Insight into the heat transfer of third-grade micropolar fluid over an exponentially stretched surface

Due to their unique microstructures, micropolar fluids have attracted enormous attention for their industrial applications, including convective heat and mass transfer polymer production and rigid and random cooling particles of metallic sheets. The thermodynamical demonstration is an integral asset for anticipating the ideal softening of heat transfer. This is because there is a decent connection between mathematical and scientific heat transfers through thermodynamic anticipated outcomes. A model is developed under the micropolar stream of a non-Newtonian (3rd grade) liquid in light of specific presumptions. Such a model is dealt with by summoning likeness answers for administering conditions. The acquired arrangement of nonlinear conditions is mathematically settled using the fourth-fifth order Runge-Kutta-Fehlberg strategy. The outcomes of recognized boundaries on liquid streams are investigated in subtleties through the sketched realistic images. Actual amounts like Nusselt number, Sherwood number, and skin-part coefficient are explored mathematically by tables. It is observed that the velocity distribution boosts for larger values of any of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1$$\end{document}α1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β, and declines for larger \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _2$$\end{document}α2 and Hartmann numbers. Furthermore, the temperature distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta (\eta )$$\end{document}θ(η) shows direct behavior with the radiation parameter and Eckert number, while, opposite behavior with Pr, and K. Moreover, the concentration distribution shows diminishing behavior as we put the higher value of the Brownian motion number.

www.nature.com/scientificreports/ Here p, I , T , and S are the pressure, identity tensor, Cauchy stress tensor, and the extra stress tensor respectively. Furthermore, α * k (k = 1, 2), β j (j = 1, 2, 3) are metallic constants and B i (i = 1, 2, 3) are the kinematic tensors defines as: T = −pI + µB 1 + α * 1 B 2 + α * 2 B 1 2 + β 1 B 3 + β 2 (B 1 B 2 + B 2 B 1 ) + β 3 B 1 (trcB 2 1 ).        www.nature.com/scientificreports/    where v and u are the parts of velocity in y and x directions respectively. C denotes concentration, N being Part of microrotation vector, orthogonal to the considered plane, and ρ is density of fluid. D T shows thermophoresis diffusion coefficient, D B is Brownian diffusion coefficient, and τ = (ρC p ) nf (ρC p ) f is the quotient of the heat capacity of the nano-particles and the heat of base fluid. C ∞ and T ∞ are free stream concentration and temperature. µ denotes dynamic viscosity whereas ν = µ ρ represents kinematic viscosity, C p is specific heat, T for temperature, γ shows spin gradient viscosity, k 1 (T) is variable thermal conductivity, j is microinertia density, k f is thermal conductivity of base fluid, k is the vortex viscosity, and (α * 1 , α * 2 , β 3 ) are material constants. Taking γ as and define K = k µ as micropolar parameter. Also, suppose that The Rosseland radiative heat flux is q r = −4σ * 3k * where mean absorption coefficient is shown as k * and σ * represents Stefan-Boltzmann constant. where ψ x, y is the stream function.

Equation (3) is now transformed as
η is similar variable, while θ(η) , f ′ (η) and φ(η) are the similarity representation of temperature, velocity profile, and concentration. h(η) is dimensionless micropolar profile respectively. Furthermore, T f and C f represent the temperature and concentration at the wall of the sheet, respectively. Mass conservation equation is automatically satisfied by putting Eq. (11) in Eq. (1). The PDEs (2), (4), (5), and (10) are transmuted to the non-linear coupled ODEs: With the altered BCs as: Here involved dimensionless variables are given as where δ T is thermal slip, B is microinertia density parameter,Ec, Pr, Ha 2 and Le are Eckert, Prandtl, Hartmann, and Lewis number respectively, Nt is Thermophoresis parameter, Nb is Brownian motion parameter, K the micropolar parameter, δ is the heat generation/absorption parameter, Rd the radiation parameter, β , α 1 and α 2 are 3rd grade, cross viscous and viscoelastic parameter.
The local Sherwood number Sh x , skin friction coefficient Cf x and local Nusselt number

Solution of the problem
The PDEs system describing the above model is transmuted into an ODEs system by using suitable similarity variables.  Figure 2 underlines the influence of fluid variable α 1 on velocity distribution, which shows the growth of f ′ (η) as rising the parameter α 1 . Physically, the viscosity of the material lowers if a larger value of α 1 is considered because of that force between the adjacent layers reduces, so in this case velocity increases. Figure 3 is the visual portrayal of α 2 on velocity. It is visualized that for a higher value of α 2 , a decline is seen in the curve of velocity distribution. This parameter causes shear-thickening of the fluid and a rise in resistance, that reduces the boundary layer flow, and originates an decrement in the size of the momentum boundary layer width. Figure 4 visualized variation of parameter β on velocity field, a rise in the velocity distribution is witnessed for a higher value of β . Actually β is inversely related to the viscosity. For greater β , viscosity declines and thus velocity rises. Figure 5 shows characteristics of Hartmann number Ha 2 on velocity distribution. It seems that velocity falls via larger Hartmann number. This is because, when magnetic field rises then Lorentz forces get stronger. Resistance (20) τ w = kN + (k + µ) ∂u ∂y + 2β 3 ∂u ∂y , pp2 = (−3 * Pr) * (3 + 3 * ∈ * p 5 + 4 * Rd * Pr) −1 * (1 + K) * Ec * p 3 * p 3 + p 1 * p 6 + δ * p 5 +Nb * p 8 * p 6 + Nt * p 6 * p 6 + ∈ Pr * p 6 * p 6 pp3 = −Nt Nb * pp2 − Le * p 1 * p 8 , pp4 = 2 * (2 + K) −1 * 4 * K * B * p 9 + 2 * K * B * p 3 + 3 * p 2 * p 9 − p 1 * p 10 , p 2 = 1, p 1 = 0, p 5 + δ T * p 6 , Nb * p 8 + Nt * p 6 = 0, p 9 = −m * p 3 , as η → 0, p 9 = 0, p 3 = 0, p 2 = 0, p 5 = 0, p 7 = 0, as η → ∞. www.nature.com/scientificreports/ to fluid flow is now more than before thus velocity is lowered. This process assists in controlling the size of the boundary layer. Figure 6 is describing that temperature of the fluid declines for boosting of Pr. Perceptions are that thermal boundary layer thickness appears to decline while the rising upsides of Pr are utilized. As a result, when the Prandtl number ascents, pace of thermal conductivity gets higher. In this way, with more noteworthy Pr, heat will disperse more rapidly from the sheet. As it's undeniably true that liquids having prevalent Prandtl number Pr will have lesser worth of thermal conduction. Therefore, Prandtl number is utilized to augment the cooling behavior in the flows. Heat exchange for different values of Radiation parameter Rd is presented in Fig. 7. A direct relation is witnessed between Rd and temperature distribution. It is concluded that with upper values of Rd results in enhancement of temperature distribution. In fact, asending value of parameter Rd means more heat is transferred to the fluid, that takes temperature distribution to go upward. Figure 8 is visual proof of the effects of micropolar parameter K on θ(η) , which is an evident that θ(η) is diminishing function of K. Figure 9 indicates the temperature θ(η) for different Ec values. By boosting Ec, an elevation is seen in the temperature distribution. It transfers kinematic energy into inner energy by overcoming viscous fluid tension and converting it to heat. Higher values of the Eckert number enhance the system's kinetic energy, which raises the temperature. As a result, increased viscous heat dissipation causes both growing heat and rising temperature.
The role of the Le on φ(η) is observed in Fig. 10. The Lewis number is stated as the rate of heat-to-mass diffusion coefficient. It is utilized to express the flow of liquid in which heat and momentum transfer occur simultaneously. The concentration distribution becomes steeper when Lewis number is increased. A bigger Le implies the lower D B (as it can be seen in Eq. 17 ) which causes a shorter penetration depth for concentration boundary layer. Figure 11 illustrates impact of Nb on the concentration field. A rise in Brownian motion variable gives a fall in φ(η) inside boundary layer. Furthermore, the progressing amounts of brownian motion coefficient reduces the micro-mixing of nanoparticles into the fluid's zone which diminishes the boundary layer thickness of concentration distribution. The consequences of Nt on φ(η) is exhibited in Fig. 12. It is seen that dimensionless concentration increases due to thermophoretic parameter. The thermophoresis constant amplifies the thermophoretic force, resulting in the transfer of nanoparticles from warm to cool locations and an increment in nanoparticle volume. In Fig. 13, the impact of Microinertia density parameter B is seen on micropolar profile. The behavior of h(η) is decreasing for increasing values of B.
In Table 1, the impacts of α 1 , α 2 , β , and Ha 2 are noticed on the skin-friction coefficient whereas keeping other parameters fixed. Fixed values used are ǫ = 0.3 , Pr = 1.5 , K = 0.2 , Ec = 0.8 , δ = 0.5 , Nt = 0.3 , Rd = 0.5 ,, B = 1.0 , δ T = 0.8 , m = 0.5,Nb = 1.0 , and Le = 0.7 . It is seen that skin-friction coefficient shows decrement for any larger value of α 1 , α 2 , and Ha 2 and shows opposite behavior for a larger value of β . Because the viscosity at the surface of the sheet increases with the evolution of the material constant, the skin friction coefficient decrease. The skin friction coefficient increases as the shear thickening coefficient β increases the thickness of the boundary layer. Increasing the Hartmann number Ha 2 , which refers to a significantly stronger axial magnetic field and an increment in magnetic Lorentz resisting forces in the x-direction has a significant impact on the magnitude of primary skin friction coefficient for all axial coordinate values.
It is seen that Nusselt number gives a rise in its values as a larger value of Pr, K, Ec, Rd, δ , and Nt is used while Nusselt number shows a decline for higher input of ǫ . The increment in the values of Prandtl number cause a decline in the thermal diffusivity, and hence resist the rise in heat transmission rate at the boundary. Boosting the Prandtl number raises the average Nusselt number at the heated surface. Higher values of the Rd improve convective heat transmission which ultimately raises the average Nusselt number.

Concluding remarks
Third-grade micropolar fluid flow is analyzed in this research article. An ODEs system is generated from the PDEs system by a similarity transformation and then analytically solved by the fourth-fifth order Runge-Kutta-Fehlberg strategy. Several graphs are presented for the temperature, velocity, concentration, and micropolar fields to observe the impact of different parameters on them. The impact of various parameters on Nusselt number, skin-friction coefficient, and Sherwood number is also concluded. The main findings are • Velocity distribution f ′ (η) boosts with α 1 β , while it has opposite behavior for α 2 and Hartmann number Ha 2 . • Temperature distribution θ(η) presents an increasing behavior for Radiation parameter Rd, and Eckert number Ec, while, the opposite behavior for Pr, and K. • Concentration distribution φ(η) shows diminishing behavior as we put a higher value of Brownian motion number Nb and Le. On the contrary, this has the opposite behavior for Nt. • Micropolar distribution h(η) displays an opposite relation with microinertia density parameter B.
• Skin-friction coefficient goes on increasing for higher values of parameters α 1 , α 2 , and Ha 2 and shows opposite behavior for bigger values of β. www.nature.com/scientificreports/ • Nusselt number shows an increasing behavior as larger values of parameters Pr, K, Ec, Rd, δ , and Nt are used, while Nusselt number depicts a decline for higher input of parameter ǫ. • Sherwood number shows increment for bigger values of parameters Le and Nt and decrement for the higher value of parameter Nb.

Data availability
All data generated or analyzed during this study are included in this article.