Thermal Improvement in Pseudo-Plastic Material Using Ternary Hybrid Nanoparticles via Non-Fourier’s Law over Porous Heated Surface

Abstract

The numerical, analytical, theoretical and experimental study of thermal transport is an active field of research due to its enormous applications and use in numerous systems. This report covers the impacts of thermal transport on pseudo-plastic material past over a horizontal, heated and stretched porous sheet. Modeling of energy conservation is based upon a generalized heat flux model along with a heat generation/absorption factor. The modeled phenomenon is derived in the Cartesian coordinate system under the usual boundary-layer approach proposed by Prandtl, which removes the complexity of the problem. The modeled rheology is obtained in the form of coupled, nonlinear PDEs. These derived PDEs are converted into ODEs with the engagement of similarity transformation. Afterwards, converted ODEs containing some emerging parameters have been approximated numerically with a powerful and effective scheme, namely the finite element approach. The obtained results are compared with the published findings as a limiting case of current research, and an excellent agreement in the obtained solution was found, which guarantees the effectiveness of the used methodology. Furthermore, it is recommended that the finite element approach is a good method among other existing methods and can be effectively applied to nonlinear problems arising in the mathematical modeling of different phenomenon.

1. Introduction

Numerous rheology relations have been proposed and developed by the researchers to study the material properties by discussing the physical significance of the involved parameters. The investigation covering the utilization of pseudo-plasticity received a lot of attention due to its wider applications. For instance, Akbar et al. [1] conducted a survey on peristaltic phenomena on a pseudo-plastic model flowing in a horizontal channel. They formulated the model under long wavelength analysis. They solved the modeled problem analytically. They have shown the flow characteristics against numerous involved parameters through graphs. They noticed the pumping phenomena against elastic parameters. They recorded that the velocity of the pseudo-plastic model is less than the Newtonian model. Salahuddin et al. [2] observed the flow of the pseudo-plastic model in a stretching horizontal cylinder by considering a temperature-dependent model for thermal conductivity and magneto-hydrodynamic effect and generated the model under the boundary-layer analysis. They transformed the boundary-layer model presenting the pseudo-plastic fluid momentum and thermal transport equations into ODEs by selecting similar transformations and then solving them numerically via the Keller box scheme (KBS). They have examined the increase in heat transfer rate against the Prandtl number. Si et al. [3] presented the mixed convective analysis of the pseudo-plastic model flowing over a stretched vertical sheet. They solved the resulting equations numerically and drafted the stability survey for its numerical solution. Their analysis showed that higher values of power law index retard the fluid flow. The unsteady rheology for a pseudo-plastic model passed over an elongated wall was investigated by Kholili et al. [4] numerically. They found that the spin friction coefficient for dilatant material is less than that of pseudo-plastic material. Hina et al. [5] used a curved configuration to examine the flow rheology of the pseudo-plastic model with wall properties in a channel obeying the peristaltic mechanism. They found a solution using a perturbation scheme, and the obtained solution has been plotted for numerous involved, influential variables. They recorded the decline in the concentration field against plastic parameters. Kawase et al. [6] discussed the turbulent flow of the pseudo-plastic model in a pipe. They reported an augmentation in velocity against higher values of the Reynolds number.

The usage of nanoparticles in industry and other fields of applied sciences and their utilization in everyday life demonstrate their importance, which no one can ignore. These particles have been frequently used for cancer therapy, and they have also been extensively used to improve thermal performance. Aamir and Khan [7] examined the involvement of the Buongiorno Model on thermal and mass transport in unsteady Williamson material towards a heated stretched sheet, numerically. They converted the derived model equations in ODEs with the use of appropriate transformation and then solved the resulting ODEs via the shooting approach coded in MATLAB 14.0 software. They repeated the increase in temperature profile against the Brownian motion parameter, whereas the viscosity ratio parameter enhances the velocity field and its associated boundary-layer thickness. The MHD flow of Newtonian fluid with the phenomena of thermal process and Brownian motion over an impulsively heated wall was examined by Mohyud-Din et al. [8]. They used a Maple package to solve the modeled equation via a shooting scheme and plotted the velocity concentration and temperature solution against numerous parameters and noticed their involvement. They monitored the enhancement in fluid temperature against the thermophoresis parameter. Lee et al. [9] studied the mixture of NH₃/H₂O and Al₂O₃ nanoparticles theoretically and experimentally and reported the extensive applications of these used particles to improve thermal performance. Prasad et al. [10] studied the comportment of heat generation in a viscous nanofluid past a porous stretching sheet. They recorded the increase in its thermal profile against higher values of the Dufour parameter. Gireesha et al. [11] examined the suspension of nanoparticles on radiated, stretched non-Newtonian material and solved the governing equations numerically. They found the increase in temperature and fluid concentration against the Biot number. Inclined MHD and volume fraction contribution on a mixture of dusty nanoparticles was reported by Sandeep et al. [12]. They solved the non-dimensional boundary value problem numerically. They recorded the increase in fluid velocity against volume fraction and monitored the fluid velocity of dust particles. Mahanthesh et al. [13] studied the non-linear radiated flow past a bi-directional stretched surface under radiation phenomena. They observed the increase in temperature and concentration fields against higher values of nanoparticle volume fraction. Kandasamy et al. [14] analyzed the influence of convective boundary conditions on mixed convective nanofluid flow. They noted the increase in concentration profile for the Soret number. Another relevant research is reported in [15,16,17,18,19,20,21,22,23]. Nazir et al. [24] discussed significant comparative analysis among hybrid nanoparticles and nanoparticles in Carreau-Yasuda material using base fluid based on ethylene glycol. They used a strong approach called the finite element approach to conduct numerical results of velocity, thermal energy and solute particles against variations in various physical parameters. Chu et al. [25] adopted the finite element approach to simulate numerical consequences for velocity, concentration and heat energy among nanomaterials and hybrid nanomaterials via ethylene glycol. Nazir et al. [26] studied non-Fourier’s law in hyperbolic tangent liquid, inserting hybrid nanostructures using the finite element scheme. Swain et al. [27] predicted significant growth in heat energy using hybrid nanoparticles via slip conditions.

Extensive research has been reported so far on heat and mass transport under several physical effects with the consideration of the single-phase model and the Buongiorno model and the involvement of different types of nanoparticles on different surfaces. Upon viewing the published articles and books, it was noticed that the contribution of ternary hybrid nanoparticles in the pseudo-plastic fluid has not been explored, so this attempt covers the inclusion of ternary hybrid nanoparticles on a pseudo-plastic model past over a heated porous stretching sheet while addressing a generalized heat flux theory.

2. Mathematical Formulation

The developing model of the heat transfer in pseudo-plastic material over the heated sheet is analyzed. The involvement of tri-hybrid nanoparticles in the motion of fluid particles is considered while the motion of fluid particles is induced with the help of wall movement along the x-direction. The characterization of heat energy is carried out within theories of non-Fourier and a Forchheimer porous medium. The concept of heat generation is implemented to know the behavior of thermal changes including tri-hybrid nanoparticles. Moreover, the composition among nanoparticles ($A{l}_2{O}_3, Ti{O}_2$ and $Si{O}_2$) is called tri-hybrid nanoparticles, and the composition among $Ti{O}_2$ and $Si{O}_2$ is known as hybrid nanoparticles, whereas ethylene glycol (EG) is called base fluid. Figure 1 reveals the schematic behavior of fluid while a magnetic field is exerted along the y-direction of the surface and motion of fluid particles is induced within the wall movement along a horizontal direction. It is observed that the induced magnetic field is neglected, whereas wall temperature and ambient temperature are considered by $T_w$ and $C_w$, respectively. The roles of the developing model and the schematic are considered in Figure 2. The power-law model related to shear stress is defined as

$${\tau }_{xy}=-n{\left|\frac{\partial \overline{U}}{\partial y}\right|}^m\frac{\partial \overline{U}}{\partial y}$$

(1)

partial differential equations (PDEs) are formulated as

$$\frac{\partial \overline{U}}{\partial x}+\frac{\partial \overline{V}}{\partial y}=0$$

(2)

$$\overline{U}\frac{\partial \overline{U}}{\partial x}+\overline{V}\frac{\partial \overline{U}}{\partial y}={\nu }_{Thnf}\frac{\partial }{\partial y}\left({\left|\frac{\partial \overline{U}}{\partial y}\right|}^{m-1}\frac{\partial \overline{U}}{\partial y}\right)-\frac{{\nu }_{Thnf}}{k^s}{F}_D\overline{U}-\frac{F_D}{{\left(k^s\right)}^{\frac{1}{2}}}{\overline{U}}^2$$

(3)

$$\begin{gathered} \overline{U}\frac{\partial T}{\partial x}+\overline{V}\frac{\partial T}{\partial y}=\frac{k_{Thnf}}{{\left(\rho C_p\right)}_{Thnf}}\frac{{\partial }^{2}T}{\partial y^2}-{\lambda }_t\left(\begin{array}{c}\overline{U}\frac{\partial \overline{U}}{\partial x}\frac{\partial T}{\partial x}+\overline{V}\frac{\partial \overline{U}}{\partial y}\frac{\partial T}{\partial x}+\overline{U}\frac{\partial \overline{V}}{\partial x}\frac{\partial T}{\partial y}+2\overline{U}\overline{V}\frac{{\partial }^{2}T}{\partial x\partial y}\\ {\overline{U}}^2\frac{{\partial }^{2}T}{\partial x^2}+{\overline{V}}^2\frac{{\partial }^{2}T}{\partial y^2}-\frac{Q}{{\left(\rho C_p\right)}_{Thnf}}\left(\overline{U}\frac{\partial T}{\partial x}+\overline{V}\frac{\partial T}{\partial y}\right)\end{array}\right) \\ +\frac{k_{Thnf}}{{\left(\rho C_p\right)}_{Thnf}}\frac{{\partial }^{2}T}{{\partial }^{2}y}+\frac{Q}{{\left(\rho C_p\right)}_{Thnf}}\left(T-T_{\infty }\right) \end{gathered}$$

(4)

where $\left(Q\right)$ is heat generation, $\left(T\right)$ is temperature, ($k$) is thermal conductivity, $\left(\rho \right)$ is fluid density, ($\overline{U},\overline{V}$) are velocities, (${\lambda }_t$) is time relaxation, ($C_P$) is specific heat capacitance, ($\nu$) is kinematic viscosity, ($m$) is the power law number, ($y,x$) are space coordinates,($x$) is the horizontal coordinate, ($y$) is the vertical coordinate, ($Thnf$) are tri-hybrid nanoparticles, ($\tau$) is tensor of fluid, ($T_{\infty }$) is ambient temperature, ($k_{*}$) is the permeability within porous medium and ($F_D$) is the inertia coefficient in terms of the porous medium described above. It is noticed that $m$ reveals the characterization of shear thinning and shear thickening behaviors of the fluid. In Equation (1), the fluid is known as Newtonian liquid for case $m=1$, the fluid becomes dilatant liquid for the case $m>1$ and the fluid is called pseudo-plastic material for case $0<m<1$. The no-slip theory is used to generate boundary conditions (BCs), and the required conditions are

$$\begin{gathered} \overline{U}=u_w,\overline{V}=-v_w,T=T_w \mathrm{at} y=0, \\ \overline{U}\to u_{\infty },T\to T_{\infty } \mathrm{when} y\to \infty . \end{gathered}$$

(5)

Transformations of the model are

$$\theta =\frac{T-T_{\infty }}{T_w-T_{\infty }},\xi =y{\left(\frac{U^{2-m}}{x{\nu }_f}\right)}^{\frac{1}{m+1}},\mathsf{\Psi }=F{\left(x{\nu }_f{U}^{2m-1}\right)}^{\frac{1}{m+1}}$$

(6)

Transformations’ delivery system of ODEs and non-linear ODEs are

$${\left({\left|F^{″}\right|}^{m-1}{F}^{″}\right)}^{\prime }+\frac{1}{m+1}{F}^{″}F-ϵF^{\prime }-\frac{{\nu }_f}{{\nu }_{Thnf}}{F}_R\left(F^{\prime }{}^2\right)=0,$$

(7)

$$\begin{gathered} {\theta }^{″}+\frac{Pr}{m+1}F{\theta }^{\prime }-\frac{k_f{\left(\rho C_p\right)}_{hnf}}{k_{hnf}{\left(\rho C_p\right)}_f}Pr{\mathsf{\Omega }}_A\left[F{F}^{\prime }{\theta }^{\prime }+F^2{\theta }^{″}+H_{h}F{\theta }^{\prime }\right] \\ +\frac{k_f}{k_{hnf}}{H}_{h}Pr\theta =0 \end{gathered}$$

(8)

.

Thermal properties for correlations among tri-hybrid nanoparticles are mentioned below and their values are mentioned in

Table 1. Thermal properties of ethylene glycol, silicon dioxide, aluminum oxide and titanium dioxide related nanoparticles in EG.

	$\mathit{K}$ (Thermal Conductivity)	$\mathit{\sigma }$ (Electical Conductivity)	$\mathit{\rho }$ (Desity)
$C_2{H}_6{O}_2$	0.253	$4.3\times {10}^{-5}$	1113.5
$A{l}_2{O}_3$	32.9	$5.96\times {10}^7$	6310
$Ti{O}_2$	8.953	$2.4\times {10}^6$	4250
$Si{O}_2$	1.4013	$3.5\times {10}^6$	2270

.

$${\rho }_{Thnf}=\left(1-{\phi }_1\right)\left\{\left(1-{\phi }_2\right)\left[\left(1-{\phi }_3\right){\rho }_f+{\phi }_3{\rho }_3\right]+{\phi }_2{\rho }_2\right\}+{\phi }_1{\rho }_1,$$

(9)

$$\frac{{\mu }_f}{{\left(1-{\phi }_3\right)}^{2.5}{\left(1-{\phi }_2\right)}^{2.5}{\left(1-{\phi }_1\right)}^{2.5}}={\mu }_{Thnf}, \frac{K_{Thnf}}{K_{hnf}}=\frac{K_1+2K_{nf}-2{\phi }_1\left(K_{nf}-K_1\right)}{K_1+2K_{nf}+{\phi }_1\left(K_{nf}-K_1\right)},$$

(10)

$$\frac{K_{hnf}}{K_{nf}}=\frac{K_2+2K_{nf}-2{\phi }_2\left(K_{nf}-K_2\right)}{K_2+2K_{nf}+{\phi }_2\left(K_{hnf}-K_2\right)},\frac{K_{nf}}{K_f}=\frac{K_3+2K_f-2{\phi }_3\left(K_f-K_3\right)}{K_3+2K_f+{\phi }_3\left(K_f-K_3\right)},$$

(11)

$$\frac{{\sigma }_{Tnf}}{{\sigma }_{hnf}}=\frac{{\sigma }_1\left(1+2{\phi }_1\right)-{\phi }_{hnf}\left(1-2{\phi }_1\right)}{{\sigma }_1\left(1-{\phi }_1\right)+{\sigma }_{hnf}\left(1+{\phi }_1\right)},\frac{{\sigma }_{hnf}}{{\sigma }_{nf}}=\frac{{\sigma }_2\left(1+2{\phi }_2\right)+{\phi }_{nf}\left(1-2{\phi }_2\right)}{{\sigma }_2\left(1-{\phi }_2\right)+{\sigma }_{nf}\left(1+{\phi }_2\right)},$$

(12)

$$\frac{{\sigma }_{nf}}{{\sigma }_f}=\frac{{\sigma }_3\left(1+2{\phi }_3\right)+{\phi }_f\left(1-2{\phi }_3\right)}{{\sigma }_3\left(1-{\phi }_3\right)+{\sigma }_f\left(1+{\phi }_3\right)}.$$

(13)

Here, volume fractions (${\phi }_1,{\phi }_2,{\phi }_3$), the wall temperature ($T_w$), the Prandtl number ($Pr$), the power-law number ($m$), the porosity number ($ϵ$), the heat generation number ($H_h$), the time relaxation number (${\mathsf{\Omega }}_A$) and the Forchheimer number ($F_r$) are mentioned above.

The coefficient related to drag force is defined as

$$C_f=\frac{2\left({\tau }_w\right)}{U^2{\rho }_f},$$

(14)

$${\left(Re\right)}^{\frac{1}{m+1}}{C}_f=-\frac{{\left(1-{\phi }_2\right)}^{-2.5}}{{\left(1-{\phi }_1\right)}^{2.5}{\left(1-{\phi }_3\right)}^{2.5}}\left[F^{″}\left(0\right){\left|F^{″}\left(0\right)\right|}^{m-1}\right].$$

(15)

The temperature gradient in view of tri-hybrid nanoparticles is modeled as

$$Nu=\frac{x{Q}_w}{\left(T_w-T_{\infty }\right)k_f},{\left(Re\right)}^{\frac{-1}{m+1}}Nu=-\frac{k_{Thnf}}{k_f}{\theta }^{\prime }\left(0\right).$$

(16)

$Re\left(=\frac{x^m{U}^{2-m}}{{\nu }_f}\right)$ is the local Reynolds number.

3. Numerical Scheme

The finite element scheme (FES) is utilized to simulate numerical results. The finite element scheme is very capable of simulating CFD problems. Detailed steps are described below.

Step-I: Equations (6) and (7) within BCs are called strong form. It is noticed that collecting each term in Equations (6) and (7) on one side and integrating them over each element of domain produces residuals. This procedure is known as the weighted residual method for the development of weak form. Residuals of the current problem are derived as

$${{\displaystyle \int }}_{{\eta }_e}^{{\eta }_{e+1}}W{t}_1\left[F^{\prime }-H\right]d\eta =0,$$

(17)

$${{\displaystyle \int }}_{{\eta }_e}^{{\eta }_{e+1}}W{t}_2\left[{\left({\left|H^{\prime }\right|}^{m-1}{H}^{\prime }\right)}^{\prime }+\frac{1}{m+1}{H}^{\prime }F-ϵH-\frac{{\nu }_f}{{\nu }_{Thnf}}{F}_R\left(H^2\right)\right]d\eta =0,$$

(18)

$${{\displaystyle \int }}_{{\eta }_e}^{{\eta }_{e+1}}W{t}_3\left[\begin{array}{c}{\theta }^{″}+\frac{Pr}{m+1}F{\theta }^{\prime }-\frac{k_f{\left(\rho C_p\right)}_{hnf}}{k_{hnf}{\left(\rho C_p\right)}_f}Pr{\mathsf{\Omega }}_A\left(FH{\theta }^{\prime }+F^2{\theta }^{″}+H_{h}F{\theta }^{\prime }\right)\\ +\frac{k_f}{k_{hnf}}{H}_{h}Pr\theta \end{array}\right]d\eta =0,$$

(19)

where $W{t}_1,W{t}_2$ and $W{t}_3$ are weight functions.

Step-II: The approach associated with Galerkin finite element is imposed to obtain a weak form in view of shape functions. The mesh free analysis is depicted in

Table 2. Mesh-free simulations for temperature and velocity within the middle of each 300 elements.

Number of Elements	${\mathit{F}}^{\prime }\left(\frac{{\mathit{\xi }}_{\mathit{m}\mathit{a}\mathit{x}}}{2}\right)$	$\mathit{\theta }\left(\frac{{\mathit{\xi }}_{\mathit{m}\mathit{a}\mathit{x}}}{2}\right)$
30	0.01798642358	0.05182213902
60	0.01648518152	0.04818508213
90	0.01599749810	0.04704279909
120	0.01575614853	0.04648531466
150	0.01561212389	0.04615529422
180	0.01551645523	0.04593716579
210	0.01544828141	0.04578231915
240	0.01539725740	0.04566668788
270	0.01535759940	0.04557702325
300	0.01532589058	0.04550548704

.

Step-III: The assembly approach is utilized for the development of stiffness elements, whereas the assembly approach is performed via the assembly procedure of FEA. The stiffness elements are

$$K_{ij}^{11}={{\displaystyle \int }}_{{\eta }_e}^{{\eta }_{e+1}}\left(\frac{d{\psi }_j}{d\eta }\right){\psi }_{i}d\eta ,K_{ij}^{13}=0,K_{ij}^{14}=0,K_{ij}^{12}=-{{\displaystyle \int }}_{{\eta }_e}^{{\eta }_{e+1}}\left({\psi }_i{\psi }_J\right)d\eta ,$$

(20)

$$K_{ij}^{22}={{\displaystyle \int }}_{{\eta }_e}^{{\eta }_{e+1}}\left[\begin{array}{c}-\left({\left(\overline{H^{\prime }}\right)}^{m-1}+\overline{H^{\prime }}\left(m-1\right){\left(\overline{H^{\prime }}\right)}^{m-2}\right)\frac{d{\psi }_j}{d\eta }\frac{d{\psi }_i}{d\eta }+\frac{\overline{F}}{m+1}{\psi }_i\frac{d{\psi }_j}{d\eta }\\ -ϵ{\psi }_i{\psi }_i-\frac{{\nu }_f}{{\nu }_{Thnf}}{F}_R\overline{H}{\psi }_i{\psi }_j\end{array}\right]d\eta ,$$

(21)

$$K_{ij}^{33}={{\displaystyle \int }}_{{\eta }_e}^{{\eta }_{e+1}}\left[\begin{array}{c}-\frac{d{\psi }_j}{d\eta }\frac{d{\psi }_i}{d\eta }+\frac{Pr}{m+1}\overline{F}{\psi }_i\frac{d{\psi }_j}{d\eta }-\frac{k_f{\left(\rho C_p\right)}_{hnf}}{k_{hnf}{\left(\rho C_p\right)}_f}Pr\overline{H}{\text{Ω}}_A\overline{F}{\psi }_i\frac{d{\psi }_j}{d\eta }\\ -\frac{k_f{\left(\rho C_p\right)}_{hnf}}{k_{hnf}{\left(\rho C_p\right)}_f}Pr{\text{Ω}}_A\overline{F^2}\left(\frac{d{\psi }_j}{d\eta }\frac{d{\psi }_i}{d\eta }\right)-\frac{k_f{\left(\rho C_p\right)}_{hnf}}{k_{hnf}{\left(\rho C_p\right)}_f}\frac{d{\psi }_j}{d\eta }{\psi }_i{\psi }_{i}Pr{\text{Ω}}_A{H}_h\overline{F}\\ -\frac{k_f}{k_{hnf}}{H}_{h}Pr{\psi }_j{\psi }_i\end{array}\right]d\eta$$

(22)

$$K_{ij}^{21}=0, K_{ij}^{23}=0,K_{ij}^{31}=0,K_{ij}^{32},b_i^2=0,b_i^1=0,b_i^3=0,$$

(23)

Step-IV: The Picard linearization approach provides a transformed algebraic system (linear equations).

Step-V: Finally, the system of linear algebraic equations is numerically solved within computational tolerance (${10}^{-5}$). The stopping condition is listed below.

$$\left|\frac{{\delta }_{i+1}-{\delta }_i}{{\delta }^i}\right|<{10}^{-5}.$$

(24)

Step-VI: Table 2 demonstrates study of mesh-free.

Step-VII: Three-hundred elements are required to obtain convergence analysis.

Table 3. Comparative values of skin friction coefficient with already published works when ${\phi }_1=0, {\phi }_3=0, {\phi }_1=0, ϵ=0$ and $F_r=0$.

$-{\left(\mathit{R}\mathit{e}\right)}^{\frac{1}{\mathit{m}+1}}{\mathit{C}}_{\mathit{f}}$		Present Work $-{\left(\mathit{R}\mathit{e}\right)}^{\frac{1}{\mathit{m}+1}}{\mathit{C}}_{\mathit{f}}$
Chen [28]	0.4438	0.44370132403
Fox et al. [29]	0.4437	0.44310243102
Sakiadis [30]	0.44375	0.44351314301

illustrates comparisons among published works and the present work considering ${\phi }_1=0, {\phi }_3=0, {\phi }_1=0, ϵ=0$ and $F_r=0$.

4. Graphical Discussion and Outcomes

The complex model of pseudo-plastic liquid with the existence of tri-hybrid nanoparticles is developed. The heat mechanism is addressed, and it is based on non-Fourier’s law involving heat generation. Flow characterizations are visualized under the impact of Forchheimer porous media. The present complex model is solved by FEA. The outcomes of heat energy, motion into particles, gradient velocity and temperature gradient versus different parameters are tabulated below.

4.1. Graphical Comparisons of Fluid Motion among Nanoparticles, Hybrid Nanostructures and Tri-Hybrid Nanoparticles

Characterizations of comparative flow are measured among nanoparticles, hybrid nanostructures and tri-hybrid nanoparticles versus the impacts of various parameters. These comparative simulations among pseudo-plastic nanoparticles, pseudo-plastic hybrid nanostructures and pseudo-plastic tri-hybrid nanoparticles are carried out by Figure 3, Figure 4 and Figure 5. It is deduced that the impact of pseudo-plastic tri-hybrid nanoparticles is carried out by solid curves, and the behavior of pseudo-plastic hybrid nanoparticles is illustrated by dot curves. Dot-dash curves are plotted for the influence of pseudo-plastic nanoparticles. Figure 3 is prepared for the comparative study between pseudo-plastic nanoparticles, pseudo-plastic tri-hybrid nanoparticles and pseudo-plastic hybrid nanoparticles versus changes in the power law number ($m$). The occurrence of power law number ($m$) is formulated because of pseudo-plastic material in the current investigation. The thickness of fluid is made more viscous using variations in the power law number ($m$). Flow distribution is declined versus $m<1$. Moreover, layers made through pseudo-plastic tri-hybrid nanoparticles are higher than layers made through pseudo-plastic nanoparticles and pseudo-plastic hybrid-nanoparticles. Moreover, moment layers related to boundary decline when $m$ is increased. The fluid is known as Newtonian liquid for the case $m=1$, the fluid becomes dilatant liquid for the case $m>1$ and the fluid is called pseudo-plastic material for case $0<m<1$. Figure 4 depicts graphical comparative outcomes on flow distribution including tri-hybrid nanoparticles versus distribution in the Forchheimer number. It is deduced that a negative term is noticed due to the impact of Forchheimer porous theory. Therefore, this negative term is responsible for the development of the retardation force into motion if fluid particles during maximum viscosity are created into the motion of fluid particles because of retardation force. Furthermore, the flow for tri-hybrid nanoparticles is the most significant flow produced by nanoparticles and hybrid nanostructures. The Forchheimer number is a dimensionless number. Physically, it is defined as a fraction among viscous force and pressure gradient. The flow of pseudo-plastic nanoparticles, pseudo-plastic hybrid nanoparticles and pseudo-plastic tri-hybrid nanoparticles slows down when $F_r$ is increased. $F_r$ is inversely proportional to the viscous force. Therefore, the fluid becomes more viscous versus higher values of $F_r$. The role of $ϵ$ is visualized in the flow distribution after inserting tri-hybrid nanoparticles, nanoparticles and hybrid nanoparticles. This variation in $ϵ$ versus flow distribution is addressed in Figure 5. The curves describing flow are reduced via variation in $ϵ$ because the impact of $ϵ$ is generated due to the porosity of the surface. It is noticed that occurrence of $ϵ$ appears in a negative term (in dimensionless momentum equation). The frictional force is created into the motion of fluid particles and this frictional force brings a declination to the motion of fluid particles. The thickness of the boundary layers of a surface is reduced for nanoparticles and hybrid-nanoparticles compared to the thickness of tri-hybrid nanoparticles. From a physics point of view, Darcy’s law states that the discharge rate is directly proportional to the temperature gradient. Velocity curves for pseudo-plastic nanoparticles, pseudo-plastic hybrid nanoparticles and pseudo-plastic tri-hybrid nanoparticles reduce the function versus the impact of $ϵ$.

4.2. Graphical Comparisons of Fluid Temperature among Nanoparticles, Hybrid Nanostructures and Tri-Hybrid Nanoparticles

The mechanism of heat production is addressed against changes in various physical parameters when inserting the role of pseudo-plastic nanoparticles, pseudo-plastic hybrid nanostructures and pseudo-plastic tri-hybrid nanoparticles, and these outcomes are plotted by Figure 6, Figure 7 and Figure 8. Figure 6 reveals the mechanism of heat production versus distribution in $F_r$. Higher production of thermal energy is produced when $F_r$ is inclined. Pseudo-plastic tri-hybrid nanostructures have a vital impact on the maximum production of thermal energy rather than the production of heat energy via pseudo-plastic nanoparticles and pseudo-plastic hybrid nanostructures (mixtures of $A{l}_2{O}_3$ and $Si{O}_2$). Meanwhile, thermal energy is enhanced via changes in $F_r$. Moreover, it is defined as fraction among viscous force to pressure gradient. Thermal layers at the boundary of the surface decreases with higher values of $F_r$. Figure 7 is prepared for the distribution of thermal energy among pseudo-plastic nanoparticles, pseudo-plastic hybrid nanostructures and pseudo-plastic tri-hybrid nanoparticles versus changes in the heat generation number. The production of thermal energy is maximized when taking a higher value of $H_h$. It is noticed that heat is generated by viscous dissipation, whereas heat is transported through molecular conduction. Therefore, $H_h$ is the ratio of heat generation and external heating source. An increment in production related to heat energy occurs because of the external heat source. In this case, the external heat source is placed at surface of wall, and heat energy among the particles is enhanced due to the external heat source. Positive numerical values represent the heat generation phenomena. Solid curves (pseudo-plastic tri-hybrid nanoparticles) are greater than dot curves (hybrid nanoparticles) and solid dot curves (pseudo-plastic nanoparticles). Figure 8 is the heighted relationship among heat energy and the time relaxation number including nanoparticles, pseudo-plastic hybrid nanostructures and pseudo-plastic tri-hybrid nanoparticles. It is revealed that the appearance of ${\mathsf{\Omega }}_A$ happened due to non-Fourier’s law. The production of thermal energy is minimized versus a variation in ${\mathsf{\Omega }}_A$. Thermal energy is maximized for pseudo-plastic tri-hybrid nanoparticles rather than hybrid nanoparticles and pseudo-plastic nanostructures. Thickness associated within thermal energy is also managed using changes in ${\mathsf{\Omega }}_A$. An enhancement in ${\mathsf{\Omega }}_A$ results in an increment in performance of the pseudo-plastic fluid to reinstate an equilibrium state, and it creates a reduction into a thermal state of fluid. Therefore, thermal boundary layers are decreased. However, thermal boundary layers for pseudo-plastic tri-hybrid nanoparticles are stronger than pseudo-plastic hybrid nanoparticles and pseudo-plastic nanoparticles.

4.3. Variation of Gradient Velocity and Rate of Heat Flux among Nanoparticles, Hybrid Nanostructures and Tri-Hybrid Nanoparticles

Table 4. Comparative values of skin friction coefficient and temperature gradient among tri-hybrid nanoparticles and hybrid nanostructures against $F_r$, $H_h$ and ${\mathsf{\Omega }}_A$.

		Hybrid Nanoparticles		Tri-Hybrid Nanoparticles
		$-{\left(\mathit{R}\mathit{e}\right)}^{\frac{1}{\mathit{m}+1}}{\mathit{C}}_{\mathit{f}}$	$-{\left(\mathit{R}\mathit{e}\right)}^{\frac{-1}{\mathit{m}+1}}\mathit{N}\mathit{u}$	$-{\left(\mathit{R}\mathit{e}\right)}^{\frac{1}{\mathit{m}+1}}{\mathit{C}}_{\mathit{f}}$	$-{\left(\mathit{R}\mathit{e}\right)}^{\frac{-1}{\mathit{m}+1}}\mathit{N}\mathit{u}$
	0.0	0.83549	0.76474	1.40604	2.6548
$F_r$	0.4	0.96224	0.56538	1.26146	2.5714
	0.8	0.98177	0.34362	1.19194	2.4360
	0.0	0.85724	0.14350	1.74575	2.4656
$H_h$	0.3	0.61123	0.12829	1.53121	2.3125
	0.6	0.43102	0.00586	1.32173	2.0878
	0.0	0.89613	0.19690	1.79476	2.9885
${\mathsf{\Omega }}_A$	0.21	0.89613	0.20090	1.79476	2.1040
	0.51	0.89613	0.30498	1.79476	2.0916

predicts the comparative study among hybrid nanoparticles and tri-hybrid nanostructures for skin friction and temperature gradient against changes in time relaxation, the Forchheimer number and $H_h$ (heat generation number). The velocity gradient decreases versus the change in $H_h$, and the velocity gradient becomes higher when $F_r$ is enhanced, but a constant impact is noticed on the velocity gradient versus distribution in ${\mathsf{\Omega }}_A$. It is noticed that there is a higher impact for tri-hybrid nanoparticles than for the nanostructures. The production of a temperature gradient declines against a variation in $H_h$ and $F_r$. However, maximum production is enhanced when ${\mathsf{\Omega }}_A$ is increased. Tri-hybrid nanoparticles are considered the most significant for the development of the production of heat energy rather than the production of thermal energy obtained by hybrid nanoparticles.

5. Main Findings

The characteristics of thermal energy and heat generation are incorporated into a stretching surface. It is observed that the motions of pseudo-plastic nanoparticles, pseudo-plastic hybrid nanoparticles and pseudo-plastic tri-hybrid nanoparticles are induced by the wall movement along the x-direction. A magnetic field is exerted along the normal to the surface. The inclusion of tri-hybrid nanoparticles in pseudo-plastic liquid is addressed past a stretching surface. Heat generation is incorporated in CCHFM (Cattaneo-Christov heat flux model). Forchheimer porous media is used to analyze flow characterizations. Such a complex model is solved by the finite element approach (FEA). The main concern is to measure the comparative analysis among pseudo-plastic nanoparticles, pseudo-plastic hybrid nanoparticles and pseudo-plastic tri-hybrid nanoparticles. FEA is adopted to solve the current model. The main outcomes of the current model are mentioned below.

• The reduction in the flow of particles is addressed via a variation in power law, porosity and Forchheimer numbers;
• The thermal energy of fluid particles is boosted when inserting higher values of Forchheimer and heat generation numbers;
• A higher value of relaxation time number is utilized for developing less production of heat energy;
• Ternary hybrid nanoparticles play a vital role in the production of heat energy rather than the production of heat energy through hybrid nanoparticles and nanostructures;
• Ternary hybrid nanomaterials are visualized most efficiently for construction of temperature gradient and surface force rather than nanoparticles.