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==Abstract==
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The aim of the present study is to investigate the Hall and ion slip currents on an incompressible free convective flow, heat and mass transfer of a micropolar fluid in a porous medium between expanding or contracting walls with chemical reaction, Soret and Dufour effects. Assume that the walls are moving with a time dependent rate of the distance and the fluid is injecting or sucking with an absolute velocity. The walls are maintained at constant but different temperatures and concentrations. The governing partial differential equations are reduced into nonlinear ordinary differential equations by similarity transformations and then the resultant equations are solved numerically by quasilinearization technique. The results are analyzed for velocity components, microrotation, temperature and concentration with respect to different fluid and geometric parameters and presented in the form of graphs. It is noticed that with the increase in chemical reaction, Hall and ion slip parameters the temperature of the fluid is enhanced whereas the concentration is decreased. Also for the Newtonian fluid, the numerical values of axial velocity are compared with the existing literature and are found to be in good agreement.
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==Keywords==
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Micropolar fluid; Chemical reaction; Soret effect; Dufour effect; Hall and ion slip; Expanding or contracting walls
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==Nomenclature==
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''t''- time
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''a''(''t'')- distance between the origin and lower/upper wall
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<math display="inline">V_1</math>- injection/suction velocity
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''p''- fluid pressure
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<math display="inline">\overline{q}</math>- velocity vector
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''c''- specific heat at constant temperature
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<math display="inline">\overline{l}</math>- microrotation vector
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''N''- microrotation component
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''Ec''- Eckert number, <math display="inline">\frac{\mu V_1}{\rho ac(T_2-T_1)}</math>
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''k''- thermal conductivity
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''k''<sub>1</sub>- viscosity parameter
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''k''<sub>2</sub>- permeability of the medium
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''k''<sub>3</sub>- chemical reaction rate
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''u''- velocity component in ''x''-direction
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''v''- velocity component in ''y''-direction
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''Pr''- Prandtl number, <math display="inline">\frac{\mu c}{k}</math>
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''Re''- suction/injection Reynolds number, <math display="inline">\frac{\rho V_1a}{\mu }</math>
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''j''- gyration parameter
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<math display="inline">\overline{J}</math>- current density
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<math display="inline">J_1</math>- nondimensional gyration parameter, <math display="inline">\frac{\rho {jaV}_1}{\gamma }</math>
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<math display="inline">\overline{B}</math>- total magnetic field
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<math display="inline">\overline{b}</math>- induced magnetic field
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<math display="inline">B_0</math>- magnetic flux density
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''D''- rate of deformation tensor
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<math display="inline">\overline{E}</math>- electric field
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''Ha''- Hartmann number, <math display="inline">B_0a\sqrt{\frac{\sigma }{\mu }}</math>
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''D''<sup>−1</sup>- inverse Darcy parameter, <math display="inline">\frac{a^2}{k_2}</math>
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''R''- nondimensional viscosity parameter, <math display="inline">\frac{k_1}{\mu }</math>
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<math display="inline">s_1</math>- nondimensional micropolar parameter, <math display="inline">\frac{k_1a^2}{\gamma }</math>
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<math display="inline">s_2</math>- nondimensional micropolar parameter, <math display="inline">\frac{\gamma c}{a^2k}</math>
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''T''- temperature
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<math display="inline">T_1</math>- temperature of the lower wall
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<math display="inline">T_2</math>- temperature of the upper wall
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<math display="inline">T^{{_\ast}}</math>- dimensionless temperature, <math display="inline">\frac{T-T_1}{T_2-T_1}</math>
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''C''- concentration
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''C''<sup></sup>- nondimensional concentration, <math display="inline">\frac{C-C_1}{C_2-C_1}</math>
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''C''<sub>1</sub>- concentration of the lower wall
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''C''<sub>2</sub>- concentration of the upper wall
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''D''<sub>1</sub>- mass diffusivity
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''Kr''- nondimensional chemical reaction parameter, <math display="inline">\frac{k_3a^2}{D_1}</math>
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''Sc''- Schmidt number, <math display="inline">\frac{\upsilon }{D_1}</math>
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''Gr''- thermal Grashof number, <math display="inline">\frac{\rho g{\beta }_T(T_2-T_1)a^3}{{\upsilon }^2}</math>
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''Gc''- solutal Grashof number, <math display="inline">\frac{\rho g{\beta }_C(C_2-C_1)a^3}{{\upsilon }^2}</math>
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''Sh''- Sherwood number, <math display="inline">\frac{\overset{\cdot}{n_A}}{a\upsilon \left(C_2-C_1\right)}</math>
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<math display="inline">\overset{\cdot}{n_A}</math>- mass transfer rate
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''Sr''- Soret number, <math display="inline">\frac{D_1k_T\upsilon V_1}{{cT}_m\overset{\cdot}{n_A}}</math>
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''Du''- Dufour number, <math display="inline">\frac{D_1k_T\overset{\cdot}{n_A\rho c}}{{\upsilon }^2V_1c_sk}</math>
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<math display="inline">T_m</math>- mean temperature
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<math display="inline">k_T</math>- thermal-diffusion ratio
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<math display="inline">c_s</math>- concentration susceptibility
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===Greek letters===
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<math display="inline">\lambda </math>- dimensionless ''y  '' coordinate, <math display="inline">\frac{y}{a}</math>
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<math display="inline">\alpha \mbox{,}\beta \mbox{,}\gamma </math>- gyro viscosity parameters
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<math display="inline">\zeta </math>- dimensionless axial variable, <math display="inline">\frac{x}{a}</math>
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<math display="inline">\upsilon </math>- kinematic viscosity
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<math display="inline">\rho </math>- fluid density
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<math display="inline">\mu </math>- fluid viscosity
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<math display="inline">{\mu }^{{'}}</math>- magnetic permeability
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<math display="inline">\sigma </math>- conductivity
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<math display="inline">\eta </math>- wall expansion ratio, <math display="inline">\frac{a\overset{\cdot}{a}}{\upsilon }</math>
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==1. Introduction==
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The flow through porous channels has many important applications, in both engineering and biophysical flows. Examples of this include cosmetic industry, petroleum industry, soil mechanics, food preservation, the mechanics of the cochlea in the human ear, blood flow and artificial dialysis. Many researchers have investigated the numerical simulation of blood flow through curved geometries in the presence of a magnetic field by considering the effect of Dean number [[#b0005|[1]]] and [[#b0010|[2]]]. The theory of micropolar fluids was introduced by Eringen [[#b0015|[3]]] which are considered as an extension of generalized viscous fluids with microstructure. Examples of micropolar fluids include lubricants, colloidal suspensions, porous rocks, aerogels, polymer blends, micro emulsions. Micropolar fluid flow in a porous channel was studied by Ashraf et al. [[#b0020|[4]]] and drawn numerical solution using the finite difference scheme. Ojjela and Naresh Kumar [[#b0025|[5]]] obtained a numerical solution by the quasilinearization method for the MHD flow and heat transfer of a micropolar fluid through a porous channel. Chamkha et al. [[#b0030|[6]]] considered the transient free convective-radiative micropolar fluid flow over the vertical porous plate with the chemical reaction and Joule heating. Aurangzaib et al. [[#b0035|[7]]] investigated the problem of thermophoresis effect on MHD micropolar fluid flow over a stretching surface with Soret and Dufour effects. Srinivasacharya and RamReddy [[#b0040|[8]]] examined the steady convective flow of a micropolar fluid over a vertical plate in a non-Darcian porous medium with Soret and Dufour effects and a numerical solution was obtained by Keller-Box method. The MHD flow of micropolar fluid through concentric cylinders with the chemical reaction and cross diffusion effects was studied by Srinivasacharya and Shiferaw [[#b0045|[9]]]. The Magnetohydrodynamic flow of a micropolar fluid with Hall and ion slip currents plays a great significance role in the real world applications in engineering. Ayano [[#b0050|[10]]] considered the mixed convective micropolar fluid flow with heat and mass transfer in the presence of Hall and ion slip currents and the reduced governing equations are solved by the Keller-box method. The MHD flow of a micropolar fluid over a vertical plate with Hall and ion slip currents was investigated numerically by Anika et al. [[#b0055|[11]]]. An analytical approximate solution HAM is applied to the effect of space porosity on mixed convection flow of micropolar fluid through a vertical channel with double diffusion and viscous dissipation was investigated by Muthuraj et al. [[#b0060|[12]]]. Vedavathi et al. [[#b0065|[13]]] illustrated the Soret and Dufour effects on the free convective flow of a viscous fluid past a vertical plate with radiation. The effects of Hall and ion slip on the mixed convection heat and mass transfer of second grade fluid with Soret and Dufour effects were investigated analytically by Hayat and Nawaz [[#b0070|[14]]]. Chamkha and Ben-Nakhi [[#b0075|[15]]] considered the mixed convection flow of a radiating viscous fluid along a permeable surface in a porous medium with Soret and Dufour effects. Soret and Dufour effects on free convective heat and mass transfer of incompressible viscous fluid from a vertical cone in a saturated porous medium with varying wall temperature and concentration were studied by Cheng [[#b0080|[16]]]. The MHD flow of a viscous fluid over a vertical porous plate with Hall current has been considered by Anika et al. [[#b0085|[17]]].
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The study of magnetohydrodynamic flow, heat and mass transfer through porous expanding or contracting channels is attracted by many authors due to great applications in science and technology, such as transport of biological fluids through contracting or expanding vessels, the synchronous pulsating of porous diaphragms, the expanding or contracting jets, transpiration cooling and gaseous diffusion, the air circulation in the respiratory system, boundary layer control, and MHD pumps. Ojjela and Naresh Kumar [[#b0090|[18]]] studied numerically the flow and heat transfer of chemically reacting micropolar fluid in porous expanding or contracting walls with slip velocity and Hall and ion slip effects. Si et al. [[#b0095|[19]]] and [[#b0100|[20]]] analyzed the problems of flow and heat transfer of micropolar fluids with expanding or contracting walls and obtained analytical solution by the Homotopy analysis method. The two dimensional unsteady viscous fluid flow between expanding or contracting walls with permeability investigated by Majdalani et al. [[#b0105|[21]]] and the problem was solved both numerically and analytically. The viscous fluid flow through expanding and contracting walls with a less permeability has been studied by Asghar et al. [[#b0110|[22]]] and obtained an analytical solution by Adomain decomposition method. The same problem with Lie-group method was considered by Boutros et al. [[#b0115|[23]]]. Uchida and Akoi [[#b0120|[24]]] analyzed the laminar incompressible flow of a viscous fluid through a semi-infinite porous pipe whose radius varied with time. The heat and mass transfer analysis for the laminar flow between expanding or contracting walls with thermal diffusion and diffusion effects was discussed by Subramanyam Reddy et al. [[#b0125|[25]]]. Hymavathi and Shanker [[#b0130|[26]]] and Hymavathi [[#b0135|[27]]] applied the quasilinearization method to solve the visco-elastic fluid flow and heat transfer through a nonisothermal stretching sheet. The convective flow and heat transfer of viscous fluid in a vertical channel was studied by Huang [[#b0140|[28]]] and applied the quasilinearization method to solve the problem.
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In the present study, the effects of Hall and ion slip on two dimensional free convection flow and heat transfer of chemically reacting micropolar fluid in a porous medium between expanding or contracting walls with Soret and Dufour are considered. The reduced flow field equations are solved using the quasilinearization method. The effects of various parameters such as inverse Darcy’s parameter, chemical reaction rate, Soret effect, Dufour effect, Hall and ion slip parameters on the velocity components, microrotation, temperature distribution and concentration are studied in detail and shown in the form of graphs and table.
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==2. Formulation of the problem==
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Consider a two dimensional laminar incompressible micropolar fluid flow through an elongated porous semi-infinite channel for which one end is closed by solid membrane. The walls of the channel are expanding or contracting at the time dependent rate <math display="inline">\overset{\cdot}{a}(t)</math>. Assume that the fluid is injected and aspirated orthogonally through the plates with injection/suction velocities – ''V''<sub>1</sub> and ''V''<sub>1</sub>. Also the non-uniform temperature and concentration at the lower and upper walls are ''T''<sub>1</sub>, ''C''<sub>1</sub> and ''T''<sub>2</sub>, ''C''<sub>2</sub> respectively. Also the heat and mass transfer processes in the presence of Soret and Dufour effects are considered. The region inside the parallel walls is subjected to porous medium and a constant external magnetic field of strength ''B''<sub>0</sub> perpendicular to the ''XY'' – plane is considered ( [[#f0005|Fig. 1]]).
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<span id='f0005'></span>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|-
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|
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[[Image:draft_Content_147589898-1-s2.0-S1110016816300059-gr1.jpg|center|355px|The schematic diagram of fluid flow through expanding or contracting walls.]]
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|-
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| <span style="text-align: center; font-size: 75%;">
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Figure 1.
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The schematic diagram of fluid flow through expanding or contracting walls.
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</span>
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|}
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The governing equations of the micropolar fluid flow, heat and mass transfer in the presence of buoyancy forces and magnetic field are [[#b0090|[18]]] and [[#b0150|[30]]]
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>\nabla \cdot \overline{q}=0</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
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|}
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<span id='e0010'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>\rho \left[\frac{\partial \overline{q}}{\partial t}+\right. </math><math>\left. (\overline{q}.\nabla )\overline{q}\right]=</math><math>-\nabla p+k_1\nabla \times \overline{l}-\left(\mu +\right. </math><math>\left. k_1\right)\nabla \times \nabla \times \overline{q}-</math><math>\frac{\mu +k_1}{k_2}\overline{q}+\overline{J}\times \overline{B}+</math><math>\overline{F_b}</math>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
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|}
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<span id='e0015'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>\rho j\left[\frac{\partial \overline{l}}{\partial t}+\right. </math><math>\left. (\overline{q}.\nabla )\overline{l}\right]=</math><math>-2k_1\overline{l}+k_1\nabla \times \overline{q}-\gamma \nabla \times \nabla \times \overline{l}</math>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
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|}
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<span id='e0020'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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| <math>\rho c\left[\frac{\partial T}{\partial t}+(\overline{q}.\nabla )T\right]=</math><math>k{\nabla }^2T+2\mu D:D+\frac{k_1}{2}{\left(curl(\overline{q})-2\overline{l}\right)}^2+</math><math>\gamma \nabla \overline{l}:\nabla \overline{l}+\frac{\mu +k_1}{k_2}{\left|\overline{q}\right|}^2+</math><math>\frac{{\left|\overline{J}\right|}^2}{\sigma }+\frac{\rho D_1k_T}{c_s}{\nabla }^2C</math>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
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|}
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<span id='e0025'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
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{| style="text-align: center; margin:auto;" 
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| <math>\left[\frac{\partial C}{\partial t}+(\overline{q}.\nabla )C\right]=</math><math>D_1{\nabla }^2C-k_3(C-C_1)+\frac{D_1k_T}{T_m}{\nabla }^2T</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
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where <math display="inline">\overline{F_b}</math> is the buoyancy force and it is defined as <math display="inline">\left(\rho g{\beta }_T(T-T_1)+\rho g{\beta }_C(C-C_1)\right)\hat{i}</math>.
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Neglecting the displacement currents, the Maxwell equations and the generalized Ohm’s law are [[#b0145|[29]]]
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
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{| style="text-align: center; margin:auto;" 
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| <math>\nabla .\overline{B}=0\mbox{,}\quad \nabla \times \overline{B}=</math><math>{\mu }^{{'}}\overline{J}\mbox{,}\quad \nabla \times \overline{E}=</math><math>\frac{\partial \overline{B}}{\partial t}\mbox{,}</math>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>\overline{J}=\sigma (\overline{E}+\overline{q}\times \overline{B})-</math><math>\frac{\beta e}{B_0}\left(\overline{J}\times \overline{B}\right)+</math><math>\frac{\beta e\beta i}{B_0^2}\left(\overline{J}\times \overline{B}\right)\times \overline{B}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
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where <math display="inline">\overline{B}=B_0\hat{k}+\overline{b}\mbox{,}\overline{b}</math> is induced magnetic field. Assume that the induced magnetic field is negligible compared to the applied magnetic field so that magnetic Reynolds number is small, the electric field is zero and magnetic permeability is constant throughout the flow field.
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The velocity and microrotation components are
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
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{| style="text-align: center; margin:auto;" 
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| <math>\overline{q}\quad =u\hat{i}+v\hat{j}\quad \mbox{and}\quad \overline{l}=</math><math>N\hat{k}\mbox{.}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
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|}
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Following Ojjela and Naresh Kumar [[#b0090|[18]]] the velocity, microrotation, temperature and concentration are,
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>u(x\mbox{,}\lambda \mbox{,}t)=-\frac{\upsilon x}{a^2}F^/(\lambda \mbox{,}t)\mbox{,}\quad v(x\mbox{,}\lambda \mbox{,}t)=</math><math>\frac{\upsilon }{a}F(\lambda \mbox{,}t)\mbox{,}\quad N(x\mbox{,}\lambda \mbox{,}t)=</math><math>\frac{\upsilon x}{a^3}G(\lambda \mbox{,}t)</math>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>T(x\mbox{,}\lambda \mbox{,}t)=T_1+\frac{\left(\mu +k_1\right)V_1}{\rho ac}\left[{\varphi }_1(\lambda )+\right. </math><math>\left. {\left(\frac{x}{a}\right)}^2{\varphi }_2(\lambda )\right]\quad \mbox{and}</math>
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<span id='e0040'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
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{| style="text-align: center; margin:auto;" 
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| <math>C(x\mbox{,}\lambda \mbox{,}t)=C_1+\frac{\overset{\cdot}{n_A}}{a\upsilon }\left[g_1(\lambda )+\right. </math><math>\left. {\left(\frac{x}{a}\right)}^2g_2(\lambda )\right]</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
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where <math display="inline">\lambda =\frac{y}{h}</math> and ''F  ''(<math display="inline">\lambda </math>''t  ''), <math display="inline">G(\lambda \mbox{,}t)\mbox{,}{\phi }_1(\lambda )\mbox{,}{\phi }_2(\lambda )\mbox{,}g_1(\lambda )</math> and <math display="inline">g_2(\lambda )</math> are to be determined.
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The boundary conditions for the velocity, microrotation, temperature and concentration are
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<span id='e0045'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>\begin{array}{rl}
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 & u(x\mbox{,}\lambda \mbox{,}t)=0\mbox{,}v(x\mbox{,}\lambda \mbox{,}t)=-V_1\mbox{,}N(x\mbox{,}\lambda \mbox{,}t)=0\mbox{,}\\
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 & T(x\mbox{,}\lambda \mbox{,}t)=T_1\mbox{,}C(x\mbox{,}\lambda \mbox{,}t)=C_1\quad \mbox{at}\quad \lambda =-1\\
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 & u(x\mbox{,}\lambda \mbox{,}t)=0\mbox{,}v(x\mbox{,}\lambda \mbox{,}t)=V_1\mbox{,}N(x\mbox{,}\lambda \mbox{,}t)=0\mbox{,}\\
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 & T(x\mbox{,}\lambda \mbox{,}t)=T_2\mbox{,}C(x\mbox{,}\lambda \mbox{,}t)=C_2\quad \mbox{at}\quad \lambda =1
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\end{array}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
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Substituting [[#e0040|(8)]] in [[#e0010|(2)]], [[#e0015|(3)]], [[#e0020|(4)]] and [[#e0025|(5)]] then we obtain,
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<span id='e0050'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>f^{IV}=\frac{1}{1+R}(-\eta (3f^{//}+\lambda f^{\left//\right/})+</math><math>Re({ff}^{\left//\right/}-f^/f^{//})+{Rg}^{//}+(1+R)D^{-1}f^{//}+</math><math>\frac{{Ha}^2\alpha e}{\alpha e^2+\beta e^2}f^{//}+</math><math>\frac{Ec\quad Gr}{\zeta \quad Re}({\phi }_1^/+{\zeta }^2{\phi }_2^/)+</math><math>\frac{Ec\quad Gc}{\zeta \quad Re}(g_1^/+{\zeta }^2g_2^/))</math>
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|}
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| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
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<span id='e0055'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>g^{//}=-J_1\eta (3g+\lambda g^/)+J_1\quad Re({fg}^/-</math><math>f^/g)-s_1(f^{//}-2g)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
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{| style="text-align: center; margin:auto;" 
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|-
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| <math>{\varphi }_1^{//}=-2{\varphi }_2-Re\quad s_2g^2-Re\quad Pr\left((1+\right. </math><math>\left. R)D^{-1}f^2+\frac{{Ha}^2}{\alpha e^2+\beta e^2}f^2+\right. </math><math>\left. 4f^{/^2}-f{\varphi }_1^/\right)</math>
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<span id='e0060'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;" 
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| <math>-Pr\quad \eta ({\phi }_1+\lambda {\phi }_1^/)-Du(g_1^{//}+</math><math>2g_2)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
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<span id='e0065'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
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| <math>{\varphi }_2^{//}=-Re\quad Pr\left(\frac{R}{2}{\left(f^{//}-2g\right)}^2+\right. </math><math>\left. \frac{s_2}{Pr}g^{/^2}+(1+R)D^{-1}f^{/^2}+{Ha}^2f^{/^2}+\right. </math><math>\left. f^{//^2}+2f^/{\varphi }_2-f{\varphi }_2^/\right)-</math><math>Pr\left(3{\phi }_2+\lambda {\phi }_2^/\right)-Du\quad g_2^{//}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
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<span id='e0070'></span>
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{| style="text-align: center; margin:auto;" 
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| <math>g_1^{//}=-2g_2+Kr\quad g_1+Sc\quad Re\quad {fg}_1^/-</math><math>Sc\quad \eta (g_1+\lambda g_1^/)-Sc\quad Sr({\varphi }_1^{//}+</math><math>2{\varphi }_2)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
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<span id='e0075'></span>
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{| style="text-align: center; margin:auto;" 
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| <math>g_2^{//}=Kr\quad g_2+Sc\quad Re({fg}_2^/-2f^/g_2)-</math><math>Sc\quad \eta (3g_2+\lambda g_2^/)-Sc\quad Sr{\varphi }_2^{//}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
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where prime denotes the differentiation with respect to <math display="inline">\lambda </math> and <math display="inline">f=\frac{F}{Re}\mbox{,}g=\frac{G}{Re}</math>.
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The dimensionless form of temperature and concentration from [[#e0040|(8)]] is
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{| style="text-align: center; margin:auto;" 
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| <math>T^{{_\ast}}=\frac{T-T_1}{T_2-T_1}=Ec({\phi }_1+{\zeta }^2{\phi }_2)</math>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>C^{{_\ast}}=\frac{C-C_1}{C_2-C_1}=Sh(g_1+{\zeta }^2\quad g_2)</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
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The boundary conditions Eq. [[#e0045|(9)]] in terms of <math display="inline">f\mbox{,}g\mbox{,}{\varphi }_1\mbox{,}{\varphi }_2\mbox{,}g_1</math> and <math display="inline">g_2</math> are
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{| style="text-align: center; margin:auto;" 
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| <math>f(-1)=-1\mbox{,}\quad f(1)=1\mbox{,}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
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{| style="text-align: center; margin:auto;" 
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| <math>f^/(-1)=0\mbox{,}\quad f^/(1)=0\mbox{,}</math>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>g(-1)=0\mbox{,}\quad g(1)=0</math>
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{| style="text-align: center; margin:auto;" 
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| <math>{\varphi }_1(-1)=0\mbox{,}\quad {\phi }_1(1)=1/Ec</math>
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{| style="text-align: center; margin:auto;" 
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| <math>{\varphi }_2(-1)=0\mbox{,}\quad {\phi }_2(1)=0\mbox{,}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>g_1(-1)=0\mbox{,}\quad g_1(1)=1/Sh</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>g_2(-1)=0\mbox{,}\quad g_2(1)=0</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (17)
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==3. Solution of the problem==
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The nonlinear Eqs. [[#e0050|(10)]], [[#e0055|(11)]], [[#e0060|(12)]], [[#e0065|(13)]], [[#e0070|(14)]] and [[#e0075|(15)]] are converted into the following system of first order differential equations by the substitution
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(<math display="inline">f\mbox{,}f^/\mbox{,}f^{//}\mbox{,}f^{\left//\right/}\mbox{,}g\mbox{,}g^/\mbox{,}{\varphi }_1\mbox{,}{\varphi }_1^/\mbox{,}{\varphi }_2\mbox{,}{\varphi }_2^/\mbox{,}g_1\mbox{,}g_1^/\mbox{,}g_2\mbox{,}g_2^/</math>) = <math display="inline">\left(x_1\mbox{,}x_2\mbox{,}x_3\mbox{,}x_4\mbox{,}x_5\mbox{,}x_6\mbox{,}x_7\mbox{,}x_8\mbox{,}x_9\mbox{,}x_{10}\mbox{,}x_{11}\mbox{,}x_{12}\mbox{,}x_{13}\mbox{,}x_{14}\right)</math>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_1}{d\lambda }=x_2\mbox{,}\quad \frac{{dx}_2}{d\lambda }=</math><math>x_3\mbox{,}\quad \frac{{dx}_3}{d\lambda }=x_4\mbox{,}</math>
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_4}{d\lambda }=\frac{R}{1+R}\left(-s_1(x_3-\right. </math><math>\left. 2x_5)+J_1(x_1x_6-x_2x_5-3\eta x_5-\lambda \eta x_6)\right)-</math><math>\frac{Re}{1+R}(x_2x_3-x_1x_4)+D^{-1}x_3+\frac{{Ha}^2\alpha e}{\left(\alpha e^2+\beta e^2\right)(1+R)}x_3-</math><math>\frac{\eta }{\left(1+R\right)}\left(3x_3+\lambda x_4\right)+</math><math>\frac{Sh\quad Gm}{(1+R)\zeta \quad Re}(x_{12}+{\zeta }^2x_{14})+</math><math>\frac{Ec\quad Gr}{(1+R)\zeta \quad Re}(x_8+{\zeta }^2x_{10})\mbox{,}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_5}{d\lambda }=x_6\mbox{,}</math>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_6}{d\lambda }=-s_1(x_3-2x_5)+J_1(x_1x_6-</math><math>x_2x_5-3\eta x_6-\lambda \eta x_7)\mbox{,}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_7}{d\lambda }=x_8\mbox{,}</math>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_8}{d\lambda }=-2x_9-\frac{Re\quad Pr}{1-Du\quad Sc\quad Sr}\left(\frac{\eta }{Re}\left(x_7+\right. \right. </math><math>\left. \left. \lambda x_8\right)+4x_2^2+(1+R)D^{-1}x_1^2+\right. </math><math>\left. \frac{{Ha}^2}{\alpha e^2+\beta e^2}x_1^2-x_1x_8+\right. </math><math>\left. \frac{s_2}{Pr}x_5^2\right)-\frac{Kr\quad Du}{1-Du\quad Sc\quad Sr}x_{11}-</math><math>\frac{Du\quad Sc}{1-Du\quad Sc\quad Sr}Re\quad x_1x_{12}+</math><math>\frac{Sc\quad Du}{1-Du\quad Sc\quad Sr}\eta \left(x_{11}+\right. </math><math>\left. \lambda x_{12}\right)\mbox{,}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_9}{d\lambda }=x_{10}\mbox{,}</math>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_{10}}{d\lambda }=-\frac{Re\quad Pr}{1-Du\quad Sc\quad Sr}\left(\frac{\eta }{Re}\left(3x_9+\right. \right. </math><math>\left. \left. \lambda x_{10}\right)+\frac{R}{2}{\left(x_3-2x_5\right)}^2+\right. </math><math>\left. x_3^2+(1+R)D^{-1}x_2^2+\frac{{Ha}^2}{\alpha e^2+\beta e^2}x_2^2+\right. </math><math>\left. 2x_2x_9-x_1x_{10}\right)-\frac{Du\quad Sc}{1-Du\quad Sc\quad Sr}Re(x_1x_{14}-</math><math>2x_2x_{13})+\frac{Du\quad Sc}{1-Du\quad Sc\quad Sr}\eta (3x_{13}+</math><math>\lambda x_{14})-\frac{Kr\quad Du}{1-Du\quad Sc\quad Sr}x_{13}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_{11}}{d\lambda }=x_{12}\mbox{,}</math>
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| <math>\frac{{dx}_{12}}{d\lambda }=-2x_{13}+\frac{Sc\quad Sr\quad Re\quad Pr}{1-Du\quad Sc\quad Sr}\left(\frac{\eta }{Re}(x_7+\right. </math><math>\left. \lambda x_8)+4x_2^2+(1+R)D^{-1}x_1^2+\frac{{Ha}^2}{\alpha e^2+\beta e^2}x_1^2-\right. </math><math>\left. x_1x_8+\frac{s_2}{Pr}x_5^2\right)+\frac{Kr}{1-Du\quad Sc\quad Sr}x_{11}+</math><math>\frac{Sc}{1-Du\quad Sc\quad Sr}Re\quad x_1x_{12}-\frac{Sc}{1-Du\quad Sc\quad Sr}\eta (x_{11}+</math><math>\lambda x_{12})\mbox{,}</math>
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_{13}}{d\lambda }=x_{14}\mbox{,}</math>
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<span id='e0090'></span>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_{14}}{d\lambda }=\frac{Sc\quad Sr\quad Re\quad Pr}{1-Du\quad Sc\quad Sr}\left(\frac{\eta }{Re}(3x_9+\right. </math><math>\left. \lambda x_{10})+x_3^2+(1+R)D^{-1}x_2^2+\frac{{Ha}^2}{\alpha e^2+\beta e^2}x_2^2+\right. </math><math>\left. \frac{s_2}{Pr}x_6^2+\frac{R}{2}{\left(x_3-2x_5\right)}^2+\right. </math><math>\left. 2x_2x_9-x_1x_{10}\right)+\frac{Kr}{1-Du\quad Sc\quad Sr}x_{13}+</math><math>\frac{Sc}{1-Du\quad Sc\quad Sr}Re(x_1x_{14}-2x_2x_{13})-</math><math>\frac{Sc}{1-Du\quad Sc\quad Sr}\eta (3x_{13}+\lambda x_{14})</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (18)
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The boundary conditions in terms of <math display="inline">x_1\mbox{,}x_2\mbox{,}x_3\mbox{,}x_4x_5\mbox{,}x_6\mbox{,}x_7\mbox{,}x_8\mbox{,}x_9\mbox{,}x_{10}\mbox{,}x_{11}\mbox{,}x_{12}\mbox{,}x_{13}\mbox{,}x_{14}</math> are
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<span id='e0095'></span>
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{| style="text-align: center; margin:auto;" 
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| <math>\begin{array}{cc}
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 & x_1(-1)=-1\mbox{,}\quad x_2(-1)=0\mbox{,}\quad x_5(-1)=0\mbox{,}\quad x_7(-1)=0\mbox{,}\quad x_9(-1)=0\mbox{,}\quad x_{11}(-1)=0\mbox{,}\quad x_{13}(-1)=0\mbox{,}\\
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 & x_1(1)=1\mbox{,}\quad x_2(1)=0\mbox{,}\quad x_5(1)=0\mbox{,}\quad x_7(1)=1/Ec\mbox{,}\quad x_9(1)=0\mbox{,}\quad x_{11}(1)=1/Sh\mbox{,}\quad x_{13}(1)=0
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\end{array}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (19)
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The system of Eq. [[#e0090|(18)]] is solved numerically subject to the boundary conditions [[#e0095|(19)]] using the quasilinearization method given by Bellman and Kalaba [[#b0155|[31]]].
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Let (<math display="inline">x_i^r</math>, ''i'' = 1, 2, … , 14) be an approximate current solution and <math display="inline">(x_i^{r+1}</math>, ''i'' = 1, 2, … , 14) be an improved solution of [[#e0090|(18)]]. Using Taylor’s series expansion about the current solution by neglecting the second and higher order derivative terms, the coupled first order system [[#e0090|(18)]] is linearized as follows:
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_1^{r+1}}{d\lambda }\quad =x_2^{r+1}\mbox{,}\quad \frac{{dx}_2^{r+1}}{d\lambda }\quad =</math><math>x_3^{r+1}\mbox{,}\quad \frac{{dx}_3^{r+1}}{d\lambda }\quad =</math><math>\quad x_4^{r+1}\mbox{,}</math>
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_4^{r+1}}{d\lambda }=\frac{Re}{1+R}(x_1^{r+1}x_4^r+</math><math>x_4^{r+1}x_1^r-x_2^{r+1}x_3^r-x_2^rx_3^{r+1})+D^{-1}x_3^{r+1}+</math><math>\frac{{Ha}^2\quad \alpha e\quad x_3^{r+1}}{\left(1+R\right)\left(\alpha e^2+\beta e^2\right)}+</math><math>\frac{R}{1+R}(-s_1(x_3^{r+1}-2x_5^{r+1})+J_1(x_1^{r+1}x_6^r+</math><math>x_1^rx_6^{r+1}-x_2^{r+1}x_5^r-x_5^{r+1}x_2^r-3\eta x_5^{r+1}-</math><math>\lambda \eta x_6^{r+1}))+\frac{E\quad Gr}{(1+R)\zeta \quad Re}\left(x_8^{r+1}+\right. </math><math>\left. {\zeta }^2x_{10}^{r+1}\right)+\frac{Sh\quad Gm}{(1+R)\zeta \quad Re}\left(x_{12}^{r+1}+\right. </math><math>\left. {\zeta }^2x_{14}^{r+1}\right)-\frac{Re}{1+R}\left(-\right. </math><math>\left. x_2^rx_3^r+x_1^rx_4^r\right)-\frac{J_1}{1+R}(x_1^rx_6^r-</math><math>x_2^rx_5^r)-\frac{\eta }{\left(1+R\right)}\left(3x_3^{r+1}+\right. </math><math>\left. \lambda x_4^{r+1}\right)\mbox{,}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_5^{r+1}}{d\lambda }\quad =\quad x_6^{r+1}\mbox{,}</math>
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_6^{r+1}}{d\lambda }=-s_1(x_3^{r+1}-2x_5^{r+1})+</math><math>J_1(x_1^{r+1}x_6^r+x_1^rx_6^{r+1}-x_2^{r+1}x_5^r-x_5^{r+1}x_2^r-</math><math>3\eta x_5^{r+1}-\lambda \eta x_6^{r+1})-J_1(x_1^rx_6^r-</math><math>x_2^rx_5^r)\mbox{,}\quad \frac{{dx}_7^{r+1}}{d\lambda }=</math><math>x_8^{r+1}</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | 
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\frac{{dx}_8^{r+1}}{d\lambda }=-2x_9^{r+1}-\frac{Re\quad Pr}{1-Du\quad Sc\quad Sr}\left(\frac{\eta }{Re}\left(x_7^{r+1}+\right. \right. </math><math>\left. \left. \lambda x_8^{r+1}\right)+8x_2^rx_2^{r+1}+\right. </math><math>\left. 2(1+R)D^{-1}x_1^rx_1^{r+1}+\frac{2{Ha}^2}{\alpha e^2+\beta e^2}x_1^rx_1^{r+1}-\right. </math><math>\left. x_1^rx_8^{r+1}-x_8^rx_1^{r+1}+\frac{2s_2}{Pr}x_5^rx_5^{r+1}\right)-</math><math>\frac{Kr\quad Du}{1-Du\quad Sc\quad Sr}x_{11}^{r+1}+</math><math>\frac{Sc\quad Du}{1-Du\quad Sc\quad Sr}\eta \left(x_{11}^{r+1}+\right. </math><math>\left. \lambda x_{12}^{r+1}\right)-\frac{Du\quad Sc}{1-Du\quad Sc\quad Sr}Re\left(x_1^rx_{12}^{r+1}+\right. </math><math>\left. x_{12}^rx_1^{r+1}-x_1^rx_{12}^r\right)+\frac{Re\quad Pr}{1-Du\quad Sc\quad Sr}\left(4x_2^rx_2^r+\right. </math><math>\left. (1+R)D^{-1}x_1^rx_1^r+\frac{{Ha}^2}{\alpha e^2+\beta e^2}x_1^rx_1^r-\right. </math><math>\left. x_1^rx_8^r+\frac{s_2}{Pr}x_5^rx_5^r\right)\mbox{,}\quad \frac{{dx}_9^{r+1}}{d\lambda }=</math><math>x_{10}^{r+1}\mbox{,}</math>
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|-
686
| <math>\frac{{dx}_{10}^{r+1}}{d\lambda }=-\frac{Re\quad Pr}{1-Du\quad Sc\quad Sr}\left(\frac{\eta }{Re}\left(3x_9^{r+1}+\right. \right. </math><math>\left. \left. \lambda x_{10}^{r+1}\right)+\frac{R}{2}(2x_3^rx_3^{r+1}+\right. </math><math>\left. 8x_5^rx_5^{r+1}-4x_3^rx_5^{r+1}-4x_5^rx_3^{r+1})+\right. </math><math>\left. 2x_3^rx_3^{r+1}+2(1+R)D^{-1}x_2^rx_2^{r+1}+\right. </math><math>\left. \frac{2\quad {Ha}^2}{\alpha e^2+\beta e^2}x_2^rx_2^{r+1}+\right. </math><math>\left. 2x_2^rx_9^{r+1}+2x_2^{r+1}x_9^r-x_1^rx_{10}^{r+1}-\right. </math><math>\left. x_1^{r+1}x_{10}^r\right)+\frac{Du\quad Sc}{1-Du\quad Sc\quad Sr}\eta (3x_{13}^{r+1}+</math><math>\lambda x_{14}^{r+1})-\frac{Du\quad Sc}{1-Du\quad Sc\quad Sr}Re(x_1^rx_{14}^{r+1}+</math><math>x_1^{r+1}x_{14}^r-2x_2^rx_{13}^{r+1}-2x_2^{r+1}x_{13}^r-</math><math>x_1^rx_{14}^r+2x_2^rx_{13}^r)+\frac{Re\quad Pr}{1-Du\quad Sc\quad Sr}\left(\frac{R}{2}(x_3^rx_3^r+\right. </math><math>\left. 4x_5^rx_5^r-4x_3^rx_5^r)+x_3^rx_3^r+(1+R)D^{-1}x_2^rx_2^r+\right. </math><math>\left. \frac{{Ha}^2}{\alpha e^2+\beta e^2}x_2^rx_2^r+\right. </math><math>\left. 2x_2^rx_9^r-x_1^rx_{10}^r\right)-\frac{Kr\quad Du}{1-Du\quad Sc\quad Sr}x_{13}^{r+1}\mbox{,}</math>
687
|}
688
| style="width: 5px;text-align: right;white-space: nowrap;" | 
689
|}
690
691
{| class="formulaSCP" style="width: 100%; text-align: center;" 
692
|-
693
| 
694
{| style="text-align: center; margin:auto;" 
695
|-
696
| <math>\frac{{dx}_{11}^{r+1}}{d\lambda }=x_{12}^{r+1}\mbox{,}</math>
697
|}
698
| style="width: 5px;text-align: right;white-space: nowrap;" | 
699
|}
700
701
{| class="formulaSCP" style="width: 100%; text-align: center;" 
702
|-
703
| 
704
{| style="text-align: center; margin:auto;" 
705
|-
706
| <math>\frac{{dx}_{12}^{r+1}}{d\lambda }=-2x_{13}^{r+1}+\frac{Sc\quad Sr\quad Re\quad Pr}{1-Du\quad Sc\quad Sr}\left(\frac{\eta }{Re}(x_7^{r+1}+\right. </math><math>\left. \lambda x_8^{r+1})+8x_2^rx_2^{r+1}+2(1+R)D^{-1}x_1^rx_1^{r+1}+\right. </math><math>\left. \frac{2{Ha}^2}{\alpha e^2+\beta e^2}x_1^rx_1^{r+1}-\right. </math><math>\left. x_1^rx_8^{r+1}-x_1^{r+1}x_8^r+\frac{2s_2}{Pr}x_5^rx_5^{r+1}\right)-</math><math>\frac{Sc}{1-Du\quad Sc\quad Sr}\eta (x_{11}^{r+1}+</math><math>\lambda x_{12}^{r+1})+\frac{Kr}{1-Du\quad Sc\quad Sr}x_{11}^{r+1}+</math><math>\frac{Sc}{1-Du\quad Sc\quad Sr}Re(x_1^rx_{12}^{r+1}+</math><math>x_1^{r+1}x_{12}^r-x_1^rx_{12}^r)-\frac{Sc\quad Sr\quad Re\quad Pr}{1-Du\quad Sc\quad Sr}\left(4x_2^rx_2^r+\right. </math><math>\left. (1+R)D^{-1}x_1^rx_1^r+\frac{{Ha}^2}{\alpha e^2+\beta e^2}x_1^rx_1^r-\right. </math><math>\left. x_1^rx_8^r+\frac{s_2}{Pr}x_5^rx_5^r\right)\mbox{,}</math>
707
|}
708
| style="width: 5px;text-align: right;white-space: nowrap;" | 
709
|}
710
711
{| class="formulaSCP" style="width: 100%; text-align: center;" 
712
|-
713
| 
714
{| style="text-align: center; margin:auto;" 
715
|-
716
| <math>\frac{{dx}_{13}^{r+1}}{d\lambda }=x_{14}^{r+1}\mbox{,}</math>
717
|}
718
| style="width: 5px;text-align: right;white-space: nowrap;" | 
719
|}
720
721
<span id='e0100'></span>
722
{| class="formulaSCP" style="width: 100%; text-align: center;" 
723
|-
724
| 
725
{| style="text-align: center; margin:auto;" 
726
|-
727
| <math>\frac{{dx}_{14}^{r+1}}{d\lambda }=\frac{Sc\quad Sr\quad Re\quad Pr}{1-Du\quad Sc\quad Sr}\left(\frac{\eta }{Re}(3x_9^{r+1}+\right. </math><math>\left. \lambda x_{10}^{r+1})+2x_3^rx_3^r+2(1+R)D^{-1}x_2^rx_2^{r+1}+\right. </math><math>\left. \frac{2{Ha}^2}{\alpha e^2+\beta e^2}x_2^rx_2^{r+1}+\right. </math><math>\left. \frac{2s_2}{Pr}x_6^rx_6^{r+1}+\frac{R}{2}(2x_3^rx_3^{r+1}+\right. </math><math>\left. 8x_5^rx_5^{r+1}-4x_3^rx_5^{r+1}-4x_5^rx_3^{r+1})+\right. </math><math>\left. 2x_2^rx_9^{r+1}+2x_2^{r+1}x_9^r-x_1^rx_{10}^{r+1}-\right. </math><math>\left. x_1^{r+1}x_{10}^r\right)+\frac{Kr}{1-Du\quad Sc\quad Sr}x_{13}^{r+1}+</math><math>\frac{Sc}{1-Du\quad Sc\quad Sr}Re(x_1^rx_{14}^{r+1}+</math><math>x_1^{r+1}x_{14}^r-2x_2^rx_{13}^{r+1}-2x_2^{r+1}x_{13}^r-</math><math>x_1^rx_{14}^r+2x_2^rx_{13}^r)-\frac{Sc}{1-Du\quad Sc\quad Sr}\eta (3x_{13}^{r+1}+</math><math>\lambda x_{14}^{r+1})-\frac{Sc\quad Sr\quad Re\quad Pr}{1-Du\quad Sc\quad Sr}(x_3^rx_3^r+</math><math>(1+R)D^{-1}x_2^rx_2^r+\frac{{Ha}^2}{\alpha e^2+\beta e^2}x_2^rx_2^r+</math><math>\frac{s_2}{Pr}x_6^rx_6^r+\frac{R}{2}(x_3^rx_3^r+4x_5^rx_5^r-</math><math>4x_3^rx_5^r)+2x_2^rx_9^r-x_1^rx_{10}^r)</math>
728
|}
729
| style="width: 5px;text-align: right;white-space: nowrap;" | (20)
730
|}
731
732
To solve for (<math display="inline">x_i^{r+1}</math>, ''i'' = 1, 2, … , 14), the solution to seven separate initial value problems, are denoted by <math display="inline">x_i^{h1}(\lambda )\mbox{,}x_i^{h2}(\lambda )\mbox{,}x_i^{h3}(\lambda )\mbox{,}x_i^{h4}(\lambda )\mbox{,}x_i^{h5}(\lambda )\mbox{,}x_i^{h6}(\lambda )\mbox{,}x_i^{h7}(\lambda )</math> (which are the solutions of the homogeneous system corresponding to [[#e0100|(20)]]) and <math display="inline">x_i^{p1}(\lambda )</math> (which is the particular solution of [[#e0100|(20)]]), with the following initial conditions are obtained by using the 4th order Runge–Kutta method.
733
734
{| class="formulaSCP" style="width: 100%; text-align: center;" 
735
|-
736
| 
737
{| style="text-align: center; margin:auto;" 
738
|-
739
| <math>\begin{array}{rl}
740
 & x_3^{h1}(0)=1\mbox{,}\quad x_i^{h1}(0)=0\quad \mbox{for}\quad i\quad \not =\quad 3\mbox{,}\\
741
 & x_4^{h2}(0)=1\mbox{,}\quad x_i^{h2}(0)=0\quad \mbox{for}\quad i\quad \quad \not =\quad 4\mbox{,}\\
742
 & x_6^{h3}(0)=1\mbox{,}\quad x_i^{h3}(0)=0\quad \mbox{for}\quad i\quad \quad \not =\quad 6\mbox{,}\\
743
 & x_8^{h4}(0)=1\mbox{,}\quad x_i^{h4}(0)=0\quad \mbox{for}\quad i\quad \quad \not =\quad 8\mbox{,}\\
744
 & x_{10}^{h5}(0)=1\mbox{,}\quad x_i^{h5}(0)=0\quad \mbox{for}\quad i\quad \quad \not =\quad 10\mbox{,}\\
745
 & x_{12}^{h6}(0)=1\mbox{,}\quad x_i^{h6}(0)=0\quad \mbox{for}\quad i\quad \not =\quad 12\mbox{,}\\
746
 & x_{14}^{h7}(0)=1\mbox{,}\quad x_i^{h7}(0)=0\quad \mbox{for}\quad i\quad \not =\quad 14\mbox{,}\\
747
 & x_1^{p1}(0)=-1\mbox{,}\\
748
 & x_2^{p1}(0)=x_3^{p1}(0)=x_4^{p1}(0)=x_5^{p1}(0)=0
749
\end{array}</math>
750
|}
751
| style="width: 5px;text-align: right;white-space: nowrap;" | 
752
|}
753
754
{| class="formulaSCP" style="width: 100%; text-align: center;" 
755
|-
756
| 
757
{| style="text-align: center; margin:auto;" 
758
|-
759
| <math>x_6^{p1}(0)=x_7^{p1}(0)=x_8^{p1}(0)=x_9^{p1}(0)=x_{10}^{p1}(0)=</math><math>x_{11}^{p1}(0)=x_{12}^{p1}(0)=x_{13}^{p1}(0)=x_{14}^{p1}(0)=</math><math>0</math>
760
|}
761
| style="width: 5px;text-align: right;white-space: nowrap;" | (21)
762
|}
763
764
By using the principle of superposition, the general solution can be written as
765
766
{| class="formulaSCP" style="width: 100%; text-align: center;" 
767
|-
768
| 
769
{| style="text-align: center; margin:auto;" 
770
|-
771
| <math>x_i^{n+1}(\lambda )=C_1x_i^{h1}(\lambda )+C_2x_i^{h2}(\lambda )+</math><math>C_3x_i^{h3}(\lambda )+C_4x_i^{h4}(\lambda )+C_5x_i^{h5}(\lambda )+</math><math>C_6x_i^{h6}(\lambda )+C_7x_i^{h7}(\lambda )+x_i^{p1}(\lambda )</math>
772
|}
773
| style="width: 5px;text-align: right;white-space: nowrap;" | (22)
774
|}
775
776
where <math display="inline">C_1\mbox{,}C_2\mbox{,}C_3\mbox{,}C_4\mbox{,}C_5\mbox{,}C_6</math> and <math display="inline">C_7</math> are the unknown constants and are determined by considering the boundary conditions at <math display="inline">\lambda =1</math>. This solution <math display="inline">(x_i^{r+1}</math>, ''i'' = 1, 2, … , 14) is then compared with solution at the previous step (<math display="inline">x_i^r</math>, ''i'' = 1, 2, … , 14) and further iteration is performed if the convergence has not been achieved.
777
778
==4. Results and discussion==
779
780
A numerical solution for the system of nonlinear differential equations Eq. [[#e0090|(18)]] subject to the boundary conditions Eq. [[#e0095|(19)]] is obtained by the method of quasilinearization. The effects of various parameters such as Soret number ''Sr'', Dufour number ''Du'', chemical reaction parameter ''Kr'', inverse Darcy’s parameter ''D''<sup>−1</sup>, Hall parameter <math display="inline">\beta e</math> and ion slip parameter <math display="inline">\beta i</math> on nondimensional velocity components, temperature distribution, microrotation and concentration are discussed through graphs in the domain [−1, 1].
781
782
The effect of ''Sr'' on temperature and concentration is presented in  [[#f0010|Fig. 2]]. From this it is evident that as ''Sr'' increases the concentration is also increasing, whereas the temperature distribution is decreasing. This is because of the mass flux created by the temperature gradient is inversely proportional to the mean temperature, this causes the loss of temperature of the fluid and the concentration of the fluid increases due to the thermal diffusion rate is increasing with the suction/injection velocity.  [[#f0015|Fig. 3]] displays the change in the temperature distribution and concentration for different values of ''Du''. From this it is observed that when ''Du'' increases, the temperature distribution and concentration are decreasing toward the upper wall. This is due to the fact that the energy flux created by the concentration gradient is inversely proportional to the suction/injection velocity.  [[#f0020|Fig. 4]] describes the behavior of the temperature distribution and concentration for various values of ''Kr''. As ''Kr  '' increases the temperature distribution of the fluid also increases, whereas the concentration decreases toward the upper wall. It means that increase in the chemical reaction rate produces a decrease in the species concentration. This causes the concentration buoyancy effects to decrease as chemical reaction increases. The effect of ion slip parameter <math display="inline">\beta i</math> on velocity components, microrotation, temperature distribution and concentration is presented in [[#f0025|Fig. 5]]. From this it is noticed that as <math display="inline">\beta i</math> increases the temperature is increasing, whereas concentration is decreasing toward the upper wall and the microrotation decreases in the region <math display="inline">-1<\lambda <0</math> and increases in <math display="inline">0<\lambda <1</math> whereas the radial velocity follows the opposite trend of microrotation. However, the axial velocity attains the maximum at the center of the walls. This is due to decrease in the effective conductivity which reduces the damping force on the flow field. [[#f0030|Fig. 6]] displays the effect of Hall parameter <math display="inline">\beta e</math> on velocity components, microrotation, temperature and concentration. As <math display="inline">\beta e</math> increases the profiles of velocity components, microrotation, temperature and concentration follow the similar trend of <math display="inline">\beta i</math>. This is because of the velocity of the fluid increases with current density. The variation of the velocity components, microrotation, temperature distribution and concentration for different values of ''D''<sup>−1</sup> is shown in [[#f0035|Fig. 7]]. From this one can analyze that the temperature and concentration are increasing with ''D''<sup>−1</sup> and the radial velocity decreases up to the center of the channel, and then increases whereas the microrotation follows the opposite trend of radial velocity. However, As ''D''<sup>−1</sup> increases the axial velocity decreases at the center of the walls and increasing near the walls due to the resistance offered by the porosity of the medium is more than the resistance due to the magnetic lines of force.
783
784
<span id='f0010'></span>
785
786
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
787
|-
788
|
789
790
791
[[Image:draft_Content_147589898-1-s2.0-S1110016816300059-gr2.jpg|center|357px|Effect of Sr on (a) temperature and (b) concentration for Kr=0.2, Gr=5, Gm=5, ...]]
792
793
794
|-
795
| <span style="text-align: center; font-size: 75%;">
796
797
Figure 2.
798
799
Effect of ''Sr'' on (a) temperature and (b) concentration for ''Kr'' = 0.2, ''Gr'' = 5, ''Gm'' = 5, ''Re'' = 2, ''Du'' = 0.2, ''Sc'' = 0.8, ''Pr'' = 0.2, ''R'' = 10, ''J''<sub>1</sub> = 0.2, ''s''<sub>1</sub> = 2, ''s''<sub>2</sub> = 2, ''Ha'' = 2, ''D''<sup>−1</sup> = 2, <math display="inline">\beta </math>''e'' = 0.2, <math display="inline">\beta </math>''i'' = 0.2, <math display="inline">\eta </math> = 2.
800
801
</span>
802
|}
803
804
<span id='f0015'></span>
805
806
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
807
|-
808
|
809
810
811
[[Image:draft_Content_147589898-1-s2.0-S1110016816300059-gr3.jpg|center|356px|Effect of Du on (a) temperature and (b) concentration for Kr=0.2, Gr=5, Gm=5, ...]]
812
813
814
|-
815
| <span style="text-align: center; font-size: 75%;">
816
817
Figure 3.
818
819
Effect of ''Du'' on (a) temperature and (b) concentration for ''Kr'' = 0.2, ''Gr'' = 5, ''Gm'' = 5, ''Re'' = 2, ''Sr'' = 0.2, ''Sc'' = 0.8, ''Pr'' = 0.2, ''R'' = 10, ''J''<sub>1</sub> = 0.2, ''s''<sub>1</sub> = 2, ''s''<sub>2</sub> = 2, ''Ha'' = 2, ''D''<sup>−1</sup> = 2, <math display="inline">\beta </math>''e'' = 0.2, <math display="inline">\beta </math>''i'' = 0.2, <math display="inline">\eta </math> = 2.
820
821
</span>
822
|}
823
824
<span id='f0020'></span>
825
826
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
827
|-
828
|
829
830
831
[[Image:draft_Content_147589898-1-s2.0-S1110016816300059-gr4.jpg|center|355px|Effect of Kr on (a) temperature and (b) concentration for Du=2, Gr=5, Gm=5, ...]]
832
833
834
|-
835
| <span style="text-align: center; font-size: 75%;">
836
837
Figure 4.
838
839
Effect of ''Kr'' on (a) temperature and (b) concentration for ''Du'' = 2, ''Gr'' = 5, ''Gm'' = 5, ''Re'' = 2, ''Sr'' = 0.02, ''Sc'' = 0.8, ''Pr'' = 0.2, ''R'' = 10, ''J''<sub>1</sub> = 0.2, ''s''<sub>1</sub> = 2, ''s''<sub>2</sub> = 2, ''Ha'' = 2, ''D''<sup>−1</sup> = 2, <math display="inline">\beta </math>''e'' = 0.2, <math display="inline">\beta </math>''i'' = 0.2, <math display="inline">\eta </math> = 2.
840
841
</span>
842
|}
843
844
<span id='f0025'></span>
845
846
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
847
|-
848
|
849
850
851
[[Image:draft_Content_147589898-1-s2.0-S1110016816300059-gr5.jpg|center|px|Effect of βi on (a) axial velocity, (b) radial velocity, (c) microrotation, (d) ...]]
852
853
854
|-
855
| <span style="text-align: center; font-size: 75%;">
856
857
Figure 5.
858
859
Effect of <math display="inline">\beta </math>''i'' on (a) axial velocity, (b) radial velocity, (c) microrotation, (d) temperature and (e) concentration for ''Du'' = 2, ''Gr'' = 5, ''Gm'' = 5, ''Re'' = 2, ''Sr'' = 0.2, ''Sc'' = 0.8, ''Pr'' = 0.2, ''R'' = 10, ''J''<sub>1</sub> = 0.2, ''s''<sub>1</sub> = 2, ''s''<sub>2</sub> = 2, ''Ha'' = 2, ''D''<sup>−1</sup> = 2, ''Kr'' = 2, <math display="inline">\beta </math>''e'' = 0.2, <math display="inline">\eta </math> = 0.2.
860
861
</span>
862
|}
863
864
<span id='f0030'></span>
865
866
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
867
|-
868
|
869
870
871
[[Image:draft_Content_147589898-1-s2.0-S1110016816300059-gr6.jpg|center|px|Effect of βe on (a) axial velocity, (b) radial velocity, (c) microrotation, (d) ...]]
872
873
874
|-
875
| <span style="text-align: center; font-size: 75%;">
876
877
Figure 6.
878
879
Effect of <math display="inline">\beta </math>''e'' on (a) axial velocity, (b) radial velocity, (c) microrotation, (d) temperature and (e) concentration for ''Du'' = 0.02, ''Gr'' = 5, ''Gm'' = 5, ''Re'' = 2, ''Sr'' = 0.2, ''Sc'' = 0.8, ''Pr'' = 0.2, ''R'' = 10, ''J''<sub>1</sub> = 0.2, ''s''<sub>1</sub> = 4, ''s''<sub>2</sub> = 2, ''Ha'' = 2, ''D''<sup>−1</sup> = 2, ''Kr'' = 2, <math display="inline">\beta </math>''i'' = 5, <math display="inline">\eta </math> = 0.2.
880
881
</span>
882
|}
883
884
<span id='f0035'></span>
885
886
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
887
|-
888
|
889
890
891
[[Image:draft_Content_147589898-1-s2.0-S1110016816300059-gr7.jpg|center|px|Effect of D−1 on (a) axial velocity, (b) radial velocity, (c) microrotation, (d) ...]]
892
893
894
|-
895
| <span style="text-align: center; font-size: 75%;">
896
897
Figure 7.
898
899
Effect of ''D''<sup>−1</sup> on (a) axial velocity, (b) radial velocity, (c) microrotation, (d) temperature and (e) concentration for ''Du'' = 0.02, ''Gr'' = 5, ''Gm'' = 5, ''Re'' = 2, ''Sr'' = 0.2, ''Sc'' = 0.8, ''Pr'' = 0.2, ''R'' = 10, ''J''<sub>1</sub> = 0.2, ''s''<sub>1</sub> = 2, ''s''<sub>2</sub> = 2, ''Ha'' = 2, <math display="inline">\beta </math>''i'' = 0.2, ''Kr'' = 2, <math display="inline">\beta </math>''e'' = 5, <math display="inline">\eta </math> = 0.2.
900
901
</span>
902
|}
903
904
The numerical values of axial velocity with <math display="inline">{\alpha }_2=0.5</math> and ''Re'' = 5 for Newtonian fluid case are presented in [[#t0005|Table 1]]. It is observed that the results are showing excellent agreement with Majdalani et al. [[#b0105|[21]]], Asghar et al. [[#b0110|[22]]] and Boutros et al. [[#b0115|[23]]].
905
906
<span id='t0005'></span>
907
908
{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
909
|+
910
911
Table 1.
912
913
The numerical values of axial velocity with present and existing results for <math display="inline">\eta </math> = 0.5 and ''Re'' = 5 (Newtonian case).
914
915
|-
916
917
! <math display="inline">\lambda </math>
918
! Present
919
! Majdalani et al. [[#b0105|[21]]]
920
! Asghar et al. [[#b0110|[22]]]
921
! Boutros et al. [[#b0115|[23]]]
922
|-
923
924
| 0
925
| 1.53785
926
| 1.536002
927
| 1.559474
928
| 1.556324
929
|-
930
931
| 0.05
932
| 1.53344
933
| 1.531846
934
| 1.554822
935
| 1.551780
936
|-
937
938
| 0.1
939
| 1.52025
940
| 1.519377
941
| 1.540888
942
| 1.538164
943
|-
944
945
| 0.15
946
| 1.49834
947
| 1.498596
948
| 1.517743
949
| 1.515522
950
|-
951
952
| 0.2
953
| 1.46782
954
| 1.469505
955
| 1.485503
956
| 1.483935
957
|-
958
959
| 0.25
960
| 1.42883
961
| 1.432114
962
| 1.444331
963
| 1.443517
964
|-
965
966
| 0.3
967
| 1.38158
968
| 1.386445
969
| 1.394435
970
| 1.394421
971
|-
972
973
| 0.35
974
| 1.32627
975
| 1.332539
976
| 1.336071
977
| 1.336839
978
|-
979
980
| 0.4
981
| 1.26318
982
| 1.270464
983
| 1.269540
984
| 1.271006
985
|-
986
987
| 0.45
988
| 1.19258
989
| 1.200325
990
| 1.195188
991
| 1.197207
992
|-
993
994
| 0.5
995
| 1.11478
996
| 1.122275
997
| 1.113403
998
| 1.115778
999
|-
1000
1001
| 0.55
1002
| 1.03011
1003
| 1.036527
1004
| 1.024617
1005
| 1.027110
1006
|-
1007
1008
| 0.6
1009
| 0.93888
1010
| 0.943364
1011
| 0.929302
1012
| 0.931656
1013
|-
1014
1015
| 0.65
1016
| 0.84142
1017
| 0.843156
1018
| 0.827971
1019
| 0.829933
1020
|-
1021
1022
| 0.7
1023
| 0.73802
1024
| 0.736373
1025
| 0.721170
1026
| 0.722523
1027
|-
1028
1029
| 0.75
1030
| 0.62892
1031
| 0.623597
1032
| 0.609480
1033
| 0.610078
1034
|-
1035
1036
| 0.8
1037
| 0.51432
1038
| 0.505538
1039
| 0.493513
1040
| 0.493322
1041
|-
1042
1043
| 0.85
1044
| 0.39429
1045
| 0.383052
1046
| 0.373909
1047
| 0.373046
1048
|-
1049
1050
| 0.9
1051
| 0.26877
1052
| 0.257149
1053
| 0.251330
1054
| 0.250109
1055
|-
1056
1057
| 0.95
1058
| 0.13751
1059
| 0.129010
1060
| 0.126461
1061
| 0.125435
1062
|-
1063
1064
| 1
1065
| 0.000000
1066
| 0.000000
1067
| 0.000000
1068
| 0.000000
1069
|}
1070
1071
==5. Conclusions==
1072
1073
The influence of Hall and ion slip currents on the free convective flow of chemically reacting micropolar fluid in a porous expanding or contracting walls with Soret and Dufour effects is considered. The numerical solution of the reduced governing equations is obtained by the method of quasilinearization. It is observed that.
1074
* The temperature distribution of the fluid is decreased with the increase of ''Sr'' and ''Du'' and the concentration of the fluid is enhanced with ''Sr'' whereas it is decreased as ''Du'' increases.
1075
* The velocity components, microrotation, temperature and concentration of the fluid have the similar effects for Hall and ion slip parameters.
1076
* The concentration of the fluid is decreased whereas the temperature is enhanced with ''Kr''.
1077
* ''D''<sup>−1</sup> exhibits the similar effect for temperature and concentration of the fluid.
1078
* The present results are compared with previously published work [[#b0105|[21]]], [[#b0110|[22]]] and [[#b0115|[23]]] and found that the axial velocity values are showing a remarkable agreement.
1079
1080
These results have possible applications in engineering and applied sciences such as the regression of the burning surface in solid rocket motors, paper manufacturing, irrigation, and transport of biological fluids through expanding or contracting vessels.
1081
1082
==Acknowledgments==
1083
1084
The authors are thankful to the editor and referees for their valuable suggestions to improve the version of the paper. Also, one of the authors (N. N. K) gratefully acknowledges the Defence Research and Development Organization (DRDO), Government of India for the financial support as a Senior Research Fellowship ([[#gp005|DIAT/F/REG(G)/1613]]).
1085
1086
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1087
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