Numerical simulation of oscillatory oblique stagnation point flow of a magneto micropolar nanofluid

The particular inquiry is made to envision the characteristics of magneto-hydrodynamic oscillatory oblique stagnation point flow of micropolar nanofluid. The applied magnetic field is assumed parallel towards isolating streamline. A relative investigation is executed for copper and alumina nanoparticles while seeing water type base fluid. To be more specific, in the presence of both weak and strong concentration, the physical situation of micropolar fluid is mathematically modeled in terms of differential equations. The transformed mixed system is finally elucidated by midpoint method with the Richardson extrapolation development and shooting mechanism with fifth order R–K Fehlberg technique. The impact of governing parameters are shown and explored graphically. The obtained results are compared with existing published literature. Moreover, it is found that the magnetic susceptibility of nanofluids shows provoking nature towards copper as compared to Alumina. Also it is perceived that Cu–water shows higher wall shear stress and heat transfer rate than Al2O3–water. Additional, the thickness of momentum boundary layer is thin for weak concentration as related to strong concentration.


Introduction
The study of nanouids claims remarkable practical applications in investigational and modern disciplines. The disclosure of nanoparticles has enhanced the prociency and decreased the budget of cooling and warming structures. As anyone might expect, analysts additionally notice the subsequent information: the nanoparticles enhance both the thermal conductivity and apparent viscosity signicantly of the nanouid. Thus, additional propelling power is needed to retain the nanouid ow over the device. The reduction in pumping cost for a cooling system because of small size particles was probed by Choi. 1 Murshed et al. 2 observed that the operative thermal conductivity and viciousness of nano-uids considerably enhanced by particle volume fraction. The variation of thermal conductivity for nanouids largely depends on the shape, size, and material of which nanoparticles are made. The effectiveness of nanoparticles is not limited towards coolants. Scientists and researchers have noticed a few other successful employments of particles examined in ref. 3. Nanouids are presently being created for therapeutic applications, including cancer treatment and safe surgery by cooling. The blockage of UV rays can be achieved by using nanoparticles (zinc oxide) into plastic packaging and also offer an anti-bacterial shield. This helps to rectify the strength and stability of the plastic sheets. A recent application of nanouid ow is nano-drug delivery. Pak and Cho 4 deliberated the characteristics of heat ux in the uid by adding metallic oxide particles (nanometer-sized) to it. This attempt reports that the inciting nature of heat transfer is actually because of the alteration in thermal conductivity of the metallic particles and common uid. Aab et al. 5 investigated the nanoconned phase change materials for thermal energy applications. Far along [6][7][8][9][10][11][12] noticed that the nanoparticles are the source of variations in heat transfer. It is important to note that during these attempts uniform distributions of nanoparticles are assumed throughout the ow regime. Buongiorno 13 identied such elements entertained velocity slips towards base uid molecules are unable to maintain uniform concentration throughout the ow regime. The mechanism of seven-slip was proposed by him who can play an active role in the heat transfer enhancement. Further, he declared that both thermal and Brownian diffusion is most active slip mechanisms. Sheikholeslami and Ganji 14 scrutinized the nanouid ow together with heat transfer by means of corresponding plates squeezing. They established the direct relation of Nusselt number with nanoparticle volume fraction, Eckert number and squeeze number towards separated plates. Recently, Khan et al. 15 explore the magneto-hydrodynamic ow under the region of oblique stagnation point with slip effect for water-based nanouid containing, and (as nanoparticles). They found that -water is the source of enrichment of heat transfer at the sheet followed by -water and -water. Nadeem et al. 16 examined the model-based study of SWCNT and MWCNT thermal conductivities effect on the heat transfer due to the oscillating wall conditions. Combined effects of viscous dissipation and Joule heating on MHD Sisko nanouid over a stretching cylinder was deliberated by Hussain et al. 17 Sheikholeslami et al. 18 described the ow of CuO nanoparticles with radiation and discharging rate.
An analysis of micropolar liquids has captivated the devotion of investigators and experts in the eld of uid science. Such consideration is due to fact that the conventional Newtonian uids cannot depict the complete description of uid ow in various biological and industrial applications. For polar uids, a distinct and special kind of microstructure material claims stress tensor which should be non-symmetric. Basically, in terms of the physical frame, it depicts those materials which consist of situated molecules (arbitrarily) cast out in a viscous liquid. The variations in couple stress, body couples, micro-rotational and disclose micro-inertial are supported by polar uids. In general, the majority of the physiological liquids treated as polar uid like suspensions of rigid or deformable particles in the viscous uid, plasma, and cervical. In short, Eringen 19 was the earliest to mention the theory of micropolar uids. Aer his study, micropolar uids have been recognized widely by researchers because of numerous engineering and industrial applications. To mention just a few, cervical ows, contaminated and clean engine lubricants, colloids and polymeric suspensions, thrust bearing technologies and radial diffusion paint rheology.
Micropolar uids do act together closely with nanouids as micropolar uids are uids with microstructure and nanouid are colloidal suspension of metallic or non-metallic nanosize particles in the base uid. Physically micropolar uids symbolize uids comprising of rigid, randomly oriented (or spherical) particles suspended in a viscous medium. These particles may be of nano-size, which actually makes the micropolar uids to behave as nanouids. Physical examples of micropolar uids can be seen in ferrouids, blood ows, bubbly liquids, liquid crystals, and so on, all of them containing intrinsic polarities. It is important to note that some of these micropolar uids do behave as nanouid with heat transfer enhancement characteristics due to the presence of nanosize particles apart from their intrinsic polarities. The addition of nanoparticles in a micropolar uid, make the mixture more complex as compare to conventional nanouids and offer investigators with a new dimension to explore the uid ow characteristics.
The uid (cerebrospinal) motion in the brain was identi-ed by Power 20 and he has shown that the Cerebrospinal uid is adequately modeled through micropolar uids. Lukaszewicz 21 explained in his book about the physical aspects of micropolar uids regarding practical applications. Das 22 examined the ow characteristics of heat-mass transfer in the micropolar uid over an inclined sheet along with both chemical reaction and thermophoresis effects. Double diffusive unsteady convective micropolar ow past a vertical porous plate moving through binary mixture using modied Boussinesq approximation was discussed by Animansaun. 23 Recently, the impact of temperature dependent viscosity on the micropolar uid ow by way of two nanouids was taken by Nadeem et al. 24 They found that both micro rotation viscosity and micro inertia density are a function of temperature dependent dynamic viscosity. More recently, Gina Nov et al. 25 scrutinized the properties of micropolar uid ow in a wavy differentially heated cavity with natural convection effect. Whereas, the impression of magnetic eld on micropolar uid ow along a vertical channel was explored by Borrelli et al. 26 Some important literature which enhances the features of micropolar uids is given in ref. 27-29. Takhar et al. 29 examined MHD ow over a moving plate in a rotating uid with the magnetic eld, Hall currents, and free stream velocity. MHD stagnation point ow with exible features was investigated by Khan et al. 30 Genuinely, in the manufacturing production of polymer uids, colloidal solutions and the uid having minor additives; there is frequently a point where the local velocity of the uid owns symmetric stress tensor and micro-rotation of particles is nil. Some of the important studies about stagnation point ow are given in ref. 31-34. The review of the above-mentioned literature reects that as yet, the impact of magnetic eld on oblique stagnation point ow of micropolar nanouid with the manifestation of copper and alumina nanoparticles is not been addressed on an oscillating surface. So, in this article, the inuence of nanoparticles is added by activating nanoparticles viscosities and thermal conductivity effective model. The problem is of more importance because, micropolar uids do interact closely with nanouids since micropolar uids are uids with microstructure and nanouid are colloidal suspension of metallic or non-metallic nanosize particles in the base uid. Physically speaking, micropolar uids represent uids consisting of rigid, randomly oriented (or spherical) particles suspended in a viscous medium. These particles may be of nano-size, which indeed makes the micropolar uids to behave as nanouids. Physical examples of micropolar uids can be seen in ferro-uids, blood ows, bubbly liquids, liquid crystals, and so on, all of them containing intrinsic polarities. It is important to note that some of these micropolar uids do behave as nanouid with heat transfer enhancement characteristics due to the presence of nanosize particles apart from their intrinsic polarities. The physical ow illustration of the problem is mathematically modeled in terms of partial differential equations. A suitable similarity transformation is used to attain ordinary differential equations. The converted differential equations (coupled system) are ultimately unraveled by using BVP solution method and shooting scheme along with h order R-K Fehlberg algorithm. A brief parametric analysis is executed to inspect the effect logs of involved physical parameters on dimensionless velocity and temperature by way of graphical attitudes. Further, the tabular structure is also design to analyze the variation of some physical quantities namely, velocity and temperature gradients adjacent to the at surface. The obtained results are compared with existing published literature. An excellent match has been found which yields the validity of the current analysis. In the last it is mentioned here that the addition of nanoparticles in a micropolar uid, make the mixture more complex as compare to conventional nanouids and provide researchers with a new dimension to explore the uid ow characteristics (Fig. 1).

Problem description and governing equations
Consider the problem of stagnation point ow of an electrically conducting micropolar nanouid over an oscillatory surface with velocity U 0 cos u t. The uid impinges obliquely to the oscillatory surface y ¼ 0. By neglecting external mechanical body force and body couple the ow rheological equations becomes where H, and E are the magnetic and electric elds, respectively, in which H ¼ H(cos we 1 + sin we 2 ) and w ¼ arctan the microrotation or angular velocity, r nf is the density of nanouid, nanouids dynamic viscosity is m nf , 8 is the vortex viscosity, j is the microinertia coefficient, g nf is the spin-gradient viscosity, m e indicates the magnetic permeability, s e represent the electrical conductivity (m e , s e ¼ constants > 0). For structure (1) we affix the boundary condition: Fig. 1 Flow description of the problem.
where a is the strength of an irrotational straining ow, a is a nondimensional constant which represent the ratio of the vorticity of a rotational shear ow to the strength of an irrotational straining ow, A, B are constants such that A is determined as part of the solution of the orthogonal ow, instead B is a free parameter. Also, B À A determines the displacement of the uniform shear ow parallel to the wall y ¼ 0. For the case of n ¼ 0, we have N$k ¼ 0 at the wall which shows strong concentration. 35 Physically it means that microelements near the surface are unable to rotate. 36 Further, for the case n ¼ 1/2, narrates the disappearing of anti-symmetric portion of the stress tensor and indicates weak concentration 37 of microelements. On the other side, at n ¼ 1, ows indicate turbulent boundary layers. 38 From conditions (2), mean that at innity, N ¼ shows that the micropolar uid behaves like a classical uid far away from the surface. Also, from free stream velocity, we can nd that the stagnation point is b a ðB À AÞ; A and stream lines are hyperbolas whose asymptotes are: Considered the magnetic eld H as dened in ref. 15 and assumed form of solutions as N ¼ xF ðyÞ þ Gðy; tÞ; v ¼ Àaf ðyÞ; u ¼ axf 0 ðyÞ þ bgðy; tÞ; Eqn (1) and (2) takes the form From (8), it is understood that the behavior of f ( y) and From eqn (7) and (8), we nd the pressure eld as 15 Thermophysical characteristics in which p 0 is the stagnation pressure. From eqn (10), it is clearly seen that maximum pressure occurs at the stagnation point in through-out the ow domain.
Making use of eqn (10) and the following similarity solutions 15 we obtain the ow eld equations and the related boundary constraints, from eqn (5)-(9), in nondimensional form as  where in which 4 displays the volume fraction of nanoparticles, r s and r f are the density of solid fractions and base uid, k nf is the thermal conductivity of nanouid, k f and k s are the thermal conductivity of base uid the and solid fractions, respectively, (rC p ) nf is the heat capacity of nanouid, K is a material parameter, Pr is the Prandtl number, M is the Hartmann number, 3 and U are the dimensionless amplitude and frequency of the wave. The surface shear stress (C f ) and heat transfer rate (Nu) in dimensionless form can be expressed as where s w , is the wall shear stress and q w the surface heat ux denes as By using of (14), (17) and (18), we may write it as ! ; where Re x ¼ ax 2 /v f is the local Reynolds number. The equation of dividing streamline is and meets the boundary y ¼ 0. Further, from eqn (12) and (18) we nd the point of maximum pressure and point of zero skin friction as ðb À aÞ; We note that x p is independent of M whereas x s depends on M. The ratio (for a xed time) is identical for all angle of incidence.

Solution procedure
Numerical solution of f, g 0 , F, and G 0 -ow have been obtained numerically by means of midpoint method with Richardson extrapolation enhancement. Furthermore, the series solutions of eqn (12) 3,6 (g 1 (y) and G 1 (y) -ow) for small value of frequency U have been obtained as We are concerned only in real part of the solution. Thus n ¼ 1; 2; 3.
The above system has been tackled numerically using midpoint method with the Richardson extrapolation enhancement.
From (12) Fig. 6 Time series of the flow of the temperature field q(y, t)at five distinct spaces from the sheet for the time period t˛[0, 10p] for Cuwater,  Making use of (12) 8 we may write 1 Pr where Q 10 (y) ¼ q 0 (y) is given in (18) The numerical integration for the above system can be executed easily with aid of any mathematical soware.

Results and discussion
Numerical assessment is carried out towards model equations of water-based micropolar nanouid containing metals and oxide ceramics nanoparticles named as alumina (Al 2 O 3 ) and copper (Cu). The range of solid volume fraction 4 for the nanoparticles is maintained as 0 # 4 # 0.2 along with the upper limit of Prandtl number 6.2 for base uid i.e. water. Table 1 is used to present the thermos-physical properties of copper, alumina, and water. The numerical scheme is validated by constructing a discrete case of Hartmann number by ignoring the effects of nanoparticles shown in Table 2, we have found that our obtained results are agreed perfectly with. 40 Fig. 8 Shows the streamlines of Cu-water nanofluid when b a The impact of relevant physical parameters on nanouid velocity distributions is identied by Fig. 2-6. Fig. 2 shows the behavior of f(y), f 0 (y), f 00 (y) for M ¼ 10 À7 , 4 ¼ 0.0, K ¼ 0.0. Fig. 3 is used to study the behavior of f 0 (y) towards unlike values of K, 4, n and different nanoparticles when base uid is water. In Fig. 3(a), it is detected that momentum boundary layer thickness increases by growing the material parameter K. There is an important fact that Al 2 O 3 -water nanouid produces a thicker velocity boundary layer than Cu-water as illustrated in Fig. 3(b). The strength of Fig. 3(c) is to draw out the impact of an imperative parameter n, the micro gyration parameter, which indicates the concentration of the micropolar uid. From this gure, we tend to recognize that the velocity boundary layer thickness is thin just in case of week concentration as compared to strong concentration. Fig. 4 is designed in order to see the impact of time t on u. It is seen that u shows an oscillation performance with maximum amplitude at the surface and gradually declines away from the surface. Fig. 5 depicts the attitude of the temperature distribution q(y, t) towards 4 and M when Pr ¼ 6.2. The inuence of increasing Hartmann number on temperature prole, the decreasing nature of temperature eld can be observed near the surface, while it shows a rise in behavior with the enhancement in nanoparticle volume fraction. The impact of time t on q(x, y, t) is shown with the aid of Fig. 6. It is noticed that q(x, y, t) exhibit waving nature and the amplitude of wave is found maximum near the surface and reduces far from the surface. Further, it is examined that the temperature is maximum at the surface, that is y ¼ 0, and decrease away from it. The oblique ows are presented by way of streamline patterns in Fig. 7 and 8. The streamline come across the wall y ¼ 0, at x s . It is concluded from these gures that their Table 5 Numerical value of b À a and g    location is governed by b À a and time t. Fig. 9. shows the bar graph comparison of both copper and aluminium oxide nanoparticles. It demonstrates that copper has a higher surface temperature gradient when contrasted with the aluminium oxide nanoparticles. As Cu has the highest value of thermal conductivity as compared to TiO 2 and Al 2 O 3 . The reduced value of thermal diffusivity leads to higher temperature gradients and, hence, higher improvements in heat transfer. More real applications of nanouids include different types of microchannels, heat exchangers, thermosyphons, heat pipes, chillers, car radiators, cooling and heating in buildings, solar collectors, air conditioning and refrigeration, cooling of electronics, in diesel electric generator as jacket water coolant, nanouids in transformer cooling oil, in drag reductions and many others.   Tables 3-10 delineate the impacts of the involved parameter on the physical quantities near the wall for both copper and aluminium oxide nanoparticles when water is preserved as a base uid. We comment that the estimations of a and f 00 (0) rely on upon M, 4 and, K, as should be obvious from Tables 3 and 4. More precisely, f 00 (0) increases and a decreases as 4 and M are increases. Moreover, increases in material parameter K cause an increase in a and decrease in f 00 (0). Tables 5 and 6 shows the numerical values of velocity gradient at the surface against M, 4, K and b À a ¼ Àa, 0, a and it is noticed that the magnitude of g 0 (y) does not depend on 4. As far as the variation of g 0 (y) against M and K are concerned, is found its magnitude shows  increments when M rises while shows decline nature for all b À a when K increases. The rapid increase is found for Cu-water nanouid as compared to Al 2 O 3 -water. The numerical variation of B 0 0 ð0Þ; B 0 2 ð0Þ; B 0 4 ð0Þ against M, 4, and K are revealed in Tables  7 and 8. Generally, magnetohydrodynamic (MHD) ow plays the main role in the manufactured products and several businesses like pumps and oil purication, etc. In Tables 9 and 10, it is found that the gradient of temperature is decreasing function of both material parameter K and nanoparticle volume fraction 4. Thus, the rate of heat transfer increase near the surface. It is important to note that Cu-water remarks higher heat transfer rate as compared to Al 2 O 3 -water nanouid. As Cu has the highest value of thermal conductivity as compared to Al 2 O 3 . The reduced value of thermal diffusivity leads to higher temperature gradients and, hence, higher improvements in heat transfer. Furthermore, it is also noticed that the temperature gradient shows inciting attitude when we increase Hartmann number M which brings enhancement in heat transfer rate near the surface. In general, micropolar uids deal excessive resistance to the uid motion rather than the Newtonian uid. This occurrence also demonstrates that the greater micropolar parameter improves the total viscosity in the uid ow. Thus, the micropolar uid is a very effective uid medium in the boundary layer for observing the laminar ow.

Concluding remarks
The properties of magneto-hydrodynamic oblique stagnation point ow of micropolar nanouid over an oscillatory plate were reported by way of parametric study. In this attempt, we have chosen alumina Al 2 O 3 and copper Cu as nanoparticles when water is treated as base a uid. The key nding of the current analysis is itemized as follows The momentum boundary layer is thicker for Al 2 O 3 -water as associated to Cu-water. In addition, Al 2 O 3 -water results show more surface temperature while Cu-water generates the lowest surface temperature.
Thickness decay is found for momentum boundary layer against increasing value of nanoparticles volume fraction while opposite attitude towards material parameter. Further, the thickness of momentum boundary layer is thin for the case of weak concentration as compared to strong concentration.
The local wall shear stress is the increasing function of Hartmann number and material parameter while it shows opposite attitude towards nanoparticles volume fraction. It was also noticed that Cu-water with the comparison of Al 2 O 3 -water gives maximum local wall shear stress.
The magnitude of rate of heat transfer is signicantly large for Cu-water as compared to Al 2 O 3 -water. On the other hand, the heat transfer rate near the plate surface is declining function of Hartmann number while the contrary trend is found for both material parameter and nanoparticles volume fraction.

Conflicts of interest
There are no conicts of interest to declare