Analytical solution of MHD micropolar nanofluid flow and forced convection heat transfer with entropy generation analysis past a linearly stretching sheet

This perusal attempts to model and interpret the entropy generation analysis and the flow field of 2-D, steady, viscous, incompressible and laminar boundary layer and forced convection heat transport of micropolar ferrofluid past a stretching sheet including suction and normal magnetic field effects. The porous sheet’s velocity and temperature are presumed to change linearly. Exact explicit solutions of the velocity, angular velocity and temperature distributions have been derived. The impacts of physical parameters on the local skin friction coefficient, the local Nusselt number, the entropy generation number further the velocities and temperature distributions are analyzed by tables and graphs. The angular velocity has more value than velocity for the least value of the magnetic and material parameters. The entropy generation number has a direct relation with material parameter and Brinkman either Reynolds numbers. Moreover, an inverse relation with the Prandtl number.

Sheikholeslami et al. [39] demonstrated Lorentz and thermophoresis forces play different roles on concentration. Golafshan et al. [40] illustrated thermophoresis parameter change slightly on the temperature profile either the local Nusselt number. Derakhshan et al. [41] demonstrated there is a direct relation between the Nusselt number and the viscosity coefficient. Lund et al. [42] depicted an increasing trend in the skin friction and mass transfer for the angle parameter. Jahan et al. [43] made linear and quadratic regressions to estimate the local Nusselt numbers by the Brownian motion and thermophoresis parameters. Li et al. [44] found out the Nusselt number for the shrinking sheet case is more than the stretching sheet case. Dzulkifli et al. [45] reported the unsteadiness parameter extends the range of solution for stretching/shrinking parameter.
Turkyilmazoglu [46] demonstrated the heat transfer rate has an inverse relation with the heat jump on the wall. Fereidoonimehr et al. [47] realized by enhancing the molecular weight the nanoparticle volume fraction raises. Turkyilmazoglu [48] presented analytical solutions induced by nonlinearly deforming porous sheet. Turkyilmazoglu [49] proved magnetic interaction parameter and Prandtl number have different effects on the momentum and temperature boundary layer thicknesses. El-Mistikawy [50] solved related ODEs analytically by using Kummer's equation for the temperature constituents. In this study, ODEs arising from physics have been solved analytically. Graphs and tables are presented and the variety of physical parameters are explained in detail. The explicit exact solutions for the dimensionless flow, velocity, angular velocity and temperature are reported in Section 3. In Section 4, the physical quantities of interest, for instance, the local skin friction coefficient, the local Nusselt number and the entropy generation number are calculated. Results are discussed in Section 5 and the final section deals with the conclusions.

Mathematical model
where u and v are velocity components in the x-and y-directions, respectively, N is the micro- where w v is a constant mass flux velocity, the positive and negative cases are related to suction and injection, respectively. m is the micro-gyration constraint with 01 m  . To convert the PDE boundary layer Eqs. (1)-(4) into the coupled ordinary differential equations, the following similarity transformations are used: Substituting (7) into (2), (3) and (4) the following ordinary differential equations are acquired: Following the similarity transformation Eqs. (5) and (6) Here prime denotes differentiation with respect to the similarity variable  , are the dimensionless stream function, the dimensionless angular velocity and the dimensionless temperature, respectively.

Exact analytic solutions
According to research conducted by A. Chakrabarti et al. [53], the physical solutions for stretching sheet come from the exponential relation  

Parameters of engineering interests
Some important physical parameters include the entropy generation number, the local skin friction coefficient and the local Nusselt number.
Where w q is the surface heat flux as given To improve the efficiency of thermal systems, it is important to calculate entropy generation. The entropy analysis displayed the regions of the system with more energy dissipation. The volumetric rate of the local entropy generation of the micropolar fluid flow containing magnetite ferrofluid with a magnetic field past the stretching sheet stated as follows [

Results and discussion
Tiwari-Das model has been employed to probe the entropy generation analysis, the 2-D boundary layer flow and the heat transfer accompanied by the magnetic field effect with micropolar fluid.
The ODEs are solved analytically by introducing the closed-form solution for the flow field and forced convection heat transfer, also the important parameters of engineering. Besides, the ODEs are calculated numerically by Runge-Kutta-Fehlberg using Maple software. The local skin friction coefficients were obtained and tabulated by analytical and numerical methods by changing the magnetic parameter and fixing other parameters such as Pr=6.2 and K=50. As can be seen, the magnitude of 12 Re x fx C enlarged significantly through increasing M.
More accurately, by changing the M from 2 to 16, the 12 Re x fx C boosted 765.707%. With the increase of the Lorentz force, the micro-gyration constraint decreases and as a consequence, the 12 Re x fx C will be enlarged. By increasing the Prandtl number, velocity, angular velocity and temperature increments.
Velocities and thermal boundary layers thicknesses increase very slightly. On the other hand, over the sheet, the angular velocity is more than velocity.

Conclusion
In this study, the fluid flow and heat transfer of Fe3O4 water-based micropolar ferrofluids over a stretching sheet has been analytically scrutinized. Exact explicit solutions for the  