Haar wavelet solution of the MHD Jeffery-Hamel flow and heat transfer in EyringPowell fluid Najeeb Alam Khan, Faqiha Sultan, Amber Shaikh, Asmat Ara, and Qammar Rubbab Citation: AIP Advances 6, 115102 (2016); doi: 10.1063/1.4967212 View online: http://dx.doi.org/10.1063/1.4967212 View Table of Contents: http://aip.scitation.org/toc/adv/6/11 Published by the American Institute of Physics AIP ADVANCES 6, 115102 (2016) Haar wavelet solution of the MHD Jeffery-Hamel flow and heat transfer in Eyring-Powell fluid Najeeb Alam Khan,1,a Faqiha Sultan,2 Amber Shaikh,1 Asmat Ara,3 and Qammar Rubbab4 Department of Mathematics, University of Karachi, Karachi 75270, Pakistan of Sciences and Humanities, National University of Computer and Emerging Sciences, Karachi 75030, Pakistan Mohammad Ali Jinnah University, Karachi 75400, Pakistan Department of Mathematics, Air University, Multan Campus, Pakistan Department (Received August 2016; accepted 24 October 2016; published online November 2016) This study deals with the numerical investigation of Jeffery-Hamel flow and heat transfer in Eyring-Powell fluid in the presence of an outer magnetic field by using Haar wavelet method Jeffery-Hamel flows occur in various practical situations involving flow between two non-parallel walls Applications of such fluids in biological and industrial sciences brought a great concern to the investigation of flow characteristics in converging and diverging channels A suitable similarity transformation is applied to transform the nonlinear coupled partial differential equations (PDEs) into nonlinear coupled ordinary differential equations (ODEs), which govern the momentum and heat transfer properties of the fluid Due to the high nonlinearity of resulting coupled ODEs, the exact solution is unlikely Thus, the solution is approximated using a numerical scheme based on Haar wavelets and the results are verified by comparing with 4th order Runge-Kutta results © 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4967212] I INTRODUCTION Flows through convergent-divergent channels gained importance in early nineteenth century after the revolutionary works of Refs and In modern eras, Jeffery-Hamel flows have various applications in fluid mechanics, aerospace, civil, bio-mechanical, mechanical, chemical, and environmental engineering along with exploring the rivers and canals Practical applications of these types of flows include flow through rivers, canals, and different biological flows such as flow through arteries and venous blood vessels Since the pioneer work of Jeffery and Hamel, many researchers have been investigating the applications of this fluid, and they have reported the variation in flow characteristics by changing the angle between the two channel.3–5 Jeffery-Hamel flows are considered with various other flow conditions and the effects of many other fluid characteristics have been investigated.6 investigated the thermal radiation effects on the conventional Jeffery-Hamel flow caused by a point source or sink in convergent/divergent channels with stretching or shrinking walls of the stationary channel.7 explored the effects of magnetic field applied transversely on Jeffery-Hamel flow using Cuwater nanofluid in the middle of two nonparallel plane walls.8 obtained the similarity solutions for the flow of Jeffery-Hamel fluid and described its relation to flow in a converging-diverging channel A number of scientific problems are inherently nonlinear such as the flow models in fluid mechanics including Jeffery-Hamel flows and as a result of this nonlinear dynamic system of the models, the exact solutions are improbable So, various numerical techniques have been presented to solve these systems.9 obtained the approximate homotopy analysis solution for the Jeffery-Hamel flows.10 presented an improved homotopy analysis solution of the nonlinear equations of Jeffery-Hamel a Corresponding author E-Mail: njbalam@yahoo.com Tel.: +923333012008 2158-3226/2016/6(11)/115102/13 6, 115102-1 © Author(s) 2016 115102-2 Khan et al AIP Advances 6, 115102 (2016) fluid flow.11 have proposed another new technique for MHD Jeffery-Hamel flow, named spectralhomotopy analysis method.12 used the Adomian decomposition method to analytically investigate the Jeffery-Hamel flow with nanoparticles and high magnetic field.13 introduced the optimal homotopy asymptotic solution of the problem Recently, a massive attention has been directed towards the artificial intelligence techniques to numerically solve the nonlinear differential equations.14 have recently introduced a numerical treatment based on stochastic algorithms for the nonlinear MHD Jeffery-Hamel problems Generally, fluids used in chemical engineering processes and polymer processing are nonNewtonian in nature Realizing the fact that these fluids possess many industrial applications, Powell and Eyring presented such a fluid model in 1944, acknowledged as Eyring-Powell fluid model15 that gives several benefits beyond the other non-Newtonian fluids Although this model is more complex mathematically, additional consideration is dedicated to it owing to its distinctive benefits above the Power-law model Mainly because it is acquired using the kinetic theory of liquid instead of the empirical relation, unlike power-law models Additionally, it truly exhibits Newtonian character when shear rate is high and low Considering the significance of this fluid, vital researches have been presented to study the impact of several fluid conditions Some of the recent studies include: a research on the effects of double diffusion on the flow of Eyring-Powell fluid over a cone conducted by,16 investigation of MHD flow, heat and mass transfer in radiating Eyring-Powell fluid with suspended nanoparticles over a stretching sheet in three dimensions by,17 the study the flow of nano-EyringPowell fluid and heat and mass transfer in a two-layer channel by,18 moreover, another research is dedicated to examine the effect of slip conditions and wall properties on MHD peristaltic flow and heat/mass transfer in Eyring-Powell fluid by.19 The analysis of the MHD Jeffery-Hamel flow of Eyring-Powell fluid with heat transfer has not been considered yet This paper aims to fill this gap and presents an investigation on the effects of magnetic field on Jeffery-Hamel flow and heat transfer in Eyring-Powell fluid The JefferyHamel problem occurs in various flow phenomena of fluid flow between two no-parallel walls The mathematical model of Jeffery-Hamel problems defines a nonlinear dynamic system, which does not possess an exact solution To overcome this issue, a numerical method based on Haar wavelets is used to find the numerical solution of the system of complicated nonlinear equations governing the flow and heat transfer characteristics Moreover, a shooting technique via NDSolve command in Mathematica 10 is also used to solve and verify the solution The results obtained by these two methods are presented through graphs and tables and discussed in detail for the variation of some important parameters Comparison of these two methods is made by comparing the numerical values of the physical quantities, for the sake of the validity of the obtained results II MATHEMATICAL MODEL The extra-stress tensor for Eyring-Powell fluid is defined as: 1 sinh−1 γ˙ β γ˙ d Γ= µ + A1 (1) where µ represents the shear viscosity, β and d are the characteristics of the Eyring-Powell model, d has the dimension of (time)−1 , γ˙ = 2 trA1 , and A1 = ∇V + (∇V )t is the kinematical tensor The 1 second order approximation of sinh−1 d1 γ˙ with γd˙ channels keeping Re = 20 115102-8 Khan et al AIP Advances 6, 115102 (2016) FIG The effect of Eyring-Powell parameter EP on velocity profiles in a divergent channel keeping α = 2.5 and Re = 20 where, Ci,2 = Pi,1 (η)dη (41) Finally, integrating Eq (39) from to η and substituting the value of g (0) provide: 2M˜ g (η) = + 2M˜ bi Pi,2 (η) − i=1 bi Ci,2 (42) i=1 IV RESULTS AND DISCUSSION With the purpose of having a clear insight of the flow and heat transfer properties of the fluid, the system of governing equations is solved by Haar wavelet and 4th order Runge-Kutta method by FIG The effect of Eyring-Powell parameter EP on temperature profiles in a divergent channel keeping α = 2.5 and Re = 20 115102-9 Khan et al AIP Advances 6, 115102 (2016) FIG The effect of Hartman number Ha on velocity profiles in a divergent channel keeping α = 2.5 and Re = 20 using computational software The influence of the magnetic field and other fluid parameters has been investigated on flow and heat transfer in a wedge shaped region The effects of pertinent parameters have been observed through Figs 2-12 and Tables I–II The results obtained by Haar wavelet are validated, by comparison of numerical values of the physical quantities, with the values obtained by NDSolve For the purpose of conciseness, the constant values of the parameters are assumed to be EP = 0.2, λ = 0.1, Ha = 5, Pr = 6.2, Ec = 0.5, and Re = 20 The overall structure and the effect of angle α on the fluid velocity and temperature distribution of Eyring-Powell fluid flow over a wedge shaped region is presented in Figs 2-3 Fig depicts the velocity profile of the flow in convergent/divergent channel In a convergent channel i.e α < and for adequately high Reynolds number, the velocity profile remains almost constant over a big portion and rapidly declines to zero near the walls, demonstrating a clear boundary layer behavior As the angle α increases on negative side, the velocity profile rises, increasing the constant portion parallel to the horizontal axis While, in a divergent channel, the velocity profile is more arched near the centerline and increment in positive angle α reduces the velocity by increasing the curve of the profile FIG The effect of Hartman number Ha on temperature profiles in a divergent channel keeping α = 2.5 and Re = 20 115102-10 Khan et al AIP Advances 6, 115102 (2016) FIG The effect of Eckert number Ec on temperature profiles in a divergent channel keeping α = 2.5 and Re = 20 Fig shows that the temperature profiles exhibit same pattern in convergent and divergent channels Also, as the angle of the channel gets larger, the profiles show the same increasing behavior, but in a convergent channel, the temperature profile increases more rapidly as compare to the profiles in a divergent channel The influence of Eyring-Powell parameter on the flow and heat transfer in a divergent channel, which is the prime interest of this study, is presented in Figs and 5, respectively The Eyring-Powell parameter decreases the velocity of the fluid, however, the heat transfer in fluid flow increases when EP increases The effects of exterior magnetic field are also observed on the flow and heat transfer in EyringPowell fluid for α > 0, depicted in Figs and The velocity of the fluid rises near the centerline decreasing the curve of the profiles as the magnetic field is strengthen in terms of Hartman number as shown in Fig Likewise, Fig shows the increase in heat transfer as the Hartman number increases Moreover, the temperature profiles are plotted against the Eckert number in Fig and the profiles are observed to be rising when Eckert number is increased FIG The comparison of velocity profiles for Haar wavelet and NDSolve in a divergent channel keeping EP = 0.2 and Re = 20 115102-11 Khan et al AIP Advances 6, 115102 (2016) FIG 10 The comparison of temperature profiles for Haar wavelet and NDSolve in a divergent channel keeping EP = 0.2 and Re = 20 Lastly, the comparison of velocity and temperature profiles is made in Figs 9-12 and a good agreement between the Haar wavelet and NDSolve results is observed Furthermore, it is observed that the velocity profile reduces with an increase in Reynolds number as depicted in Fig 11 and an increase in Prandtl number enhances the heat transfer in the flow over a diverging channel as shown in Fig 12 The validity of the Haar wavelet results is further confirmed by the comparison of the numerical values of local skin friction coefficient in Table I and local Nusselt number in Table II for different parameters The numerical values of both the physical quantities show a good agreement between both the methods and more accuracy of the Haar wavelet results can be obtained ˜ Along with the comparison, the effects of EP , Re, and Pr are also by increasing the value of M investigated It is observed in Table I that the Eyring-Powell parameter and the Reynolds number increase the skin friction reducing the flow at wall While, the temperature distribution in the flow of Eyring-Powell fluid over a wedge shaped region increases when the values of EP , Re, and Pr rise FIG 11 The comparison of velocity profiles for Haar wavelet and NDSolve in a divergent channel keeping EP = 0.2 and α = 2.5 115102-12 Khan et al AIP Advances 6, 115102 (2016) FIG 12 The comparison of temperature profiles for Haar wavelet and NDSolve in a divergent channel keeping EP = 0.2, α = 2.5, and Re = 20 TABLE I Numerical values of local skin friction coefficient r Re Cf , for different physical parameters while keeping α = 2.5, Re = 20, Ha = 5, Pr = 6.2, Ec = 0.5, and λ = 0.1 EP Re 0.0 0.2 0.4 0.8 0.2 0.2 2.3059 2.8772 3.4022 4.3072 2.2641 2.7893 3.3162 4.3216 1.5592 2.3684 2.8772 1.3801 2.2490 2.7893 10 15 20 TABLE II Numerical values of local Nusselt number −Nu = − α = 2.5, Re = 20, Ha = 5, Ec = 0.5 and λ = 0.1 0.0 0.2 0.4 0.8 NDSolve 20 0.2 EP Haar wavelet Re Pr g (1) α for different physical parameters while keeping Haar wavelet NDSolve 21.1049 26.5188 32.5135 46.0569 21.0178 26.0160 31.6500 45.3897 20 6.2 10 15 20 6.2 22.9508 24.2502 26.5188 22.3801 23.7886 26.0160 4.2772 12.8317 21.3861 29.9405 4.19612 12.5884 20.9806 29.3729 20 V CONCLUDING REMARKS In this study, a novel numerical technique based on Haar wavelets was employed to examine the Jeffery-Hamel flow of Eyring-Powell fluid and heat transfer with an external magnetic field applied 115102-13 Khan et al AIP Advances 6, 115102 (2016) transversely The governing equations of the flow and heat transfer properties of the fluid were solved for several pertinent parameters and it was observed that the Eyring-Powell parameter reduce the fluid velocity while heat transfer in the flow over wedge shaped region was increased with increasing values of Eyring-Powell parameter On the other hand, the presence of magnetic file enhanced both the fluid velocity and temperature distribution The present technique of Haar wavelets is easily applicable and consistent to understand the flow and heat transfer characteristics of the governing equations At low mode validity of the Haar wavelet results is proved by graphically comparing the profiles of fluid velocity and temperature 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flow and heat transfer in Eyring- Powell fluid The JefferyHamel problem occurs in various flow phenomena of fluid flow between