Significance of Thermal Phenomena and Mechanisms of Heat Transfer through the Dynamics of Second-Grade Micropolar Nanofluids

Abstract

Due to their unique microstructures, micropolar fluids have attracted enormous attention due to their potential for industrial application, including convective heat and mass transfer polymer production and the rigid and random cooling of particles for metallic sheets. In this context, a micropolar second-grade fluid flow over a vertical Riga plate is investigated for hidden microstructures. The novelty of the flow model allows us to explore the significance of Brownian motion and thermophoresis on the dynamics of non-Newtonian fluid. A mathematical model is developed under the flow assumptions for micropolar second-grade fluid over a vertical Riga plate of PDEs, reducing them into ODEs by invoking similarity techniques. The acquired system of non-linear ODEs is elucidated numerically using bvp4c methodology. Furthermore, comparative tables are generated to confirm the bvp4c technique, ensuring the accuracy of our numerical approach. This rheological study of micropolar second-grade fluid suggests that temperature distribution increases due to variations in the micropolar parameter (K), Eckert number (Ec), and the thermophoresis parameter (Nt), and the concentration distribution (Φ(η)) keeps rising against the boosting values of Brownian motion (Nb); however, the inverse trend is noted against thermophoresis (Nt).

1. Introduction

The examination of fluid flow over expanding sheets has countless applications in engineering and industries such as manufacturing plastic and metal spinning, rubber sheets, wire drawing, crude oil purification, glass blowing, paper production, polymer extrusion, etc. Sakiadis pioneered discussions on boundary layer flow for expanding sheets [1,2]. He discussed boundary layer behavior on a continuous solid and flat sheet. Crane [3] contributed to extending the idea of Sakiadis for boundary layer problems. He obtained an analytical solution for a viscous incompressible fluid over the expanding surface. Magyari and Keller [4] studied heat transfer influence over an exponentially expanding sheet by varying wall temperature. Under similar circumstances, different investigators discussed thermal and dissipation effects over the extending sheets by utilizing various methodologies [5,6,7]. Ali et al. [8,9,10] explored the MHD boundary layer flow for a Casson fluid across a porous extending surface. Currently, nanofluids have numerous applications, including in heat exchangers, jets, electrical chips, medicines, and so on. Nanofluids were introduced for the first time by Choi and Eastman [11] in 1995. Mabood et al. [12] numerically analyzed three-dimensional nanofluid flow over an extending surface by taking the base liquid as water and three different nanoparticles: alumina, copper, and titanium dioxide. Nanofluids are very useful to cool automobile engines, high flux devices, air conditioners, microwave ovens, and many other devices used in daily life [13,14]. Advanced nanotechnology has offered new useful tools to decrease environmental crises with nanomaterials that can boost the capacity of fossil fuel consumption [15]. Several recent studies of the extending surface problem have been reported for various fluid models [16,17,18,19]. Mazaheri et al. [20] examined the two-phase microchannel flow subject to tiny particles to observe the enhancement of heat transfer rate, and Bahiraei et al. [21] also examined the significance of nanoparticle shape on the dynamics of thermohydraulic performance.

Hypotheses around micropolar liquids have received special consideration for a long time since conventional Newtonian liquids cannot accurately depict the character of liquid with pendulous particles. A micropolar liquid complies with the constitutive equations of the supposed non-Newtonian liquid model. Within micropolar liquids, separated from classical speed field, micro-rotation vectors and gyration parameters are presented to examine the kinematics of micro-rotation. Such model may be connected to clarify the stream of colloidal arrangements, fluid gems, liquids with added substances, suspension arrangements, creature blood, etc. The existence of tidy smoke specific in gas may be modeled utilizing micropolar liquid elements. Unlike other liquids, micropolar liquids have microstructures located in liquids with non-symmetrical stress tensors. Eringen [22] investigated the flow of micropolar liquids under the continuum hypothesis. In our everyday lives, the manufacturing of non-Newtonian fluids, including physiological liquids, liquid materials, engine lubricants, paints, colloidal liquids, and food, plays a significant role. Guram and Smith [23] studied micropolar liquid flow by taking different interactions. Gorla and Takhar [24] used Eringen’s theory to study micropolar liquid over a rotating axisymmetric sheet with concentrated heat. Gorla et al. [25] explored the buoyancy impact on the forced convection of a micropolar stagnation flow. Nazar et al. [26,27] examined the micropolar boundary layer viscous liquid stream past an extended surface. Eldahab et al. [28] contemplated the radiation consequences of a warm move in a micropolar liquid course through a porous medium. Ishak [29] scrutinized the radiation effects on thermal flow instigated by an extending surface immersed in micropolar liquid. Nadeem et al. [30] deliberated the MHD stagnation point of micropolar fluid flow in a 2D surface. Yacob and Ishak [31] examined micropolar liquid flow over a squeezing surface on a plane. Interested readers are referred to some of the recent contributions by researchers [32,33,34].

The solutions of non-linear differential equations have attracted significant attention in the last few decades. Several engineers and mathematicians have solved various problems by utilizing different techniques. Several techniques have been developed to achieve accurate results, for instance, the Adomian decay technique [35], the finite difference method [36], the finite element technique [37], homotopy perturbation strategies [38,39], iterative strategies [40], MATLAB-bvp4c, etc. Recently, the MATLAB-bvp4c approach has become popular with various researchers to solve non-linear ordinary differential equations due to its remarkable accuracy. Several studies have been reported on the solutions using the bvp4c method [41,42,43,44,45].

In the previously mentioned literature, less consideration is paid towards the dynamics of second-grade micropolar fluid flow over an expanding Riga surface subject to thermal radiation, heat source, nanoparticles, a zero-order slip, thermal slip, viscous dissipation, and multi-buoyancy. In most articles already published [8,12,30,33], the dynamics of non-Newtonian fluid are controlled with the utilization of Lorentz force perpendicular to the x-axis, but present elaborated fluid problems controlled the Lorentz force parallel to the sheet. Motivated by the above-mentioned numerous applications of non-Newtonian fluids and nanofluids in the literature, an elaborated fluid model is selected. As far as the authors are aware, these aspects of the problem are not considered in the existing studies. The main target of this comprehensive study is the upgrade the host fluid temperature subject to tiny particles (nanoparticles). The developing mathematical model under the flow assumption is solved through numerical procedures after converting PDEs to non-linear ODEs. Governing equations are solved through the bvp4c numerical technique. The physical effects and numerical results are presented in tables and graphs. This report is related to various manufacturing applications, including plastic and metal spinning, paints, liquid materials, crude oil purification, heat exchangers, polymer extrusion, etc.

2. Mathematical Analysis

Consider a 2D steady incompressible second-grade micropolar liquid flow over a fixed stretching surface subject to thermal radiation, a heat source, nanoparticles, zero-order slip, thermal slip, viscous dissipation, and multi-buoyancy, as depicted in Figure 1. Furthermore, the dynamics of non-Newtonian fluid are controlled with the utilization of the Lorentz force parallel to the sheet (see Figure 1a. The positive y-axis is perpendicular to the stretchable surface, and the positive x-axis is the along the stretched surface. The sedimentation of tiny particles is ignored, and the suspension of nanoparticles is assumed to be a stable fluid. $C_{\infty }$ and $T_{\infty }$ are the concentration and free stream temperature. Under the above assumptions, the equations overseeing the problem are composed as in [12,22,28,34]:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(1)

$$\begin{gathered} v\frac{\partial u}{\partial y}+u\frac{\partial u}{\partial x}=\left(\nu +\frac{k}{\rho }\right)\frac{{\partial }^{2}u}{\partial y^2}+\frac{k}{\rho }\frac{\partial N}{\partial y}+\frac{\alpha }{\rho }\left(\frac{\partial u}{\partial x}\frac{{\partial }^{2}u}{\partial y^2}+u\frac{{\partial }^{3}u}{\partial x\partial y^2}+\frac{\partial u}{\partial y}\frac{{\partial }^{2}v}{\partial y^2}+v\frac{{\partial }^{3}u}{\partial y^3}\right) \\ +g\frac{\beta }{\rho }\left(T-T_{\infty }\right)+g\frac{\beta }{\rho }\left(C-C_{\infty }\right)+\frac{M_{°}{j}_{°}}{8\pi }{e}^{-\frac{\pi }{a}y}, \end{gathered}$$

(2)

$$u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y}=\frac{\gamma }{j\rho }\left(\frac{{\partial }^{2}N}{\partial y^2}\right)-\frac{k}{j\rho }\left(2N+\frac{\partial u}{\partial y}\right),$$

(3)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{1}{\rho C_{\rho }}\frac{\partial }{\partial y}\left(k\left(T\right)\frac{\partial T}{\partial y}\right)+\left(\frac{\mu +k}{\rho C_{\rho }}\right){(\frac{\partial u}{\partial y})}^2-\frac{1}{\rho C_{\rho }}\frac{\partial q_r}{\partial y}+Q\left(T-T_{\infty }\right)+\tau [D_B\frac{\partial C}{\partial y}\frac{\partial T}{\partial y}+\frac{D_T}{T_{\infty }}\left(\frac{\partial T}{\partial y}{)}^2\right],$$

(4)

$$\left(u\frac{\partial }{\partial x}+v\frac{\partial }{\partial y}\right)C=D_B\frac{{\partial }^{2}C}{\partial y^2}+\frac{D_T}{T_{\infty }}\frac{{\partial }^{2}T}{\partial y^2},$$

(5)

and the corresponding boundary conditions are given as in [28,45]

$$\begin{gathered} v=0,u=u_w+L_2\frac{\partial u}{\partial y},D_B\frac{\partial C}{\partial y}+\frac{D_T}{T_{\infty }}\frac{\partial T}{\partial y}=0,N=-m\frac{\partial u}{\partial y},T=T_f+L_1\frac{\partial T}{\partial y}asy=0, \\ u\to U,C\to 0,T\to T_{\infty },N\to 0,asy\to \infty , \end{gathered}$$

(6)

where $u$ and $v$ are parts of velocity in the x and y directions, respectively; $N$ is a micro rotation vector component perpendicular to the plane; $\nu$ is the kinematic viscosity; $\mu$ is dynamic viscosity; $\rho$ is the density of the fluid; $k$ is vortex viscosity; $u\left(x\right)$ is free stream velocity at the edge of the boundary layer;$j$ is micro-inertia density; $D_B$ is the coefficient of Brownian diffusion; $D_T$ is coefficient of thermophoresis diffusion; $T$ is the temperature; $C$ is concentration; $\alpha$ is thermal diffusivity; $\tau$ is a ratio of heat capacity of nano-particles to the heat of the base fluid; and $g$ is the gravitational acceleration. Additionally, $J_0,M_0,a,\beta ,Q$, $L_1,\mathrm{and}{L}_2$ are the electrodes applied to current density, the magnetization of permanent magnets, electrode and magnet width, expansion coefficient, heat generation/absorption coefficient, and slip parameters, respectively.

Figure 1. Flow configuration at slandering surface.

Some common examples of thermal slip are thermal boots, thermal footers, thermal jackets, thermal slipper socks, thermal slip printers, etc. There are typical examples in real life of velocity slip, such as the contact between two surfaces. Supposing $\gamma$ as:

$$\gamma =\left(\mu +\frac{\kappa }{2}\right)j=\mu \left(1+\frac{K}{2}\right)j,$$

(7)

where $K=\frac{k}{\mu }$ is the micropolar boundary, the micropolar liquid field can foresee the proper conduct in the restricting situation when the microstructure impacts become immaterial. The all-out turn N diminishes to the rakish stream velocity or stream vortices. Additionally, we have

$$k\left(T\right)=k\left(1+ϵ\theta \left(\eta \right)\right).$$

(8)

It is also cited that $q_r$ is Rosseland radiative heat flux

$$q_r=\frac{-4\sigma }{3k}\frac{{\partial }^{2}T}{\partial y^2}{T}^4\approx 4T_{\infty }^{3}T-3T_{\infty }^4,$$

(9)

and

$$\frac{\partial q_r}{\partial y}=\frac{-16T_{\infty }^3\sigma }{3k}\frac{{\partial }^{2}T}{\partial {y^2}^{\prime }}$$

(10)

The following suitable similarity transformations are introduced to simplify the analysis:

$$\begin{array}{c}\psi =\sqrt{\frac{4U\upsilon x}{3}}f\left(\eta \right),\eta =\sqrt{\frac{3U}{4\upsilon x}}y,N=U\sqrt{\frac{3U}{4\upsilon x}}h\left(\eta \right),\hfill \\ \varnothing \left(\eta \right)=\frac{C-C_{\infty }}{C_f-C_{\infty }},\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_f-T_{\infty }},\hfill \end{array}$$

(11)

where $\psi \left(x,y\right)$ represents the stream function with $u=\frac{\partial \psi }{\partial y}$ and $v=-\frac{\partial \psi }{\partial x}$,$\eta$ is the dimensionless variable, $f$ is the dimensionless velocity profile, $\varnothing$ and $\theta$ are the dimensionless portrayals of concentration and temperature, and $h$ is the dimensionless micropolar profile. Equation (1) is satisfied identically, and by utilizing Equation (11) in Equations (2)–(5), the accompanying arrangement of non-linear ODEs can be acquired:

$$\left(1+K\right)f^{‴}+K{h}^{\prime }+\frac{2}{3}f{f}^{″}-{\alpha }_1\left[2f^{\prime }{f}^{‴}-{f^{″}}^2+f{f}^{\left(iv\right)}\right]+{\lambda }_1\Phi +{\lambda }_2\Theta +M{e}^{-\eta {\lambda }_3}=0,$$

(12)

$$\left(1+\frac{K}{2}\right)h^{″}+\frac{2}{3}\left(f^{\prime }h+fh’\right)-\frac{4}{3}K\left(2h+f^{″}\right)=0,$$

(13)

$$\frac{1}{Pr}\left(ϵ{{\Theta }^{\prime }}^2+{\Theta }^{″}+ϵ\Theta {\Theta }^{″}\right)+\frac{2}{3}f{\Theta }^{\prime }+\left(1+K\right)Ec{f}^{″}+\frac{4}{3}Rd{\Theta }^{″}+\delta \Theta +Nb{\Theta }^{\prime }{\Phi }^{\prime }+Nt{{\Theta }^{\prime }}^2=0,$$

(14)

$${\Phi }^{″}+\frac{Nt}{Nb}{\Theta }^{″}+\frac{2}{3}Lef{\Phi }^{\prime }=0$$

(15)

with corresponding boundary conditions as

$$\begin{array}{c}f\left(0\right)=0,f^{\prime }\left(0\right)=1+\Omega f^{″}\left(0\right),f^{\prime }(\infty )=1,\hfill \\ h\left(0\right)=-m{f^{\prime }}^{\prime \left(0\right)},h(\infty )=0,\Theta \left(0\right)=1+\lambda {\Theta }^{\prime }\left(0\right),\hfill \\ Nb{\Phi }^{\prime }\left(0\right)+Nt{\Theta }^{\prime }\left(0\right)=0,\Theta (\infty )=\Phi (\infty )=0,\hfill \end{array}$$

(16)

where the dimensionless parameters used above are

$$\begin{array}{c}Pr=\frac{\mathit{k}}{\mu {\mathit{C}}_{\mathit{\rho }}},\mathit{K}=\frac{\mathit{k}}{\mathit{\mu }},\mathit{E}\mathit{c}=\frac{{\mathit{U}}^{\mathbf{2}}}{{\mathit{C}}_{\mathit{\rho }\left({\mathit{T}}_{\mathit{f}}-{\mathit{T}}_{\infty }\right)}},\mathit{N}\mathit{b}=\frac{\mathit{\tau }}{\mathit{\upsilon }{\mathit{D}}_{\mathit{B}}}\left({\mathit{C}}_{\mathit{f}}-{\mathit{C}}_{\infty }\right),\mathit{N}\mathit{t}=\frac{\mathit{\tau }}{\mathit{\upsilon }}\frac{{\mathit{D}}_{\mathit{T}}}{{\mathit{T}}_{\infty }}\left({\mathit{T}}_{\mathit{f}}-{\mathit{T}}_{\infty }\right),\hfill \\ \mathit{L}\mathit{e}=\frac{{\mathit{D}}_{\mathit{B}}}{\mathit{\upsilon }},\mathit{\delta }=Q\left(\frac{\mathbf{4}\mathit{x}}{\mathbf{3}\mathit{\upsilon }}\right),{\mathit{\alpha }}_{\mathbf{1}}=\frac{\mathit{\alpha }}{\mathit{\mu }}\frac{\mathit{U}}{\mathbf{2}\mathit{x}},\lambda =\mathit{k}\sqrt{\frac{\mathbf{3}\mathit{U}}{\mathbf{4}\mathit{\upsilon }\mathit{x}}},{\mathit{\lambda }}_{\mathbf{2}}=\frac{\mathbf{4}\mathit{x}}{\mathbf{3}{\mathit{U}}^{\mathbf{2}}}\mathit{g}{\mathit{\beta }}_{\mathit{T}}\left({\mathit{T}}_{\mathit{f}}-{\mathit{T}}_{\infty }\right),\hfill \\ {\mathit{\lambda }}_{\mathbf{1}}=\frac{\mathbf{4}\mathit{x}}{\mathbf{3}{\mathit{U}}^{\mathbf{2}}}\mathit{g}\mathit{\beta }\left({\mathit{C}}_{\mathit{f}}-{\mathit{C}}_{\infty }\right),\mathit{M}=\frac{\mathbf{4}\mathit{x}}{\mathbf{3}{\mathit{U}}^{\mathbf{2}}}\frac{{\mathit{M}}_{°}{\mathit{j}}_{°}}{\mathrm{8}\mathit{\pi }},{\mathit{\lambda }}_{\mathbf{3}}=\frac{\mathit{\pi }}{\mathit{a}}\sqrt{\frac{\mathbf{3}\mathit{U}}{\mathbf{4}\mathit{\upsilon }\mathit{x}}},\mathit{R}\mathit{d}=\frac{\mathbf{4}{T}_{\infty }^{\mathbf{3}}\mathit{\sigma }}{\mathit{\mu }\mathit{k}{\mathit{C}}_{\mathit{\rho }}}.\hfill \end{array}$$

(17)

Here, $\mathrm{Pr}$ is the Prandtl number, $K$ is the micropolar parameter, $Ec$ is the Eckert number, $Nb$ and $Nt$ are the Brownian motion and thermophoresis parameters, $Le$ is the Lewis number, $Rd$ represents the thermal radiative parameter, $M$ is the Hartmann number, and ${\lambda }_1\mathrm{and}{\lambda }_2$ are the modified and local Grashof numbers, respectively. Furthermore, ${\lambda }_3,\lambda ,\mathrm{and}\delta$ are dimensionless parameters.

3. Solution Procedure

The governing PDEs of the mathematical system are transformed into ODEs by suitable transformations. The obtained system of ODEs is solved numerically by utilizing the built-in bvp4c solver in the MATLAB software by setting relative error tolerance at le-4. The solution procedure is given as

$$\begin{gathered} y\left(1\right)=f\left(\eta \right); \\ y\left(2\right)=f^{\prime }\left(\eta \right); \\ y\left(3\right)=f^{″}\left(\eta \right); \\ y\left(4\right)=f^{‴}\left(\eta \right); \\ yy1=f^{\left(iv\right)}\left(\eta \right); \\ y\left(5\right)=h\left(\eta \right); \\ y\left(6\right)=h^{\prime }\left(\eta \right); \\ yy2=h^{″}\left(\eta \right); \\ y\left(7\right)=\Theta \left(\eta \right); \\ y\left(8\right)={\Theta }^{\prime }\left(\eta \right); \\ yy3={\Theta }^{″}\left(\eta \right); \\ y\left(9\right)=\Phi \left(\eta \right); \\ y\left(10\right)={\Phi }^{\prime }\left(\eta \right); \\ yy4={{\Phi }^{″}}^{\left(\eta \right)}. \\ yy1={\left({\alpha }_{1}y\left(1\right)\right)}^{-1}\left(\left(1+K\right)y\left(4\right)+Ky\left(6\right)+\frac{2}{3}y\left(1\right)y\left(3\right)+{\lambda }_{1}y\left(9\right)+{\lambda }_{2}y\left(7\right)+M\exp \left(-x{\lambda }_3\right)-2{\alpha }_{1}y\left(2\right)y\left(4\right)+{\alpha }_{1}y\left(3\right)y\left(3\right)\right); \\ yy2=\frac{2}{3}{\left(1+\frac{K}{2}\right)}^{-1}\left(K\left(2y\left(3\right)+4y\left(5\right)\right)-y\left(1\right)y\left(6\right)-y\left(2\right)y\left(5\right)\right); \\ yy3=-{\left(\frac{1+ϵy\left(7\right)}{Pr}+\frac{4}{3}Rd\right)}^{-1}\left(\left(1+K\right)Ecy\left(3\right)+\frac{2}{3}y\left(1\right)y\left(8\right)+\delta y\left(7\right)+Nby\left(8\right)y\left(10\right)+\left(Nt+\frac{ϵ}{Pr}\right)y\left(8\right)y\left(8\right)\right); \\ \mathit{y}\mathit{y}\mathbf{4}=-\frac{\mathit{N}\mathit{t}}{\mathit{N}\mathit{b}}\mathit{y}\mathit{y}\mathbf{3}-\frac{\mathbf{2}}{\mathbf{3}}\mathit{L}\mathit{e}\mathit{y}\left(\mathbf{1}\right)\mathit{y}\left(\mathbf{10}\right); \\ y\mathbf{0}\left(\mathbf{1}\right);y\mathbf{0}\left(\mathbf{2}\right)-\mathit{\Omega }\mathit{y}\mathbf{0}\left(\mathbf{3}\right);yinf\left(\mathbf{2}\right)-\mathbf{1};yinf\left(\mathbf{3}\right);y\mathbf{0}\left(\mathbf{5}\right)+my\mathbf{0}\left(\mathbf{3}\right);yinf\left(\mathbf{5}\right); \\ \mathrm{y}0\left(7\right)-1-\lambda \mathrm{y}0\left(8\right);\mathrm{yinf}\left(7\right);\mathrm{Nby}0\left(10\right)+\mathrm{Nty}0\left(8\right);\mathrm{yinf}\left(9\right). \end{gathered}$$

4. Results Validation

Table 1 and Table 2 show the results validation of the RK method with Bvp4c for physical constraints ramified in the skin friction coefficient and Nusselt number. The values of parameters other than those used in Table 1 and Table 2 are taken as constants $\mathrm{Pr}=0.9,Rd=0.3,ϵ=0.3,Ec=0.5,\delta =0.5,Nb=1.2,Nt=0.7,M=0.4,{\alpha }_1={\lambda }_1={\lambda }_2={\lambda }_3=0.5,Le=0.5,\mathrm{and}\lambda =0.5$. From Table 1 and Table 2, it can be observed that our results are valid.

Table 1. Comparison of skin friction coefficient for different inputs of K.

$\mathit{K}$	Skin Friction Coefficient (RK 4th Order)		Skin Friction Coefficient (Bvp4c)
	(m = 0.5)	(m = 0.0)	(m = 0.5)	(m = 0.0)
0.5	1.1948	0.9913	1.19476	0.99127
0.6	1.1423	0.9257	1.14227	0.92568
0.7	1.0948	0.8692	1.09476	0.86915
0.8	1.0518	0.8200	1.05175	0.81996

Table 2. Comparison of Nusselt number for different inputs of K.

$K$	Nusselt Number (RK 4th Order)		Nusselt Number (Bvp4c)
	(m = 0.5)	(m = 0.0)	(m = 0.5)	(m = 0.0)
0.5	0.3434	0.3524	0.34336	0.35235
0.6	0.3829	0.3918	0.38285	0.39177
0.7	0.4222	0.4304	0.42217	0.43036
0.8	0.4611	0.4685	0.46108	0.46847

5. Simulated Results

This article investigates the micropolar fluid flow over a fixed wedge. By the fluid flow assumptions under consideration, the mathematical model is constructed. After appropriate transformations, the system of PDEs is converted into ODEs. Equations (12)–(15) along with boundary condition Equation (16) is solved numerically by using MATLAB. This MATLAB software uses Runge–Kutta–Fehlberg’s fourth–fifth-order technique to acquire numerical solutions of BVP. Next, the influence of the involved main parameters on the concentration profile $\Phi \left(\eta \right)$, temperature profile $\Theta \left(\eta \right)$, velocity profile $f^{\prime }\left(\eta \right)$, and micropolar profile $h\left(\eta \right)$ is examined through graphical portrayal.

5.1. Velocity Profile

Figure 2 displays the influence of ${\alpha }_1$ on the velocity profile; it shows the growth of $f^{\prime }\left(\eta \right)$ as rising the parameter ${\alpha }_1$. The result is that the boundary layer thickness is enhanced for rising material parameter values because strain is resisted and shear flow is linear to the applied stress. These phenomena enhance the velocity graph as the fluid’s thickness is increased. Figure 3 shows evidence that by raising the micropolar parameter $K,$ a decline is seen in the velocity profile. When the micropolar parameter is improved, the velocity declines due to strengthening the vertex viscosity of the fluid. Figure 4 highlights the consequences of ${\lambda }_1$ on the velocity profile. This makes it clear that the decreasing trend for velocity $f^{\prime }\left(\eta \right)$ is due to the increasing values of ${\lambda }_1$. A direct relation between the velocity field $f^{\prime }\left(\eta \right)$ and parameter ${\lambda }_2$ is depicted in Figure 5. In Figure 6, the influence of Hartmann number M on $f^{\prime }\left(\eta \right)$ is observed. This clarifies that the velocity is an increasing function of $M$. The Hartmann number increases the intensity of the outer electric field, which amplifies the parallel wall Lorentz force and improves the velocity profile. This signifies that a similar Lorentz force sustains the flow in the x-direction. Furthermore, the wall velocity gradient increases while the thickness of the boundary layer declines. The relation between the velocity profile and the parameter $\Omega$ is seen in Figure 7. It is observed that $\Omega$ and $f^{\prime }\left(\eta \right)$ have the same behavior of increasing when the velocity obtains higher values by enlarging the values of the slip parameter. Physically, the slip between the surface and fluid is enhanced, resisting improvement to the velocity of the fluid.

Figure 2. Impact of ${\alpha }_1$ on the velocity profile.

Figure 3. Impact of $K$ on the velocity profile.

Figure 4. Impact of ${\lambda }_1$ on the velocity profile.

Figure 5. Impact of ${\lambda }_2$ on the velocity profile.

Figure 6. Impact of $M$ on the velocity profile.

Figure 7. Impact of $\Omega$ on the velocity profile.

5.2. Temperature Profile

The dependence of the temperature profile $\Theta \left(\eta \right)$ on the micropolar parameter $K$ is highlighted in Figure 8. The regime is significantly heated, and the thickness of the thermal boundary layer increases because the micro-rotation enlarges the temperature of the fluid. The increased viscosity of the vortex promotes thermal diffusion and serves as an agitator. This boosts the efficiency of thermal convection within the fluid’s body from the microscopic to the macroscopic scale and effectively transmits heat from the fluid regime with greater intensity. The relation between $\Theta \left(\eta \right)$ and Prandtl number $Pr$ is represented in Figure 9. Enhancing the Prandtl number $Pr$ reduces the temperature profile. It should be pointed out that a higher value of the Prandtl number effects the thermal diffusivity. Thus, thermal diffusivity decreases as does the thermal boundary layer thickness. Physically, advancing the Prandtl number values diminishes the thermal boundary layer thickness. The Prandtl number is the ratio of momentum diffusivity to heat diffusivity. It controls the heat flux zones, the heat transfer problems, and the relative thickening of the motion. Figure 10 visualizes the role of radiative parameters on the temperature function. It is seen that thermal flux dies out for greater values of $Rd.$ Physically, the radiation parameter has a direct influence on the ambient temperature of the fluid. Thus, the stronger impact of growing strength on the radiation parameters enhances the temperature.

Figure 8. Impact of $K$ on the temperature profile.

Figure 9. Impact of $Pr$ on the temperature profile.

Figure 10. Impact of $Rd$ on the temperature profile.

It is noticed that a higher temperature and a thicker thermal boundary layer are related to a higher thermal radiation constant. As a result of the augmented temperature field, the increased radiation transfers a significant amount of heat to the liquid. Physically, due to amplifying $Rd$, more heat is produced in the fluid flow, which results in improved temperature distribution and related boundary layer thickness. Figure 11 indicates the temperature profiles for various values of the Eckert number $Ec$. The importance of $Ec$ is increased, which enhances the values of the temperature profile. Physically, $Ec$ is the thermal diffusivity and mass diffusivity ratio. It is utilized to elucidate the fluid flow in which heat and mass transfer coincide.

Figure 11. Impact of $Ec$ on the temperature profile.

Figure 12 presents the relation between $\Theta \left(\eta \right)$ and $ϵ$. Thermal thickness rises by incrementing values of ϵ. Figure 13 reveals the influence of $\delta$ on temperature profiles. The temperature profile enlarges due to growing values of $\delta$. Physically, the higher input of heat source causes a more induced flow towards the sheet because of the thermal buoyancy impact, so the temperature profile of the fluid is enhanced against the growing value of the heat source. The graphical behavior of the temperature profile for the growing of thermophoresis parameter $Nt$ is provided in Figure 14. The figure delineates that the temperature profile is an increasing function of $Nt$. By boosting the value of $Nt,$ an enlargement in thermophoretic force occurs that causes the temperature to be improved. The thermophoresis is the transport force that arises owing to the occurrence of a temperature gradient. Thermophoresis offers a wide range of applications, including radioactive particle deposition in silicon thin films, nuclear reactors, particles striking gas turbine blade surfaces, and aerosol technologies. Thermophoresis is a tool for examining temperature distribution; an increase in Nt causes nanoparticles to migrate from a hotter zone to a colder region, improving temperature distribution.

Figure 12. Impact of $ϵ$ on the temperature profile.

Figure 13. Impact of $\delta$ on the temperature profile.

Figure 14. Impact of $Nt$ on the temperature profile.

5.3. Concentration Profile

Figure 15 depicts the variation of the Brownian motion parameter $Nb$ in the concentration profile. It delineates that concentration is an increasing function of the Brownian motion parameter. The ϕ(η) thickness boosts with the increase in the value of $Nb$. Figure 16 demonstrates the thermophoresis parameter belonging to the concentration profile $\Phi \left(\eta \right)$. It is evident that the concentration profile $\Phi \left(\eta \right)$ displays opposite behavior for that needed for $Nt$ to increase. The concentrations are dispersed in further non-uniform manner. Similarly, thermophoresis improves non-uniformity in the dispersal of the concentration, with a patent impact at significantly higher concentrations. In Figure 17, the impact of $\Phi \left(\eta \right)$ with the variation in the Lewis number $Le$ is examined. It is highlighted that whenever a higher value of $Le$ is taken, the concentration distribution goes upward. Physically, the Lewis number comprises the thermal and mass diffusivity ratio. The Lewis number puts the thickness of the thermal boundary layer to the concentration boundary layer. It describes the fluid flow in which heat and mass transfer coincide.

Figure 15. Impact of $Nb$ on the concentration profile.

Figure 16. Impact of $Nt$ on the concentration profile.

Figure 17. Impact of $Le$ on the concentration profile.

5.4. Numerical Analysis

It is crucial to examine the impacts of all dimensionless parameters used in modeled equations on skin friction coefficient and Sherwood number, and Nusselt number for both weak and intense concentrations. In Table 3, the results of the physical constraints are ramified in relation to the skin friction coefficient. The values of the parameters other than those used in Table 3 are taken as constants $\mathrm{Pr}=0.9,Rd=0.3,ϵ=0.3,Ec=0.5,\delta =0.5,Nb=1.2,Nt=0.7,Le=0.5,\mathrm{and}\lambda =0.5.$ The value of ${\alpha }_1$ enhances as the skin friction is enhanced. The effects of the material parameters are enhanced, reducing the skin friction coefficient. The fluid’s viscosity near the surface increased due to improvement in the material parameter, resisting the reduction of the skin friction coefficient. The impact of skin friction and $M$ is revealed in Table 3. This phenomenon enhances the velocity graph as the fluid thickness is boosted due to the strengthening the skin friction. The variation in ${\lambda }_2$ and skin friction is shown in Table 3. The values of skin friction are enhanced due to the improved slip parameters. When the values of ${\lambda }_1$ and ${\lambda }_3$ are boosted the values of skin friction decline. The velocity slip is enhanced, which reduces the skin friction because the velocity slip is enhanced and the surface becomes smooth, so the skin friction is reduced.

Table 3. Numerical analysis for the skin friction coefficient for diversified values of physical parameters.

$\mathit{K}$	${\mathit{\alpha }}_1$	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	$\mathit{M}$	$\mathit{\Omega }$	Skin Friction Coefficient
							(m = 0.5)	(m = 0)
0.5	0.5	0.5	0.5	0.5	0.4	0.5	1.1948	0.9913
0.6							1.1423	0.9257
0.7							1.0948	0.8692
0.8							1.0518	0.8200
0.5	0.3						1.0070	0.8592
	0.4						1.0980	0.9236
	0.5						1.1948	0.9913
	0.6						1.2928	1.0598
	0.5	0.5					1.1948	0.9913
		0.7					1.1184	0.9264
		0.9					1.0394	0.8596
		1.1					0.9577	0.7905
		0.5	0.4				1.0701	0.8893
			0.45				1.1325	0.9404
			0.50				1.1948	0.9913
			0.55				1.2569	1.0420
			0.5	0.1			1.4001	1.1549
				0.3			1.2827	1.0615
				0.5			1.1948	0.9913
				0.7			1.1278	0.9377
				0.5	0.0		0.7635	0.6420
					0.2		0.9747	0.9913
					0.4		1.1948	0.9924
					0.6		1.4235	1.1732
					0.4	0.2	1.4670	1.2111
						0.3	1.3787	1.1403
						0.4	1.2879	1.0670
						0.5	1.1948	0.9913

Table 4 expresses the impacts of several parameters on the Nusselt number. The parameters taken were not used in the table below, and were unchanged as ${\alpha }_1=0.5,{\lambda }_1=0.5,{\lambda }_2=0.5,{\lambda }_3=0.5,M=0.4,\mathrm{and}\Omega =0.2.$ It is noted that values of $-{\theta }^{\prime }\left(\eta \right)$ rise in a Brownian motion, which enhances the rate of heat transfer. The values of $-{\theta }^{\prime }\left(\eta \right)$ increase the thermophoresis impact of the fluid, which expands the heat transfer rate. The impacts of $Ec$ on $-{\theta }^{\prime }\left(\eta \right)$ are presented in Table 4. The skin friction declines due to boosted values of $Rd$. The Eckert number and Brownian motion parameter enlargement show a justification for both Nusselt numbers. In contrast, the radiation parameter $Rd$ displays the opposite nature, increasing for various values of $Rd$. The values of $Pr$ enhance, resisting the enhancement of heat transfer rate at the surface. Increasing the Prandtl number enhances the average Nusselt number at the heated surface and increases the drag force. The values of $Le$ number are enhanced due to the enhanced values of the Nusselt number. $Ec$ number increases, which enhances the Nusselt number, as is revealed in Table 4. The values of $ϵ$ and $\lambda$ are enhanced, which reduces the Nusselt number at the surface.

Table 4. Numerical analysis for the Nusselt number for diversified values of physical parameters.

$\mathit{K}$	$Pr$	$\mathit{R}\mathit{d}$	$\mathit{ϵ}$	$\mathit{E}\mathit{c}$	$\mathit{\delta }$	$\mathit{N}\mathit{b}$	$\mathit{N}\mathit{t}$	$\mathit{\lambda }$	$\mathit{L}\mathit{e}$	Nusselt Number
										(m = 0.5)	(m = 0)
0.5	0.9	0.3	0.3	0.5	0.5	1.2	0.7	0.5	0.5	0.3434	0.3524
0.6										0.3829	0.3918
0.7										0.4222	0.4304
0.8										0.4611	0.4685
0.5	0.7									0.1784	0.1810
	0.9									0.3434	0.3524
	1.4									0.7401	0.7754
	2.1									1.2515	1.3415
	0.9	0.3								0.3434	0.3524
		0.5								0.3422	0.3502
		0.6								0.3386	0.3460
		0.7								0.3332	0.3400
		0.3	0.2							0.3591	0.3692
			0.25							0.3511	0.3606
			0.3							0.3434	0.3524
			0.35							0.3359	0.3444
			0.3	0.5						0.3434	0.3524
				0.55						0.3991	0.4069
				0.6						0.4550	0.4616
				0.65						0.5110	0.5164
				0.5	0.4					0.2117	0.2149
					0.45					0.2750	0.2809
					0.5					0.3434	0.3524
					0.55					0.4174	0.4300
					0.5	1.0				0.3453	0.3541
						1.1				0.3442	0.3532
						1.2				0.3434	0.3524
						1.3				0.3426	0.3517
						1.2	0.7			0.3434	0.3524
							0.8			0.3509	0.3596
							0.9			0.3585	0.3669
							1.0			0.3661	0.3741
							0.7	0.3		0.3538	0.3621
								0.5		0.3434	0.3524
								0.6		0.3384	0.3478
								0.7		0.3336	0.3432
								0.5	0.4	0.3373	0.3463
									0.5	0.3434	0.3524
									0.6	0.3490	0.3581
									0.7	0.3544	0.3635

Table 5 presents the impacts of physical parameters on the Shorewood number, showing the variation in $Nb$ and $-{\varphi }^{\prime }\left(0\right).$ The values of $-{\varphi }^{\prime }\left(0\right)$ boost due to larger values of $Nb$ as the particle’s motion increases, boosting $-{\varphi }^{\prime }\left(0\right).$ $\varphi \left(\eta \right)$ thickness improves due to strengthening the values of $Nb$. It is noted that the values of $-{\varphi }^{\prime }\left(0\right)$ decline due to higher values of $Nt$ due to the more pronounced effect as the average concentration enhances. The relation of $Ec$ with $-{\varphi }^{\prime }\left(0\right)$ is presented in Table 5. It is noted that the values of $Pr$ are enhanced, which enhances the values of $-{\varphi }^{\prime }\left(0\right).$ The values of $K$ are enhanced, which improves the values of $-{\varphi }^{\prime }\left(0\right).$ The values of are $Rd$ enhanced, resisting a decline in the values of $-{\varphi }^{\prime }\left(0\right).$ It is seen that the values of ϵ improved by reducing the values of $-{\varphi }^{\prime }\left(0\right).$ The Sherwood number increases with enhanced values of δ. The Sherwood number declines due to incrementing values of λ, while the Sherwood number improves due to boosting values of $Le.$

Table 5. Numerical analysis for the Sherwood number for diversified values of physical parameters.

$\mathit{K}$	$\mathit{P}\mathit{r}$	$\mathit{R}\mathit{d}$	$\mathit{ϵ}$	$\mathit{E}\mathit{c}$	$\mathit{\delta }$	$\mathit{N}\mathit{b}$	$\mathit{N}\mathit{t}$	$\mathit{\lambda }$	$\mathit{L}\mathit{e}$	Sherwood Number
										(m = 0.5)	(m = 0)
0.5	0.9	0.3	0.3	0.5	0.5	1.2	0.7	0.5	0.5	0.1170	0.1201
0.6										0.1303	0.1332
0.7										0.1433	0.1461
0.8										0.1562	0.1587
0.5	0.7									0.0613	0.0622
	0.9									0.1170	0.1201
	1.4									0.2474	0.2588
	2.1									0.4086	0.4363
	0.9	0.3								0.1170	0.1201
		0.5								0.1011	0.1035
		0.6								0.0938	0.0958
		0.7								0.0869	0.0886
		0.3	0.2							0.1301	0.1337
			0.25							0.1233	0.1266
			0.3							0.1170	0.1201
			0.35							0.1112	0.1140
			0.3	0.5						0.1170	0.1201
				0.55						0.1356	0.1382
				0.6						0.1542	0.1564
				0.65						0.1727	0.1745
				0.5	0.4					0.0726	0.0737
					0.45					0.0941	0.0960
					0.5					0.1170	0.1201
					0.55					0.1417	0.1459
					0.5	1.0				0.1412	0.1448
						1.1				0.1280	0.1313
						1.2				0.1170	0.1201
						1.3				0.1078	0.1106
						1.2	0.7			0.1170	0.1201
							0.8			0.1366	0.1400
							0.9			0.1570	0.1606
							1.0			0.1781	0.1819
							0.7	0.3		0.1214	0.1242
								0.5		0.1170	0.1201
								0.6		0.1150	0.1181
								0.7		0.1130	0.1162
								0.5	0.4	0.1150	0.1180
									0.5	0.1170	0.1201
									0.6	0.1189	0.1220
									0.7	0.1207	0.1238

In Table 6, the impacts of the physical parameters involved in $\left(1+\frac{K}{2}\right)h^{\prime }\left(0\right)$ are examined. The values of the parameters other than those used in Table 6 are taken as constants $\mathrm{Pr}=0.9,Rd=0.3,ϵ=0.3,Ec=0.5,\delta =0.5,Nb=1.2,Nt=0.7,Le=0.5,$ and $\lambda =0.5.$ The values of $\left(1+K/2\right)h^{\prime }\left(0\right)$ enhance the boosting values of the material parameter ${\alpha }_1$, but the values of $\left(1+K/2\right)h^{\prime }\left(0\right)$ decline due to larger values of $K$. The importance of $\left(1+K/2\right)h^{\prime }\left(0\right)$ enhances for higher values of ${\lambda }_2$, while $\left(1+K/2\right)h^{\prime }\left(0\right)$ declines due to boosting values of ${\lambda }_1$. $M$ has an effect on $\left(1+K/2\right)h^{\prime }\left(0\right)$. It is noted that the values of $\left(1+K/2\right)h^{\prime }\left(0\right)$ enhance due to higher values of $M$. The values of $\left(1+\frac{K}{2}\right)h^{\prime }\left(0\right)$ decline for larger values of ${\lambda }_3$ and $\Omega$, which is revealed in Table 6.

Table 6. Numerical analysis for $\left(1+\frac{K}{2}\right)h^{\prime }\left(0\right)$ for various values of physical parameters.

$\mathit{K}$	${\mathit{\alpha }}_1$	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	$\mathit{M}$	$\mathit{\Omega }$	$\left(1+\frac{\mathit{K}}{2}\right){\mathit{h}}^{\prime }\left(0\right)$
							(m = 0.5)	(m = 0)
0.5	0.5	0.5	0.5	0.5	0.4	0.2	1.2953	−0.8505
0.6							1.1323	−0.8278
0.7							1.0053	−0.8066
0.8							0.9034	−0.7868
0.5	0.3						0.9691	−0.8197
	0.4						1.1228	−0.8359
	0.5						1.2953	−0.8505
	0.6						1.4808	−0.8632
	0.5	0.5					1.2953	−0.8505
		0.7					1.1682	−0.8239
		0.9					1.0363	−0.7947
		1.1					0.8990	−0.7624
		0.5	0.4				1.1123	−0.8103
			0.45				1.2037	−0.8308
			0.50				1.2953	−0.8505
			0.55				1.3868	−0.8697
			0.5	0.1			1.5504	−0.9253
				0.3			1.4042	−0.8825
				0.5			1.2953	−0.8505
				0.7			1.2126	−0.8264
				0.5	0.0		0.7539	−0.7302
					0.2		1.0172	−0.7915
					0.4		1.2953	−0.8505
					0.6		1.5877	−0.9075
					0.4	0.2	1.2953	−0.8505
						0.3	1.2382	−0.7683
						0.4	1.1805	−0.6844
						0.5	1.1223	−0.5989

6. Conclusions

This computational and theoretical work addresses the implications of variable thermal conductivity on the dynamics of micropolar second-grade nanofluid subject to the Lorentz force. A system of non-linear ODEs is obtained from the system of PDEs by appropriate transformation and then solved numerically by the bvp4c technique. The graphs are plotted for the temperature profile Θ(η), concentration profile Φ(η), velocity profile f’(η), Sherwood number, skin friction coefficient, and Nusselt number to see the impacts of several physical parameters. The key conclusions of the study are:

• Velocity distribution $f^{’}\left(\eta \right)$ rises with higher values of any second-grade material (${\alpha }_1$), local Grashof number (${\lambda }_2$), modified Hartmaan number (M), or $\Omega ,$ and it falls for higher values of the micropolar material parameter ($K)$ or the thermal local Grashof number (${\lambda }_1)$;
• The thickness of the thermal boundary layer boosts with thermophoretic value (Nt), Eckert number (Ec), heat source (δ), and micropolar material (K) parameters, but an opposite trend is reported against Prandtl number (Pr);
• The concentration distribution Φ(η) keeps rising against the boosting values of Brownian motion (Nb), but an inverse trend is noted against thermophoresis (Nt);
• Skin friction coefficient keeps increasing for larger values of ${\alpha }_1,{\lambda }_2,$ and M, and an opposite behavior is seen for the parameters $K,{\lambda }_1,{\lambda }_{3,}$ and $\Omega$;
• Sherwood number increases for the increasing parameters $K,Pr,Ec,\delta ,Le$, and $Nt,$ and it decreases if any value of $Rd,Nb,ϵ,$ or $\lambda$ is increased;
• $\left(1+\frac{K}{2}\right)h^{\prime }\left(0\right)$ keeps increasing for larger values of ${\alpha }_1,{\lambda }_2,$ and M, and an opposite behavior is observed for the parameters $K,{\lambda }_1,{\lambda }_3$ and $\Omega .$

Through this numerical computational effort, we have successfully elucidated the parametric impact on the dynamics of second-grade micropolar nanofluids. This study may be extended for second-grade Maxwell nanofluids, second-grade Oldroyd-B nanofluids, or second-grade Casson nanofluid.