Effects of the induced magnetic field, thermophoresis, and Brownian motion on mixed convective Jeffrey nanofluid flow through a porous channel

The main purpose of this research is to explore a comparative study of viscous and Jeffrey nanofluid flows through a parallel channel embedded in a porous medium under the influence of an induced magnetic field, Brownian motion, and thermophoresis. The convective boundary conditions are employed to study the heat and mass transfer at the lower plate. The system of transport constituent relations is reduced into coupled nondimensional ordinary differential equations through similar variables with appropriate boundary conditions. The resulting equations are analyzed for flow characteristics, heat and mass transfers, and magnetic diffusivities throughout the channel with various physical nondimensional parameters via shooting technique along with Runge‐Kutta fourth‐order scheme. It is observed that the temperature and concentration decrease with increasing Brownian motion parameters for both the viscous and Jeffrey fluid. The velocities decrease with increasing of the inverse Darcy parameter for Jeffrey fluid whereas velocities increase for a viscous fluid. The profiles of temperature and concentration for both fluids decrease with increasing of the Brownian motion parameter. The velocity profiles rise with suction/injection parameter for the both fluids. Finally, the numerical results of the present method are compared with a work available in the literature for the Newtonian case. An excellent agreement is found between the present and published numerical results.


INTRODUCTION
heat and mass transfer in a non-Newtonian fluid through porous boundaries have received great interest due to their applications in engineering and industry, such as electrostatic precipitation, aerodynamic heating, dying of paper and textile, solidifications of liquid crystals, food preservation, petroleum industries, cooling of metallic sheet in the bath, grain regression, magnetohydrodynamic (MHD) pumps, and magenetohydrodynamic generators. 4 A steady three-dimensional flow of an incompressible Jeffrey fluid past bidirectional stretching surface have reported by Hayat et al. 5 Ahmed et al 6 discussed the dilating and the squeezing porous channel flow of an incompressible Jeffrey fluid and the flow is generated due to suction/injection at the walls. Shehzad et al 7 have disclosed the nature of the thermophoresis and Brownian motion on a three-dimensional channel flow of an incompressible, MHD Jeffrey nanofluid in the presence of thermal radiation. An investigation explores the characteristics of thermophoresis and Brownian motion with the MHD Jeffrey fluid flow over a stretching surface by Khan et al. 8 Murthy et al 9 surveyed the significance of thermophoresis and Brownian motion effects on MHD slip flow of Casson nanofluid through an exponentially stretching surface. Shah et al 10 inspected the influence of thermophoresis and Brownian motion over a rotating third grade fluid flow between two parallel plates. Khan et al 11 have discussed of MHDs and radiative heat transportation in convectively heated stratified flow of Jeffrey nanofluid.
A laminar incompressible flow through the parallel channel with suction or injection at the wall have been attending many researchers for the last few decades. Berman 12 was the first investigator for the laminar incompressible flow of a viscous fluid with suction/injection at the uniform porous walls with different permeability, and later, the work was extended with consideration MHD and heat and mass transfers by Terrill, 13 Walker and Davies, 14 Terrill and Shresth, 15 Nigam and Singh, 16 Cox, 17 Bujurke et al, 18 and Ganesh and Krishnambal. 19 A study on viscous fluid through a parallel channel with bottom injection and top suction was explained by Hafeez and Ndikilar. 20 Rao and Moizuddi 21 analyzed a mathematical model with a perturbation technique for a steady, incompressible flow of micropolar fluid through a parallel channel with suction/injection at the walls. The MHD flow of upper-convected Maxwell fluid through a parallel channel that is embedded in a porous medium has been reported by Hayat et al. 22 In recent years, most of the researchers have been modeling theoretically to investigate the heat enhancement process by suspending high thermal conductivity nanoparticle in the base fluid. Choi and Eastman 23 was first who proposed the word nanofluids. From there onwards, many researchers have extended this work most recent, and Dogonchi et al 24 explored the heat transfers of MHD squeezing flow of viscous nanofluid between two parallel plates in the presence of thermal radiation. Sheikholeslami et al 25 examined the influence Brownian and thermophoresis characteristics over a steady laminar an incompressible nanofluid flow in a rotating horizontal channel. Hayat et al 26,27 investigated the impact of Cattaneo-Christov heat flux model in flow and stagnation point flows, respectively. MHD stagnation point flow of Casson fluid toward a stretching sheet is addressed by Khan et al. 28 A numerical solution is obtained to reveal effective thermal conductivity of the copper-water nanofluid through a parallel channel with consideration of Brownian motion, which was studied by Khan et al. 29 Makinde and Aziz 30 analyzed the significance of Brownian motion and thermophoresis diffusivity on a steady two-dimensional boundary layer flow of nanofluid over a stretching sheet. The effects of thermophoresis and Brownian motion of a nanofluid flow between two vertical plates embedded in a porous medium are reported by Matin and Ghanbari. 31 Babu et al 32  Many authors have considered the effect of the induced magnetic field on an incompressible laminar peristaltic flow of nanofluids, [47][48][49][50][51][52] and very few reports are available other than the study of the induced magnetic field with peristaltic flow. Denno and Fouad 53 have investigated the impact induced magnetic field due to strong nonuniform applied magnetic field on an inviscid flow through the parallel channel. Singh and Singh 54 addressed the effect of induced magnetic fluid on an incompressible, MHD flow of viscous fluid in a rotating channel bounded by nonconducting plates. A numerical report by Ibrahim 55 delights that the impact of induced magnetic field on an MHD flow of an incompressible nanofluid toward a stagnation point over a stretching sheet with Brownian movement and thermophoresis and furthermore utilizing convective boundary conditions. This numerical report was extended by Sandeep et al 56 with employing chemical reaction. Raju and Ojjela 57 have studied with the numerical technique significance of the induced magnetic field and variable thermal conductivity on convective Jeffrey fluid flow with nth-order chemical reaction.
The main objective of this article is to address the comparative study of viscous and Jeffrey nanofluids flow between two parallel plates with periodic injection/suction at the walls with convective boundary conditions embedded in porous medium under the influence of induced magnetic field, Brownian, and thermophoretic diffusivity. The flow is assumed to be a laminar, incompressible, and unsteady. The governing flow field equations are reduced into highly nonlinear coupled equations by using similarity transformation, and then, the results are elaborated with the graphs and tables through the numerical technique, namely, shooting method.

MATHEMATICAL FORMULATION
Consider an unsteady laminar, incompressible MHD flow of Jeffrey fluid through a porous medium between two parallel porous plates at y = 0 and y = h. A Cartesian coordinate system is chosen such that the axial (u) and transverse (v) velocity components are along X-and Y-directions, respectively, as shown in Figure 1. Here, assume that, at the lower plate, the fluid injected into the channel with the velocity of V 1 e i t and the suctioning fluid with the velocity V 2 e i t from the upper plates and subject to the condition |V 2 | ≥ |V 1 |. The lower and upper plates are maintained at two different temperatures T 1 e i t and T 2 e i t concentrations C 1 e i t and C 2 e i t , respectively. A uniform magnetic field is applied of strength B 0 along Y-direction, which leads to an induces the magnetic field B x and B y in X-and Y-directions, respectively. Therefore, the total magnetic field vector becomes B(B x , B 0 + B , 0).
Under the above assumptions, the governing equations for conservative momentum, energy, concentration, and induction equations with thermophoresis and Brownian motion are 24,26,46,47 The Maxwell equations and Ohm's law by neglecting the displacement current are From Equation (6), we reduced into an induction equation as where is the dynamic viscosity, k is the thermal conductivity, k 1 is the wall permeability, e is the magnetic permeability, is the density of the fluid, T and c are coefficients of thermal and solutal expansions, respectively, g is the acceleration due to gravity, c is the specific heat at constant temperature, is the electric conductivity of the fluid, B = B 0 +b magnetic field vector, and b = B x̄+ B is an induced magnetic field vector.
The above model has been reduced for the viscous nanofluid by assuming 1 = 2 = 0. The associated boundary conditions are given as follows at the lower and upper plates: Let us choose the similarity transformations as follows 11,12,14,18,47 : where = y/h and f, 1 , 2 , g 1 , g 2 , and are the dimensionless unknown functions to be computed. The skin friction coefficient and the heat and mass transfer rates are also calculated through the Nusselt and Sherwood numbers at upper and lower the plates Substitute the above similar variable Equation (10) into the governing of momentum, energy, concentration, and induction Equations (2) to (5) and then eliminating the pressure gradient. We obtain a system of the nondimensional coupled nonlinear ordinary differential Equations (14 cos + Ec Nt Nondimensional skin friction is given by where the prime represents the differentiation of unknown function with respects to the nondimensional variable. From Equation (10), the nondimensional temperature and concentrations are given by The corresponding dimensionless form of the boundary conditions in terms of unknown function f, 1 , 2 , g 1 , g 2 , and are

NUMERICAL SOLUTION OF THE PROBLEM
The model that has been transformed into a highly nonlinear coupled ordinary differential Equations (14) with an associate dimensionless boundary conditions (17) is not possible to solve analytically so that we adopt a numerical technique, namely, shooting technique along with the Runge-Kutta fourth-order integration scheme. In order to solve Equations (14), we have converted the system of equations into sixteen first order simultaneous ordinary differential equations with sixteen unknowns. In this context, we required 16 initial conditions, but we have nine initial conditions that are known and the rest of the unknowns are calculated with generalized Newton-Raphson technique with Runge-Kutta fourth-order scheme that satisfies the end conditions and repeated the technique until to get the results within the tolerance limit 10 −5 .
The whole calculation has done with MATLAB software by taking step size as = 0.01.

NUMERICAL RESULTS AND DISCUSSION
In this discussion, we have studied comparatively for viscous and Jeffrey nanofluids with various dimensionless involved key parameters, namely, Eckert number (Ec), inverse Darcy parameter (Da          Figures 14 and 15 represent the temperature and concentration of the fluid are decreasing with increasing of Nb for both viscous and Jeffrey nanofluids. It is interesting to note that the Brownian motion of nanoparticles at molecular and nanoscale levels is a key nanoscale mechanism governing their thermal and solute behaviors, and Figure 16 explores the ′ on Nb and we noticed that the profile of ′ is increases with increasing of Nb for Jeffrey fluid, whereas it is decreasing for viscous fluid.   Table 1 for different values the parameters a, Nb, Nt, and St. It is witness that, by varying Nb, the skin friction and mass transfer rate are decreased at both the plates, whereas heat transfer rate is increasing at upper plate and decreasing at lower plate. For different values of the suction/injection ratio (a), the mass transfer rates are increasing, whereas heat transfer rates are decreasing at the lower and upper plates. However, the skin friction at upper plate increases, whereas it decreases at the lower plate. Table 2 shows that the validation of the present shooting method with perturbation series solution for the Newtonian case. It is clear that the present shooting technique along with Runge-Kutta fourth-order scheme has good agreement with the analytical method to the skin friction values at lower and upper plates by neglecting the

CONCLUDING REMARKS
In the present article, a comparative analysis has been performed to investigate the effects of the induced magnetic field, Brownian motion and thermophoresis on an incompressible, and laminar flow of viscous and Jeffrey nanofluids through a porous medium between two parallel porous walls with convective boundary conditions. The governing flow field equations are transformed into a set of nonlinear ordinary differential equations through similarity variables. The results are numerically analyzed for various nondimensional functions, which govern the flow, energy, mass, and magnetic diffusivity with pertain nondimensional parameters. The shooting technique along with the Runge-Kutta fourth-order algorithm has been employed. Therefore, the following observations can be drawn.
• The profiles of temperature and concentration for both fluids decrease with increasing of the Brownian motion parameter. • The velocity profiles raise with suction/injection parameter for the both fluids.
• The tendency of axial induced magnetic field is enhanced with the Ec and Nb nondimensional parameter for Jeffrey fluid, and at the same time, it decreases for the viscous fluid. • The velocities decrease with increasing of inverse Darcy parameter for Jeffrey fluid, whereas for viscous fluid, it is increasing. • A comparison of a limiting case as made for the present numerical method with published article in the literature has better compatibility. 15 The results have many feasible applications in engineering, applied science, and biological flows such as transpiration cooling, electrostatic precipitation, aerodynamic heating, polymer technology, preservation of food, filtration, boundary layer control, petroleum industry, aerodynamic heating, MHD pumps, artificial dialysis, blood flow through the arteries, and adsorption.