Simultaneous Features of CC Heat Flux on Dusty Ternary Nanofluid (Graphene + Tungsten Oxide + Zirconium Oxide) through a Magnetic Field with Slippery Condition

Abstract

The purpose of this work is to offer a unique theoretical ternary nanofluid (graphene/tungsten oxide/zirconium oxide) framework for better heat transfer. This model describes how to create better heat conduction than a hybrid nanofluid. Three different nanostructures with different chemical and physical bonds are suspended in water to create the ternary nanofluid (graphene/tungsten oxide/zirconium oxide). Toxic substances are broken down, the air is purified, and other devices are cooled thanks to the synergy of these nanoparticles. The properties of ternary nanofluids are discussed in this article, including their thermal conductivity, specific heat capacitance, viscosity, and density. In addition, heat transport phenomena are explained by the Cattaneo–Christov (CC) heat flow theory. In the modeling of the physical phenomena under investigation, the impacts of thermal nonlinear radiation and velocity slip are considered. By using the right transformations, flow-generating PDEs are converted into nonlinear ordinary differential equations. The parameters’ impacts on the velocity and temperature fields are analyzed in detail. The modeled problem is graphically handled in MATLAB using a numerical technique (BVP4c). Graphical representations of the important factors affecting temperature and velocity fields are illustrated through graphs. The findings disclose that the performance of ternary nanofluid phase heat transfer is improved compared to dusty phase performance. Furthermore, the magnetic parameter and the velocity slip parameter both experience a slowing-down effect of their respective velocities.

1. Introduction

Nanofluids are liquids that include ultrafine particles contained in a dilute solution. These particles are suspended in the liquid. These fluids have thermophysical qualities that are far better than those of pure fluids. It has been established that increasing the amount of carbon, copper, or other nanoparticles with high thermal conductivity that is added to water, ethylene glycol, or oil may enhance the materials’ ability to transport heat. The thermal conductivity of the nanofluids is significantly raised, and the particle size and volume fraction of suspended particles have an impact on this phenomenon. Choi [1] invented the term “nanofluids”, which defines a liquid suspension of ultrasmall particles (less than 50 nm in diameter). With the fast development of nanofabrication, various affordable combinations of fluid/particles are instantly accessible. A complete assessment by Buongiorno [2] investigated convective transport in nanofluids, who argued that an appropriate explanation for the phenomenon exists, although unexpected growth in heat conductivity and viscosity is yet to be found. Recently, Reddy et al. [3] investigated hybrid dusty fluid flow through a Cattaneo–Christov heat flux model. Waqas et al. [4] discussed the heat transfer analysis of hybrid nanofluid flow with thermal radiation through a stretching sheet. Gurdal et al. [5] studied the turbulent flow and heat transfer characteristics of Ferro-nanofluid flowing in a dimpled tube under a magnetic field effect. Many researchers [6,7,8,9] have addressed the convective heat transfer during the flow of nanoliquid.

The passage of heat energy through the particles of a liquid may be described as a sort of heat transfer known as thermal radiation. It is a condition that must be met in a wide variety of applications, such as the planning of propulsion devices for space spacecraft, nuclear power plants, and gas turbines. In space applications, where devices are required to operate at high temperatures to obtain the required thermal efficiency, the impact of thermal radiation on controlling the heat transfer in certain applications and on estimating the thermal effects for processes with high temperatures is unavoidable. This is because thermal radiation has an impact on both controlling the heat transfer in certain applications and on estimating the thermal effects for processes with high temperatures. Duwairi and Duwairi [10] studied gray viscous fluid flow with magnetic and radiation parameters influences. Cortell [11] analyzed radiation and viscous dissipation impacts for the flow of the thermal boundary layer along a nonlinear stretching surface. Recently, Bakar and Soid [12] studied the MHD stagnation-point flow and heat transfer over an exponentially stretching/shrinking vertical sheet in a micropolar fluid with a buoyancy effect. Ali et al. [13] analyzed the flow and heat transfer over stretching/shrinking and porous surfaces. Azam et al. [14] discussed the transient bioconvection and activation energy impacts on Casson nanofluid with gyrotactic microorganisms and nonlinear radiation. Reddy et al. [15] studied the unsteady absorption flow and dissipation heat transfer over a non-Newtonian fluid. Gnaneswara et al. [16] investigated the effect of thermal conductivity on Blasius–Rayleigh–Stokes flow and heat transfer over a moving plate by considering the magnetic dipole moment. Some recent studies related to the consideration of heat transfer theory can be found in [17,18,19,20,21,22].

Fourier’s law of heat conduction describes heat transfer in a given medium due to temperature differences [23]. It is considered the basis for the investigation of thermal phenomena. Using this law, the temperature field equation is developed into a parabolic-type equation stating that heat transfer with initial disruption having infinite speed propagates throughout the medium under certain conditions. This heat conduction paradox requires modification of Fourier’s law. Cattaneo [24] presented a modified form of Fourier’s law known as the Maxwell–Cattaneo (MC) model by introducing the thermal relaxation time parameter. This modification converted the temperature field equation from a parabolic- to a hyperbolic-type equation, describing that heat transfer has finite speed in the entire medium. Christov [25] investigated the frame invariance of the MC model and replaced the time derivative with Oldroyd’s upper-convected time derivative. Recently, Reddy et al. [26] studied zero-mass flux and Cattaneo–Christov heat flux through a Prandtl non-Newtonian nanofluid in Darcy–Forchheimer porous space. Machireddy et al. [27] investigated the impact of Cattaneo–Christov heat flux on the hydromagnetic flow of non-Newtonian fluids filled with a Darcy–Forchheimer porous medium. Khan et al. [28] discussed Cattaneo–Christov (CC) heat and mass fluxes in the stagnation-point flow of Jeffrey nanoliquids by a stretching surface. Tausif et al. [29] investigated the modified homogeneous and heterogeneous chemical reaction and flow performance of Maxwell nanofluid with the Cattaneo–Christov heat flux law. Some recent studies related to the consideration of CC theory can be found in [30,31,32].

In this study, we investigated dusty fluid flow subjected to a magnetic field over a stretching sheet. Impacts of thermal radiation, velocity slip, and conditions are considered in the modeling of the physical phenomena. The use of CC heat flux theory is carried out in the energy equation for analyzing the heat transfer phenomena. Considering the appropriate variables, the governing PDEs are converted to nonlinear ODEs. MATLAB software is used for investigating the numerical solution. Impacts of fluid velocity and temperature are analyzed through different pertinent parameters in the considered problem.

Specifically, this work will address the research questions mentioned below:

• What will be the influence of the magnetic parameter on fluid heat transfer rate, temperature, and velocity?
• What is the significance of the CC theory on the dusty ternary fluid model?
• What will be the impact of thermal radiation on heat transfer rate and temperature?
• What will be the influence of the velocity slip parameter on fluid velocity, temperature, and heat transfer rate?

We organized this article as follows. In Section 2, we formulate the physical phenomena of the problem and transformations of PDEs. Result analyses are provided in Section 3. Concluding remarks of the present article are given in Section 4.

2. Mathematical Formulation

Consider a continuous 2D boundary layer flow and the heat transfer of a dusty ternary nanofluid across a stretching sheet moving at a constant speed $U_w=bx$. With the slot serving as the point of origin, the $x$-axis is drawn along the surface that is being stretched in the direction that the motion is going, and the $y$-axis is drawn perpendicular to the sheet in the direction that is facing away from the fluid. It is presumable that the flow will not extend beyond the area where $y$ is greater than 0. Convective heat transfer is responsible for keeping the temperature of the sheet’s surface at a constant value known as $T_f$.

Some of the assumptions taken into consideration are listed below:

• The dust particles are taken to be small enough and of sufficient number to be treated as a continuum and allow concepts such as density and velocity to have physical meaning.
• The dust particles are assumed to be spherical in shape, having the same radius and mass, and are undeformable.
• The Cartesian coordinate system is located in such a way that the $x$-axis and $y$-axis are taken along (and normal to) the surface, respectively, while the origin of the system is located at the leading edge.
• The dust particles are assumed to be uniform in size, and the density number of the dust particle is taken as constant throughout the flow.

The equations that regulate the flow of dusty ternary nanofluid can be written as follows, according to the standard boundary layer approximations (Reddy et al. [3]):

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

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\mu }_{thnf}}{{\rho }_{thnf}}\frac{{\partial }^{2}u}{\partial y^2}-\frac{\sigma B_0^2}{{\rho }_{thnf}}u$$

(2)

$$\begin{array}{c}{\left(\rho c_p\right)}_{thnf}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)+{\lambda }_1\left(\begin{array}{l}{u}^2\frac{{\partial }^{2}T}{\partial x^2}+v^2\frac{{\partial }^{2}T}{\partial y^2}+2uv\frac{{\partial }^{2}T}{\partial x\partial y}+\\ \left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)\frac{\partial T}{\partial x}+\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)\frac{\partial T}{\partial y}\end{array}\right)=k_{thnf}\frac{{\partial }^{2}T}{\partial y^2}+\\ \frac{N{c}_{pf}}{{\tau }_t}\left(T_p-T\right)+\frac{N}{{\tau }_v}{\left(u_p-u\right)}^2+\frac{1}{\rho c_p}\frac{\partial q_r}{\partial y}+{\mu }_{thnf}{\left(\frac{\partial u}{\partial y}\right)}^2\end{array}$$

(3)

Dust Phase

$$\frac{\partial u_p}{\partial x}+\frac{\partial v_p}{\partial y}=0$$

(4)

$$u_p\frac{\partial u_p}{\partial x}+v_p\frac{\partial u_p}{\partial y}=\frac{K}{m}(u-u_p)$$

(5)

$$\left(u_p\frac{\partial T_p}{\partial x}+v_p\frac{\partial T_p}{\partial y}\right)=\frac{c_{thpf}}{c_{thmf}{\tau }_T}\left(T-T_P\right)$$

(6)

By using the Rosseland approximation to the radiative heat flow expression found in Equation (3), one obtains the following:

$$q_r=-\frac{4{\sigma }^{*}}{3k^{*}}\frac{\partial T^4}{\partial y}=-\frac{16{\sigma }^{*}}{3k^{*}}{T}^3\frac{\partial T}{\partial y}$$

(7)

where $x$ and $y$, respectively, represent coordinate axes along the continuous surface in the direction of motion and perpendicular to it. $\left(u, v\right)$ and $\left(u_p, v_p\right)$ denote the velocity components of the nanofluid and dust phases along the $x$- and $y$-directions, respectively. $N$ is the number density of dust particles. ${\lambda }_1$ is the thermal relaxation time. $m$ is the mass concentration of dust particles. $K=6\pi {\mu }_{nf} r$ is the Stokes drag constant. $r$ is the radius of dust particles. $A$ is the slip constant. ${\sigma }^{*}$ is the Stefan–Boltzmann constant. $k^{*}$ is the mean absorption coefficient.

Boundary Conditions:

$$\begin{gathered} u=U_w+A{\nu }_f\frac{\partial u}{\partial y}, v=0, -k_f\frac{\partial T}{\partial y}=h_f\left(T_f-T\right) \mathrm{at} y=0, \\ u\to 0, u_p\to 0, v_p\to v, T\to T_{\infty }, T_p\to T_{\infty } as y\to \infty , \end{gathered}$$

(8)

The expression for viscosity (${\mu }_{thnf}$), density (${\rho }_{thnf}$), specific heat (${(\rho Cp)}_{thnf}$), and thermal conductivity ($k_{thnf}$) of the ternary nanofluid is as follows (Prakash et al. [9]):

$$\begin{gathered} {\mu }_{thnf}=\frac{{\mu }_f}{{\left(1-{\varphi }_{1 }\right)}^{2.5}{\left(1-{\varphi }_{2 }\right)}^{2.5}{\left(1-{\varphi }_{3 }\right)}^{2.5}}, {(\rho Cp)}_{thnf}=\left[\begin{array}{c}\left(1-{\varphi }_2\right)[\left(1-{\varphi }_1\right){\left(\rho Cp\right)}_f\\ +{\varphi }_1{\left(\rho Cp\right)}_{s1}]+{\varphi }_2{\left(\rho Cp\right)}_{s2}\end{array}\right], \\ {\rho }_{thnf}=\left(1-{\varphi }_1\right)\left[\left(1-{\varphi }_2\right)\left\{\begin{array}{c}\left(1-{\varphi }_3\right){\rho }_f\\ +{\varphi }_3{\rho }_3\end{array}\right\}+{\varphi }_2{\rho }_2\right]+{\varphi }_1{\rho }_1, \\ \frac{k_{thnf}}{k_{hnf}}=\frac{k_1+2k_{hnf}-2{\varphi }_1\left(k_{hnf}-k_1\right)}{k_1+2k_{hnf}+{\varphi }_1\left(k_{hnf}-k_1\right)},\frac{k_{hnf}}{k_{nf}}=\frac{k_2+2k_{nf}-2{\varphi }_2\left(k_{nf}-k_2\right)}{k_2+2k_{nf}+{\varphi }_2\left(k_{hnf}-k_2\right)},\frac{k_{nf}}{k_{nf}}=\frac{k_3+2k_f-2{\varphi }_3\left(k_f-k_3\right)}{k_3+2k_{nf}+{\varphi }_3\left(k_f-k_3\right)} \end{gathered}$$

where ${\varphi }_1, {\varphi }_2$, and ${\varphi }_3$ are the nanoparticle volume fraction of copper/alumina/zirconium oxide, respectively. $C_p$ is the specific heat. $k_f$ denotes the thermal conductivity of the regular fluid.

The nonlinear PDEs in the model problem are converted to nonlinear ODEs by way of a similarity transformation, as shown below.

$$\begin{gathered} u=bx{f}^{\prime }\left(\eta \right),v=-\sqrt{b{v}_f}f(\eta ),u_p=bx{F}^{\prime }\left(\eta \right),v_p=-\sqrt{b{v}_f}F(\eta )\eta =\sqrt{\frac{b}{v_f}}y,\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_f-T_{\infty }}, \\ {\theta }_p\left(\eta \right)=\frac{T_p-T_{\infty }}{T_f-T_{\infty }} \end{gathered}$$

(9)

with $T=T_f\left(1+({\theta }_w-1\right)\theta ) \mathrm{and}$ ${\theta }_w=\frac{T_f}{T_{\infty }}$, ${\theta }_w>1$ the temperature ratio parameter.

Equations (1) and (4) are completely satisfied by these transformations, and the other equations in the model, [2,3,5,6], together with the boundary conditions, are translated as follows:

Fluid Phase:

$$\frac{{\mu }_{thnf}}{{\rho }_{thnf}}{f}^{‴}+\left(f{f}^{″}-f^{\prime 2}\right)+{\mu }_{thnf}\left[l{\beta }_v\left(F^{\prime }-f^{\prime }\right)-Q{F}^{\prime }\right]=0.$$

(10)

$$\begin{gathered} \frac{k_{thnf}}{k_f}\left({\theta }^{″}+R\left[{\left(1+\left({\theta }_w-1\right)\theta \right)}^3{\theta }^{″}+3\left({\theta }_w-1\right){\theta }^{{}^{\prime }2}{\left(1+\left({\theta }_w-1\right)\theta \right)}^2\right]\right)+ \\ \mathrm{Pr}\frac{{\left(\rho cp\right)}_f}{{\left(\rho cp\right)}_{thnf}}\left[\begin{array}{c}\Gamma \left(ff{}^{\prime }{\theta }^{\prime }-f^2{\theta }^{\prime }\right)\\ +f\theta {}^{\prime }+l{\beta }_T\left[{\theta }_p-\theta \right]\\ +lEc{\beta }_v{\left[F^{\prime }-f^{\prime }\right]}^2\end{array}\right]+PrEc{f{}^{″}}^{{}^2}=0. \end{gathered}$$

(11)

Dust Phase

$$FF{}^{″}-F{{}^{\prime }}^2+{\beta }_{\upsilon }\left(f{}^{\prime }-F{}^{\prime }\right)=0$$

(12)

$${\theta }_p^{\prime }F+\alpha {\beta }_T\left({\theta }_p-\theta \right)=0$$

(13)

$$\begin{gathered} f\left(0\right)=0, f^{\prime }\left(0\right)=1+\frac{\delta }{{\left(1+{\varphi }_1\right)}^{2.5}{\left(1+{\varphi }_2\right)}^{2.5}{\left(1+{\varphi }_2\right)}^{2.5}}{f}^{″}\left(0\right), {\theta }^{\prime }\left(0\right)=-B_i\left(1-\theta \left(0\right)\right), \\ {f}^{\prime }(\infty )\to 0, F^{\prime }(\infty )\to 0, F(\infty )\to f(\infty ), \theta (\infty )\to 0, {\theta }_p(\infty )\to 0. \end{gathered}$$

(14)

where $Q=\frac{\sigma B_0^2}{{\rho }_f},\mathrm{Pr}=\frac{{\left(\mu c_p\right)}_f}{k_f},Ec=\frac{U_w^2}{\left(T_f-T_{\infty }\right)c_{pf}},$$R=\frac{4{\sigma }^{*}{T}_{\infty }^3}{3k_{thnf}{k}^{*}},$ ${\beta }_T=\frac{1}{{\tau }_{T}c}$, ${\beta }_v=\frac{1}{{\tau }_{v}c}$, ${\tau }_v=\frac{m}{K}$, $l=\frac{mN}{{\rho }_f}$, $\gamma =\frac{c_{pf}}{c_{mf}}$ $\Gamma =b{\lambda }_1$, $\delta =A\frac{U_w}{{\nu }_f}$ and $Bi=\sqrt{\frac{{\nu }_{f }}{c}}\frac{h_f}{k_f}$.

The physical quantities of $C_f$ and $Nu$ are;

$$C_f=\frac{{\tau }_w}{{\rho }_f{U}_w^2}, Nu=\frac{x{q}_w}{k_f\left(T_f-T_{\infty }\right)} \mathrm{at} y=0,$$

where ${\tau }_w$ and $q_w$ are given by

$${\tau }_w={\mu }_{thnf}\left(\frac{\partial u}{\partial y}\right)q_w=-k_{thnf}\left(\frac{\partial T}{\partial y}\right) \mathrm{at} y=0.$$

In terms of the nondimensional variables, we express them as

$$C_{f}R{e}^{-\left(\frac{1}{2}\right)}=\left(\frac{{\mu }_{thnf}}{{\mu }_f}\right)f^{″}\left(0\right) \mathrm{and} Nu{\left(Re\right)}^{-\left(\frac{1}{2}\right)}=-\frac{k_{thnf}}{k_f}\left[1+R{\theta }_w^3\right]{\theta }^{\prime }\left(0\right).$$

where $\mathrm{Re}=\frac{U_w^2}{b{v}_f}$ is the local Reynolds number.

3. Numerical Method

Solving the nonlinear system of ODEs generated by a numerical technique (BVP4c) in MATLAB (see Equations (10)–(13)) may be achieved numerically with the aid of boundary constraints using the approach described in Equation (14).

Table 1. Thermophysical properties of a ternary nanofluid (graphene + sirconium oxide + tungsten oxide) (Prakasha et al. [9]) and Muhammad et al. [33]).

Thermophysical Properties	Base Fluid	Ternary Nanofluid
	$H_{2}O$	Graphene	Zirconium Oxide	Tungsten OXIDE
$\rho$$(\mathrm{kg}/{\mathrm{m}}^2$)	997.1	2200	5680	$7160$
$C_p \left(\mathrm{j}/\mathrm{kgK}\right)$	4179	5000	502	$96.15$
$k$$\left(\mathrm{w}/\mathrm{mk}\right)$	0.613	790	1.7	$1.63$

shows the physical properties of the ternary nanofluid. Equations of a higher level of order differential type are produced. A variety of physical influences on the velocity and temperature profiles for both dusty and ternary nanofluid cases are analyzed graphically. Calculated values for $C_f{\left(Re\right)}^{\frac{1}{2}}$, and $Nu{\left(Re\right)}^{-\frac{1}{2}}$ are tabulated below (Tables 3 and 4). At the outset, we reduce these ODEs to the first-order form by simplifying the higher-order terms.

Let $f=y_1, f^{\prime }=y_2, f^{″}=y_3, f^{‴}=y_4, \theta =y_5, {\theta }^{\prime }=y_6, {\theta }^{″}=y_7 F=y_8, F^{\prime }=y_9, F^{″}=y_{10}, {\theta }_p=y_{11}, {\theta }_p^{\prime }=y_{12}$.

Fluid Phase:

$$y_4=-\frac{{\rho }_{thnf}}{{\mu }_{thnf}}\left[\left(y_1{y}_3-y_2^2\right)+{\mu }_{thnf}\left[l{\beta }_v\left(y_9-y_2\right)-Q{y}_2\right]\right] ,$$

(15)

$$y_7=-\frac{k_f}{k_{thnf}}\left[\begin{array}{c}\left(1+R\left[\begin{array}{c}{\left(1+\left({\theta }_w-1\right)y_5\right)}^3+\\ 3\left({\theta }_w-1\right)y_6^2{\left(1+\left({\theta }_w-1\right)y_5\right)}^2\end{array}\right]\right)+\\ \mathrm{Pr}\frac{{\left(\rho cp\right)}_f}{{\left(\rho cp\right)}_{thnf}}\left[\begin{array}{c}\Gamma \left(y_1{y}_2{y}_6-y_1^2{y}_6\right)\\ +y_1{y}_6+l{\beta }_T\left[y_{11}-y_5\right]\\ +lEc\beta {\left[y_9-y_2\right]}^2\end{array}\right]+PrEc{y}_3^2.\end{array}\right],$$

(16)

Dust Phase:

$$y_{10}=\frac{1}{y_8}\left[y_9^2-{\beta }_v\left(y_2-y_9\right)\right]$$

(17)

$$y_{12}=\frac{1}{y_8}\left[-\alpha {\beta }_T\left(y_{11}-y_5\right)\right]$$

(18)

Boundary Conditions:

$$\begin{gathered} \left(0\right)=0, y_2\left(0\right)=1+\frac{\delta }{{\left(1+\varnothing \right)}^{2.5}}{y}_3\left(0\right), y_6\left(0\right)=-B_i\left(1-y_5\left(0\right)\right), y_2(\infty )\to 0, y_9(\infty )\to 0, \\ {y}_8(\infty )\to y_1(\infty ), y{ }_5(\infty )\to 0, y_{11}(\infty )\to 0 \end{gathered}$$

(19)

4. Results and Discussion

In this paper, our study is focused on the dusty ternary nanofluid flow with the Cattaneo–Christov model over a stretching sheet in the presence of a magnetic field, nonlinear radiation, and convective heat transfer. We analyzed the effects of the physical parameters on the velocity and thermal fields through figures and tables by using the fixed parametric values $\mathrm{Pr}=6.9, Ec=0.5, R=0.6, {\theta }_w=1.2, \delta =0.7, Q=0.5, Bi=0.8, {\varphi }_1={\varphi }_2={\varphi }_3=0.005$, and $\mathsf{\Gamma }=0.4$. Additionally, the results of the comparison of the Prandtl number to previously published studies are shown in

Table 2. Validation of the results in comparison with those found in the literature.

$\mathbf{Pr}$	Ghadikolaei et al. [34]	Hosseinzadeh et al. [35]	Reddy et al. [3]	Present Results
0.7	0.4538	0.4541	0.4539	0.45415
2.0	0.9113	0.9114	0.9113	0.91133
7.0	1.8954	1.8954	1.8954	1.89545

. We can conclude that the current results are in excellent agreement with previous findings in the literature. Furthermore, $C_f{\left(Re\right)}^{\frac{1}{2}}$ and $Nu{\left(Re\right)}^{-\frac{1}{2}}$ values for the present analysis are computed and given in

Table 3. The friction factor values for various pertinent parameters.

$\mathit{Q}$	$\mathit{\delta }$	${\mathit{\varphi }}_1$	${\mathit{\varphi }}_2$	${\mathit{\varphi }}_3$	Friction Factor
					Dusty Phase	Ternary Phase
0.5					0.785239	1.511657
1.0					0.613042	1.447579
1.5					0.508094	1.406297
	0.5				0.620207	1.470950
	1.0				0.541187	1.464362
	1.5				0.446056	1.453310
		0.005			0.740501	1.477960
		0.001			0.688252	1.475486
		0.015			0.640870	1.472762
			0.005		0.967495	1.480191
			0.001		0.828302	1.475561
			0.015		0.549498	1.460959
				0.005	0.740501	1.477961
				0.001	0.688252	1.475486
				0.015	0.549498	1.460959

and

Table 4. The Nusselt number values for various pertinent parameters.

$\mathbf{Pr}$	$\mathit{E}\mathit{c}$	$\mathit{B}\mathit{i}$	$\mathbf{\Gamma }$	$\mathit{R}$	${\mathit{\theta }}_{\mathit{w}}$	${\mathit{\varphi }}_1$	${\mathit{\varphi }}_2$	${\mathit{\varphi }}_3$	Nusselt Number
									Dust Phase	Ternary Phase
0.5									1.44697	1.56233
1.0									1.46193	2.69770
1.5									1.48274	3.51973
	0.1								1.50243	1.94796
	0.3								1.43731	1.93508
	0.5								1.39414	1.92532
		0.1							1.51166	2.23259
		0.2							1.44758	1.74300
		0.3							1.40634	1.44461
			0.1						1.57142	1.97687
			0.2						1.84996	2.01510
			0.3						2.08391	2.01747
				0.5					1.45866	1.91198
				1.0					1.53613	1.88219
				1.5					1.57934	1.86686
					0.8				1.28682	1.48919
					1.2				1.32862	1.64629
					1.4				1.36865	1.79380
						0.005			1.06239	1.88550
						0.001			1.16455	1.89615
						0.015			1.27872	1.91104
							0.005		1.39414	1.92532
							0.001		1.43731	1.93508
							0.015		1.51166	2.23259
								0.005	1.17455	1.87615
								0.001	1.22872	1.89104
								0.015	1.30414	1.92532

.

Figure 1 shows how $Q$ affects the velocity field ($f{}^{\prime }\left(\eta \right)$ and $F{}^{\prime }\left(\eta \right)$) for both ternary nanofluid and dusty fluid cases. A decreasing trend can be observed in $f{}^{\prime }\left(\eta \right)$ and $F{}^{\prime }\left(\eta \right)$ scenarios for boosting values of $Q$. Physically, the magnetic field depends on Lorentz force, which is stronger for a larger magnetic field. Therefore, the velocity profile declines with more magnetic fields. In addition, the influence of the ternary nanofluid is examined more thoroughly than in the case of dusty fluid. In addition, the appropriate layer thickness is scaled back in both scenarios when the $Q$ value is increased.

Figure 2, Figure 3 and Figure 4 depict the instances of dusty fluid and ternary nanofluid and the influence of ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$ on the $F{}^{\prime }\left(\eta \right)$ and $f{}^{\prime }\left(\eta \right)$ profiles, respectively. A decrease in the velocity curve that occurs as a consequence of the incorporation of solid nanoparticles (${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$) into the nanomaterial was seen in both dusty and ternary nanofluid cases. This outcome is due to collisions between nanoparticles that have been extensively dispersed across the environment.

In order to explore the impact of the $\delta$ parameter on the velocity distributions of the $F{}^{\prime }\left(\eta \right)$ and $f{}^{\prime }\left(\eta \right)$ profiles, Figure 5 is displayed. It is clear from these results that changing the value of the parameter results has a significant reduction of the velocity distribution for both the $F{}^{\prime }\left(\eta \right)$ and $f{}^{\prime }\left(\eta \right)$ situations. This is due to the fact that the fluid’s near-surface velocity does not sustain correctly for the extended surface velocity when taking slip effect into consideration. Therefore, if the value of the slip velocity parameter is increased, the slide speed will be increased accordingly. Following this, there is a decrease in the liquid velocity as a result of the stretching surface deformation, which can only be communicated to the fluid under slip conditions.

In Figure 6, we can see how $R$ affects the temperature field for both ternary nanofluid phase and dusty phase ($\theta \left(\eta \right)$ and ${\theta }_p\left(\eta \right)$) profiles. It seems to be enhanced when $R$ improves. In accordance with the physical principle, as the radiation parameter increases, heat is radiated at the fluid at a much higher rate than when the mean absorption coefficient decreases. Physically, the effect of thermal radiation is to boost the transfer, as by increasing thermal radiation, the thermal boundary layer increases. Therefore, it has been reported that the process of thermal radiation reduction should proceed at a faster rate. The use of radiation can regulate the distribution of temperature and flow, and such applications can be used in pseudoscience to monitor blood pressure via the magnetotherapy process.

Figure 7 depicts the analysis of the characteristics of the function $\theta \left(\eta \right)$ and ${\theta }_p\left(\eta \right)$ in relation to rising values of ${\theta }_w$ for a ternary nanofluid and dusty fluid condition, respectively. It is witnessed that the thermal field and thermal layer thickness of the ternary nanofluid and dusty fluid condition both increase at an exponential rate when the ${\theta }_w$ parameter is increased. In addition, for high values of ${\theta }_w$, a rise in fluid temperature enriches the whole fluid case. This is the situation when the temperature of the fluid is increased. In addition, the enhancement of fluid temperature is optimal in the case of the ternary nanofluid, followed by the case of dusty fluid.

Figure 8 displays the changes in the thermal field for the ternary nanofluid and dusty fluid instances as a function of the $Ec$ parameter. To improve the $Ec$ parameter, it is evident that the thermal field rises in both ternary nanofluid and dusty fluid phases. Physically, the Eckert number is the ratio of kinetic energy to the specific enthalpy difference between wall and fluid. Therefore, an increase in the Eckert number causes the transformation of kinetic energy into internal energy by work that is done against the viscous fluid stresses. Due to this, increasing $Ec$ enhances dramatically the temperature of the fluid.

In Figure 9, we see how varying the $\mathsf{\Gamma }$ parameter affects the thermal distribution for both $\theta \left(\eta \right)$ and ${\theta }_p\left(\eta \right)$ cases. It is inferred that when the $\mathsf{\Gamma }$ parameter increases, the fluid temperature decreases. An increase in the $\mathsf{\Gamma }$ parameter decreases the temperature field as it decelerates the relaxation of heat flow.

The thermal field for the $\theta \left(\eta \right)$ and ${\theta }_p\left(\eta \right)$ instances is shown to be affected by the ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$ parameters in Figure 10, Figure 11 and Figure 12. The fact that the nanoparticle is so tiny may be deduced from the amount of heat that it generates when it releases the accumulated energy. It is possible that in both cases ($\theta \left(\eta \right)$ and ${\theta }_p\left(\eta \right)$), extra energy will be needed for the mixing of additional nanoparticles. In turn, this will cause an increment in the temperature and the thickness of the associated layer.

Figure 13 shows the effect of the $Bi$ parameter on thermal field for both $\theta \left(\eta \right)$ and ${\theta }_p\left(\eta \right)$ instances. It is noted that the thermal field for both $\theta \left(\eta \right)$ and ${\theta }_p\left(\eta \right)$ instances increases with an increase in the $Bi$ parameter. Furthermore, the interrelated thickness of the thermal layer also enhances by rising the values of the $Bi$ parameter.

In Table 3 and Table 4, the impact of various substantial factors over skin friction and the Nusselt number for the fluid flow system has been portrayed in numerical manners. The fluid motion decreases with expansion in magnetic effects and slip factor. The friction factor decreases with augmenting values of the nanoparticle volume friction parameter (${\varphi }_1, {\varphi }_2$, and ${\varphi }_3$), as shown in Table 3. The Nusselt number is maximized for higher values of the nanoparticle volume friction parameter (${\varphi }_1, {\varphi }_2$, and ${\varphi }_3$), as shown in Table 4. An increment of the values of the Eckert number, Prandtl number, and Biot number enhances the Nusselt number. Additionally, the Nusselt number rises with the rising values of thermal relaxation. One can conclude that the Nusselt number is enhanced for higher values of radiation and temperature parameters.

5. Conclusions

By using a CC heat flux model, we investigated the movement of a dusty ternary nanofluid over a stretching sheet. The shown partial differential equations are converted to ordinary differential equations and numerically solved via the BVP4c technique. Graphs depict the concentration, the velocity, and the temperature. Overall, we are able to draw the following conclusions:

• The magnetic field depends essentially on Lorentz force, which is predominant for a larger magnetic field. Due to this, the velocity profiles decrease by further increasing the values of the magnetic parameter.
• The velocity profiles are decreasing by increasing the values of the slip parameter. However, once the slip velocity parameter is increased, the slide speed will also be decreased accordingly.
• The Eckert number is defined as the ratio between kinetic energy and the specific enthalpy difference between the wall and the fluid. Thus, increasing the Eckert number causes an increment in the temperature profile for both dusty and ternary phases.
• The thermal radiation role is to boost heat transfer by enlarging the thermal radiation and thermal boundary layers for both dusty and ternary phases.
• The tiny nanoparticle effect may be deduced from the amount of heat generated by the accumulated energy. Due to this, temperature profiles are enhanced by increasing values of the nanoparticle volume friction parameter.
• An increment in the value of thermal relaxation generates a decrement in the temperature distribution.
• Temperature profiles increase by increasing the values of the temperature ratio parameter.
• The performance of the dusty phase heat transfer has significantly enhanced compared to the performance of the ternary nanofluid phase.