21589, Saudi Arabia. Keywords: 3D Sisko liquid, generalized heat flux and mass diffusion relations, temperature dependent thermal conductivity.

Transcription

1 Non-Fourier s and Fick s laws applications in flow of temperature dependent thermal conductivity generalized Newtonian liquids: A 3D computational study Masood Khan a, Latif Ahmad a,,, Waqar Azeem Khan a,c and Ali Saleh Alshomrani d a Department of Mathematics, Quaid-i-Azam University, Islamaad 44, Pakistan Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal Upper Dir 8, Pakistan c Department of Mathematics and Statistics, Hazara University, Mansehra 23, Pakistan d Department of Mathematics, Faculty of Science, King Adulaziz University, Jeddah 2589, Saudi Araia Astract: The current article presents a numerical study for local solutions of Sisko liquid flow y idirectional stretched surface. Additionally, the analysis of relaxation times for the heat and mass transfer mechanisms are made y utilizing modified heat flux and mass diffusion models. These improved relations are the generalized form of Fourier s and Fick s laws in which the time space upper-convected derivative are employed to portray the heat conduction and mass diffusion mechanisms. Appropriate transformations lead to a strongly nonlinear differential system of equations which is then solved numerically y employing the applications of vp4c package in Matla software. Another numerical method namely shooting technique with RK45 Fehlerg and Newton- Raphson method is utilized to authenticate the results. Graphical illustrations demonstrating the impacts of sundry physical parameters with required discussion highlighting their physical effect are also a part of this exploration. It is perceived that the temperature and concentration of Sisko liquid are diminishing functions of relaxation times for the heat and mass transfer mechanisms. It is also fascinating to perceive that the temperature and concentration of Sisko liquid are higher in case of classical form in comparison with an improved constitutive relations. Keywords: 3D Sisko liquid, generalized heat flux and mass diffusion relations, temperature dependent thermal conductivity. Introduction Heat conduction and mass diffusion are road mechanisms in nature and have gained prodigious interest from the researchers and engineers owing to their extensive range of applications in industrial and engineering like nuclear reactors cooling, heat conduction in tissues and electronic devices, power generation etc. Characteristics of heat and mass transfer phenomena have een explored adequately y utilizing the Fourier s and Fick s laws for the heat conduction and mass diffusion, respectively, in recent years. Several researchers presented the work related to flow and heat and mass transfer. In this regard, Liao and Pop [] discussed explicitly an analytical elucidation for similarity oundary layer equations. Alam and Ahammad [2] addressed the features of variale heat and mass transfer and fluxes over an inclined stretching sheet. Isaa et al. [3] considered exponential variation of ounded permeale shrinking sheet during the oundary layer flow of Casson fluid with magnetic and Corresponding author latifahmad@math.qau.edu.pk

2 mixed convection effects. Ramly et al. [4] investigated the impacts of thermal radiation on oundary layer flow over a radially stretching surface with zero and non-zero fluxes. Khan [5] examined a series solution of visco-elastic fluid flow over shrinking/stretching surface with magnetic field effect. Furthermore, many researchers [6 8] given their attention toward such types work to fill the gap. However, the aforementioned studies consisting with a drawack that these classical laws contradict the principle of causality and produce paraolic energy and concentration equations. The propagation speed of heat and mass disturance are finite y introducing the thermal and concentration relaxation times in the Fourier s and Fick s models. Cattaneo [9] revised the Fourier s law y using relaxation time dependent term in the energy equation to evade the paradox of heat conduction appliance. It is estimated that the addition of time dependent relaxation in the energy equation convert the paraolic equation into hyperolic energy equation due to which heat energy is transported with limited speed in the form of propagation of thermal waves. Furthermore, these waves have numerous applications in areas such as nanofluids and skin urns. Christov [] modified the Cattaneo relation which is efficiently preserves the material-invariant formulation. Ciarletta and Straughan [] demonstrated the uniqueness and staility for the modified heat conduction and mass diffusion relation. Hayat et al. [2] inspected the features of the developed heat conduction relation on an Oldroyd-B fluid y using chemical processes. They perceived from their oservations that the temperature of Sisko liquid falloff for oosted values of relaxation time parameter. Khan and Khan [3] analyzed the characteristics of 3D flow of Burgers liquid for developed heat conduction relation. They detected from their oservations that the temperature of the Burgers liquid is extensively affected for the augmented values of relaxation time parameter. Waqas et al. [4] investigated the characteristics of variale thermal conductivity and developed heat conduction relation for generalized Burgers liquid. Sui et al. [5] studied the impact of developed doule diffusion relations for the Maxwell nanofluid. The features of developed heat conduction relation on magneto viscoelastic liquid over a stretched surface is explored y Li et al. [6]. The effect of improved heat conduction and mass diffusion models on 3D Burgers fluid flow utilizing chemical processes has een investigated y Khan et al. [7]. Malik et al. [8] presented a numerical study of magneto-casson fluid flow with variale viscosity and Cattaneo-Christove heat flux model. Muhammad et al. [9] scrutinized the features of Cattaneo-Christove heat and mass flux models in squeezing nanofluid flow. The characteristics of non-fourier s and Fick s laws in the flow of Jeffery nanofluid flow over an inclined stretching surface is illustrated y Khan et al. [2]. The massive utility of nonlinear materials has sustantially involved the scientists and engineers in their study during the past few decades. Especially such nonlinear liquids are involved in paper production, oil reservoir engineering, geophysics, ioengineering, chemical and nuclear industries, polymer solution and cosmetic processes etc. As a consequence, different types of non-linear relations have een suggested according to the features of nonlinear liquids. Amongst these nonlinear materials, Sisko liquid relation is considered to predict the shear thinning and shear thickening properties of fluids. Munir et al. [2] scrutinized the characteristics of convective flow with heat conduction mechanism over a idirectional stretching surface for Sisko liquid flow. Khan et al. [22] explored the features of convectively 2

3 heated surface for the Sisko liquid. Malik et al. [23] inspected the features of random motion and thermophoresis for the Sisko liquid flow. Khan et al. [24] studied the numerical analysis of an improved heat conduction and mass diffusion models for flow of Sisko liquid. They concluded from their graphical oservations that the temperature and concentration profiles are significantly affected y the power-law index when it intensifies from n < to n >. The numerical investigation of radiative magneto-sisko nanofluid flow is reported y Khan et al. [25]. Furthermore, large numer of attempts were made to study aout such fluids flow, heat and mass transfer phenomena and many of them are addressed in [26 28]. To study the impact of thermal relaxation parameters on 3D steady Sisko liquid flow the Cattaneo-Christove model is utilized. Likewise, this fluid model is considered to incorporate the shear thinning < n < and shear thickening n > properties of fluids in the presence of generalized Fourier s and Fick s laws. Fourier [29] suggested a well-known law of heat conduction which is commonly used for heat conduction aspects from time it revealed up in literature. Later on this phenomenon is commonly recognized with the name of paradox of heat conduction. Throughout in the literature present study is never considered to explore the characteristics of thermal and concentration diffusions introducing y Cattaneo-Christov fluxes. However, the phenomena of heat and mass transfer arises when there exists a temperature and concentration differences etween the components of a similar ody. These phenomena have massive technological and industrial use, for example, in cooling of atomic reactors, microelectronics, pasteurization of food, fuel cells, power generation, energy production etc. This impact on the fluid flow is responsile for the temperature alance in the flow of fluid. Every flow parameter which shows a significant influence is studied through the graphs of temperature and concentration. For the modeling of internal temperature profiles, an accurate determination and distriution of heat source along with thermal conductivities of various components is needed. Due to this source, the cooling system is maintained in the external side of the attery or external side of the attery pack. In addition, enhanced thermal managing will increase the conception of neighorhood maturing instrument and might speedy superior attery outlines and upgraded lifetimes. At last, knowing interior temperature profiles at high current densities give the likelihood to anticipate and stay away from conditions for thermal escaped, which is a key welleing constraint for LIBs amid procedure at high existing densities. The second important influence of this physical property utilized in this work for the first time in the flow of 3D steady Sisko liquid. Regarding the aforesaid studies and their enormous applications, the intention of the current article is to intensely explore the impact of modified heat flux and mass diffusion models for 3D Sisko liquid flow. The prolem under consideration is modeled in the form of set of highly nonlinear partial differential equations PDEs which are then transformed into coupled nonlinear ordinary differential equations ODEs y utilizing appropriate transformations. The numerical results are achieved y utilizing the vp4c function in Matla. Additionally, graphical illustrations emphasizing various prominent parameters effects with 3

4 their physical importance are given. Nomenclature u, v, w velocity components a,, n physical quantities of Sisko liquid x, y, z cartesian coordinate system U w, V w stretching velocities S extra stress tensor nala V velocity field p fluid pressure T, C temperature and concentration of fluid ρ f, c f fluid density and specific heat α thermal diffusivity D diffusion coefficient T, C amient temperature and concentration of fluid c p specific heat at constant pressure f, g dimensionless stream functions ρc f heat capacity of ase fluid θ, ϕ dimensionless temperature and concentration dimensionless variale ε thermal conductivity of fluid k thermal conductivity of fluid at infinite distance ν kinematic viscosity c constant q, J normal heat and mass fluxes t time δ E, δ C relaxation times of heat and mass fluxes 2 Mathematical formulation Let us consider the 3D flow of Sisko liquid in the region z >. The flow is induced y a idirectional stretched surface with velocities u = cx and v = dy, where c and d are taken as constants as shown in Fig.. The velocity, temperature, concentration and stress fields are assumed to e dependent on x, y and z. The features of relaxation times of Heat conduction and mass diffusion are taken in energy equation to visualize impact of these time dependent terms on the energy and concentration involving Sisko liquid. At stretched surface temperature and concentration of Sisko liquid takes the value T w, C w while far away from stretched surface takes these values T, C, respectively. Under the overhead assumptions, the governing equations [5, 24] for Sisko liquid flow can e expressed as follows: divv =, ρ f V. V = p +.S, 2 4

5 Figure : Flow geometry for idirectional stretching surface. where ρc f V. T =.q, 3 V. C =.J, 4 n S = a + 2 tr A 2 A. 5 In Eq. 5, A = gradv + gradv is the first Rivilin-Ericksen tensor. The general form of modified Fourier s and Fick s laws considered in [5] and are written as: q+δ E q t J+δ C J t + V q q V + Vq = k T T, 6 + V J J V + VJ = D C, 7 where k T is the temperature dependent thermal conductivity and given as T T k T = k + ε. 8 T w T Eq. and 2 reduces in the form of Eq. 9 to. While energy and concentration conservation laws defined in Eqs. 3 and 4 will takes the form defined through Eqs. 2 and 3 y eliminating q from Eqs. 3 and 6 and J from Eqs. 4 and 7 and will takes the following form u x + v y + w z u u x + v u y + w u z = a ρ f 2 u z 2 ρ f 5 =, 9 u n, z z

6 u v x + v v y + w w z = a ρ f 2 v z 2 + u T x + v T y + w T z + δ E = ρc p f z u C x + v C y + w C z + δ C ρ f u 2 2 T x 2 u n v z z + v 2 2 T y 2 + 2vw 2 T x y + w 2 T z 2 +2uv 2 T + 2uv 2 T y z x z + u u + v u + w u T x y z x + u v x + v v + w v T y z y + u w + v w + w w T x y z z k T T z u 2 2 C x 2 z,, 2 + v 2 2 C y 2 + w 2 C z 2 +2uv 2 C + 2vw 2 C + 2uv 2 C x y y z x z + u u + v u + w u C x y z x + u v x + v v + w v C y z y + u w + v w + w w C x y z z The corresponding oundary conditions are stated as follows = D ρc f 2 C z 2. 3 u = U w = cx, v = V w = dy, w =, T = T w, and C = C w at z =, 4 u, v, T T, C C as z. 5 The final flow equations for such fluid model are otained while using suitale transformations as defined elow: [ c u = cxf, v = cyg n 2 n+ 2n, w = c n + f + n ] + n f + g x n n+, θ = T T T w T, ϕ = C C C w C, ρ f = z c 2 n ρ f n+ x n +n. 6 Using transformations defined in Eq.6, which identically satisfied Eq. 9 and Eqs. to 3 along with the imposed oundary conditions defined through 4 and 5 with respect to the dimensionless variale can e otained as follows: Ag + 2n Af + ff + n f n f + gf f 2 =, 7 n + 2n fg + f n g g 2 n g f f n 2 + gg =, 8 n + 6

7 2n + εθ θ + εθ 2 + Pr fθ + Pr gθ n + [ 2 ] 2n 2n 2n Pr λ E n + f + g θ + n + f + g n + f + g θ =, 9 [ 2 ] 2n 2n 2n ϕ Scλ C n + f + g ϕ + n + f + g n + f + g ϕ 2n +Sc fϕ + Scgϕ =, 2 n + f =, f =, g =, g = α, θ =, ϕ =, 2 f, g, θ, ϕ as. 22 Here all the flow parameters are defines as: Re a = Uwxρ f a 2 A = Re n+ Re a α = d c Pr = ρcpxuw k Re 2 2 Sc = xuwre n+ D λ E = Uwδ E x, Re = U w 2 n x n ρ f n+, λ C = Uwδ C x local Reynold numers material parameter of Sisko liquid stretching ratio parameter generalized Prandtl numer generalized Schmidt numer relaxation parameters 3 Physical quantities Physical quantities involved in the flow prolem have a special interest as presented in this illustration. These physical quantities are the resistive forces and the rate of heat and mass transfer and can e expressed in the form of local skin friction, local Nusselt and local Sherwood numers. Moreover, the local Nusselt and Sherwood numers can e otained in the asence of relaxation parameters λ E and λ C.These are further defined as: C fx = τ xz ρu, C 2 w 2 fy = τ yz ρu, 23 2 w 2 x T Nu x = T f T x C Sh x = C w C The aove expressions in dimensionless variales are given as, 24 z z=. 25 z z= 2 Re n+ C fx = Af f n, 26 7

8 2 Re n+ C fy = V w U w [ Ag + f n g ], 27 Re n+ Nux = θ, 28 Re n+ Shx = ϕ Numerical scheme The approach used for finding the computational results of the prolem is one of the wellknown collocation technique known as vp4c. If the computational results otained during each iteration process is close to the exact results, then such perception of closeness is stated as numerical staility. This closeness of the approximations is due to the use of symmetric points in each suinterval of the whole interval. Moreover, single solution rather than dual solutions generated y this technique will always e a more stale solution y keeping the step size smaller in the interval. The transformed prolem through Eqs. 7 to 2 along with the oundary conditions 2 and 22 are first converted into first order ODEs y using some new variales. In this work, the tolerance level is fixed up to 6. Main procedures of the aforementioned method are as follows. then x 6 = f = x, f = x 2, f = x 3, f = x 3, 3 g = x 4, g = x 5, g = x 6, g = x 6, 3 θ = x 7, θ = x 8, θ = x 8, 32 ϕ = x 9, ϕ = x, ϕ = x, 33 x 3 = 2n +n x x 3 + x 2 2 x 4 x 3 A + n x 3 n, 34 [ n x 3 x 3 n 2 ] 2n +n x x 4 x6 + x 2 5 x 8 = εx2 8 Pr [ ] [{ 2n x + x +n 4 x8 + Pr λ 2n E x = Scλ C [{ 2n +n A + x 3 n, 35 } { x + x 2n } ] +n 4 x2 + x +n 5 x8 { + εx 7 Pr λ 2n } 2, 36 E x + x +n 4 } ] [ x2 + x 5 x + Sc 2n ] +n x + x 4 x { Scλ 2n } 2, 37 C x + x +n 4 x + x 4 } { 2n +n with the associated oundary conditions x =, x 2 =, x 2 =, 38 x 4 =, x 5 = α, x 5 =, 39 x 7 =, x 9 =, 4 x 7 =, x 9 =. 4 8

9 5 Testing of code The graphical outcomes as shown in Figs. 2a, and 3a, otained during the implementation of two different techniques namely vp4c package and shooting technique with RK45 Fehlerg and Newton-Raphson method are in excellent agreement. In the asence of thermal relaxation times parameter, the resistive forces and the rat of heat and mass transfer are calculated with the help of vp4c, shooting technique an homotopy analysis method HAM as shown in Tales to 6 and the results are found in excellent correlation. Furthermore, the taular values as shown in Tale 7 is a more confidence to validated the present work in the form of rate of heat conduction with the result reported y [2]. 6 Results and discussion The emphasis of current research work is to inspect the feature of developed heat conduction and mass diffusion relations for Sisko liquid over a idirectional stretched surface. In this research work the Matla package vp4c is implemented to find the numerical solution of the prolem. Furthermore, the prime emphasis of the following deate is to investigate the impact of the governing physical parameters on the heat and mass transfer mechanisms for the Sisko liquid. Such parameters are power-law index n, material parameter A, stretching ratio parameter α, variale thermal conductivity ε, Prandtl numer Pr, thermal and concentration relaxation time parameters λ E, λ C and Schmidt numer Sc. 7 Temperature profile 7. Effect of n and ε on temperature profiles Figs. 4a, represents the impacts of n and ε on temperature profile of Sisko liquid flow. From these plots one may recognized that the temperature of Sisko liquid decays for the two cases, i.e., pseudoplatic as well as dilatant fluids with the increasing values of n. A significant result is noticed for pseudoplastic < n < liquid as compared to dilatant n > liquid. Then again, the temperature profile of the Sisko liquid enhances with the raising values of variale thermal conductivity. The numerical value ε = speaks that the Sisko liquid have constant conductivity. Besides, low temperature is detected for the constant thermal conductivity and higher thermal conductivity for variale thermal conductivity. Escalation in ε causes enhancement in temperature profile of the liquid along with the thermal oundary layer thickness. 7.2 Effect of A and α on temperature profiles The impact of material parameter A on temperature profile is depicting through Figs. 5a,. From these Figs. it is presumed that the temperature of the liquid is declining with associated thermal oundary layer thickness, while using the growing values of A and α. The power-law index in this plotting is considered y pseudoplastic < n < and dilatant n > liquids condition. From physical perspective the diminishing impact of 9

10 the temperature of the liquid is ecause of higher shear rate causing low viscosity and high viscosity causing low shear rate with the rising values of A. Another variation in parameter α demonstrates a critical physical outcome, i.e., at whatever point the stretching in the surface uilds, the separation etween the liquid particles are expanding and this is the reason that the temperature of the liquid aatements and thickness of thermal oundary layer is additionally decreasing. 7.3 Effect of λ E and Pr on temperature profiles Figs. 6a, have een plotted to exhiit the effect of the thermal relaxation parameter λ E and Prandtl numer Pr on the temperature profile. From these representations, it is accounted for that raise in λ E prompts decrease in the temperature profile and related thermal oundary layer thickness. Physically, this is a direct result of the way that fluid particles will required more opportunity for the exchange of heat to the closest adjoining particles of the liquid. The increase in λ E demonstrates a non-conductor property of the liquid which causes a diminishing in temperature profile. The growing values of Pr causes a diminishment, while plotting temperature profile and related thermal oundary layer thickness is also reduced. Physically, this is a direct result of modified rate of heat dissemination with the raising values of Pr. For the higher values of Prandtl numer the diffusivity of liquid is appeared small and as a result a reduction in temperature field curves depicted. 8 Concentration profile 8. Effect of n and A on concentration profiles The vacillation ecause of n and A in concentration profile and the related oundary layer thickness is plotted through Figs. 7a,. It is examined from these graphs that concentration and related concentration oundary layer thickness reduces as the values of n increasing. An extremely significant outcome is seen during the variation in power-law index n etween and. In this plotting a reducing conduct in the dimensionless concentration profile and related oundary layer thickness for particular values of A is illustrated. From physical viewpoint, this is a direct result of the low share rate with the higher viscosity of the liquid and accordingly low rate of mass diffusion is oserved. 8.2 Effect of Sc and λ C on concentration profiles For various higher values of Sc and λ C on concentration profile is shown through Figs. 8a,. A decreasing ehavior of concentration profile and related concentration oundary layer thickness is seen ecause of the rising values of Sc and λ C. The aforementioned Figs. are plotted while keeping n as indicated y the physical properties of power-law liquids, i.e., pseudoplastic and dilatant liquids. Physically, an immediate connection etween Sc and molecular diffusivity is found and at whatever point, the values of Sc uplifting, the increase in molecular diffusivity of the liquid arises and accordingly concentration profile decreases.

11 .9 a n =.,.3,.5,.7 n =.5, 2.5, 3.5, n =.5, n = θ.5 θ.5.3. A =., Sc =.7, α =.5, ε =, λ E = λ c =, Pr = ε =.,,, A =., Sc =.7, α =.5, Pr =., λ E = λ c = Figure 2: Comparison etween vp4c and shooting method..9 a n =.5, n =.5.9 n =.5 n = ε =, Sc =.7, ε =, Sc =.7, θ.5 α =.5, Pr =., θ.5 A =., Pr =., λ E = λ c =. λ E = λ c =..3 A =.,.5,.,.5.3 α =.,.3,, Figure 3: Comparison etween vp4c and shooting method..9 a n =.5 n =.5.9 n =.5 n = ε =, Sc =.7, ε =, Sc =.7, θ.5 A =., Pr =., θ.5 A =., λ E =, α =.5, λ c =. α =.5, λ c =..3.3 λ E =.,.,,.3 Pr =,.,.2, Figure 4: Impact of the power-law index and thermal conductivity parameter on temperature θ.

12 .9 a n =.,.3,.5,.7 n =.5, 2.5, 3.5, n =.5 n = φ.5.3. A =., Sc =.7, α =.5, ε =, λ E = λ c =, Pr = φ.5.3. A =.,.5,.,.5 ε =, Sc =.7, α =.5, Pr =., λ E = λ c = Figure 5: Impact of the material parameter and stretching ratio parameter on temperature θ..9 a n =.5 n =.5.9 n =.5 n = α =.5, Pr =., α =.5, Pr =., φ.5 ε =, A =., φ.5 ε =, A =., λ E = λ c =. λ E =, Sc =.7..3 Sc =.7,.9,.,.3.3 λ C =.7,.9,., Figure 6: Impact of the relaxation time parameter and Prandtl numer on temperature θ. α =.5, ε =, λ E = λ C =, Pr =., A =., Sc =.7. a n =.5 α =.5, ε =, λ E = λ C =, Pr =., A =., Sc =.7. n =.5 vp4c package Shooting method vp4c package Shooting method θ θ Figure 7: Impact of the power-law index and material parameter on concentration ϕ. 2

13 α =.5, ε =, λ E = λ C =, Pr =., A =., Sc =.7. a n=.5 α =.5, ε =, λ E = λ C =, Pr =., A =., Sc =.7. n =.5 vp4c package Shooting method vp4c package Shooting method φ φ Figure 8: Impact of the Schmidt numer and relaxation parameter on concentration ϕ. 3

14 Tale : A comparison for different values of parameters α and A, when λ E = λ C =, ε =, Pr =. and Sc =.7. A α Re n+ 2 C fx n =.5 vp4c results shooting results Re n+ 2 C fx n =.5 vp4c results shooting results Tale 2: A comparison for different values of parameters α and A when λ E = λ C =, ε =, Pr =. and Sc =.7. A α Re n+ U 2 C w fy n =.5 vp4c results shooting results Vw Re n+ U 2 C w fy n =.5 vp4c results shooting results Vw 4

15 Tale 3: A comparison for different values of parameters α, A, ε, Pr and Sc, when λ E = λ C =. α A ε Pr Re n+ Nu x n =.5 vp4c results shooting results Re n+ Nu x n =.5 vp4c results shooting results Tale 4: A comparison for different values of parameters α, A and Sc, when λ E = λ C =, ε = and Pr=.. α A Sc Re n+ Sh x n =.5 vp4c results shooting results Re n+ Sh x n =.5 vp4c results shooting results

16 Tale 5: A comparison for different values of parameters α, when λ E = λ C =, A =, ε =, Pr =. and Sc =.7. α Re n+ 2 C fx n = vp4c results HAM results Re n+ 2 C fy U w Vw n = vp4c results HAM results Tale 6: A comparison for different values of parameters ε and Sc, when λ E = λ C =, A =, α =.5 and Pr =.. Re n+ Nu x Re n+ Sh x ε n = Sc n = vp4c results HAM results vp4c results HAM results Tale 7: A comparison of the present work with the pulished data for different values of α, when λ E = λ C = Sc = ε =, A =.5 and Pr =.. α Re n+ Nu x n =.5 Present results Munir et al. [2] Re n+ Nu x n =.5 Present results Munir et al. [2] Concluding remarks In the present exploration, we have scrutinized the characteristics of oundary layer flow of 3D Sisko liquid over a idirectional stretched surface. Additionally, the impacts of Cattaneo- Christov heat and mass flux models were also considered here. The numerical technique namely Matla function vp4c was engaged to solve the modeled prolem. Moreover, another computational method namely shooting technique was implemented to figure out the results. We have extracted interesting insights regarding the influence of various physical parameters that govern the prolem. Significant findings of the prolem are: 6

17 The power-law index caused a reduction in the temperature distriution. The temperature distriution was declined with the larger value of thermal relaxation time. Temperature distriution was declining function of the generalized Prandtl numer. It was anticipated that the concentration profile diminished with an increase of the concentration relaxation time. A significant growing effect was noticed during the increasing values of variale thermal conductivity, while plotting temperature profile. Acknowledgements The authors acknowledge with thanks the Deanship of Scientific Research DSR at Kin Adulaziz University, Jeddah, Saudi Araia for technical and financial support. The authors would also like to thank referee for the valuale comments which enhanced the quality of the paper. References [] Liao, S.J., Pop.I. Explicit analytic solution for similarity oundary layer equations, Int. J. Heat Mass Transf., 47, pp [2] Alam, M.S. and Ahammad, M.U. Effects of variale chemical reaction and variale electric conductivity on free convective heat and mass transfer flow along an inclined stretching sheet with variale heat and mass fluxes under the influence of Dufour and Soret effects, Nonlinear Anal. Model. Control, 6, pp [3] Isaa, S.S.P.M., Arifin, N.M., Nazarc, R., Bachok, N., Ali, F.M. and Pop, I. MHD mixed convection oundary layer flow of a Casson fluid ounded y permeale shrinking sheet with exponential variation, Scientia Iranica B, 242, pp [4] Ramly, N.A., Sivasankaran, S. and Noor, N.F.M. Zero and nonzero normal fluxes of thermal radiative oundary layer flow of nano fluid over a radially stretched surface, Scientia Iranica B, 246, pp [5] Khan, Y. Magnetohydrodynamic flow of linear visco-elastic fluid model aove a shrinking/stretching sheet: A series solution, Scientia Iranica B, 245, pp [6] Niranjan, H., Sivasankaran, S. and Bhuvaneswari, M. Chemical reaction, Soret and Dufour effects on MHD mixed convection stagnation point ow with radiation and slip condition, Scientia Iranica B, 242, pp [7] Hussain, Q., Asghar, S. and Alsaedi, A. Heat transfer analysis in peristaltic slip flow with Hall and ion-slip currents, Scientia Iranica C, 236, pp

18 [8] Heydari, M.M. Investigation of fluid flow and heat transfer of compressile flow in a constricted microchannel, Scientia Iranica B, 235, pp [9] Cattaneo, C. and Sulla conduzionedelcalore. AttiSemin. Mat. Fis. Univ. Modena Reggio Emilia, 3, pp [] Christov, C.I. On frame indifferent formulation of the Maxwell-Cattaneo model of finite speed heat conduction, Mech. Res. Commun., 36, pp [] Ciarletta, M. and Straughan, B. Uniqueness and structural staility for the Cattaneo Christov equations, Mech. Res. Commun., 37, pp [2] Hayat, T., Imtiaz, M., Alsaedi, A. and Almezal, S. On Cattaneo Christov heat flux in MHD flow of Oldroyd-B fluid with homogeneous heterogeneous reactions, J. Mol. Liq., 4, pp [3] Khan, M. and Khan, W.A. Three-dimensional flow and heat transfer to Burgers fluid using Cattaneo-Christov heat flux model, J. Mol. Liq., 22, pp [4] Waqas, M., Hayat, T., Farooq, M., Shehzad, S.A. and Alsaedi, A. Cattaneo-Christov heat flux model for flow of variale thermal conductivity generalized Burgers fluid, J. Mol. Liq., 22, pp [5] Sui, J., Zheng, L. and Zhang, X. Boundary layer heat and mass transfer with Cattaneo- Christov doule-diffusion in upper-convected Maxwell nanofluid past a stretching sheet with slip velocity, Int. J. Therm. Sci., 4, pp [6] Liu, L., Zheng, L., Liu, F. and Zhang, X. Anomalous convection diffusion and wave coupling transport of cells on com frame with fractional Cattaneo-Christov flux, Commun. Nonlinear Sci. Numer. Simulat., 38, pp [7] Khan, W.A., Khan, M. and Alshomrani, A. S. Impact of chemical processes on 3D Burgers fluid utilizing Cattaneo-Christov doule-diffusion: Applications of non-fourier s heat and non-fick s mass flux models, J. Mol. Liq., 223, pp [8] Malik, M.Y., Khan, M., Salahuddin, T., Khan, I. Variale viscosity and MHD flow in Casson fluid with Cattaneo Christov heat flux model: Using Keller ox method, Engi. Sci. Tech., Int. J., 94, pp [9] Muhammad, N., Nadeem, S. and Mustafa, T. Squeezed flow of a nanofluid with Cattaneo-Christov heat and mass fluxes, Res. Phys., 7, pp [2] Khan, M., Shahid, A., Malik, M.Y. and Salahuddin, T. Thermal and concentration diffusion in Jeffery nanofluid flow over an inclined stretching sheet: A generalized Fourier s and Fick s perspective, J. Mol. Liq., 25, pp [2] Munir, A., Shahzad, A. and Khan, M. Convective flow of Sisko fluid over a idirectional stretching surface, PLOS ONE, 6, e

19 [22] Khan, M., Malik, R., Munir, A. and Khan, W.A. Flow and heat transfer to Sisko nanofluid over a nonlinear stretching sheet, PLOS ONE, 5, e [23] Malik, R., Khan, M., Munir, A. and Khan, W.A. Flow and heat transfer in Sisko fluid with convective oundary condition, PLOS ONE, 9, e [24] Khan, W.A., Khan, M., Alshomrani, A.S. and Ahmad, L. Numerical investigation of generalized Fourier s and Fick s laws for Sisko fluid flow, J. Mol. Liq., 224, pp [25] Khan, M., Ahmad, L. and Khan, W.A. Numerically framing the impact of radiation on magnetonanoparticles for 3D Sisko fluid flow, J. Braz. Soc. Mech. Sci. Eng., 39, pp [26] Awais, M., Malik, M.Y., Bilal, S., Salahuddin, T. and Hussain, A. Magnetohydrodynamic MHD flow of Sisko fluid near the axisymmetric stagnation point towards a stretching cylinder, Res. Phys., 7, pp [27] Hussain, A., Malik, M.Y., Salahuddin, T., Bilal, S. and Awais, M. Comined effects of viscous dissipation and Joule heating on MHD Sisko nanofluid over a stretching cylinder, J. Mol. Liq., 23, pp [28] Hussain, A., Malik, M.Y., Bilal, S., Awais, M. and Salahuddin, T., Computational analysis of magnetohydrodynamic Sisko fluid flow over a stretching cylinder in the presence of viscous dissipation and temperature dependent thermal conductivity, Res. Phys., 7, pp [29] Fourier, J.B.J. Théorie Analytique De La Chaleur, Paris,

20 Masood Khan is orn in 973 in Haripur, Khyer Pakhtunkhwa, Pakistan, He is currently a Professor at Department of Mathematics, Quaid-i-Azam University, Islamaad, Pakistan. He received his BSc and MSc degrees in 993 and 997, respectively from University of Peshawar, Khyer Pakhtunkhwa, Pakistan. He received his PhD degree in 24 in Applied Mathematics, from Quaid-i-Azam University, Islamaad, Pakistan. His suject of interest is Fluid Mechanics as a researcher and teacher. Latif Ahmad is orn in 984 in Upper Dir, Khyer Pakhtunkhwa, He received his BS HONS degree in 28 from University of Malakand at Chackdara, Khyer Pakhtunkhwa, Pakistan. He is Gold Medalist in BS. He received his MPhil degree in 24 from University of Peshawar, Khyer Pakhtunkhwa, Pakistan in Applied Mathematics. He is also a permanent faculty memer at Department of Mathematics Shaheed Benazir Bhutto University, Sheringal Upper Dir, Khyer Pakhtunkhwa, Pakistan. He is on study leave for his PhD studies at Department of Mathematics, Quaid-i-Azam University, Islmaad, Pakistan. His research interests includes mathematical modeling and numerical simulation of Newtonian and non-newtonian fluids flow. He is pulished thirteen papers in ISI Journals. Waqar Azeem Khan is orn in 985 in Karachi, Pakistan. He received his MPhil degree in 22 from Department of Mathematics, Quaid-i-Azam University, Islmaad, Pakistan. He is also a faculty memer at Department of Mathematics and Statistics at Hazara University, Mansehra, Pakistan. He received his PhD degree in 27 from Department of Mathematics, Quaid-i-Azam University, Islmaad, Pakistan. His research interests includes mathematical modeling and numerical simulation of non-newtonian fluids flow. He has pulished forty papers in ISI Journals. Ali Saleh Zuhair Al Shamrani is orn in 982 in the city of Baha, Saudi Araia. He is currently a Professor at Department of Mathematics, Faculty of Science, King Adul Aziz University, Jeddah Saudi Araia. His research interests are in the general areas of numerical analysis. He has also an excellent grade with second honor G. He remain a teacher for three years in the Ministry of Education Saudi Araia.