IJOER-JUN-2017-4

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 53 Numerical treatment and Global Error Estimation of a MHD Flows of an Oldroyd 6-Constant Nano-Fluid through a non- Darcy Porous medium with Heat and Mass Transfer Abeer A. Shaaban Department of Mathematics Faculty of Education Ain Shams University Roxy Cairo Egypt Department of Management Information Systems Faculty of Business Administration in Rass Qassim University Qassim KSA Abstract — Explicit Finite-Difference method was used to obtain the solution of the system of the non-linear ordinary differential equations which transform from the non-linear partial differential equations. These equations describe the steady magneto-hydrodynamic flow of an oldroyd 6-constant non-Newtonian nano-fluid through a non-Darcy porous medium with heat and mass transfer. The numerical formula of the velocity the temperature the concentration and the nanoparticles concentration distributions of the problem were illustrated graphically. The effect of Darcy number D a Forchheimer number F s magnetic field parameter M local temperature Grashof number G r local nanoparticle Grashof B r Prandtl number P r Dufour number N d Brownian motion parameter N b Thermophoresis parameter N t Lewis number L e Sort number L d Chemical reaction parameter R c and Chemical reaction order m on those formula were discussed at the values of material parameters specially in the case of pure Coutte flow. Then the effects of modified pressure gradients on those formulas were discussed in the case of pure Poiseuille flow and the generalized Couette flow. Also an estimation of the global error for the numerical values of the solutions is calculated by using Zadunaisky technique. Keywords — Finite-difference method Heat and Mass transfer MHD flows Non-Darcy Porous medium Oldoryd 6- Constant non-Newtonian nano-Fluid. I. NOMENCLATURE Chemical reaction parameter defined by Eq. 42 Chemical Reaction rate constant A The dimensionless concentration Local nanoparticle Grashof number defined by Eq. 42 B r The time t Forchheimer number c The fluid temperature T The concentration of the fluid C The temperature at lower plate The concentration at lower plate The temperature at upper plate The concentration at upper plate The velocity vector Nanoparticle susceptibility Darcy number defined by Eq. 42 D a Brownian diffusion coefficient D B Greek symbols Thermophoretic diffusion coefficient D T The nanoparticles phenomena Electrical field E The dissipation function The external force F The dimensionless nanoparticles Forchheimer number defined by Eq. 42 F s The dimensionless temperature Gravitational acceleration G The non-Newtonian parameters defined by Eq. 36 Local temperature Grashof number defined by Eq. 42 G r The magnetic field Gradient operator The current density J Laplacian operator Thermal conductivity

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 54 the dynamic viscosity of fluid Permeability constant the magnetic permeability Thermal diffusion ratio K T The kinematic viscosity Lewis number defined by Eq. 42 The specific heat capacity at constant pressure Sort number defined by Eq. 42 The density of the fluid f  Chemical Reaction order M The density of the particle p  Magnetic field parameter defined by Eq. 42 M heat capacity of the fluid f c  Brownian motion parameter defined by Eq. 42 N b effective heat capacity of the nanoparticle material p c  Dufour numbe defined by Eq. 42 N d Electrical conductivity of the fluid The thermophoresis parameter defined by Eq. 42 N t The Cauchy Stress tensor The fluid pressure P Volumetric thermal and solute expansion coefficients of the base fluid Prandtl number defined by Eq. 42 P r II. INTRODUCTION The study of non-Newtonian fluids has gained much attention recently in view of its promising applications in engineering and industry. Such fluids exhibit a non-linear relationship between the stresses and the rate of strain. Due to non-linear dependence the analysis of the behavior of the non-Newtonian fluids tends to be much more complicated and subtle in comparison with that of Newtonian fluids. Flow of fluids with complex microstructure e. g. molten polymer polymer solutions blood paints greases oils ketchup etc. cannot be described by a single model of non-Newtonian fluids. Many models that exist are based either on natural modifications of established macroscopic theories or molecular considerations. In general the equations of motion for non-Newtonian fluids are of higher order than the Navier-Stokes equations and thus one need conditions in addition to the usual adherence boundary condition 28. Guillope and Saut 2 has established existence results for some shearing motions of viscoelastic fluids of Oldroyd type. Some exact solutions of an Oldroyd 3-constant fluid are studied in 4 5 11 13. Baris 21 investigated the steady flow of an Oldroyd 6-constant fluid between intersecting planes using the series expansion method. Hayat et al. 23 studied the Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field by using the Homotopy analysis method. Hayat et al. 25 studied the steady flow of a magneto-hydrodynamic Oldroyd 6-constant fluid by the motion of an infinite plate using the Homotopy analysis method. Wang et al. 30 investigated the non-linear magnetohydrodynamic problems of an Oldroyd 6-constant fluid by using analytical method and the finite-difference discretization method. Hayat et al. 24 studied the effect of the slip condition on flows of an Oldroyd 6-constant fluid. Rana et al. 14 studied the Hall effects on hydromagnetic flow of an Oldroyd 6-constant fluid between concentric cylinders by the finite difference method. Hayat et al. 26 investigated the exact solution of a thin film flow of an Oldroyd 6-constant fluid over a moving belt by the Homotopy perturbation. Investigation of nanofluid flow has received special focus in the past due to its relevance in numerous industrial applications. The researchers not only discovered unexpected thermal properties of nanofluids but also proposed mechanisms behind the enhanced thermal properties of nanofluids and thus identified unusual opportunities to develop them as next generation coolants for computers and safe coolants for nuclear reactors. A combination of nanofluid with biotechnological components can provide potential applications in agriculture pharmaceuticals and biological sensors. Various types of nanomaterials including nanoparticles nanowires nanofibers nanostructures and nanomachines are used in biotechnological applications. The commercialization of nano-biotechnological products seems to have a potential future and within next a few years many new products of this nature are likely to be used. Nano and micro-fluidics is a new area which has potential for engineering applications especially for the development of new biomedical devices and procedures 1.

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 55 The magnetic nanofluids possess both liquid and magnetic properties. These fluids have key importance in modulators optical switches optical gratings and tunable optical fiber filters. The magnetic nanoparticles are significant in medicine construction of loud speakers sink float separation cancer therapy and tumor analysis. Nowadays the sustainable energy generation is one of the most important issues across the globe. Perhaps the solar energy is also one of the best sources of renewable energy with minimal environment impact. Solar power is a direct way obtaining heat water and electricity from the nature. Researchers concluded that heat transfer and solar collection processes can be improved through the addition of nanoparticles in the fluids. Fluid heating and cooling are required in many industrial fields such as power manufacturing and transportation. Effective cooling techniques are needed for cooling any sort of high energy device. Common heat transfer fluids such as ethylene glycol water and engine oil have limited heat transfer capabilities due to their low heat transfer properties and thus cannot meet the modern cooling requirements 1. The terminology of nanofluid was first used by Choi 22 when he experimentally discovered a significant improvement in the heat transfer performance through the addition of small amount nanometer sized particles in the base fluids. This addition also causes scattering of the incident radiation which allows a deeper absorption within the fluid. Recently Trieb and Nitsch 3 proposed the idea of solar thermal collectors by using nanofluids to directly absorb the solar radiation. The phenomenon of thermal conductivity enhancement by dispersing nanoparticles in the liquid was also observed by Masuda et al. 7. Buongiorno 9 recognized that the two main effects namely the Brownian motion and thermophoretic diffusion of nanoparticles contribute to the massive increase in the thermal conductivity of the liquids. He also proposed the modifications in the transport equations due to these effects. Vajravelu et al. 12 investigated heat and mass transfer properties of three-layer fluid flow in which nano-fluid layer is squeezed between two clear viscous fluid. Farooq et al. 27 studied heat and mass transfer of two-layer flows of third-grade nano-fluids in a vertical channel. Abou-zeid et al. 16 obtained numerical solutions and Global error estimation of natural convection effects on gliding motion of bacteria on a power-law nano-slime through a non-Darcy porous medium. El-Dabe et al. 17 investigated magneto-hydrodynamic non-Newtonian nano-fluid flow over a stretching sheet through a non-Darcy porous medium with radiation and chemical reaction. Magneto-hydrodynamic MHD is concerning the mathematical and physical scaffold that introduces magnetic-dynamics in electrically conducting fluids e.g. in plasmas and liquid metals . The applications of Magneto-hydrodynamic incompressible viscous flow in science and engineering involving heat and mass transfer under the influence of chemical reaction is great importance to many areas of science and engineering. This frequently occurs in agriculture engineering plasma studies and petroleum industries 15. Flow through porous media plays an important role in countless practical applications such as ground water flows enhanced oil recovery processes contamination of soils by hazardous wastes pollution movement 20. An understanding of the dynamics of fluids in a porous medium is of principal interest because these flows are quite prevalent in nature. Such flows have attracted the attention of number of scholars due to their applications in many branches of science and technology viz. in the fields of agricultural engineering to study the underground water resources seepage of water in river beds in petroleum technology to study the movement of natural gas oil and water through the oil reservoirs. In the widely used continuum approach to transport processes in a porous media the differential equation governing the macroscopic fluid motion is based on the experimentally established Darcy law 8 which accounts for the drag exerted on the fluid by porous medium 29. The objective of this work is to investigate the numerical solution by using Explicit Finite Difference method 6 for the system of non-linear differential equations which arises from magneto-hydrodynamic flows of an Oldroyd 6-constant Nano- fluid with a magnetic field through a non-Darcy porous medium with heat and mass transfer. We obtained the distributions of the velocity the temperature the concentration and the Nanoparticles. Numerical results are found for different values of various non-dimensional parameters in the case of pure Coutte flow . The effects of modified pressure gradients on those formulas were discussed in the case of pure Poiseuille flow and the generalized Couette flow. The results are shown graphically and discussed in detail. Also the global error estimation for the error propagation is obtained by Zadunaisky technique 19. III. FLUID MODEL For an Oldroyd 6-constant fluid the Cauchy stress tensor is given by 30

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 56 1 . 2 Where . 3 is the indeterminate part of the stress due to the constraint of incompressibility S is the extra stress tensor is the first Rivlin-Ericksen tensor L is the velocity gradient is the velocity vector and are the six material constants. The contra variant convected derivative satisfying the principle of material frame in difference in terms of the material is defined by the following equation . 4 in which the superscript T is used for the transpose operation. It should be noted that when the model reduces to the classical linearly viscous model of Newtonian fluid. For a 3-constant model of an Oldroyd-B fluid is described while for a Maxwell model is formulated. For the model describes a second-grade fluid. IV. MATHEMATICAL FORMULATION We consider magnetohydrodynamic flows of an Oldroyd 6-constant Nano-fluid through a non-Darcy porous medium between two parallel plates see FIG. 1. We take in our consideration the presence of heat and mass transfer with chemical reaction. FIGURE 1: MAGNETOHYDRODYNAMIC FLOWS OF AN OLDROYD 6-CONSTANT NANO-FLUID THROUGH A NON- DARCY POROUS MEDIUM BETWEEN TWO PARALLEL PLATES V. BASIC EQUATIONS The basic equations governing the flow of an incompressible fluid are the following equations 10 18 The continuity equation 5 The momentum equation 6

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 57 The energy equation 7 The concentration equation 8 The nanoparticles concentration equation . 9 Where denotes the material time derivative and . For a simple one-dimensional shearing flow considered in this work the extra stress tensor and the velocity are of the form . 10 Also the constant magnetic field the temperature the concentration and the nanoparticles are of the form 11 For steady motion it is clear that with the simplified dependence 10-11 the continuity equation 5 is satisfied identically and equations 1-3 and 6 become 12 13 14 15 16 17 18 19 . 20 by addition of two equations 15 and 18 we have . 21 from equations 17 and 19 we get

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 58 . 22 Substituting 18 – 22 in 14 and 16 we have 23 . 24 where 25 . 26 Defining the modified pressure 27 and using 12 – 14 and 24 we arrive at 28 . 29 The dissipation function can be written as follows . 30 By using the definitions 10 we have . 31 Then equations 7 - 9 can be written as follows: 32 33 . 34 The boundary conditions of the problem are given by: 35 We shall now write the field equations 28 32 – 34 and the boundary conditions 35 in terms of a set of dimensionless variables and for this purpose we choose H and U as the characteristic length and velocity and introduce the following

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 59 dimensionless variables of both: . 36 Thus the system of our non-linear differential equations 28 32 – 34 and the boundary conditions 35 can be rewritten in a non-dimensional forms: 37 38 39 . 40 with the dimensionless boundary conditions: 41 Where the dimensionless parameters are defined by . 42 For convenience we shall drop the bars that identify the dimensionless quantities. The system of non-linear ordinary differential equations 37 – 40 with the boundary conditions 41 will be solved numerically by using the explicit finite- difference method. And we computed the global error for the solutions of the problem. VI. NUMERICAL SOLUTION The equations 37 – 40 can be written after applied explicit finite difference schemes 6 as: 43 44

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 60 45 . 46 Where the index refers to and the . According to the boundary conditions 41 we can solved equations 43 – 46 numerically then a Newtonian iteration method continues until either of goals specified by accuracy goal or precision goal is achieved. VII. ESTIMATION OF THE GLOBAL ERROR We used Zadunaisky technique 19 for calculating the global error which can be explained in the following steps: 1. We fined the interpolating polynomial of from the values of them which came from the explicit finite-difference method. Then we named it and we found the interpolating functions of and we named them as: . 47 2. We calculate the defect functions which can be written as follows: 48 3. We add the defect functions to the original problem. 4. We solved the pseudo-problem new problem by the same method which is used for solving the main problem and we will have the new solutions. 5. We calculate an estimation of the global error from the formulas 49 In this relation is the approximate solutions of the new problem the pseudo-problem at the point and is the exact solutions of pseudo-problem at . The values of global error for the solutions of the problem which solved by the explicit finite difference method are shown in table 1. The error in table 1 based on using 26 points to find interpolating polynomials of degree 25. In order to achieve the above task we used the Mathematica package.

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 61 TABLE 1 DIFFERENT VALUES OF THE DIMENSIONLESS PHYSICAL QUANTITIES VIII. NUMERICAL RESULTS AND DISCUSSION In this paper we generalized the problem of MHD flows to include the non-Newtonian Nano-fluid obeying Oldroyd 6- constant model through a porous medium of non-Darcy type with heat and mass transfer. The system of non-linear ordinary differential equations 37 – 40 with the boundary conditions 41 was solved numerically by using an Explicit finite Difference method. The functions are obtained and illustrated graphically as shown in figures a-1 – a-5 b-1 – b-13 c-1 – c-13 and d-1 – d-13 for different values of the parameters of the problem in the case of pure Couette flow at the values of non-Newtonian parameters . And figures e-1 – e-4 show the effects of modified pressure gradients on those formula in the case of pure Poiseuille flow at the values of non- Newtonian parameters . And figures f-1 – f-4 show the effects of modified pressure gradients on those formula in the case of generalized Couette flow at the values of non-Newtonian parameters . FIG. A-1: Profiles of the velocity uy for a pure Couette flow with various values of Br for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The dimensionless velocity u The dimensionless distance y Br0.1 Br10 Br15 Br30 a18 a22 The global error by using finite difference technique y u Y1 error e1n  Y3 error e3n SY5 error e 5n Y7 error e7n 0 0.00E+00 0.00E+00 1.00E+00 0.00E+00 1.00E+00 0.00E+00 1.00E+00 0.00E+00 0.08 0.084677 2.45938E-06 1.0654 1.06548E-05 0.877711 4.73414E-06 0.832759 9.45491E-06 0.16 0.167804 4.11468E-06 1.10363 1.91716E-05 0.765806 8.26993E-06 0.681824 1.69842E-05 0.24 0.249488 5.13257E-06 1.11623 2.58221E-05 0.662794 1.08326E-05 0.546264 2.28215E-05 0.32 0.329882 5.63641E-06 1.10429 3.07583E-05 0.567513 1.25695E-05 0.425427 2.71033E-05 0.4 0.40918 5.70977E-06 1.06848 3.40436E-05 0.479077 1.35718E-05 0.318913 2.98941E-05 0.48 0.487619 5.39798E-06 1.00906 3.56739E-05 0.39684 1.38901E-05 0.226562 3.12032E-05 0.56 0.565479 4.70768E-06 0.925913 3.55924E-05 0.320364 1.3545E-05 0.148452 3.09962E-05 0.64 0.64308 3.60483E-06 0.818507 3.37017E-05 0.249402 1.25357E-05 0.0848959 2.92048E-05 0.72 0.720789 2.019E-06 0.685922 2.98767E-05 0.183879 1.0847E-05 0.0364469 2.57363E-05 0.8 0.799017 9.48686E-08 0.526812 2.39868E-05 0.123881 8.45801E-06 0.00391268 2.04927E-05 0.88 0.878223 2.40736E-06 0.339383 1.59365E-05 0.0696485 5.37175E-06 -0.0116299 1.34196E-05 0.96 0.958919 2.76915E-06 0.121352 5.76248E-06 0.0215698 1.761E-06 -0.00881092 4.69398E-06 1 1.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 62 FIG. A-2: Profiles of the velocity uy for a pure Couette flow with various values of Gr for a system have the particulars M0.1 Da1 Fs0.5 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. A-3: Profiles of the velocity uy for a pure Couette flow with various values of Da for a system have the particulars M0.1 Fs0.5 Nt0.3 Gr0.1 Br 0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. A-4: Profiles of the velocity uy for a pure Couette flow with various values of Fs for a system have the particulars M0.1 Da1 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The dimensionless velocity u The dimensionless distance y Gr1 Gr3 Gr5 Gr10 a18 a22 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 The dimensionless velocity u The dimensionless distance y Da0.01 Da0.04 Da0.1 Da0.4 a18 a22 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 The dimensionless velocity u The dimensionless distance y Fs1 Fs20 Fs50 Fs100 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 63 FIG. A-5: Profiles of the velocity uy for a pure Couette flow with various values of M for a system have the particulars Da1 Fs0.5 Nt0.3 Gr0.1 Br 0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. 8.1 Couette flow Considering the Couette flow of an Oldroyd 6-constant fluid between two infinite parallel plates for which the lower plate is fixed and the top plate moves with the velocity it is assumed that the flow is driven only by the motion of the top plate. The modified pressure gradient in the flow direction vanishes instant . Figures a-1 – a-5 b-1 – b-13 c-1 – c- 13 and d-1 – d-13 will show the effect of the problem parameters on the problem solutions in this case. Figures a-1 – a-5 show the distributions of the velocity profile u at different values of some parameters of the problem. It is clear that the velocity increases by increasing each of local temperature Grashof number G r local nanoparticle Grashof B r and Darcy number D a . But the velocity decreases by increasing each of Forchheimer number F s magnetic field parameter M. Figures b-1 – b-13 describe the distributions of the temperature profile θ at different values of some parameters of the problem. It is noted that as the local nanoparticle Grashof B r local temperature Grashof number G r and Darcy number D a increase the temperature increases in the region and it returns decrease to . The temperature increases by increasing each of Sort number L d Lewis number L e Brownian motion parameter N b Dufour number N d Thermophoresis parameter N t Prandtl number P r and Chemical reaction parameter R c . And as Forchheimer number F s increases the temperature decreases in the region and it returns increase to . Also as magnetic field parameter M increases the temperature decreases in the region and it returns increase to . Finally the temperature decreases by increasing Chemical reaction order m. FIG. B-1: Profiles of the temperature θy for a pure Couette flow with various values of Br for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 The dimensionless velocity u The dimensionless distance y M10 M30 M50 M100 a18 a22 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y Br0.1 Br10 Br15 Br30 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 64 FIG. B-2: Profiles of the temperature θy for a pure Couette flow with various values of Gr for a system have the particulars M0.1 Da1 Fs0.5 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. B-3: Profiles of the temperature θy for a pure Couette flow with various values Ld for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Rc0.5 m2. FIG. B-4: Profiles of the temperature θy for a pure Couette flow with various values of Le for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Ld0.1 Rc0.5 m2. -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y Gr1 Gr3 Gr5 Gr10 a18 a22 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y Ld1 Ld2 Ld3 Ld6 a18 a22 -0.2 0.3 0.8 1.3 1.8 2.3 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y Le0.01 Le1 Le3 Le5 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 65 FIG. b-5: Profiles of the temperature θy for a pure Couette flow with various values of Nb for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Le2 Nt0.3 Ld0.1 Rc0.5 m2. FIG. b-6: Profiles of the temperature θy for a pure Couette flow with various values of Nd for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nb0.5 Le2 Nt0.3 Ld0.1 Rc0.5 m2. FIG. b-7: Profiles of the temperature θy for a pure Couette flow with various values of Nt for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nb0.5 Le2 Nd2 Ld0.1 Rc0.5 m2. -0.2 0.3 0.8 1.3 1.8 2.3 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y Nb0.1 Nb0.5 Nb2 Nb5 a18 a22 -0.2 0.3 0.8 1.3 1.8 2.3 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y Nd0.1 Nd1 Nd3 Nd5 a18 a22 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y Nt0.1 Nt0.5 Nt1 Nt1.5 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 66 FIG. b-8: Profiles of the temperature θy for a pure Couette flow with various values of Da for a system have the particulars M0.1 Fs0.51 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. b-9: Profiles of the temperature θy for a pure Couette flow with various values of Fs for a system have the particulars M0.1 Da1 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. b-10: Profiles of the temperature θy for a pure Couette flow with various values of M for a system have the particulars Da1 Fs0.51 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y Da0.01 Da0.04 Da0.1 Da0.4 a18 a22 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y Fs1 Fs20 Fs50 Fs100 a18 a22 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y M10 M30 M50 M100 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 67 FIG. b-11: Profiles of the temperature θy for a pure Couette flow with various values of Pr for a system have the particulars Da1 Fs0.51 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. b-12: Profiles of the temperature θy for a pure Couette flow with various values of Rc for a system have the particulars Pr0.7 Da1 Fs0.51 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 m2. FIG. b-13: Profiles of the temperature θy for a pure Couette flow with various values of m for a system have the particulars Pr0.7 Da1 Fs0.51 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5. -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y Pr0.1 Pr0.5 Pr0.7 a18 a22 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y Rc0.5 Rc2 Rc10 Rc20 a18 a22 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y m1 m2 m3 m4 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 68 Figures c-1 – c-13 illustrate the distributions of the concentration profile S at different values of some parameters of the problem. It is seen that the concentration decreases by increasing each of local nanoparticle Grashof B r local temperature Grashof number G r Sort number L d Lewis number L e Brownian motion parameter N b Dufour number N d Thermophoresis parameter N t Prandtl number P r and Chemical reaction parameter R c . And as Darcy number D a increases the concentration decreases in the region and it returns increase to . But as Forchheimer number F s increases the concentration increases in the region and it returns decrease to . The effect of magnetic field parameter M on the concentration disappears in the region the concentration returns decrease to at large values of M. Finally the concentration increases by increasing Chemical reaction order m. FIG. c-1: Profiles of the concentration Sy for a pure Couette flow with various values of Br for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. c-2: Profiles of the concentration Sy for a pure Couette flow with various values of Gr for a system have the particulars M0.1 Da1 Fs0.5 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. c-3: Profiles of the concentration Sy for a pure Couette flow with various values of Ld for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Rc0.5 m2. -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 0.2 0.4 0.6 0.8 1 The dimensionless concentration S The dimensionless distance y Br0.1 Br10 Br15 Br30 a18 a22 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 0.2 0.4 0.6 0.8 1 The dimensionless concentration S The dimensionless distance y Gr1 Gr3 Gr5 a18 a22 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 0 0.2 0.4 0.6 0.8 1 The dimensionless concentration S The dimensionless distance y Ld1 Ld2 Ld3 Ld6 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 69 FIG. c-4: Profiles of the Concentration Sy for a pure Couette flow with various values of Le for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Ld0.1 Rc0.5 m2. FIG. c-5: Profiles of the concentration Sy for a pure Couette flow with various values of Nb for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Le2 Nt0.3 Ld0.1 Rc0.5 m2. FIG. c-6: Profiles of the Concentration Sy for a pure Couette flow with various values of Nd for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nb0.5 Le2 Nt0.3 Ld0.1 Rc0.5 m2. FIG. c-7: Profiles of the Concentration Sy for a pure Couette flow with various values of Nt for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nb0.5 Le2 Nd2 Ld0.1 Rc0.5 m2. -0.4 0.1 0.6 1.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The dimensionless Concentration S The dimensionless distance y Le0.01 Le1 Le3 a18 a22 -0.2 0.3 0.8 1.3 1.8 2.3 0 0.2 0.4 0.6 0.8 1 The dimensionless concentration S The dimensionless distance y Nb0.1 Nb0.5 Nb2 a18 a22 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The dimensionless Cocentration S The dimensionless distance y Nd0.1 Nd1 Nd3 a18 a22 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The dimensionless concentration S The dimensionless distance y Nt0.1 Nt0.5 Nt1 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 70 FIG. c-8: Profiles of the concentration Sy for a pure Couette flow with various values of Da for a system have the particulars M0.1 Fs1 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. c-9: Profiles of the concentration Sy for a pure Couette flow with various values of Fs for a system have the particulars M0.1 Da1 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. c-10: Profiles of the concentration Sy for a pure Couette flow with various values of M for a system have the particulars Da1 Fs1 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless concentration S The dimensionless distance y Da0.01 Da0.04 Da0.1 Da0.4 a18 a22 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless concentration S The dimensionless distance y Fs1 Fs20 Fs50 Fs100 a18 a22 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless concentration S The dimensionless distance y M10 M30 M50 M100 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 71 FIG. c-11: Profiles of the concentration Sy for a pure Couette flow with various values of Pr for a system have the particulars Da1 Fs1 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. c-12: Profiles of the concentration Sy for a pure Couette flow with various values of Rc for a system have the particulars Pr0.7 Da1 Fs1 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 m2. FIG. c-13: Profiles of the concentration Sy for a pure Couette flow with various values of m for a system have the particulars Pr0.7 Da1 Fs1 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5. Figures d-1 – d-13 display the distributions of the nanoparticle profile at different values of some parameters of the problem. It is clear that the nanoparticle decreases by increasing each of local nanoparticle Grashof B r local temperature Grashof number G r Sort number L d Lewis number L e Dufour number N d Thermophoresis parameter N t Prandtl number P r and Chemical reaction parameter R c . But the nanoparticle increases by increasing Brownian motion parameter N b. And as Darcy number D a increases the nanoparticle decreases in the region and it returns increase to . But as Forchheimer number F s increases the nanoparticle increases in the region and it returns decrease to . The effect of magnetic field parameter M on the nanoparticle disappears in the region and the 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless concentration S The dimensionless distance y Pr0.1 Pr0.5 Pr0.7 Pr1 a18 a22 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless concentration S The dimensionless distance y Rc0.5 Rc2 Rc10 Rc20 a18 a22 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The dimensionless concentration S The dimensionless distance y m1 m2 m3 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 72 nanoparticle returns decrease to at large values of M. Finally the nanoparticle increases by increasing Chemical reaction order m. FIG. d-1: Profiles of the nano-particle y for a pure Couette flow with various values of Br for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. d-2: Profiles of the nano-particle y for a pure Couette flow with various valuesof Gr for a system have the particulars M0.1 Da1 Fs0.5 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. d-3: Profiles of the nano-particle y for a pure Couette flow with various values of Ld for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Rc0.5 m2. -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle The dimensionless distance y Br0.1 Br10 Br15 Br30 a18 a22 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle The dimensionless distance y Gr1 Gr3 Gr5 Gr10 a18 a22 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle The dimensionless distance y Ld1 Ld2 Ld3 Ld6 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 73 FIG. d-4: Profiles of the nano-particle y for a pure Couette flow with various values the Le for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Ld0.1 Rc0.5 m2. FIG. d-5: Profiles of nano-particle y for a pure Couette flow with various values of Nb for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Le2 Nt0.3 Ld0.1 Rc0.5 m2. FIG. d-6: Profiles of the nano-particle y for a pure Couette flow with various values of Nd for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nb0.5 Le2 Nt0.3 Ld0.1 Rc0.5 m2. -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle The dimensionless distance y Le0.01 Le1 Le3 Le5 a18 a22 -1.2 -0.7 -0.2 0.3 0.8 1.3 1.8 2.3 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle Nb0.1 Nb0.5 Nb2 Nb5 a18 a22 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle The dimensionless distance y Nd0.1 Nd1 Nd3 Nd5 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 74 FIG. d-7: Profiles of the nano-particle y for a pure Couette flow with various values of Nt for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nb0.5 Le2 Nd2 Ld0.1 Rc0.5 m2. FIG. d-8: Profiles of the nano-particle y for a pure Couette flow with various values of Da for a system have the particulars M0.1 Fs0.5 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. d-9: Profiles of the nano-particle y for a pure Couette flow with various values of Fs for a system have the particulars M0.1 Da1 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle Nt0.1 Nt0.5 Nt1 Nt1.5 a18 a22 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle The dimensionless distance y Da0.01 Da0.04 Da0.1 Da0.4 a18 a22 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle The dimensionless distance y Fs1 Fs20 Fs50 Fs100 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 75 FIG. d-10: Profiles of the nano-particle y for a pure Couette flow with various values of M for a system have the particulars Da1 Fs0.5 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. d-11: Profiles of the nano-particle y for a pure Couette flow with various values of Pr for a system have the particulars Da1 Fs0.5 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. d-12: Profiles of the nano-particle y for a pure Couette flow with various values of Rc for a system have the particulars Pr0.7 Da1 Fs0.5 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 m2. -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle The dimensionless distance y M10 M30 M50 M100 a18 a22 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle The dimensionless distance y Pr0.1 Pr0.5 Pr0.7 Pr1 a18 a22 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle Rc0.5 Rc2 Rc10 Rc20 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 76 FIG. d-13: Profiles of the nano-particle y for a pure Couette flow with various values of m for a system have the particulars Pr0.7 Da1 Fs0.5 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5. 8.2 Poiseuille flow Figures e-1 – e-4 describe the effect of pressure gradient on the velocity the temperature the concentration and the nanoparticle distributions. It is clear that the velocity the concentration and the nanoparticle decrease with increasing while the temperature increases with increasing . FIG. e-1: Profiles of the velocity uy for a pure Poiseuille flow with various values of dP/dx for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. e-2: Profiles of the temperature θy for a pure Poiseuille flow with various values of dP/dx for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle The dimensionless distance y m1 m2 m3 m4 a18 a22 -0.8 -0.6 -0.4 -0.2 0 0 0.2 0.4 0.6 0.8 1 The dimensionless velocity u dP/dx2 dP/dx5 dP/dx10 dP/dx20 a18 a22 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y dP/dx2 dP/dx5 dP/dx10 dP/dx20 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 77 FIG. e-3: Profiles of the concentration Sy for a pure Poiseuille flow with various values of dP/dx for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. e-4: Profiles of the nano-particle y for a pure Poiseuille flow with various values of dP/dx for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. 8.3 Generalized Couette flow If the flow of an Oldroyd 6-constant fluid between two parallel plates is driven by both the motion of the top plate and a constant pressure gradient in the direction parallel to the plates a generalized couette flow is formed for ≠0 U₀1. Numerical results for such a generalized couette flow are illustrated in figures f-1 – f-4. FIG. f-1: Profiles of the velocity uy for a generalized Couette flow with various values of dP/dx for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The dimensionless Concentration S The dimensionless distance y dP/dx2 dP/dx5 dP/dx10 a18 a22 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle The dimensionless distance y dP/dx2 dP/dx5 dP/dx10 dP/dx20 a18 a22 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The dimensionless velocity u The dimensionless distance y dP/dx2 dP/dx5 dP/dx10 dP/dx20 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 78 FIG. f-2: Profiles of the temperature θy for a generalized Couette flow with various values of dP/dx for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. f-3: Profiles of the concentration Sy for a generalized Couette flow with various values of dP/dx for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. FIG. f-4: Profiles of the nano-particle y for a generalized Couette flow with various values of dP/dx for a system have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2. Figures f-1 – f-4 display the variation of the velocity the temperature the concentration and the nanoparticle distributions for several values of the pressure gradient for . It is observed that the velocity the concentration and the nanoparticle decrease with increasing while the temperature increases with increasing . -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 0 0.2 0.4 0.6 0.8 1 The dimensionless temperature θ The dimensionless distance y dP/dx2 dP/dx5 dP/dx10 a18 a22 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 0 0.2 0.4 0.6 0.8 1 The dimensionless Concentration S The dimensionless distance y dP/dx2 dP/dx5 dP/dx10 dP/dx20 a18 a22 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 0 0.2 0.4 0.6 0.8 1 The dimensionless nano-particle The dimensionless distance y dP/dx2 dP/dx5 dP/dx10 dP/dx20 a18 a22

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International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017 Page | 79 IX. CONCLUSION In this work we have studied magnetohydrodynamic flows of an Oldroyd 6-constant nano-fluid through a non-Darcy porous medium with heat and mass transfer. The governing boundary value problem was solved numerically by an Explicit Finite- Difference method. We concentrated our work on obtaining the velocity the temperature the concentration and the nanoparticle distributions which are illustrated graphically at different values of the parameters of the problem in three cases pure Couette flow pure Poiseuille flow and generalized Couette flow. Global error estimation is also obtained using Zadunaisky technique. We used 26 points to find the interpolating polynomial of degree 25 in interval 01 and the results are shown in table 1. We notice that the error in table 1 is good enough to justify the use of resulting numerical values. The main findings from the current study can be summarized as follows: 1- By increasing B r G r and D a the velocity u increases whereas it decreases by increasing each of F s and M in the case of Couette flow at . And the other parameters doesn’t have any effect on the velocity. 2- By increasing B r G r and D a the temperature θ increases at the first part while it decreases at the second part . And by increasing L d L e N b N d N t P r and R c the temperature increases while by increasing m the temperature decreases. By increasing F s the temperature decreases at the first part while it increases at the second part . Also By increasing M the temperature decreases at the first part while it increases at the second part . All these results are given in the case of Couette flow at . 3- By increasing Br Gr Ld Le Nb Nd Nt Pr and Rc the concentration S decreases while by increasing m the concentration increases. By increasing Da the concentration decreases at the first part while it increases for . But by increasing Fs the concentration increases in the first part while it decreases at the second part . And by increasing M the concentration doesn’t have any change at the first part while it decreases at the second part at large values of M. All these results are given in the case of Couette flow at . 4- By increasing B r G r L d L e N d N t P r and R c the nanoparticle decreases while by increasing N b and m the nanoparticle increases. By increasing Da the nanoparticle decreases at the first part while it increases for . But by increasing Fs the nanoparticle increases in the first part while it decreases at the second part . And by increasing M the nanoparticle doesn’t have any change at the first part while it decreases at the second part at large values of M. All these results are given in the case of Couette flow at . 5- By increasing the velocity the concentration and the nanoparticle decrease while the temperature increases in the case of Poiseuille flow at . 6- By increasing the velocity the concentration and the nanoparticle decrease while the temperature increases in the case of generalized Couette flow at . REFERENCES 1 A. Mushtaq M. Mustafa T. Hayat and A. Alsaedi “Nonlinear radiative heat transfer in the flow of nanofluid due to solar energy: A numerical study” J. Taiwan Inst. Chem. Eng. Vol. 45 Pp. 1176 -1183 2014. 2 C. Guillope and JC. Saut “Global existence and one-dimensional non-linear stability of shearing motions of viscoelastic fluids of Oldroyd type RAIRO Model” Math Anal Numer Vol. 24 Pp. 369 -401 1990. 3 F. Trieb and J. Nitsch “Recommendations for the market introduction of solar thermal power st ations Renew Energy” Vol. 14 Pp. 17-22 1998. 4 G. Pontrelli “Longitudinal and torsional oscillations of a rod in an Oldroyd -B fluid with suction or injection” Acta Mech. Vol. 123 Pp. 57-68 1997. 5 G. Pontrelli and K. Bhatnagar “Flow of a viscoelastic fluid between two rotating circular cylinders subject to suction or injection” Int. J. Numer. Maths. Fluids Vol. 24 Pp. 337-349 1997. 6 H. C. Saxena “Examples in finite differences and numerical analysis: S. Chand C ompany Ltd Ram Nagar New delhi 1991. 7 H. Masuda A. Ebata K. Teramae and N. Hishinuma 1993 Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles Netsu Bussei Vol. 7: 227-233.

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