Simultaneous solutions for first order and second order slips on micropolar fluid flow across a convective surface in the presence of Lorentz force and variable heat source/sink

This report presents the flow and heat transfer characteristics of MHD micropolar fluid due to the stretching of a surface with second order velocity slip. The influence of nonlinear radiation and irregular heat source/sink are anticipated. Simultaneous solutions are presented for first and second-order velocity slips. The PDEs which govern the flow have been transformed as ODEs by the choice of suitable similarity transformations. The transformed nonlinear ODEs are converted into linear by shooting method then solved numerically by fourth-order Runge-Kutta method. Graphs are drowned to discern the effect of varied nondimensional parameters on the flow fields (velocity, microrotation, and temperature). Along with them the coefficients of Skin friction, couple stress, and local Nussel number are also anticipated and portrayed with the support of the table. The results unveil that the non-uniform heat source/sink and non-linear radiation parameters plays a key role in the heat transfer performance. Also, second-order slip velocity causes strengthen in the distribution of velocity but a reduction in the distribution of temperature is perceived.

beds cooling, glass forming approaches, hot rolling, plastic sheets extraction, paper production, etc. Based upon these applications the feature of the desired products will be controlled by the measure of thermal transfer. In 1970, the first study on the motion over a stretching surface was scrutinized by Crane 9 . Micropolar fluid motion induced by a strained surface was described by Chiam 10 . Hayat et al. 11 reported the impact of Nusselt number on non-Newtonian liquid motion via a non-linear surface. The influence of organic response on the stagnated motion of Newtonian liquid across a cylinder was reported by Najib et al. 12 and presented dual solutions for shrinking and stretching cases. Later, the work of Hayat et al. 11 was then extended by Babu et al. 13 with injection/ suction. Soid et al. 14 analyzed the heat transport attribute on time-dependent flow of Newtonian liquid induced by a shrinking sheet.
The analysis of magnetohydrodynamics has ample significances in the fields of cooling of the reactor, astrophysics, accelerators, design of heat exchangers, power generators, geophysics, plasma studies, and cancer research. Micropolar fluid motion across a nonlinear stretchable sheet under the impact of drag force was scrutinized by Hayat et al. 15 and concluded that angular velocity has an inverse relationship with material parameter. Nadeem and Hussain 16 explored the Lorentz force essence on viscous Newtonian fluid through the porous stretched surface. The MHD flow driven by a wedge or a cone with the aid of non-Fourier heat conduction was reported by Kumar et al. 17 and concluded that temperature field is suppresses with a hike in the magnitude of relaxation parameter but enhances with magnetic field parameter. Impact of Lorentz force on forced convection Jeffery fluid through a permeable surface was reported by Ahmad and Ishak 18 . They implemented the finite difference numerical scheme to obtain the solution. The influence of thermophoresis and Brownian motion on MHD flow of nanoliquid across a solid surface in the attendance of linear Rossland approximation was investigated by Mabood et al. 19 . Mabood et al. 20 presented an analytical solution for the problem of time-dependent flow of nanoliquid past a convectively heated surface in the attendance of Lorentz force and concluded that the magnetic field parameter has a proclivity to enhance the curves of heat function. The influence of variable viscosity on conducting flow of nanoliquid across a rotating surface in the presence of a porous medium was reported by Mabood et al. 21 . Recently, Kumar et al. 22 investigated the flow characteristics of micropolar with radiation and frictional heating. They found that friction factor has an inverse relationship with the magnetic field parameter.
The impact of heat transport has sufficient significance in the sub-disciplines of paramedical and some engineering. Power creation, Oceanography, heat exchangers, magnetic drug targeting, thermal conduction in tissues, convection in earth's mantle, electronic devices, boilers, missiles, fuel cells, etc are some applications of heat transfer. Polymer processing, gas turbines, production of paper, space vehicles, hypersonic fights, space technology, production of glass, gas cooled nuclear reactors are certain beneficial claims of radiation. Ziabakhsh et al. 23 examined the heat generation and microrotation impacts on non-Newtonian liquid motion past a surface and concluded that a rise in blowing constant origins an improvement in the thermal fields. The influence of Newtonian heating on micropolar fluid motion driven by a stretched geometry was described by Qasim et al. 24 and established that an augmentation in both the local Nusselt number and temperature distribution with larger Newtonian parameter. Cortell 25 discussed the influence of the rate of heat transport on radiative liquid over a solid surface. Brownian motion influence on magnetohydrodynamic radiative flow of shear thickening liquid via stretchable geometry was reported by Farooq et al. 26 . Recently, the authors 27,28 have paid their attention to investigate heat transport with nonlinear radiation.
The phenomenon of uneven heat sink/source has countless solicitations in medicine and many engineering happenings like cooling of metallic sheets, the intention of a thrust bearing, unpolished oil retrieval, etc. Pal 29 studied the thermal transport attributes of time-dependent fluid flow across a stretched sheet with irregular heat sink/source. Some notable data about the irregular heat fall or raise effect on electrically conducting liquid flow due to stretching surface can be view in the ref. 30 . It was founded that uneven heat constraints have done a key character in the performance of thermal transport. RamReddy et al. 31 numerically reported the behavior of Biot and Soret numbers on the forced convective motion of viscous liquid through a vertical plate and finalized that all the physical quantities have an inverse relationship. Patil et al. 32 bestowed an inexact result for the forced convective motion subjected to the convective sheet. Kumar et al. 33 presented the features of thermal transport on ferrofluid motion past a convective sheet with radiation. Impact of variable heat source/sink on MHD flow of micropolar liquid across a stretching surface with thermo-diffusion was reported by Mabood et al. 34 . The influence of heat source/sink and radiation on MHD flow of nanofluid past a nonlinear surface was scrutinized by Makinde et al. 35 and Mabood et al. 36 and concluded that the heat source/sink parameter has a tendency to enhance the temperature.
In the aforesaid investigations the impact of no slip condition is presumed. Mainly first order slip is important when the fluid is particulate like bubbles, mixtures (grease, egg yolk, the combination of oil and liquid), and polymer solutions. Many years ago, Navier 37 recommended the momentum slip condition to liquid motion across a sheet. Behavior of secondary slip on Newtonian fluid flow due to a convective surface was analyzed by Fang et al. 38 . The impact of radiation on magnetohydrodynamic mixed convective slip motion was reported by Beg et al. 39 . Martin and Boyd 40 investigated the influence of velocity slip on viscous liquid over a stretching sheet. With the aid of self-similar transformations, the flow equations are mutated and hence numerical results for the flow fields are presented. Attributes of slip motion and heat transport on magnetohydrodynamic micropolar liquid due to stretched sheet was reported by Ibrahim 41 . Recently, Mabood et al. 42 scrutinized the influence of primary slip on boundary layer motion of radiative liquid across a melting surface. They noticed that higher values of the slip parameter cause an increment in the fluid temperature but the friction factor is inversely proportional to the slip parameter. Ibrahim et al. 43 presented numerical scrutiny to examine the influence of chemical reaction and velocity slip on electrically conduction flow of non-Newtonian liquid across a convectively heated surface and found that Biot number has a propensity to heighten the heat transfer rate.
All the afore believed studies, the scientists put their struggles to gain the knowledge in the heat transfer of MHD flows under distinct physical aspects such as heat source/sink, linear Rossland approximation and slip

Mathematical formulation
Flow geometry of the model is exposed in Fig With the above-declared assumptions, the flow equations will be (See refs 24,26,41 ), Here (u, v) are the constituents of velocity in the ways of (x, y) correspondingly, (μ, ρ) correspondingly, the viscosity (dynamic) and density, κ is the vortex viscosity, N is the microrotation velocity, electrical conductivity and micro-inertia density are σ = υ ( ) j , p respectively, (C p , k) correspondingly, the heat capacitance and conductivity (thermal). Consider is the material parameter. www.nature.com/scientificreports www.nature.com/scientificreports/ In Eq. (4) the second term in the R.H.S, ‴ q is defined as (See ref. 26 )   Table 2. Influence of sundry flow parameters on C F , C S and Nu for δ = 0 and δ = 1.  www.nature.com/scientificreports www.nature.com/scientificreports/ Here T s and T ∞ are the nearby and ambient temperatures of the sheet correspondingly, the lessening and swelling values of A* and B* corresponds to heat fall or raise. Consider www.nature.com/scientificreports www.nature.com/scientificreports/ Consider (See refs 28,41 ),   www.nature.com/scientificreports www.nature.com/scientificreports/

C F C S Nu
e Consider the second-order velocity slip model (Ibrahim 41 )    www.nature.com/scientificreports www.nature.com/scientificreports/ Here Knudsen number is the pivotal factor, which is a rate of the molecular free mean path of characteristic length. For very small K n no-slip is noticed between the surface and fluid. However, K n lies between 10 −3 to 0.1, first order slip arises near the fluid-surface interaction.
Consider the transformations in order to get the dimensional less expressions of the flow equations: (See refs 24,26,41 ), Here χ is the stream function and η is the similarity variable, η η θ η ′ f g ( ), ( )& ( ) are the dimensionless flow fields and θ w is the temperature ratio parameter.
From Eq. (13), Eq. (1) satisfied trivially and the Eqs (2), (3) and (8) becomes www.nature.com/scientificreports www.nature.com/scientificreports/ The corresponding mutated boundary conditions are www.nature.com/scientificreports www.nature.com/scientificreports/ here α λ γ δ M N Bi , , Pr, , , , & r are the micropolar parameter, magnetic field parameters, Prandtl number, nonlinear radiation parameter, Biot number, stretching ratio parameter, first and second order slip parameter respectively and these are defined by p r f 0 2 3 Calculated friction factor, couple stress coefficient and heat transfer rate are is the confined Reynolds number.

Deliberation of Results
This section sightsees the inspiration of flow variable parameters on the flow fields. The scheme of nonlinear and coupled ODEs (14)-(16) with the restrictions of the boundary (17)- (18) have resolved numerically with the sequential solicitation of shooting and R.K. methods. The effect of varied dimensionless parameters on the fluid temperature, micro-rotation, and velocity field is exposed via plots. Further, we scrutinize the influence of the same variables on the physical quantities and the consequences are exhibited in the table. We prescribed the values of α θ γ = = = .
= .  Tables 2 and 3. From the table, it is noticed that a rise in the values of Lorentz force results a drop in all the quantities ( ″ ′ f g (0), (0) and θ − ′(0)) however a reverse consequence is perceived for material parameter. Swelling values of micropolar constant upshots a hike in ′ g (0) but a reduction is noticed for a larger magnetic field parameter. Larger M r yields an increment in the couple stress coefficient, friction factor, and the measure of heat transport. Also, the measure of thermal transport is supreme for larger Bi for both the cases. Heat generation or absorption and temperature ratio parameters cause a reduction in heat transfer rate while an opposite outcome is noticed with nonlinear radiative energy for both the slips (first and second order) cases. From Table 3 it is noticed that an increase in γ results a hike in both ″ ′ f g (0), (0) but a contrary development is noticed for second order slip parameter. Also, the thermal transport rate is a cumulative function of second order slip parameter but an inverse outcome is discerned for γ. Figure 2 renders for the variation of magnetic field parameter (M) on the velocity, microrotation and thermal fields. Figure 2(a),(b) divulge that, both the distributions of linear and angular momentum suppresses as the values of magnetic field variable enhances. This contest the physical interpretation on hiring the magnetic force to an electrically conducting fluid, and this provides an elevation in the drag force, which consequences in the decelerating strength on velocities. Owing to this, a reduction in the fields of velocity and microrotation is noticed. But in case of thermal fields, it's quite reverse to fluid velocities as exposed in Fig. 2(c). From the plots, it is spotted that both the velocity and the angular velocity are attained maximum due to second order slip and the maximum temperature is noticed for first order slip. Figure 3 is plotted to know the essence of the micropolar parameter (α) on the velocity, microrotation and thermal fields. It is interesting to note that, an enhancement in the values of α boosts the linear velocity, however, a contrary consequence is detected for microrotation and thermal fields. The results specify that the momentum transfer layer-by-layer is boosted expressively owing to the escalation of viscosity caused by the collective micro-rotation of particles, i.e., large values of the micropolar parameter; on the other hand, the thermal diffusions are weakened slightly. Hence, there is a reduction in both the micro-rotation and thermal fields are noticed for larger material constant. Figure 4(a) is deliberate to discuss the influence of radiation parameter (N r ) on the distribution of temperature. From the figure, it is noticed that the impression of radiation parameter N r on θ(η) is increasing. It is familiar that the mechanism of radiation and is the heat transference phenomenon which releases the energy via fluid particles such that some additional heat is produced in the flow. It is worth mention that the influence of radiation becomes more significant as → ∞ N r and the influence of radiation can be neglected when = N 0 r . Moreover, high heat transfer is attained in the presence of first-order slip than that of second-order slip.
The essence of the temperature ratio parameter on fluid temperature is investigated through Fig. 4(b). It is worth mention that thermal field enhances with increasing values of θ w . Mathematically, θ = ∞ w T T w is the ratio of temperature at the surface to the temperature at a free stream. The value of θ w must be greater than 1 for the nonlinear radiation. Also, an increase in temperature ratio parameter causes a hike in the temperature along the (2019) 9:14706 | https://doi.org/10.1038/s41598-019-51242-5 www.nature.com/scientificreports www.nature.com/scientificreports/ surface. As a result, the distribution of thermal field and the corresponding layer thickness enhances. It is significant to remark that as θ → 1 w , the temperature of nonlinear Rosseland and linear Rosseland approximation are the same. We observed an interesting result that the distribution of temperature (θ η ( )) is high for δ = 0 when compared to that of δ ≠ 0.
The influence of Biot number (Bi) on the fluid temperature is investigated through Fig. 4(c). We see that the ascending values of the Biot number improve the fluid temperature. Biot number occurs in the current investigation owing to the assumption of convective boundary condition and signifies the ratio of diffusive resistance within the sheet to the convective resistance at the surface of the sheet. Thus smaller values of the Biot number provides high convective resistance at the surface, which leads to low heat transfer rate from the sheet to the fluid. Hence the fluid temperature is an increasing function of Biot number. We observed a motivating consequence that the distribution of temperature (θ(η)) is high for δ = 0 when compared that of δ ≠ 0. Figure 5 is sketched to know the influence of microrotation parameter on the velocity, microrotation and thermal fields. From Fig. 5(a), it is spotted that fluid velocity is an increasing function of the microrotation parameter. As a result, we glimpse a decrement in temperature and microrotation profiles from Fig. 5(b),(c) respectively. Figures 6 and 7 are drawn to see the influence of γ and δ on η η ′ f g ( ), ( ) and θ η ( ) respectively. From Fig. 6(a),(b), we eye that an accelerating values of δ results a hike in dimensionless velocity but the thermal field is a decelerating function of δ. Hence Fig. 6(c) says that the distribution of angular momentum increases with an increment in δ. The impression of γ on θ η η g ( ), ( ) and η ′ f ( ) is investigated via Fig. 7(a)-(c) respectively. For increasing values of γ results a hike in both the thermal field and the corresponding boundary thickness. Moreover, the fluid velocity and microrotation profiles are decelerating functions of γ. So we conclude that the primary and the secondary slip parameters are proportional to each other. Hence the maximum temperature is noticed for second-order slip and maximum velocity is noticed for first-order slip parameter. Figure 8(a),(b) reveal the nature of fluid temperature for different values of non-uniform heat source/sink parameters (A*, B*). It is detected that swelling values of irregular heat source/sink parameters results a hike in the fluid temperature. Actually, an increasing values of irregular heat parameters act as an agent to produce temperature in the flow. Due to this, we observed that a rise in the fluid temperature for swelling values of A*, B*. It is prominent that the more heat transfer is attained in the absence of second-order slip.

findings of the problem
This paper narrates the flow and thermal transport attributes of shear thinking fluid across a stretching surface with Biot number and drag force. Simultaneous solutions were presented for first and second order slips. The principal outcomes are listed below • Microrotation profile is a decelerating function of micropolar parameter.
• A rise in the microrotation parameter causes the same in g(η) and θ(η) but a reverse trend is noticed for η ′ f ( ).
• Fluid temperature is enhanced due to an increment in the values of both N r and θ w .
• The Velocity distribution upturns for rising values of magnetic field parameter.
• In all the figures supreme velocity is achieved due to second order velocity slip however the supreme temperature is achieved due to first order slip. • The irregular heat raise/fall parameters plays key role in the heat transfer performance.
• The nature of all the physical quantities is reverse for slip parameters.