Hydromagnetic Flow of Micropolar Nanofluid

: Similar to other fluids (Newtonian and non ‐ Newtonian), micropolar fluid also exhibits symmetric flow and exact symmetric solution similar to the Navier–Stokes equation; however, it is not always realizable. In this article, the Buongiorno mathematical model of hydromagnetic micropolar nanofluid is considered. A joint phenomenon of heat and mass transfer is studied in this work. This model indeed incorporates two important effects, namely, the Brownian motion and the thermophoretic. In addition, the effects of magnetohydrodynamic (MHD) and chemical reaction are considered. The fluid is taken over a slanted, stretching surface making an inclination with the vertical one. Suitable similarity transformations are applied to develop a nonlinear transformed model in terms of ODEs (ordinary differential equations). For the numerical simulations, an efficient, stable, and reliable scheme of Keller ‐ box is applied to the transformed model. More exactly, the governing system of equations is written in the first order system and then arranged in the forms of a matrix system using the block ‐ tridiagonal factorization. These numerical simulations are then arranged in graphs for various parameters of interest. The physical quantities including skin friction, Nusselt number, and Sherwood number along with different effects involved in the governing equations are also justified through graphs. The consequences reveal that concentration profile increases by increasing chemical reaction parameters. In addition, the Nusselt number and Sherwood number decreases by decreasing the inclination.


Introduction
In the last couple of decades, fast advances in nanotechnology have prompted the development of new age coolants called nanofluids. Nanofluids are designed by interrupting nanoparticles with 2 Symmetry 2019, 11, x; doi: FOR PEER REVIEW www.mdpi.com/journal/symmetry normal sizes of less than 100 nm in conventional heat exchange liquids such as ethylene glycol, water, and oil. These are the current heat exchange specialists that trigger the thermal conductivity of the base fluids, and they have been a fundamental subject for experts and researchers in the course of the last few years due to their fluctuated design and modern applications Choi [1]. Eastman et al. [2] reported in a study that thermal conductivity is enhanced up to 60% when CuO nanoparticles are added into a base liquid (water) with volume fraction 5%. Furthermore, they reported that the thermal conductivity increased up to 40% by adding the copper nanoparticles with volume fraction 1% into the conventional fluid. Buongiorno [3] discussed in his study that there are seven mechanisms, which are important to enhance the thermal conductivity of the base fluid. Among these mechanisms, Brownian motion and thermophoresis are the more substantial. Currently, nanofluids are used relatively commonly in electronic engineering, physical processing, communication, etc. Several research articles on the flow of nanofluids are reported in the relevant literature. Further, nanofluids have prominent application in welding equipment, in cooling high energy flux devices including high power wave tubes and in cooling automobile engines, etc. [4]. The heat transfer of nanofluid over a nonlinear porous sheet is numerically discussed by Zaimi et al. [5]. Moreover, the flow of nanofluid over a porous slanted surface is discussed by Sandeep and Kumar [6]. Chakraborty et al. [7] examined the nanofluid flow over a permeable inclined plate with solar radiations. Rashad [8] studied nanofluid flow by considering convective boundaries and the anisotropic slip effect. Mitra [9] investigated computational modeling of nanofluid flow over a heated inclined plate. The energy and mass transport with chemical reactions over an inclined stretching plate has garnered considerable interest because of its many applications in engineering. In processes such as drying, evaporation of surface water, the flow in a desert cooler, and heat transport in a wet cooling tower, energy and mass transport occur. Shit and Majee [10] scrutinized the flow of magnetohydrodynamic (MHD) fluid on a slanted surface. Eid [11] discussed the mixed convection flow by incorporating effects of chemical reactions. Jain and Bohra [12] scrutinized the effect of chemical reactions on three dimensional incompressible flows over an inclined surface. The flow of free convection fluid through a slanted sheet was elucidated by Sheri and Modugula [13]. Recently, different researchers studied the heat and mass exchange phenomenon with different impacts, see [14][15][16][17][18].
The idea of micropolar fluid has attained great importance, because conventional Newtonian liquids cannot precisely match the properties of fluid flow in numerous industrial uses, namely, polymeric fluids, biology, colloidal solutions, and paints. Fluids that are rigid, accidentally concerned with elements adjourned in a sticky medium, and with microstructure atoms are termed as micropolar fluids, in which the element distortion is unnoticed, see Eringen [19]. Rahman et al. [20] studied the heat exchange procedure in micropolar fluid flow by incorporating different fluid properties over a permeable inclined plate. Micropolar fluid flow, by taking different effects over an inclined sheet, was studied by Das [21]. Kasim et al. [22] investigated magnetohydrodynamic (MHD) micropolar fluid flow towards an inclined plate. Srinivasacharya and Bindu [23] discussed entropy generation of micropolar fluid flow on an inclined channel with parallel plates. Anwar et al. [24] conducted a numerical study of micropolar nanofluid flow by incorporating the thermophoresis effects. Magnetomicropolar flow over a slanted channel with parallel plates was investigated by Hazbavi and Sharhani [25]. Srinivasacharya et al. [26] explored the double diffusive effect on the flow of micropolar fluid on a slanted sheet. Other similar studies on the different aspects of fluids are given in [27][28][29][30][31]. The flow symmetry of fluids in general has been studied using theoretical methods, various experiment methods, and different numerical techniques, in various fields such as the field of fluid mechanics, the field of thermal engineering, etc., due to their several benefits and relevance in terms of flow control and heat transfer enhancement. However, symmetric flows of non-Newtonian fluids have not been sufficiently elucidated due to their nonlinear characteristics.
Energy and mass exchange has attracted the attention of contemporary researchers due to its diverse uses such as in cancer therapy, sensors, drying processes, MHD generators, the cooling of 3  [32] investigated the magnetohydrodynamic flow of nanofluid with entropy generation numerically. Moreover, numerical simulation of MHD mixed convection nanofluid flow was conducted by Khan et al. [33]. Motivated by the above cited literature, we have decided to study the flow of micropolar nanofluid over an inclined surface with Brownian motion and thermophoretic impacts. In view of industrial and engineering applications, in this study, the chemical reaction and heat generation or absorption effects are considered. Moreover, a lot of work on non-Newtonian nanofluids' flow by considering different geometries has been done, but no one scholar to date has considered the inclined surface geometry combined with micropolar nanofluid by incorporating the considered effects, i.e., chemical reaction and suction or injection, with the Keller-box technique. Therefore, in order to fill this gap in the literature, we conducted this research. The inclined stretching geometry with MHD impacts have important applications in automobiles, MHD generators, extrusion of sheets, etc.

Problem Formulation
Our main purpose is to study the micropolar-type nanofluid flow over a permeable inclined plate with an angle . The velocities taken in account are, and 0, respectively; here, 'x' is the coordinate measured along the stretching surface and 'a' is a constant. The transverse magnetic field is taken perpendicular to the extending surface. It is supposed that the magnetic and electric field effects are very minor as the magnetic Reynolds number is less, see Mishra et al. [27].The micropolar finite size particles along with nanoparticles are constantly distributed in the base fluids. The fluid particles have extra space to travel about, formerly hitting other fluid particles, where these particles revolve in the fluid field and fallouts for spinning effects in the micropolar nanofluid. The temperature and nanoparticle fraction take the constant values and whereas and are the nanofluid temperature and mass fractions, respectively.
The flow equations for current problem are given by Khan and Pop [28]: , where the components of velocity along and are denoted by and , is the heat generation or absorption coefficient, * is the chemical reaction coefficient, the thermal diffusivity parameter is denoted by , and the ratio between the effective heat capacity of the nanoparticle and heat capacity of the liquid is represented by .
The boundary settings are , , , * , 0, The stream function , for that purpose is given as where Equation (1) is fulfilled identically. For this problem, transformations are demarcated as On substituting Equation (8), the system of Equations (2)- (5) is 0 , where , , , Here, denotes the local Grashof number, signifies the modified local Grashof number, the chemical reaction parameter is denoted by , heat generation or absorption factor, and is the dimensionless vertex viscosity. To achieve the true similarity solution, the parameters must be constant. This condition is achieved if the thermal expansion coefficient and concentration expansion coefficient are proportional to . Hence, we assume that (see [34][35][36][37] where and are constants. Substituting Equation (14) into the parameters and , we get and , The transformed boundary conditions are It is found that by eliminating the vertex viscosity, i.e., = 0, the nanofluid model is deprived of micropolar impacts. The Sherwood number, skin friction, and Nusselt number are defined by The associated terms for 0 1 0 , ′ 0 , and ′ 0 are defined as where is the local Reynolds number. This section deals with the mathematical results of converted nonlinear ordinary differential Equations (9)-(12) with boundary settings (16) elucidated via finite difference scheme (Keller-box scheme). This method is very fast, easy to program, and easy to use for highly nonlinear ordinary differential equations. In this problem, the step size ∆ 0.01 with boundary layer thickness 10 is considered. Moreover, convergence criteria in this problem depend on the difference between current and previous iterations. When the convergence criterion 10 is satisfied for all points in the -direction, the iteration process was terminated. For statistical outcomes of physical factors such as (Brownian motion factor), (thermophoresis factor), chemical reaction constraint , magnetic parameter , local Grashof number , (heat generation or absorption parameter), i.e., , modified local Grashof number , inclination factor , (Prandtl number), (Lewis number), suction or injection constraint , and material parameter , different figures are plotted.

Results and Discussion
In Table 1, due to the lack of , , , , , , and with = 90 o , and 10 outcomes for ′ 0 and ′ 0 are compared with published outcomes of Khan and Pop [38]. Table 2 shows that ′ 0 decreases for a growth in the magnitudes of , , , , , , and on decreasing the factor , whereas it increases with the increase of , , , and . Further, ′ 0 is boosted by various magnitudes of , , , , , , , , , and drops for greater magnitudes of , and on decreasing magnitudes of . In addition, 0 rises with increasing magnitudes of , , , , , , , , and drops with the higher magnitudes of , , , , , and on decreasing .   Figure 1 indicates that the velocity profile decelerated by enhancing the values of the inclination parameter . This is because enhancing the value of the inclination parameter retards the strength of the bouncy force by a factor due to thermal variation. We also found that the influence of the bouncy force (which is highest for 0) exceeds the main stream velocity significantly. Figure 2 indicates that the velocity profile increases for large values of modified local Grashof number .

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Symmetry 2019, 11, x; doi: FOR PEER REVIEW www.mdpi.com/journal/symmetry Physically, the viscosity of the liquid falls with the growth of , and the concentration upturns which causes the increment in the velocity contour. Moreover, Figure 3 displays that ′ enlarges with increasing values of . Here, presents the features of the buoyancy forces, which offers a direct relation with ′ . Increase of buoyancy forces diminishes the viscous force, which is favorable to the fluid motion. Figure 4 depicts impacts of on ′ . It is found that ′ retards as we upturn the factor . This is because the application of a magnetic field produces Lorentz force, which create retardation in the speed of the fluid. The same effect is seen in the case of angular velocity for changed values of in Figure 5. The angular velocity shows an inverse relation with the magnetic field strength, which matches with the result of Rafique et al. [31].     It is noticed in Figure 6 that ′ increases with the increment of . The angular velocity profile upsurge corresponding with higher magnitudes of is indicated in Figure 7, and it emulates the result of Rafique et al. [31]. The boundary layer thickness declines by enhancing the magnitudes of . Figure 8 indicates the result of the suction factor on the velocity profile. The velocity contour decreases against the growth of . It is noted that the enrichment of porosity decreases the permeability, which causes the retardation of the fluid velocity. As we increase porosity, the velocity will be increasingly retarded. Figure 9 presents the effect of heat generation on . The temperature profile increases by growing . The velocity of the liquid is enhanced by increasing ; therefore, heat is generated in the flow area and the temperature increases within the thermal boundary layer. Figures 10 and 11 depict the outcome of Brownian motion on temperature and concentration outline. It is observed that increases versus ; the opposite reaction is shown in the case of concentration profile. The movement of the nanoparticles increases due to the irregular motion which acts to heat up the boundary layer. Hence, the concentration of the nanoparticles decreases. The effects of the thermophoresis constraint on and are presented in Figures 12 and 13 respectively. It is observed that and are directly proportional to . Figure 14 depicts the effect of the Prandtl number on . It indicates that decreases the higher the Prandtl number. The boundary layer thickness shortens by enhancing the values of the Prandtl number. Figure 15 shows that the concentration profile decreases for higher values of . The Lewis number reduces the boundary layer thickness. Figure 16 indicates a direct correspondence between the concentration profile and chemical reaction parameter .