A 3D Sisko fluid flow with Cattaneo-Christov heat flux model and heterogeneous-homogeneous reactions: A numerical study

    A 3D Sisko fluid flow with Cattaneo-Christov heat flux model and heterogeneous-homogeneous reactions: A numerical study M. Khan, L. Ahmad, W.A. Khan, A.S. Alshomrani, A.K. Alzahrani, M.S. Alghamdi PII: DOI: Reference:

S0167-7322(17)30374-4 doi:10.1016/j.molliq.2017.04.059 MOLLIQ 7215

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

27 January 2017 11 March 2017 15 April 2017

Please cite this article as: M. Khan, L. Ahmad, W.A. Khan, A.S. Alshomrani, A.K. Alzahrani, M.S. Alghamdi, A 3D Sisko fluid flow with Cattaneo-Christov heat flux model and heterogeneous-homogeneous reactions: A numerical study, Journal of Molecular Liquids (2017), doi:10.1016/j.molliq.2017.04.059

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ACCEPTED MANUSCRIPT A 3D Sisko fluid flow with Cattaneo-Christov heat flux model and heterogeneous-homogeneous reactions: A numerical study M. Khana , L. Ahmada,b,1 , W.A. Khana , A.S. Alshomranic , A.K. Alzahranic and M.S. Alghamdid a

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Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan Department of Computer Science, Shaheed Benazir Bhutto University, Sheringal Upper Dir, Pakistan c Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia d Department of Mathematics, Faculty of Sciences, Jazan University, Saudi Arabia

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Abstract: In this paper, an analysis is made for the steady 3D flow of Sisko fluid over a bidi-

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rectional stretching sheet. Heat transfer analysis is illustrated while utilizing the effect of CattaneoChristov heat flux model. Besides these the impact of homogeneous-heterogeneous chemical reactions on the fluid flow is investigated. Results of the numerical computations are described in terms of the temperature and concentration plots. Different behaviors of temperature and concentration distributions for shear-thinning and shear-thickening cases are reported. The mathematical formulation consisting of nonlinear governing equations and boundary conditions is first casted into dimensionless form by suitable transformations. Interpretation of various emerging parameter is given through groups. The results indicate that a rise in the thermal relaxation parameter demonstrate a decline in the fluid temperature and thermal boundary layer thickness.

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Keywords: 3D Sisko fluid, variable thermal conductivity, Cattaneo-Christov model, homogeneous-heterogeneous reactions.

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Introduction

Most of the text deals with the flow of viscous fluid described with the help of classical Newtonian model. Majority of the fluids in industry do not hold frequently established assumption of a linear relationship between the stress and the rate of strain and, thus are characterized as non-Newtonian fluids. There are rheological complex fluids such as biological fluids, lubricating oils, polymeric liquids, paints, drilling mud, liquid crystals, etc., which have viscoelastic behavior and cannot be described merely as Newtonian fluids. As compared with Newtonian fluid flows, the constitutive behavior of non-Newtonian fluids flow is generally more complex and extremely non-linear, which may bring more difficulties in using numerical methods [1] to study such flows. Because of the reliance of consistency on shear rate, exact and effective numerical strategies are required for assessment of speed angles, which might be substantial and have sharp changes in a few conditions. Numerous chemical and biochemical procedures like gibberellic corrosive creation, and bikaverin generation [2, 3] are completed in a gooey medium that for the most part contains impetuses or biomass indicating non-Newtonian conduct. Moreover, some recent work has been carried out to explore the non-Newtonian fluid flows like [4-7]. In addition, for example the addition 1

Corresponding author: E-mail: [email protected]

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of polymers with lubricating oils to get better their viscosity index and make them less temperature reliant when such oils demonstrate non-Newtonian behavior [8]. Non-Newtonian fluid flows are frequently encountered in many physical and industrial processes [9], for example, ethanol generation from lignocelluloses [10]. The constrained convection of nonNewtonian nanofluids in a micro tube under the consistent temperature boundary condition was introduced by Hojati et al. [11]. It is for the reason that of such broad reporting in the study of lubricants simultaneously with its mathematical effortlessness that the Sisko fluid model [12] has been preferred for appliance in the current study. The study of such fluids has gained very much interest in most recent years because of their plenteous technological and industrial applications. Such fluids are frequently referred to as non-Newtonian fluids. Classic non-Newtonian flow individuality includes shear-thinning, shear-thickening, viscoelasticity and so onward. For the flows of non-Newtonian fluids there is not a solitary model that portrays the greater part of their properties as is accomplished for the Newtonian fluid. The flows of such fluids can be characterized with the assistance of a power-law model. However, for integer values of the power index, Siddiqui et al. [13] deliberate the problem for the first time. They considered the free surface condition at x = d (where d is the film thickness). However, the authors used a boundary condition in their solution which holds only for viscous fluids. Asghar et al. [14] obtained exact analytical solutions of the same problem with accurate free surface condition for numeral values of the power index. Wang et al. [15] illustrated the numerical solutions of an electrically conducting Sisko fluid flow by employing the lubrication approach. Unsteady flow of Sisko fluid instigated by an all of a sudden stimulated plate is tended to by Abelman et al. [16]. Molati et al. [17] demonstrated the unsteady unidirectional Sisko fluid flow under transversely connected magnetic field. The impact of suction and injection on unsteady flow of Sisko fluid is elucidated by Hayat et al. [18]. Recently Hayat et al. [19] addressed the flow of the Sisko with nanofluid with magnetic field property. Be that as it may, now notwithstanding consistency, another parameter, specifically the power-law list (or type) is utilized to portray the flow of such fluid can be handle with the help of a power-law model [20]. To predict the attributes of non-Newtonian fluids that are utilized as a part of oils, a power law fluid model is utilized. This model represents together with dilatants and pseudoplastic fluids relying upon their shear thickening and shear thinning properties. Some of its industrial applications include cement slurries, drilling fluids, waterborne coatings,and most pseudoplastic fluids. Sisko fluid model is a more generalized up variant of the power law model. This representation of the model carries both behaviors that is gooey and power law model. The marvel of heat transfer has across the board modern and biomedical applications such as cooling of electronic gadgets, atomic reactor cooling, control era, heat conduction intissues and numerous others. The heat flux demonstrates proposed by Fourier [21] has been the best demonstrate for comprehension heat transfer mechanism in various circumstances. One of the restrictions of this model is that it regularly prompts to an illustrative vitality condition which demonstrates that beginning unsettling influence is right away experienced by the medium under thought. Both the intellectual and technological interest of heat transfer at the molecular stage is perceptive. Many of the researchers like [22-27] addressed the 2

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heat phenomenon and thermal conductivity in their study. Truly, a finite time is included in heat transfer in solids, which happens through impacts between vitality bearer’s free electrons and phonons, furthermore, by dispersing of these carries at boundaries and material defects. The finite speed of heat transfer was perceived by many including Chester [28], Joseph and Preziosi [29] in their essential paper on ”heat waves”. The gratis electrons are dominant carriers in metals while the Phonons are foremost heat carriers for insulators and semiconductors as presented in [30]. Further, Waqas et al. [31] addressed the impact of Cattaneo-Christov heat flux model for flow of variable thermal conductivity with generalized Burgers fluid flow. Malik et al. [32] explored the heat flux model for Sisko fluid flow past a permeable non-linearly stretching cylinder. Khan et al. [33] illustrated the numerical computations with Fourier’s and Fick’s laws for the Sisko fluid flow. Fu et al [34] illustrated the heat transference in a sandwich panel with a fractured foam core. Daneshjou et al. [35] reported the non-Fourier heat transfer analysis of infinite two dimensional orthotropic FG hollow cylinders subjected to time dependent heat cause. Recently Liu et al. [36] analyzed heat conduction with fractional Cattaneo-Christov upper-convective derivative flux model. Very recently, Hayat et al. [37] elucidated about the thermally stratified stretching flow with Cattaneo–Christov heat flux. Chemical reactions are a necessary piece of innovation, of culture, and in fact of life itself. Blazing fills, refining iron, making glass and stoneware, blending brew, and making wine and cheddar are amongs numerous cases of exercises consolidating chemical reactions that have been known and utilized for a large number of years. The physical change is absolutely distinguished from chemical reaction. Physical changes incorporate changes of state, for example, ice dissolving to water and water dissipating to vapor. In the event that a physical change happens, the physical properties of a substance will change, however its chemical identity will continue as before. Accordingly, analysis and hypothesis, the two foundations of chemical science in the current world, together characterized the idea of chemical reactions. Chemical reactions are further classified as homogeneous and heterogeneous reactions. Homogeneous reaction is chemical reaction in which the reactants and products are in a similar stage, while heterogeneous reaction has reactants in at least two stages. Reactions that occur on the surface of an impetus of an alternate phase are additionally heterogeneous. The collaborations between the homogeneous and heterogeneous reactions are exceptionally extremely difficult and have gotten increasing consideration, but it is still ambiguous. Pizza et al. [38, 39] showed the capability of catalytically-coated walls in moderating characteristic fire hazards of meso/smaller scale channels. Additionally, Hayat et al. [40, 41] have done a lot of work related to chemical reactions. Khan et al. [42] also reported the outcomes for chemically reactive aspect in flow of tangent hyperbolic material. Wang et al. [43] reported about the reaction intensity between catalytic and non-catalytic combustors. It has been shown that these associations incorporate the advancement of homogeneous reaction due to chemically incited exothermicity and the control of heterogeneous reaction on the homogeneous reaction for the most part brought about by rivalry of energizes, what’s more, oxidizers of the heterogeneous reactions versus homogeneous reactions is illustrated by Li et al. [44]. The outcomes demonstrated that the catalytic combustor showed a high stability and powerless reaction intensity. Yasmeen et al. [45] examined the homogeneous-heterogeneous reactions 3

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of Ferrofluid flow by a stretched surface in the incidence of magnetic dipole. In this study, they were found a significant behavior of the fluid flow like the dminsinhing of velocity, concentration and Nusselt number. On the other hand the temperature and skin friction were found in the enhancing conduct in their studies while using different increasing values of the embedded parameters. Moreover, Khan et al. [46] illustrated the characteristics of homogenous-heterogeneous processes in the three dimensional Burgers fluid flow. The motivation for taking into account Sisko fluid flow in the current paper is that the Sisko fluid can express much typical distinctiveness of Newtonian and non-Newtonian fluids by choosing different material parameters. In view of the above discussion, the intention of this examination is to focus on the impact of heat with chemical reaction in Sisko fluid flow. The problem is initially modeled and the formulation of the non-dimensional leading equations is performed. The steady three-dimensional Sisko fluid flow with Cattaneo-Christov heat flux model is considered. The homogeneous-heterogeneous chemical reactions is also utilized to figure out the solution of the problem. A literature outline exposed that no past accomplishments were completed to concentrate this problem for non-integer value of the power index n. The solution of the non-dimensional governing equations with associated boundary conditions is addressed while using Matlab package bvp4c which uses the Collocation method.

Problem formulation

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We consider steady 3D laminar boundary layer flow of an incompressible Sisko fluid past a bidirectional stretching sheet as illustrated through figure 1. The sheet is assumed to be stretched along x− and y − axes with velocities u = cx and v = dy, respectively, where c, d > 0 are the stretching rates and flow takes place in domain z > 0. Let Tf and T∞ are the temperatures of the hot and ambient fluids, respectively, with Tf > T∞ . The temperature at the stretching surface takes the constant value Tw . Heat transfer scrutiny is conceded in the presence of thermal diffusion with relaxation of heat flux. Moreover, the heat transfer mechanism is investigated in the incidence of temperature dependent thermal conductivity. Further, the present study is carried out while taking the effect of homogeneous-heterogeneous reactions. For cubic autocatalysis the homogeneous reaction is expressed as follows: rate = k1∗ a1 b21 ,

A + 2B → 3B,

(1)

while introduced the first order isothermal reaction on the surface of the catalyst is as A → B,

rate = k2∗ a1 ,

(2)

where a1 and b1 represent the concentrations of chemical species A and B, respectively, and (k1∗ , k2∗ ) are the rate of chemical species constants. Here it is understood that such types of reactions processes are isothermal and far-away from the sheet in the ambient fluid, there is a consistent concentration a0 of reactant A and there is no autocatalyst B. Thus continuity, momentum, energy and concentration equations governing the steady threedimensional Sisko fluid flow [47] under the usual boundary layer approximations are expressed as follows: 4

ACCEPTED MANUSCRIPT ∂u ∂v ∂w + + = 0, ∂x ∂y ∂z

(3) (4)

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 n ∂u ∂u ∂u ∂u a ∂ 2u b ∂ − , u +v +w = − ∂x ∂y ∂z ρf ∂z 2 ρf ∂z ∂z  n−1 ∂v ∂v ∂v ∂u a ∂ 2v b ∂ ∂v u +v +w = − , + ∂x ∂y ∂z ρf ∂z 2 ρf ∂z ∂z ∂z  2 2 2 ∂2T u2 ∂∂xT2 + v 2 ∂∂yT2 + w2 ∂∂zT2 + 2uv ∂x∂y    ∂T ∂T ∂T ∂2T ∂2T ∂u ∂u ∂u ∂T  +2vw + 2uw + u + v + w u +v +w −δ E  ∂y∂z ∂x∂z ∂y ∂z ∂x     ∂x ∂x ∂y ∂z ∂v ∂v ∂v ∂T ∂w ∂w ∂w ∂T + u ∂x + v ∂y + w ∂z ∂y + u ∂x + v ∂y + w ∂z ∂z  ∂ 2 a1 − k1∗ a1 b21 , ∂z 2  2  ∂b1 ∂b1 ∂b1 ∂ b1 + k1∗ a1 b21 , u +v +w = DB 2 ∂x ∂y ∂z ∂z



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∂a1 ∂a1 ∂a1 u +v +w = DA ∂x ∂y ∂z

v → 0,

T → T∞ ,

a 1 → a0 ,

b1 → 0 as z → ∞,

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 k(T ) ∂ 2 T = ,  ρcp ∂z 2 (6)

(7) (8)

(9) (10)

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u → 0,

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u = Uw (x) = cx, v = Vw (y) = dy, w = 0, T = Tw ,

∂b1 1 DA ∂a = k = −k2 a1 at z = 0, a , D 1 1 B ∂z ∂z

(5)

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where (u, v, w) represent the components of velocity along x−, y−and z-directions, respectively, (a, b, n) the material constants of the Sisko fluid, T the temperature, δ E the ratio diffusion coefficient, ρf the density of the base fluid with cf the specific heat of fluid at constant temperature and (DA , DB ) the diffusion species coefficients of A and B, respectively. The thermal conductivity of the fluid is assumed to vary linearly with temperature in form of    T − T∞ k(T ) = k∞ 1 + ε , (11) Tf − T∞ where k∞ denotes the thermal conductivity of the fluid far away from the sheet surface and ε a small parameter known as the thermal conductivity parameter. We now use the following dimensionless variables:

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u = cxf (η) ,

θ (η) =

T − T∞ , T f − T∞

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v = dyg (η),

a1 = a0 φ (η) ,

w = −c

cn−2 ρf b

b1 = a0 h (η) ,

1 ! n+1 

 n−1 1−n 0 2n f+ ηf + g x n+1 , n+1 1+n 1 ! n+1 1−n c2−n η=z x 1+n . (12) b ρf

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f (0) = 0,

f 0 (0) = 1,

g(0) = 0,

g 0 → 0,

g 0 (0) = α,

θ(0) = 1,

φ0 (0) = k2 φ(0),

θ → 0, φ → 1 as η → ∞.

(17) (18)

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f 0 → 0,

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Making use of the transformations (12), Eq. (3) is identically satisfied and Eqs. (4) to (10), having in mind Eq (11), lead to the following forms   2n 2 000 00 n−1 000 f f 00 − (f 0 ) + gf 00 = 0, Af + n (−f ) f + (13) n+1   2n 2 000 00 n−1 000 00 000 00 n−2 g − (n − 1)g f (−f ) + (14) f g 00 − (g 0 ) + gg 00 = 0, Ag + (−f ) n+1 "
2n 0
0 #   2n 0 f + g f + g θ 2n 00 0 2 n+1 n+1
f θ 0 +Pr gθ 0 = 0, (15) +Pr (1+εθ)θ +ε(θ ) −Pr λE 2 00 2n n + 1 + n+1 f + g θ   2n 00 f + g φ0 − Sck1 (1 − φ)2 φ = 0, (16) φ + Sc n+1

Here in Eq. (16) we assumed that the diffusion coefficients of the chemical species A and B are of comparable size such that

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φ(η) + h(η) = 1.

(19)

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In the above equations, prime denotes the differentiation with respect to η. Moreover, A is the material parameter of the Sisko fluid, Re a and Reb denote the local Reynolds numbers, Pr the generalized Prandtl number, α the stretching ratio parameter, Sc the Schmidt number, λE the relaxation time of the heat flux, k1 measures the strength of homogeneous reaction and k2 the strength of heterogeneous reaction. These parameters are stated as follows: 2

2 Uw xρf Uw2−n xn ρf (ρc)f xUw − n+1 Rebn+1 , Reb = , Pr = , Rea = Reb , A = Rea a b k∞ 2 Uw δ E a2 k ∗ x k ∗ cx − 1 DA Sc = Rebn+1 , λE = , k1 = 0 1 , k2 = 2 Reb n+1 . Uw DA z Uw x x

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d α= , c (20)

Solution methodology

The nonlinear ordinary differential equations (13) to (16) along with the boundary conditions as given in Eqs. (17) to (18) are considered for the numerical solution. The bvp4c Matlab package is implemented to find the numerical results. According to the major requirements of this numerical technique, the main steps of the method are given as follows. Let f = y1, f 0 = y10 = y2 , f 00 = y20 = y3 , (21) g = y4,

g 0 = y40 = y5 ,

g 00 = y50 = y6 ,

(22)

θ = y7 ,

θ 0 = y70 = y8 ,

(23)

φ = y9 ,

φ0 = y90 = y10 ,

(24)

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2n y 1 y3 − y4 y 3 y22 − n+1 f = yy1 = , A + n(−y3 )n−1
2n 2 y1 y6 − y4 y6 + (n − 1)y6 yy1 (−y3 )n−2 y − 5 n+1 000 , g = yy2 = A + (−y3 )n−1  2n
2n

2n y1 + y4 n+1 y2 + y5 y8 − Pr n+1 y1 y8 − Pr y4 y8 − εy82 Pr λE n+1 00 h i , θ = yy3 =
2 2n 1 + εy7 + n+1 y1 + y4   2n 00 2 0 y1 + y4 y10 φ = yy4 = Sck1 (1 − y9 ) y9 − Sc , n+1

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000

(25) (26) (27)

(28)

and the relevant boundary conditions then becomes y2 (0) = 1,

y4 (0) = 0,

y5 (0) = α,

y10 (0) = k2 y9 (0),

y9 (∞) = 1.

(29) (30) (31) (32)

Numerical results and discussion

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y5 (∞) = 0,

y7 (∞) = 0,

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y7 (0) = 1,

y2 (∞) = 0,

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y1 (0) = 0,

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Numerical solutions to the nonlinear differential equations (13) to (16) with the boundary conditions (17) to (18) are obtained by using the Matlab routine bvp4c that uses Collocation method. The numerical results obtained are used to carry out a parametric study showing the influence of governing parameters on the temperature and concentration profiles. These include the power-law index n, Sisko fluid parameter A, thermal conductivity parameter ε, relaxation time parameter λE , homogeneous reaction parameter k1 and heterogeneous reaction parameter k2 . Impact of the power law index n on the temperature and concentration distributions are plotted through figures 2 and 3 for shear thinning as well as shear thickening fluids. The effect in both cases of shear thinning and shear thickening are observed which are of significance importance. From these figures, it is noted that with the increasing of the power law index the temperature profile is reported to be decline and the associated boundary layer thickness also reduced. While, the impact of power-law index on the temperature distribution is more outstanding for the shear thinning fluid as compared to the shear thickening fluid. Meanwhile, with the enhancement of power law index an increase is observed in the concentration profile and the associated boundary layer thickness (see figure 3). The outcome got for shear thinning is more conspicuous as contrast with shear thickening fluids. Figure 4 is plotted to see the effect of the Sisko fluid parameter A on temperature. It shows that with the enhancement of Sisko fluid parameter the temperature profile and the associated boundary layer thickness are reduced. In the shear thinning case, the result is prominent than shear thickening case. The Sisko fluid parameter describes the high viscosities at low shear rates and low viscosities at high shear rates. An increment in Sisko fluid parameter corresponds to a low viscosity at high shear rate which has caused a decline 7

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in the temperature profile and associated thermal boundary layer thickness. Figure 5 is depicted which showing an enhancement conduct of the concentration profile and the related boundary layer while varying the values of the Sisko fluid parameter in both instance of shear thinning and shear thickening fluids. Through figure 6 the enhancement of temperature profile is reported for larger values of thermal conductivity parameter ε. This increase is a direct result of thermal conductivity of the fluids for higher values of the small scalar parameter ε arised in the variable thermal conductivity. In addition, more heat is transferred from sheet to the liquid and eventually the temperature dispersion is expanded. The temperature profile of the Sisko fluid flow is seen to decrease with the addition of thermal relaxation parameter λE as depicted in figure 7. In physical sense, additional time is important for the heat transfer for molecule to molecule of the liquid. In the outcome, the temperature profile and the boundary layer are diminished for both shear thinning and shear thickening fluids. Moreover, for λE = 0, that is for traditional Fourier’s law, where the temperature is higher when contrasted with the Cattaneo-Christov model. This is because of heat transfer through material right away. Figure 8 outlines the impact of the homogeneous reaction parameter k1 on the concentration profile. It is revealed from these plots that the concentration profile and the related boundary layer are diminished for increasing values of k1 . It may be a result of the way that the reaction rates lead dissemination coefficients. A similar conduct is seen in figure 9 for various values of heterogeneous reaction parameter k2 , wherein this figure is plotted for both cases, that is, shear thinning as well as shear thickening fluids.

Conclusions

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In this study, we have examined nearly and completely the attributes of heat transfer mechanism in the boundary layer of Sisko fluid flow over a bidirectional stretching sheet within the sight of heterogeneous-homogeneous chemical reactions. Additionally, we have studied the characteristics of Cattaneo-Christove heat flux model in steady 3D flow of Sisko fluid. The numerical computations were performed by using the Matlab function bvp4c that uses the Collocation method. The major conclusions are listed below: • The temperature and related thermal boundary layer were escalating with increasing thermal conductivity parameter for both shear thinning and shear thickening fluids. However the effects of the thermal relaxation parameter were quite the opposite to those of thermal conductivity parameter. • The temperature distribution was reported to be higher in the case of Fourier’s law as compare to Cattaneo-Christov heat flux model. • The concentration distribution was reduced with the enhancement of the homogeneous and heterogeneous reactions strength parameters in both cases of shear thinning and shear thickening fluids. Acknowledgement: 8

ACCEPTED MANUSCRIPT The authors acknowledge with thanks the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah for technical and financial support.

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[47] A. Munir, A. Shahzad and M. Khan, Convective flow of Sisko fluid over a bidirectional stretching surface, PLOS ONE, 10 (6) (2015): e0130342.

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Figure captions Figure 1: Schematic diagram of the problem. Figure 2: Temperatur profile θ(η) for different values of the power-law index n. Figure 3: Concentrtion profile φ(η) for different values of the power-law index n. Figure 4: Temperature profile θ(η) for different values of the Sisko parameter A. Figure 5: Concentrtion profile φ(η) for different values of the Sisko parameter A. Figure 6: Temperature profile θ(η) for different values of the thermal conductivity parameter ε. Figure 7: Temperature profile θ(η) for different values of the relaxation time parameter λE . Figure 8: Concentrtion profile φ(η) for different values of the homogeneous parameter k1 . Figure 9: Concentrtion profile φ(η) for different values of the heterogeneous parameter k2 .

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ACCEPTED MANUSCRIPT Highlights • The steady 3D flow of Sisko fluid over a bidirectional stretching sheet is investigated.

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• The heat transfer analysis is carried out in the presence of Cattaneo-Christov heat flux model.

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• The impact of homogeneous-heterogeneous chemical processes on the fluid flow is analyzed.