electric parameter Ea is examined in 11

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Fig. 6(a-b). The presence of electrical field reduces the temperature of nanoparticles which is more effective to cool down the desired object after extrusion process. Similarly, mass transfer

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efficiency is also reduced for Ea . The importance of n on  and  is analyzed in Fig. 7(a-b). It is remarked that n  0 represents the case of shear thinning while n  1 is associated with

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viscous fluid while n  1 stands for shear thickening case. The temperature and concentration are strongly depressed for shear thickening case as compared to viscous fluid and shear thinning case. In order to under the role of Gr on  and  , Fig. 8 is dispatched. Fig. 8(a) confirmed that,

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as Gr increases, the curve of both temperature and concentration profiles decreases. However, alter in  is marginal as compared to  . Fig. 9(a) yield the significance of most important parameter namely, thermophoresis parameter N t on temperature distribution  . In fact, thermophoresis phenomenon is associated with the movement of nanoparticles due to temperature distribution difference. The temperature of nanoparticles efficiently boosts up as

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thermophoresis parameter increases. We observed in Fig. 9(b) that, for given values of Nt , the

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concentration profile reached at peak. Fig. 10(a-b) depicts the consequence of N b on  and  . The increasing values of N b results an increment in nanoparticles temperature. The physical aspect of such trend as justified as, when N b is varied, the particles of fluids moves freely and unsystematic way due to which more heat is engendered and subsequently a raise in the

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temperature of nanoparticles is examined. Therefore, N b can play an effective role in various manufacturing processes. The heat transfer phenomenon can make better by selection appropriate values of Nb. The output of N b on concentration distribution is considered in Fig. 10b. The concentration profile decreases by increasing N b . Mathematically, N b is materialized 12

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in 1/ Nb form. The physical consequences of such increasing trend can be justified as appearance of N b in the energy and concentration equation is due to the involvement of nanofluid which

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termed to enhancing the Brownian motion. The implication of most important parameter Preff on

 is suggested in Fig. 11a. This parameter is the combination of Prandtl number and radiation

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parameter. Moreover, Preff is directly related to Pr which means that larger values of Preff results larger Pr. Further, Prandtl number measures the ratio of thermal conductivity to thermal diffusivity. The appropriate Preff values are fruitful to cool down the thermal extrusion system.

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The graph of  for variable thermal conductivity parameter  is shown in Fig. 11(b). The variation in  brings an enhancement of temperature of nanoparticles gradually. The fluid conductivity of nanoparticles upsurge as  increases. As a result, more heat is exchanges from the sheet to the fluid particles and as a result the temperature of nanoparticles enhanced. Therefore, it is concluded that variable thermal conductivity results enchantment of fluid

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temperature and ultimately improve the associated boundary layer.

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The impact of distinct flow constraints on  y  y,  and y  y,  are directed in Table 2. It is observed that effective Prandtl number enhanced the dimensionless local Nusselt number and local Sherwood number. These dimensionless quantities decrease by increasing combined parameter material parameter. Larger values of mixed convection parameter enhanced these

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dimensionless quantities; however, the rate of change is minimal. With increase of thermophoresis number, effective local Nusselt number decreases but local Sherwood number opposite behavior.

6. Conclusions 13

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The physical significance of flow of Carreau nanofluid with thermal electrical field, MHD, radiation, mixed convection and thermal radiation effects is studied. The nanofluid is specified

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by thermophoresis and Brownian motions aspects. The problem is formulated and confined effects of various parameters are underlined. We important observations are digested as follows.

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The analytical results reported here are compared with other results and found decent match. In case of linear thermal radiation, the radiation parameter can be merged in Prandtl number and expressed as a combination, known as effective Prandtl number.

The presence of both magnetic field and porous medium slow down the movement of particles.

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



The heat transportation rate can be enhanced by increasing N b and N t .



An increment in variable thermal conductivity yield high conduction process.



The larger values of concentration diffusion may be accomplished by varying

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The heat transportation process can be effectively enhanced by utilizing nanoparticles.

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

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thermophoresis parameter.

Fig 1: Schematic representation of flow problem 14

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Fig. 2: h  curves for profiles of velocity, temperature and concentration

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Fig. 3: Velocity with time for (a) We (b) H a (c) Ee (d) Gr .

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Fig. 4: Graph of f y for (a) We (b) H a , (c) Gr (d) n

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Fig. 5: Graph of  and  for We

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Fig. 6: Graph of  and  for Ee

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Fig. 7: Graph of  and  for n

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Fig. 8: Graph of  and  for Gr

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Fig. 9: Graph of  and  for N t

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Fig. 10: Graph of  and  for N b

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Fig. 11: Graph of  for (a) Preff (b) 

Table 2: The illustration of effective local Nusselt number and local Sherwood number with S  0.5, n  2, Gc  0.2,   0.1.

We

0.0

Ee

0.2

Preff

Nt

Nb

 y  y, 

y  y, 

0.7

0.1

0.1

0.86487

0.73332

0.86522

0.73341

0.875437

0.743465

0.864879

0.733322

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*Declaration of Interest Statement

Journal Pre-proof Dear Editor,

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The authors have declared that they have no conflict of interest for this submission.