Nonlinear Convection Flow of Micropolar Nanofluid due to a Rotating Disk with Multiple Slip Flow

In this analysis, steady, laminar, and two-dimensional boundary layer flow of nonlinear convection micropolar nanofluid due to a rotating disk is considered.+emathematical formulation for the flow problem has beenmade. Bymeans of appropriate similarity transformation and dimensionless variables, the governing nonlinear boundary value problems were reduced into coupled highorder nonlinear ordinary differential equations with numerically solved. +e equations were calculated using method bvp4c from matlab software for various quantities of main parameters. +e influences of different parameters on skin friction coefficients f′′(0) and G′(0), wall duo stress coefficients H1′(0), -H2′(0), and -H3′(0), the Nusselt number -θ′(0), and Sherwood number Ω′(0), as well as the velocities, temperature, and concentration are analysed and discussed through tables and plotted graphs. +e findings indicate that an increase in the values of thermal and solutal nonlinear convection parameters allow to increase the value of velocities f′(η) and G(η) near surface of the disk and reduce at far away from the disk as well as thermal and solutal Grashof numbers tolerate to increase in the value of radial velocity f′(η) near surface of the disk.


Introduction
Eringen [1] was the first researcher who presents the theory of micropolar fluid for which the classical Navier-Stokes theory is has a limitation for its full description. Consequently, Hamzeh [2] evaluated the behaviour of micropolar Casson fluid on natural convective flow past a solid sphere. It was found that material parameter diminishes the values of the narrow skin friction coefficient, but grows both values of the local Nusselt number and angular velocity profiles as its value rises. Wubshet and Chaluma [3] also reported the variation of density on micropolar nanofluid flow about an isothermal sphere. Moreover, wall shear stress and angular velocity gradient at the wall enhance with an enlargement in the material parameter as illustrated by Mandal and Mukhopadhyay [4]. Furthermore, the flow of MHD and heat transfer due to stretching rotating disk were examined by Mustafa [5] and Akhter et al. [6]. Mostafa and Shimaa [7] reported the influence of magnetic field on flow and heat transfer of a micropolar fluid about a stretching surface in the presence of heat generation (absorption). Also, the effects of microrotation parameter on flow between a rotating and stationary disk was examined by Anwar and Guram [8]. Moreover, Rashidi and Freideonimehr [9] presented the effects of velocity and temperature slip on the entropy generation past rotating disk. Also, the effects of diffusion-thermo and thermodiffusion on radially stretching disk have been evaluated by Khan et al. [10]. e impact of Prandtl number on radiative flow due to stretchable rotating disk with variable thickness was computed by Tasawar et al. [11]. Shamshuddin et al. [12] have evaluated the influence of this parameter on numerical study of heat transfer and viscous flow in a dual rotating extendable disk system employing a non-Fourier heat flux model. ey indicated that, with an improvement in the Prandtl number, there is a strong decrease in temperature of fluid as well as radial skin friction. Moreover, the Stefan blowing effect on bioconvective flow and heat transfer of nanofluid over a rotating stretchable disk was reported by Lafiff et al. [13] and Yin et al. [14]. Muhammad and Naeem [15] and Noor et al. [16] have examined the influences of velocity slip with magnetic field in micropolar nanofluid flow along rotating disk and in mixed convection lower flow of a micropolar nanofluid along vertically elongating surface accordingly. Also, the flow and heat transfer computations for nanofluid fluid flow and the impact of heat generation/consumption and thermal radiation over a rotating disk were analysed by Narfifah et al. [17], Mushtaq and Mustafa [18], and Anwer et al. [19]. Moreover, Tasawar et al. [20] had discussed the impact of thermal slip condition on MHD flow of Cu-water nanofluid due to a rotating disk. ey found that thermal boundary thickness increased for lower thermal slip parameter values, but increasing the values of it reduced heat transfer from the disk to the adjacent fluid. Moreover, the numerical study of nanofluid flow and heat transfer over a rotating disk was examined by Ahmad et al. [21]. MHD mixed convection movement of a nanofluid over nonlinear enlarging sheet including variable Brownian and thermophoretic diffusion coefficient have been evaluated by Sumalatha and Shanker [22], and the results illustrated that the influence of nonlinear stretching parameter drops both the flow of the fluid as well as temperature distribution.
All the above research articles have been on the flow over a plane, over a stretching surface, on MHD boundary flow, or in many other areas. In this paper, we evaluate numerically the nonlinear convection flow of micropolar nanofluid due to a rotating disk in the presence of multiple slip conditions, using bvp4c from matlab. e outcomes of physical parameters on fluid velocity, temperature, and concentration were discussed and indicated in graphs and tables as well.

Mathematical Formulation
Let us choose the cylindrical system (r, ω, z) in the direction component of the flow velocity (u, v, w) and the angular velocity components (H1, H2, H3) correspondingly. e study considers incompressible, laminar, and nonlinear convection flow of micropolar nanofluid over a circular disk at z � 0. e disk rotates with uniform microrotation rM 0 about the z-axis. As the result of revolving symmetry the end products in the azimuthal direction may be considered. e wall of the gyrating disk has constant temperature T w and concentration C w despite the fact the ambient temperature and concentration are represented by T ∞ and C ∞ correspondingly, as shown in Figure 1. By means of Mandal and Mukhopadhyay [4], Anwer and Guram [8], Tasawar et al. [11], Noor et al. [16], and Sajjad et al. [23] the governing differential equations of the flow are given as follows: where (N 1 , N 2 , N 3 ) are angular velocity in the r, ω, and z axes, correspondingly, (κ, a, b, c) are material constants (viscosity coefficients), L stands for momentum slip factor, n stands for angular slip factor, h represents thermal factor, m is solutal jump factor, μ stands for the coefficient of fluid viscosity, ρ is the density, c p stands for the specific heat, g represents the gravity, j � (χ/M 0 ) is the microinertia per unit mass, κ is the vortex viscosity, T stands for temperature, K is the thermal conductivity of the fluid, T w � (T ∞ + ΔT) and C w � (C ∞ + ΔC) represent the variable temperature and concentration at the surface, where ΔT and ΔC being constants give the rate of growth of temperature and concentration alongside the surface and T ∞ and C ∞ stand for the uniform temperature, concentration of the free , and σ and σ * are constants, σ t and σ s are the constant coefficients of thermal and volumetric expansion, respectively. is relation will be nonlinear density, temperature, and concentration (NDTC) variation. Y � ((ρc) p /(ρc) f ) is the ratio between the effective heat capability of the nanoparticle material and the heat capability of the fluid and D B and D T stand for the Brownian and the thermophoretic diffusion coefficient, respectively. Let r(x) � d sin(x/d) be the radial distance from the symmetrical axis to the surface.
By using the nondimensional variables such as r ω Figure 1: Physical model and coordinate system.
It can appear that, due to the rotational symmetry, the derivative in the radial direction may be neglected and equations (14)- (21) are reduced to the next nonlinear system of ordinary differential equations: Mathematical Problems in Engineering 5 with the boundary conditions. At e physical measures of awareness in this problem are the narrow skin friction coefficients C fr,ω , surface couple stresses m wr,ω , the Nusselt number Nu, and the Sherwood Sh number, and they can be written as follows: where Using the dimensionless variables (12) and the boundary conditions (20), the narrow skin friction coefficients surface couple stresses, the Nusselt number and the Sherwood number are obtained:

Numerical Solution
Pairs of seven harmonized high order ordinary differential equations (14)- (20), subjected to the boundary conditions, equation (21), are answered numerically using the function bvp4c from matlab software for various values of physical parameters and numbers. Statistical results are found using Matlab BVP solver bvp4c from matlab which is a finite difference code that realize the three-stage Lobatto IIIa formulation. To apply bvp4c from matlab, first, equations (14)- (20) are converted into a system of first-order equations.
Second, assemble a boundary value problem (bvp) and using the bvp solver in matlab to numerically solve this system, including the above boundary condition and income on a suitable finite value for the far field boundary condition, that is, η ⟶ ∞, say η ∞ � 1 and the step size is taken as △η � 0.01, the numerical result is obtained; it is exact to the fifth decimal place as the measure of convergence. In solving the BVP by means of matlab, bvp4c has only two point of views: a function ODEs for calculation of the residual in the boundary conditions and a building solint that provides a guess for a mesh. e 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. [24].

Results and Discussion
In this section, the results of different governing physical parameters on nondimensional velocity, temperature, concentration, skin friction and wall couple stress coefficients and confined Nusselt and Sherwood numbers have been discussed. In this study, the comparison with the literature value is not compared, since there is no related article.

Temperature and Concentration
Profiles. e influences of thermal and solutal jump parameters s2 and s3, nonlinear convection parameters (λ, s), and thermal and solutal Grashof numbers (Gr, Gm) on temperature and concentration sketches are prearranged in Figures 12-19.
ese figures indicate that the values of temperature and concentration disseminations and their boundary layer thickness decrease with increase of thermal and solutal jump parameters s2 and s3, nonlinear convection parameters (λ, s), and thermal and solutal Grashof numbers (Gr, Gm). ese effects happen due to the decline in the kinematic viscosity of the fluid which reduces thermal diffusion, that leads to a decrease in thermal and solutal boundary layer thicknesses.         Table 1 demonstrates that increase in velocity slip parameter s0 reduces skin friction coefficients f ″ (0), G ′ (0), whereas the