Numerical solution of micropolar nanofluids with Soret, Dufor effects and multiple slip conditions

The objective of this study is to explore the effects of Dufour and Soret with multiple slip conditions on 3D boundary layer flow of a micropolar nanofluid. The mathematical modeling for the flow problem has been created. Using appropriate similarity transformation and non-dimensional variable, the governing non-linear boundary value problem is reduced to a coupled high order non-linear ordinary differential equations which is numerically solved. The solutions were using method bvp4c from matlab software for different quantities of governing parameters. The influences of different parameters on skin friction coefficients − f ″ 0 , − g ″ 0 , wall duo stress coefficients − ϕ 1 ′ 0 , − ϕ 2 ′ 0 the Nusselt number − θ ′ 0 , Sherwood number − s ′ 0 as well as the velocities, temperature, and concentration are examined and discussed through tables and graphs. The finding results indicate the skin friction coefficients, wall duo stresses coefficients (along both x- and y-axes), the Nusselt number and the Sherwood number demonstrate decreasing behavior for an increase in slip parameters but the opposite effects are detected with the values of stretching ratio parameters. A comparison with previous studies available in the literature has been done and a very good agreement is obtained.


Abstract
The objective of this study is to explore the effects of Dufour and Soret with multiple slip conditions on 3D boundary layer flow of a micropolar nanofluid. The mathematical modeling for the flow problem has been created. Using appropriate similarity transformation and non-dimensional variable, the governing non-linear boundary value problem is reduced to a coupled high order non-linear ordinary differential equations which is numerically solved. The solutions were using method bvp4c from matlab software for different quantities of governing parameters. The influences of different parameters on skin friction coefficients ( ( ) ( )) - - f g 0 , 0 , wall duo stress coefficients s 0 as well as the velocities, temperature, and concentration are examined and discussed through tables and graphs. The finding results indicate the skin friction coefficients, wall duo stresses coefficients (along both x-and y-axes), the Nusselt number and the Sherwood number demonstrate decreasing behavior for an increase in slip parameters but the opposite effects are detected with the values of stretching ratio parameters. A comparison with previous studies available in the literature has been done and a very good agreement is obtained.

Introduction
Eringen [1] is a pioneer in conceptualization of the magnificent non-Newtonian fluid called micropolar fluid in the history of fluid dynamics. Then after two years, Eringen [2] himself has developed the ground-breaking theory on micropolar fluid. Using this theory, the mathematical model of many non-Newtonian fluids were developed for which the convectional Navier-Stokes theory unable determines it. Following that, the computation of microplar fluid has received considerable attentions by different researches in the area of fluid dynamics. Accordingly, Xinhui et al [3] have examined the boundary layer flow of microplar fluid past expanding/contracting wall with porosity. Furthermore, Uddin and Kumar [4], Kucaba [5] and Ali and Alam [6] have analyzed the flow of microplar fluid under the conditions such as magnetic field, thermo diffusion and diffusion thermo and hall effect past a surface in different shape such as wedge, stretching sheet and other. Moreover, Noor et al [7] have computed the effect of density variation on microplar fluid with nonzero velocity slip condition. Furthermore, Haque et al [8] have examined the influences of viscous dissipation, Haartmann effect, convection, and joule heating on the flow of microplarfluid.
Alam et al [9] discussed the influence of time variation, second order slip flow, convective heating over a porous wedge with contracting or expanding characteristics. The result their study shows that the slip velocity parameter considerable affects the flow field.
Also, Nabwey et al [10] have presented the analysis of non-Newtonian nanofluid with the combined effect of natural and free convection flow past a vertically aligned extensible surface. The result signifies that Brownian motion and thermophoresis parameters pay a great role in deterring heat transfer rate at the surface. Moreover, Rasool et al [11] have examined the contribution of Laurenth force, chemical reaction and thermal radiation on the flow of nanofluid with Marangoni convective condition. Further, Rasool et al [12] studied the Marangoni and magnetic field effect on non-Newtonian nanofluid. The results indicate that Marangoni ratio significantly reduces the velocity profiles.
Subhakar and Gangadhar [13] have discussed the importance of thermo-diffusion, diffusion thermo and heat generation/suction on the flow Newtonian fluid. The result of the study reflects that heat source enhances the velocity and temperature of the flow. The outcomes of the study signify that Dufour and Soret numbers are to raise the skin friction.
Similarly, the effects of Dufour and Soret were reported on different papers by (EI-Kabeir et al [14], Chamkha and Rashad [15], Rashad and Chamkha [16], EL-Kabeir et al [17] and Al-Mudhaf et al [18]. It was found that an enhancement in the Dufour number produced a reduction in the local Nusselt number but boosts the local Sherwood number. However, the opposite behavior was predicted as Soret number raised for which the local Nusselt number was enlarged while the local Sherwood number was reduced. Hayat et al [19] have examined the 3D flow of couple stress fluid with effects of chemical reaction and time variation over a stretching sheet.
Sumitra [20] and Rashad et al [21] also reported the problem of mixed convection heat and mass transfer near an exponentially stretching surface in combination with magnetic flied, and Soret and Dufour effects. It was found that increasing material parameter resulted in enhancement of all the skin friction coefficient, Nusselt and Sherwood numbers. Moreover, Ravi et al [22] presented the effects of time variation and buoyance force on convective flow of micropolar fluid. It is indicated that vortex viscosity, material parameter and Prandtl number have a significant impact on flow field. The problem of heat and mass transfer on MHD boundary layer flow over a moving a vertical porous plate with suction, Soret and Dufour effects is investigated by Srinivasa et al [23].
Moreover, Uddin et al [24] addressed the impact of various physical parameters such as viscous and joule heating, multiple wall slip phenomena, mass diffusivity, variable viscosity, thermal conductivity in porous media. Furthermore, RamReddy et al [25] analyzed the effects of soret parameter on mixed convection flow along a vertical plate in nanofluid under the convective boundary conditions. Also Award and Sibanda [26] studied the effects of the Dufour and Soret numbers on the heat and mass transfer on a micropolar fluid through a horizontal channel.
Ibrahim [27] and Ibrahim [28] presented the investigation on micropolar fluid taking into account various determing parameters like Hartmann number, second order slip and passive control of nanoparticles. The results designated that the flow field and the engineering aspects of the study are sturdily affected by slip and material parameters.
Rahman [29] and Ishak et al [30] have examined the boundary layer flow of micropolar fluid over isothermal surface in the presence thermal radiation, viscous dissipation and Joule heating. Further, the examination of micropolar fluid on bi-directional stretching surface has been carried out by Ali et al [31] and Ahmad et al [32]. The result shows that the boundary layer configuration is significantly affected by the material parameter.
The above cited research papers have discussed the flow over a plane, a vertical stretching surface, with MHD boundary flow in other area. In this paper, we study numerically the combined effects of Dufour, Soret and multiple slip conditions on three dimensional boundary layer flows to ward stretching sheet in micropolar nanofluid, using bvp4c from matlab. The effects of physical parameters on fluid velocity, skin friction coefficients, microrotation, wall couple stresses, temperature, Nusselt number, Sherwood number and concentration were considered and shown in graphs and tables as well.

Mathematical formulation
Consider a steady, laminar, three dimensional boundary layer flow of a viscous, incompressible micro polar nanofluid past a stretching sheet with Soret, Dufour effects, and multiple slip conditions. The fluid, which occupies a semi-infinite domain bounded by an infinite surface remains in rest. Presumed the flat surface is stretched in two lateral directions. The sheet is stretched in such a way that, the stretching velocity ( ) = u x ax w along x-z plane, ( ) = v x by w along y-z plane and = w 0. The flow is limited to  z 0. The induced firstorder slips flow at the wall are given by U slip ,V slip , T slip and C slip . The wall temperature = + D ¥   T  T  T. w Similarly, the wall concentration, , 0,ΔT and ΔC are constants. The harmonized frame chosen in such that x-z and y-z planes are extending along the extending sheet and z-axis is normal to them as shown in figure 1 below.
Following Ziaul et al [13], Ali et al [32] and Ahmad et al [33] the governing equations of boundary layer flow can be defined as: The boundary conditions at the surface and far from the plate appropriate to the flow are defined as: Where u, v and w are velocity components along the x-, y-and z-axes, correspondingly; ρ represents the fluid density; T and C are the fluid temperature and the species concentration, is change in concentration. μ is dynamic coefficient viscosity, N * 1 and N * 2 stand for the components of the microrotation vector N normal to the planes x-z, and y-z in that order. n = j a is the microinertia density, κ stands for coefficient of vortex viscosity, K is thermal conductivity, represents the microrotation(the spin gradient ) viscosity, β is the material parameter, ¡ = r r c c p f is the ratio between the effective heat capacity of the nano particle material and the heat capacity of the fluid, D B and D T stand for the Brownian and the thermophoretic diffusion coefficient respectively, c c T T , , , p s K m are specific heat capacity at constant pressure, the concentration susceptibility, the thermal -diffusion ratio,and the fluid mean temperature, correspondingly, D TC and D CT represent the Dufour and the Soret-type diffusivity, in that order. U w , V w , are wall velocity along x-, and y-axes, respectively, N 1 , N 2 stand wall microrotations x-, and y-axes, in that order, n is microrotation parameter which is constant such that 0n1. When n=0 that is, N 1 =N 2 =0, the flow particle concentration is strong in which the microelements closed to the wall surface are unable to go around. When = n , 1 2 the anti-symmetric part of the stress tensor is vanishing which means weak concentration whilst the case = n 1 is used for developing the turbulent boundary layer flows. Following Noor et al [8] and Uddin et al [25], the U slip , V slip velocities and the T slip temperature, C slip concentration beside the stretching surfaces at the wall presented and defined as: where, α w is the dimensional slips along stretching surfaces, h and d are constants. The similarity variable is defined by: The dimensionless variables f f q f g s , , , , , 1 2 are introduced as:

* *
The equation of continuity is satisfied if the we choose a stream function ψ(x, y) such that By means of the above similarity transformation and dimensionless variables, the main equations (2)-(7) are reduced into the ordinary differential equations as follow: With the boundary conditions: Here the governing parameters are defined by:  Where f and g are the dimensionless stream functions along the x-and y-axis respectively, f′ and g′ stand for the dimensionless velocities, f 1 , f 2 , q and s, the component of angular velocity along x-and y-axes, temperature, and concentration correspondingly. η is the similarity variable, the primes (",",') show differentiation with respect to η. The Prandtl number, Pr, gives a measure of relative value of momentum and heat transport by diffusion in the velocity and thermal boundary layers, correspondingly. The value = Pr 2indicates that the relative growth of velocity boundary layer is twice the growth of the thermal boundary layer. D f , Sr, Sc, Nb, and Nt represent the Dufour number, the Soret number, the Schmidt number, the Brownian motion parameter, and the thermophoresis parameter in that order.
The engineering quantities of curiosity in this study are skin friction coefficients, Nusselt and Sherwood numbers, wall couple stresses M zx and M zy . These can be defined as:  Using the above equations, we get:

Numerical solution
The coupled six harmonized high order ordinary differential equations (13)- (18), subjected to the boundary conditions, equation (19), are explained numerically using the function bvp4c from matlab software for different values of physical parameters, namely slip parameters (A, B, D, H), Prandtl number Pr, Dufour number Df, Soret number Sr, Brownian motion parameter Nb, thermophoresis parameter Nt, Schmidt number Sc, stretching ratio parameter λ, material parameter β and micro rotation parameter n. Numerical results are found using Matlab BVP solver bvp4c from matlab which is a finite difference code that applies the three-stage Lobatto IIIa formula. To apply bvp4c from matlab: equations (13)-(18) are converted into a system of first-order equations. Second, set up a boundary value problem (bvp) and use the bvp solver in matlab to numerically solve this system, with the above boundary condition and take on a suitable finite value for the far field boundary condition, that is, h  ¥, say η ∞ =50. The step-size is taken as Δη=0.01 and accuracy to the fifth decimal place as the measure of convergence. In solving the BVP by using matlab, bvp4c has only three arguments: a function ODEs for calculation of the residual in the boundary conditions, and a structure solint that provides a guess for a mesh. The ODEs are handled exactly as in the Matlab IVP solvers. Further clarification on the procedure of bvp4c is found in the book by Shampine et al [33] Letting

Results and discussion
In this section, the influences of various governing physical parameters on non-dimensional velocities, temperature, concentration, microrotations, skin friction coefficients, wall couple stress coefficients, local heat and mass transfer rates have been considered.

Velocity profiles
The dimensionless velocity profile graphs of f′(η) and g′(η) for different values of the first order slip parameters A, B, stretching ratio λ and material parameter β are represented in figures 2-4. Figure 2 illustrates that an increase in values of slip parameters (A or B), causes a reduction in the fluid velocity profiles. However, figure 3 shows that rise in the magnitude of material parameters β, reduces the viscosity of the fluid flow that results increase in the velocity boundary layer thicknesses. Next, figure 4 reveals that an enlargement in the values of stretching ratio parameters λ, upsurge the fluid velocity g′(η). On the other hand, an enhancement in the magnitude of stretching ratio parameter λ, results in reduction of the velocity boundary layer thickness along x axis. Figures 5, 6 show the upshots of material parameter β and microrotation parameter n on an angular velocity f 1 (η) and f 2 (η). Figure 5 indicates that arises in the quantities of β, reduce the absolute values of the angular velocities. These effects occur due to the fact that an increase in the values of material limitation, raise the vortex viscosity of the fluid that causes the above effects. However, figure 6 shows that rise in the values of microrotation parameter, falls the resistance to rotate the fluid that results raise the absolute values of angular velocities.

Temperature profiles
The consequences of the slip parameter H, Dufour number Df, Soret number Sr, Prandtl number Pr, Brownian motion parameter Nb on temperature sketches are given in figures 7-9. Figure 7 indicates that boost in the values of slip parameter and Prandlt number, decline the Nusselt number, the temperature distribution. These effects are happen toward enhancing resistance to diffuse the temperature that causes decrease the Nusselt number, temperature dissemination. However, figure 8 show that as values of Dufour number, Soret number grow, result in reducing the kinematic viscosity (or increasing the thermal conductivity) of the fluid which causes increase thermal boundary layer thicknesses as shown in figure 8 (a), and by same reasoning, it correlate with decrease Nusselt number plus its boundary layer thicknesses, figure 8 (b). Moreover, figure 9 illustrates that grow in the amount of Brownian motion constraint, drop the kinematic viscosity (or enhance the Brownian  diffusion coefficient) of the fluid and resistance to diffuse the temperature that decrease in temperature profile and Nusselt number.    Table 1 drawn to assess the exactness of the method used association with previously presented data feasible in the literature has been made. From table 1, it can be seen that the numerical values of the skin friction coefficients,

Skin frictions
in this paper for various values of β and λ when n=0 is in an excellent agreement with the pervious outcomes of published paper by Ali J [31].
The outcomes indicate that the numerical used in the report is appropriate and highly accurate. Table 2 points out that an increase in values of slip parameters (A and B), causes reduction in the skin friction coefficients along both axes. Moreover, as the magnitudes of material β and microrotation parameters n increase, reduces opposition to flow the fluid that results reduction of skin friction coefficients. However, an increase in the magnitude of stretching ratio parameter λ, results an increase in the values of skin friction coefficients along both axes.      Table 2 illustrates that an increase in values of slip parameters (A and B), causes reduction in wall couple stress coefficients along both axes. However, rise in the magnitudes of material parameter β, drops the viscosity of the fluid flow that results increase(it upsurge the values of ) in wall couple stress coefficients along both axes. Moreover, an increase in the values of stretching ratio parameter λ, results boost in wall couple stress coefficients nearby both axes. Table 2 indicates that an enhancement in quantities of slip parameters (A and B), causes reduction in the Nusselt number. However, rise in the magnitude of material parameters β and microrotation n, drops the viscosity of the fluid flow that results in rise the Nusselt number. What's more, an increase in the magnitude of stretching ratio parameter λ, results in rise the value of the Nusselt number. Table 3 it can be seen that increase in the values of slip parameters (H, D) and Prandlt number, reduce the local Nusselt number. Moreover, as the absolute values of Dufour and Schmidt numbers, and thermophoresis parameter grow, result in reducing the kinematic viscosity of the fluid that causes decrease local Nusselt number. The computed values of skin friction coefficients -f′(0) and −g′(0), −f 1 (0), f 2 (0), Nusselt number q′(0), Sherwood number s′(0) when H=D=Nb=Nt=Df=0.5, Pr=2, Sc=Sr=0.2 for different values of A, B, n, β and λ.   Table 2 point out that an enlargement in values of slip parameters (A and B), causes reduction in the Sherwood number. Moreover, rise in the magnitude of material parameters β and microrotation n, reduces resistance to flow the fluid that results decrease in the value of Sherwood number. However, arise in the magnitude of stretching ratio parameter λ, results in rise the value of the Sherwood number. Table 3 it can be seen that as H and D are increasing, the absolute values of local Sherwood number is decreasing. These outcomes are happen due to the fact that fall the mass diffusion, associate in reducing the local Sherwood number. However, as the values of Df and Sc grow, drop the kinematic viscosity of the fluid results the local Sherwood number growth with Df and Sc. Moreover, boost in the amount of Prandtl number, increases resistance to diffuse the temperature that results increase in the value of local Sherwood number. On contrary, enhance in absolute value of thermphoresis parameter, causes in reduce the value of local Sherwood number.

Conclusions
This study considers the effects of Dufour and Soret with multiple slip conditions on three dimensional boundary layer flow of a micropolar nanofluid over a stretching sheet were discussed. The boundary layer equations of the flow problem are reduced into a twosome of high order non-linear ordinary differential equations using the similarity transformation. Then, the obtained differential equations are solved numerically using bvp4c from matlab software. Numerical results are obtained for different main parameters of the flow problem. Comparison with previously published work was performed and very good agreement was obtained. The effects of governing parameters are presented using figures and tables. The main findings are: 1. The velocity profiles reveal decreasing performance for raising values of slip parameters (A or B) while an inverse outcome is seen when material parameter β is improved.
2. The existence of the stretching parameter λ allows to decline the velocity boundary layer thickness near wall for f′(η) and to enhance the velocity profile g′(η) near the surface of plane.
3. The temperature profile distribution within the boundary layer can be increased by the raise in the values of Dufour number Df; but it can be decreased by the increase in either the values of prandtl number Pr or Brownian motion parameter Nb.
4. An enhancement in either the values of Prandtl number Pr or Soret number Sr or Brownian motion parameter Nb allows reducing the Nusselt number as well as its boundary layer thickness near the surface of the wall.
5. Boost in the values of solutal slip parameter D agrees to reduce concentration boundary layer thickness.
6. Sherwood number exhibits increasing behavior for improve either Soret Sr or Dufour Df number. But it exhibits opposite behavior for enhance in the values of Schmidt number Sc.
7. The skin friction coefficients ( ) - f 0 and ( ) - g 0 demonstrate decreasing behavior for the increase in either velocity slip parameters ( A or B) or material parameter β or micro rotation parameter n while an inverse outcome is observed when the values of stretching parameter λ is raised.

The wall couple stress coefficients
reveal declining performance for growth velocity slip parameters (either A or B) whereas opposite result is pragmatic for increase either material parameter β or microrotation parameter n or stretching parameter λ.