Thermally Radiative Darcy–Forchheimer Flow of Cu/Ag Nanoliquid in Water Past a Heated Stretchy Sheet with Magnetic and Viscous Dissipation Impacts

Abstract

This communication predominately discusses the rheological attributes of the Darcy–Forchheimer flow of a nanoliquid over a stretchy sheet with a magnetic impact. The present model considers the two diverse nanoparticles, such as $Cu$ and $Ag$, and water as a base liquid. The heat equation accounts for the consequences of thermal radiation and a nonlinear heat sink/source when evaluating heat transmission phenomena. The current mechanical system is represented by higher-order PDEs, which are then remodeled into nonlinear higher-order ODEs that employ appropriate symmetry variables. The current mathematical systems are numerically computed by implementing the bvp4c technique. The characteristic attitudes of the related pertinent factors on the non-dimensional profiles are sketched via the figures, tables, and charts. The analysis predicts that the speed of the nanoliquid particles becomes slower when there is more presence of a magnetic field and injection/suction parameters. The growing amount of radiation is also pointed out, and the Eckert number corresponds to enriching the thermal profile.

1. Introduction

The liquid thermic conductivity acts as a key part in many mechanical and industrial procedures, such as cooling microelectronic devices, heat exchangers, solar collectors, mining processes, etc. The common base liquids such as $H_{2}O$ and $C_2{H}_6{O}_2$ transmit a lower quantity of heat due to their poor thermic conductivity. Many researchers are working on perfecting the process through which base fluids transfer more heat. To improve the thermic conductivity of a base liquid, one of the most straightforward approaches is to suspend the nano-scale materials ($Cu$, $Cuo$, $SiC$, $Au$, $Ag$, $Ti{O}_2$, and $A{l}_2{O}_3$) inserted in the base liquid. Choi and Eastman [1] were the first to initiate the nanoliquid idea. The MHD flow of water-based nanoliquid past an SS with entropy generation was analyzed by Govindaraju et al. [2]. They proved that the larger EG number was observed in $Ag$ nanoparticles and a lesser EG number was observed in $Ti{O}_2$ nanoparticles. Malvandi et al. [3] probed the nanoliquid flow on an SS. They identified that the HBL (hydrodynamic boundary layer) becomes thinner when the nanoparticles are present. The effect of water-based CNTs on a superlinear stretching/shrinking sheet was reviewed by Mahabaleshwar et al. [4]. Sandeep and Sulochana [5] explored the flow of a nanoliquid with injection/suction. They revealed that the temperature profile upsurges when increasing the Brownian motion parameter. The influence of the Lorentz force in a nanoliquid flow was interpreted by Iqbal et al. [6]. Waini et al. [7] examined the uniform shearing flow on hybrid nanoliquids past a shrinking/stretching sheet. They observed that the HTG increases when the NPVF increases. Some pertinent research on this subject is emphasized in Refs. ([8,9,10,11,12,13,14,15,16,17]).

Many researchers have recently become interested in liquid flowing past a permeable space and the particles of various shapes enclosed in a porous medium. Darcy’s law is commonly used to understand the flow of porous gaps. However, this law is applicable only for poor porosity and low-velocity cases. Nowadays, the non-uniform porosity distribution and the greater flow transport occur in many problems. In this situation, Forchheimer [18] overcame this limitation by incorporating a square speed factor into the Darcian model. The numerical solution of the DF flow of a nanoliquid by a rotating disc was presented by Ullah et al. [19]. They identified that the liquid temperature enriches for a greater quantity of the Forchheimer number. Sarkar and Kundu [20] examined the $Cu$-water-based nanoliquid past a vertical sheet. They found that increasing the amount of the porosity parameter caused the heat transmission rate to progress. The entropy generation of the DF flow of a nanoliquid with radiation was probed by Khan et al. [21]. They noted the liquid velocity decrepitude by raising the porosity parameter and Forchheimer number. Sureshkumar Raju et al. [22] examined the DF flow of a moving needle. They observed that the TBL thickness slumps when elevating the Forchheimer number. Hayat et al. [23] discussed the nonlinear mixed convective DF flow of viscous liquid past a nonlinear SS. They noted that the nanoliquid velocity dwindles by improving the porosity parameter. Khan et al. [24] scrutinized the DF flow of viscous liquid past an SS with homogeneous–heterogeneous reactions. The liquid velocity decrepitude was observed when upturning the quantity of the reverse Darcy number. Some essential studies on these concepts are collected in Refs. ([25,26,27,28]).

The study of viscous dissipation on heat transfer is essential, notably for high viscous flows with mild velocities. In a viscous dissipation procedure, the adhesiveness of the liquid and energy transforms from an initial to a final form. Viscous dissipation in the porous medium can be classified into two terms: the first type, the Darcy term, arises from internal heating, and the second type, the Brinkman term, arises due to frictional heating. The MHD nanoliquid flow upon a nonlinear stretched flat plate with viscous-Ohmic dissipation was studied by Pal and Mandal [29]. They noted that the Eckert number plays an opposite behavior in stretching and shrinking sheets. Das et al. [30] examined the MHD flow over a porous surface subject to viscous dissipation. They found that the surface shear stress enhances when improving the Eckert number. Bhukta et al. [31] examined how thermal dissipation affects the MHD flow through a porous medium. The Carreau liquid flow with viscous dissipation was studied by Salahuddin et al. [32]. They proved that the liquid temperature upsurges for higher values of the Eckert number. The influence of a viscously dissipated energy flow of an MHD nanoliquid past an SS was numerically investigated by Gopal et al. [33]. They noticed that the velocity profile improved when the viscous dissipation quantity was enhanced. Hussain et al. [34] discovered the effect of viscous dissipation on an MHD hyperbolic liquid over a convective boundary condition. They noted that the temperature of the liquid upsurges when raising the Eckert number. The viscous dissipative flow of an MHD nanoliquid with dual stratification was illustrated by Daniel et al. [35]. They concluded that the liquid thermal profile enhances when raising the Eckert number.

The heat source can supply an enormous amount of energy without a change in its temperature, and the heat sink can receive a tremendous amount of energy without a change in its temperature. Therefore, the heat sink/source effect is also an essential aspect of heat transmissions in many real-world problems. A few applications are the power cycle, refrigeration, and heat pump cycle. Hakeem et al. [36] explained a non-uniform heat sink/source with heat radiation in Walter’s B liquid. They proved that the liquid temperature enhanced when improving the non-uniform heat sink parameter. The reactions of a heat sink/source of an Oldroyd-B nanoliquid past an SS were presented by Sandeep and Sulochana [37]. They noticed that the LNN slumps when upturning the non-uniform heat sink parameter. The flow of an MHD Eyring–Powell liquid over an SS with a non-uniform heat sink/source was delved into by Manvi et al. [38]. They noted that the heat sink parameter causes the liquid temperature to diminish. Pal and Chatterjee [39] analyzed the convective flow of micropolar liquid past an SS with a non-uniform heat sink/source. The radiative flow of an MHD micropolar liquid on an SS with a non-uniform heat sink/source was addressed by Mabood et al. [40]. Ramandevi et al. [41] inspected an MHD flow with a CCHF with a combination of viscous dissipation and a non-uniform heat sink/source. They discovered that the irregular heat parameters had an inversely proportional relationship to the heat transmission gradient. The time-dependent flow of the MHD Williamson nanoliquid past a stretched cylinder with a non-uniform heat source/sink phenomenon was deliberated by Song et al. [42]. They proved that the space-dependent heat source parameter enriches the liquid thermal profile. Mumraiz et al. [43] numerically presented the MHD flow of a hybrid nanoliquid past a vertical annulus. They saw that the temperature-reliant heat sink/source parameter upsurges the liquid thermal profile.

The primary purpose of this research communication is to examine the MHD DF flow of the $Cu/Ag$ nanoliquid in water past a heated stretchy sheet. We derive the energy equations by reimplementing the Cattaneo–Christov theory, a nonlinear heat sink/source, and viscous dissipation. The flow-through stretchy sheet is significant in many disciplines, including crystal growing, paper production, glass fiber, wire drawing, etc. Our computational outcomes are unique and novel, and it is applied in the manufacturing industry to build new thermal equipment. Our conclusions are compared to those of previously published publications for confirmation intent. The MATLAB bvp4c methodology is used to make all of the representative estimations.

2. Mathematical Formulation

Consider the 2D flow of water-based $Cu/Ag$ nanoliquid on a heated stretchy sheet. Let $\mathcal{X}-$ direction be taken along the sheet and $\mathcal{Y}-$ direction be perpendicular to the sheet. Then, the equation ${\mathcal{U}}_w=a\mathcal{X}(a>0)$ is used to explain the sheet’s wall’s expansion in a symmetric plane about the origin. The magnetic field $B_0$ is applied in $\mathcal{Y}-$ direction and the induced magnetic field is not considered because of the small Reynolds number. The sheet is kept at a uniform temperature ${\mathcal{T}}_w$ which is higher than the free stream temperature ${\mathcal{T}}_{\infty }$. The heat sink/source, radiation, and Cattaneo–Christov theory are added to energy transfer equations. The down part of the sheet is heated by hot liquid with temperature ${\mathcal{T}}_f$, which creates a heat transfer coefficient $h_c$. The physical scaling of the flow model is depicted in Figure 1. The governing equations are defined with the help of the above assumptions and presented as follows, see Hamad [44], Kameswaran et al. [45], and Shankar and Gorfie [46]:

$$\begin{array}{ccc}& & {\mathcal{U}}_{\mathcal{X}}+{\mathcal{V}}_{\mathcal{Y}}=0\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & \mathcal{U}{\mathcal{U}}_{\mathcal{X}}+\mathcal{V}{\mathcal{U}}_{\mathcal{Y}}=\frac{{\mu }_{nf}}{{\rho }_{nf}}{\mathcal{U}}_{\mathcal{YY}}-\frac{{\nu }_{nf}}{k_1^{*}}\mathcal{U}-\frac{c_b}{\sqrt{k^{*}}}{\mathcal{U}}^2-\frac{{\sigma }_{nf}}{{\rho }_{nf}}{B}_0^2\mathcal{U}\hfill \\ & & \mathcal{U}{\mathcal{T}}_{\mathcal{X}}+\mathcal{V}{\mathcal{T}}_{\mathcal{Y}}=\frac{k_{nf}}{{\left(\rho c_p\right)}_{nf}}{\mathcal{T}}_{\mathcal{YY}}+\frac{16{\sigma }^{*}{\mathcal{T}}_{\infty }^3}{3k^{*}{\left(\rho c_p\right)}_{nf}}{\mathcal{T}}_{\mathcal{YY}}+\frac{{\mu }_{nf}}{{\left(\rho c_p\right)}_{nf}}{\left({\mathcal{U}}_{\mathcal{Y}}\right)}^2\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & +\frac{{\sigma }_{nf}}{{\left(\rho c_p\right)}_{nf}}{B}_0^2{\mathcal{U}}^2+\frac{k_{nf}}{{\left(\rho c_p\right)}_{nf}}\frac{{\mathcal{U}}_w}{\mathcal{X}{\nu }_{nf}}\left[A{A}^{*}({\mathcal{T}}_f-{\mathcal{T}}_{\infty })f^{\prime }+B{B}^{*}(\mathcal{T}-{\mathcal{T}}_{\infty })\right]\hfill \\ \hfill & & -\lambda \left[{\mathcal{U}}^2{\mathcal{T}}_{\mathcal{XX}}+{\mathcal{V}}^2{\mathcal{T}}_{\mathcal{YY}}+2\mathcal{UV}{\mathcal{T}}_{\mathcal{XY}}+\left(\mathcal{U}{\mathcal{U}}_{\mathcal{X}}+\mathcal{V}{\mathcal{U}}_{\mathcal{Y}}\right){\mathcal{T}}_{\mathcal{X}}+\left(\mathcal{U}{\mathcal{V}}_{\mathcal{X}}+\mathcal{V}{\mathcal{V}}_{\mathcal{Y}}\right){\mathcal{T}}_{\mathcal{Y}}\right]\hfill \end{array}$$

(3)

The corresponding boundary conditions are

$$\begin{array}{ccc}& & \mathcal{U}={\mathcal{U}}_w;\mathcal{V}=-{\mathcal{V}}_w;-k_{nf}{\mathcal{T}}_{\mathcal{Y}}=h_c[{\mathcal{T}}_f-\mathcal{T}]\mathrm{at}\mathcal{Y}=0\hfill \\ & & \mathcal{U}\to 0;\mathcal{V}\to 0,\mathcal{T}\to {\mathcal{T}}_{\infty }as\mathcal{Y}\to \infty \hfill \end{array}$$

(4)

The flow variable details are all in the nomenclature section.

Define the variables

$$\begin{array}{c}\hfill \mathcal{U}=a\mathcal{X}{f}^{\prime }\left(\mathsf{{\rm Y}}\right);\mathcal{V}=-\sqrt{a{\nu }_f}f\left(\mathsf{{\rm Y}}\right);\mathsf{{\rm Y}}=\sqrt{\frac{a}{{\nu }_f}}\mathcal{Y};\theta =\frac{\mathcal{T}-{\mathcal{T}}_{\infty }}{{\mathcal{T}}_f-{\mathcal{T}}_{\infty }}\end{array}$$

(5)

Apply Equation (5) in Equations (2)–(4), we obtain

$$\begin{array}{ccc}& & \frac{1}{A_1{A}_2}{f}^{‴}\left(\mathsf{{\rm Y}}\right)+f\left(\mathsf{{\rm Y}}\right)f^{″}\left(\mathsf{{\rm Y}}\right)-{f^{\prime }}^2\left(\mathsf{{\rm Y}}\right)-\frac{1}{A_1{A}_2}\lambda f^{\prime }\left(\mathsf{{\rm Y}}\right)-Fr{f^{\prime }}^2\left(\mathsf{{\rm Y}}\right)-\frac{A_4}{A_2}M{f}^{\prime }\left(\mathsf{{\rm Y}}\right)=0\hfill \\ & & \left[\frac{A_5}{A_3}\frac{1}{Pr}\right]{\theta }^{″}\left(\mathsf{{\rm Y}}\right)+f\left(\mathsf{{\rm Y}}\right){\theta }^{\prime }\left(\mathsf{{\rm Y}}\right)+\frac{4}{3}Rd\frac{1}{Pr}\frac{1}{A_3}{\theta }^{″}\left(\mathsf{{\rm Y}}\right)+\frac{A_1{A}_2{A}_5}{A_3}\frac{1}{Pr}[A{A}^{*}{f}^{\prime }\left(\mathsf{{\rm Y}}\right)+B{B}^{*}\theta \left(\mathsf{{\rm Y}}\right)]\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& & +\frac{1}{A_1{A}_3}Ec{f^{″}}^2\left(\mathsf{{\rm Y}}\right)+\frac{A_4}{A_3}MEc{f}^{\prime }{}^2\left(\mathsf{{\rm Y}}\right)-{\mathsf{\Gamma }}_1\{f^2\left(\mathsf{{\rm Y}}\right){\theta }^{″}\left(\mathsf{{\rm Y}}\right)+f\left(\mathsf{{\rm Y}}\right)f^{\prime }\left(\mathsf{{\rm Y}}\right){\theta }^{\prime }\left(\mathsf{{\rm Y}}\right)\}=0\hfill \end{array}$$

(7)

The reduced boundary conditions are

$$\begin{array}{ccc}& & f\left(0\right)=fw;f^{\prime }\left(0\right)=1;f^{\prime }(\infty )=0;\hfill \\ & & {\theta }^{\prime }\left(0\right)=-\frac{Bi}{A_5}\left[1-\theta \left(0\right)\right];\theta (\infty )=0\hfill \end{array}$$

(8)

Here,

$$\begin{array}{ccc}\hfill A_1& =& \frac{{\mu }_f}{{\mu }_{nf}}={\left(1-\varphi \right)}^{2.5};\hfill \\ \hfill A_2& =& \frac{{\rho }_{nf}}{{\rho }_f}=\left(1-\varphi +\varphi \frac{{\rho }_{nf}}{{\rho }_f}\right);\hfill \\ \hfill A_3& =& \frac{{\left(\rho Cp\right)}_{nf}}{{\left(\rho Cp\right)}_f}=\left(1-\varphi +\varphi \frac{{\left(\rho Cp\right)}_{nf}}{{\left(\rho Cp\right)}_f}\right);\hfill \\ \hfill A_4& =& \frac{{\sigma }_{nf}}{{\sigma }_f}=1+\frac{3\left(\frac{{\sigma }_{nf}}{{\sigma }_f}-1\right)\varphi }{\left(\frac{{\sigma }_{nf}}{{\sigma }_f}+2\right)-\left(\frac{{\sigma }_{nf}}{{\sigma }_f}-1\right)\varphi };\hfill \\ \hfill A_5& =& \frac{k_{nf}}{k_f}=\frac{k_{nf}+(m-1)k_f-(m-1)\varphi (k_f-k_{nf})}{k_{nf}+(m-1)k_f+\varphi (k_f-k_{nf})}\hfill \end{array}$$

The skin friction coefficient and local Nusselt number are expressed as

$$\begin{array}{ccc}& & C_f\sqrt{Re}=A_1{f}^{″}\left(0\right);\frac{Nu}{\sqrt{Re}}=-\left[A_5+\frac{4}{3}Rd\right]{\theta }^{\prime }\left(0\right)\hfill \end{array}$$

3. Numerical Solution

The reformatting ODE Equations (6) and (7) and corresponding boundary conditions (8) are numerically solved using the MATLAB bvp4c technique (see Eswaramoorthi et al. [47,48]). In this case, higher-order ODEs are converted into 1st ODEs.

Let $f=H_1,f^{\prime }=H_2,f^{″}=H_3,\theta =H_4,{\theta }^{\prime }=H_5$.

The system of equations are

$$\begin{array}{ccc}& & H_1^{\prime }=H_2\hfill \\ & & H_2^{\prime }=H_3\hfill \\ & & H_3^{\prime }=\frac{H_2^2-H_1{H}_3+\frac{1}{A_1{A}_2}\lambda H_2+Fr{H}_2^2+\frac{A_4}{A_2}M{H}_2}{A_1{A}_2}\hfill \\ & & H_4^{\prime }=H_5\hfill \\ & & H_5^{\prime }=\frac{-H_1{H}_5-\frac{1}{A_1{A}_3}Ec{H}_3^2-\frac{A_1{A}_2{A}_5}{A_3}\frac{1}{Pr}[A{A}^{*}{H}_2+B{B}^{*}{H}_4]-\frac{A_4}{A_3}MEc{H}_2^2+{\mathsf{\Gamma }}_1\left\{H_1{H}_2{H}_5\right\}}{\frac{1}{A_{3}Pr}\left[A_5+\frac{4}{3}Rd\right]-{\mathsf{\Gamma }}_1\left(H_2^2\right)}\hfill \end{array}$$

The reformatting boundary constraints are

$$\begin{array}{ccc}& & H_1\left(0\right)=fw;H_2\left(0\right)=1;H_2(\infty )=0;H_5\left(0\right)=-\frac{B_i}{A_5}(1-H_4\left(0\right));H_4(\infty )=0\hfill \end{array}$$

The aforementioned problems are solved numerically by incorporating the MATLAB bvp4c function with an error of ${10}^{-5}$ and a step size of $0.05$.

4. Results and Discussion

The main goal of this segment is to examine the nanoliquid velocity, temperature, SFC, and LNN for various flow parameters. The physical properties of the nanoparticles and water are portrayed in

Table 1. Physical characteristics, see Rahim et al. [49].

Base Liquid and Nanoparticles	Copper $\left(\mathit{Cu}\right)$	Silver $\left(\mathit{Ag}\right)$	Water $\left({\mathit{H}}_2\mathit{O}\right)$
$\rho /(\mathrm{kg}/{\mathrm{m}}^{-3})$	8933	10,500	$997.1$
$C_p$/(J.kg${}^{-1}$.K${}^{-1}$)	385	235	4179
$\sigma /{(\mathsf{\Omega }.\mathrm{m})}^{-1}$	$59.6\times {10}^6$	$6.3\times {10}^7$	$5.5\times {10}^{-5}$
$\mathrm{k}/(\mathrm{W}.{\mathrm{m}}^{-1}.{\mathrm{K}}^{-1})$	401	429	$0.613$

.

Table 2. Comparison of $-f^{″}\left(0\right)$ for disparate values of $\varphi$ with $\lambda =Fr=M=0$ to Hamad et al. [44].

M	$\mathsf{\varphi }$	Present Study $\left(\mathit{Cu}\right)$	Ref. [44]	Present Study $\left(\mathit{Ag}\right)$	Ref. [44]
0	$0.05$	$1.10892$	$1.10892$	$1.13966$	$1.13966$
	$0.1$	$1.174746$	$1.17475$	$1.225068$	$1.22507$
	$0.15$	$1.208862$	$1.20886$	$1.272153$	$1.27215$
	$0.2$	$1.218044$	$1.21804$	$1.289788$	$1.28979$

presents our $-f^{″}\left(0\right)$ results which are compared with Hamad et al. [44], and it is noticed that our results perfectly coincided with Hamad et al. [44]. The contrast of the SFC and LNN for various values of $\lambda ,\mathrm{Fr},\mathrm{M}$, and $\mathrm{fw}$ is portrayed in

Table 3. The contrast of SFC and LNN for different values of $\lambda$, Fr, M, and fw.

$\mathsf{\lambda }$	$\mathit{Fr}$	M	$\mathit{fw}$	Cu
				$\mathit{Cf}$	$\mathit{Nu}$
0	$0.4$	$0.5$	$0.5$	$-1.554420$	$0.210191$
$0.3$				$-1.642577$	$0.181753$
$0.6$				$-1.725257$	$0.154570$
$0.9$				$-1.803389$	$0.128464$
$1.2$				$-1.877660$	$0.103298$
$0.2$	0	$0.5$	$0.5$	$-1.524807$	$0.209924$
	$0.5$			$-1.635348$	$0.186450$
	1			$-1.738750$	$0.163731$
	$1.5$			$-1.836185$	$0.141708$
	2			$-1.928539$	$0.120326$
$0.2$	$0.4$	0	$0.5$	$-1.456414$	$0.317903$
		$0.3$		$-1.553291$	$0.239720$
		$0.6$		$-1.643103$	$0.167643$
		$0.9$		$-1.727240$	$0.100357$
		$1.2$		$-1.806677$	$0.036960$
$0.2$	$0.4$	$0.5$	$-0.6$	$-1.045042$	$-1.307431$
			$-0.3$	$-1.173861$	$-0.591965$
			0	$-1.322095$	$-0.155272$
			$0.3$	$-1.490534$	$0.087582$
			$0.6$	$-1.679017$	$0.231798$

and

Table 4. The contrast of SFC and LNN for different values of $\lambda ,Fr,M,$ and $fw$.

$\mathsf{\lambda }$	$\mathit{Fr}$	M	$\mathit{fw}$	Ag
				$\mathit{Cf}$	$\mathit{Nu}$
0	$0.4$	$0.5$	$0.5$	$-1.598207$	$0.189668$
$0.3$				$-1.684590$	$0.161214$
$0.6$				$-1.765799$	$0.133988$
$0.9$				$-1.842682$	$0.107819$
$1.2$				$-1.915876$	$0.082575$
$0.2$	0	$0.5$	$0.5$	$-1.564458$	$0.190343$
	$0.5$			$-1.678612$	$0.165687$
	1			$-1.785342$	$0.141836$
	$1.5$			$-1.885871$	$0.118724$
	2			$-1.981127$	$0.096293$
$0.2$	$0.4$	0	$0.5$	$-1.502479$	$0.296300$
		$0.3$		$-1.597102$	$0.218869$
		$0.6$		$-1.685107$	$0.147233$
		$0.9$		$-1.767749$	$0.080185$
		$1.2$		$-1.845921$	$0.016891$
$0.2$	$0.4$	$0.5$	$-0.6$	$-1.068062$	$-1.361699$
			$-0.3$	$-1.20429$	$-0.627094$
			0	$-1.360782$	$-0.181279$
			$0.3$	$-1.538322$	$0.065680$
			$0.6$	$-1.736862$	$0.211710$

for the $Cu$ and $Ag$ nanoparticles. It is detected from these tables that the SFC and LNN decrease when improving the values of $\lambda ,\mathrm{Fr},\mathrm{and}\mathrm{M}$ for both cases. The greater presence of the $\mathrm{fw}$ values improves the LNN and suppresses the SFC.

Table 5. LNN variation for various combinations of $Rd,Ec,A{A}^{*},B{B}^{*},{\mathsf{\Gamma }}_1,$ and $Bi$.

$\mathit{Rd}$	$\mathit{Ec}$	${\mathit{AA}}^{*}$	${\mathit{BB}}^{*}$	${{\Gamma }}_1$	$\mathit{Bi}$	$\mathit{Cu}$	$\mathit{Ag}$
0	$0.5$	$0.1$	$0.1$	$0.1$	$0.5$	$0.103018$	$0.089884$
$0.5$						$0.176371$	$0.157032$
1						$0.249297$	$0.224123$
$1.5$						$0.319769$	$0.289008$
2						$0.386971$	$0.350814$
$0.6$	0	$0.1$	$0.1$	$0.1$	$0.5$	$0.711517$	$0.709465$
	$0.4$					$0.295169$	$0.278335$
	$0.8$					$-0.121179$	$-0.152796$
	$1.2$					$-0.537526$	$-0.583925$
	$1.6$					$-0.953874$	$-1.015056$
$0.6$	$0.5$	0	$0.1$	$0.1$	$0.5$	$0.201849$	$0.181840$
		$0.2$				$0.180316$	$0.159265$
		$0.4$				$0.158783$	$0.136690$
		$0.6$				$0.137249$	$0.114115$
		$0.8$				$0.115716$	$0.091540$
$0.6$	$0.5$	$0.1$	0	$0.1$	$0.5$	$0.198968$	$0.179297$
			$0.2$			$0.182950$	$0.161509$
			$0.4$			$0.165886$	$0.142452$
			$0.6$			$0.147656$	$0.121970$
			$0.8$			$0.128133$	$0.099880$
$0.6$	$0.5$	$0.1$	$0.1$	0	$0.5$	$0.177007$	$0.156358$
				$0.05$		$0.183864$	$0.163269$
				$0.1$		$0.191082$	$0.170552$
				$0.15$		$0.198676$	$0.178223$
				$0.2$		$0.206652$	$0.186291$
$0.6$	$0.5$	$0.1$	$0.1$	$0.1$	$-1$	$-0.679703$	$-0.612545$
					$-0.5$	$-0.269825$	$-0.242062$
					$0$	$0$	$0$
					$0.5$	$0.191082$	$0.170552$
					$1$	$0.333503$	$0.297204$

shows the upshot of the $\mathrm{Rd},\mathrm{Ec},A{A}^{*},B{B}^{*},{\mathsf{\Gamma }}_1$, and Bi on the LNN. It is realized that the greater amount of $\mathrm{Rd},\mathrm{Bi},\mathrm{and}$${\mathsf{\Gamma }}_1$ leads to enriching the HTG and the reverse behavior attains larger quantities of $\mathrm{Ec},A{A}^{*}$, and $B{B}^{*}$.

Figure 2a–d draw the consequences of $\lambda ,Fr,M$, and $fw$ on the nanoliquid velocity profile for both nanoparticles. It is detected that the nanoliquid velocity declines as the quantity of $\lambda ,Fr,M$, and $fw$ heightens for both nanoparticles. It is also noted that the larger MBL occurs in the $Cu$ nanoparticles than the $Ag$ nanoparticles because the $Cu$ nanoparticles have a lower density than the $Ag$ nanoparticles. Physically, the larger porosity parameter enhances the nanoliquid resistance and this causes the nanoliquid motion to slow down. Moreover, the larger magnetic field parameters create a drag force called the Lorentz force, which slows down the liquid motion. The effectuates of the $Rd$, $A{A}^{*}$, Ec, and M on the nanoliquid temperature for both nanoparticles are illustrated in Figure 3a–d. It is seen that the nanoliquid temperature enhances when it exalts the values of the $Rd$, $A{A}^{*}$, Ec, and M. Physically, the higher quantity of the radiation parameter improves the energy transport rate of the liquid. Thus, the liquid temperature improves. Figure 4a–d give the effect of the ${\mathsf{\Gamma }}_1,fw,Bi$, and $\varphi$ on the nanoliquid temperature for both nanoparticles. It is observed that the liquid temperature suppresses when enhancing the ${\mathsf{\Gamma }}_1$ and $fw$ values, and the opposite trend is observed when changing the $Bi$ and $\varphi$ values. Physically, the greater presence of a Biot number leads to strengthening the liquid thermal state which escalates the liquid temperature. The contrast between the SFC for distinct combinations of $Fr,fw,M$, and $\lambda$ is plotted in Figure 5a–d. It is discovered that the surface shear stress declines when heightening the $fw,Fr$, $\lambda ,andM$ values. In addition, the larger SFC occurs in the $Cu$ nanoparticles than in the $Ag$ nanoparticles. This is because the $Ag$ nanoparticles have a larger density than the $Cu$ nanoparticles. Figure 6a–d show the contrast between the LNN for various combinations of $Rd,Ec,M,fw,and\lambda$ for both nanoparticles. The graph shows that the LNN grows as the values of the $fwandRd$ increase, but it slumps when enhancing the values of the $Ec,M,and\lambda$ for both nanoparticles.

Figure 7a–d show the conversions of the SFC for a different combination of values of $\lambda$, $Fr$, M, and $fw$. In the case of the porosity parameter, the highest decimating percentage (6.78%) was exhibited in the viscous nanoliquid when modifying $\lambda$ from 0 to 0.3, and the lowest decimating percentage (3.97%) occurred in the $Ag-$based nanoliquid when changing $\lambda$ from 0.9 to 1.2. In the case of the Forchheimer number, the highest decimating percentage (7.29%) was exhibited in the $Ag-$based nanoliquid when modifying $Fr$ from 0 to 0.5, and the lowest decimating percentage (4.93%) occurred in the viscous nanoliquid when modifying $Fr$ from 1.5 to 2. In the magnetic field parameter case, the highest decimating percentage (7.99%) was exhibited in the viscous nanoliquid when changing M from 0 to 0.3, and the lowest decimating percentage (4.42%) occurred in the $Ag-$based nanoliquid when modifying M from 0.9 to 1.2. In the case of the injection/suction parameter, the highest decimating percentage (12.74%) was exhibited in the $Cu-$based nanoliquid when modifying $fw$ from 0 to 0.3, and the lowest decimating percentage (10.76%) occurred in the viscous nanoliquid when $fw$ was changed from −0.6 to −0.3. The ameliorating percentage of the LNN on $Rd$, ${\mathsf{\Gamma }}_1$, $A{A}^{*}$, and $B{B}^{*}$ for all the cases are deliberated in Figure 8a–d. In the case of the Rd, the highest ameliorating percentage (74.70%) was exhibited in the $Ag-$based nanoliquid when changing $Rd$ from 0 to 0.5, and the lowest ameliorating percentage (19.99%) occurred in the viscous nanoliquid when modifying $Rd$ from 1.5 to 2. In the case of the heat relaxation time parameter, the highest ameliorating percentage (4.52%) was exhibited in the $Ag-$based nanoliquid when modifying ${\mathsf{\Gamma }}_1$ from 0.15 to 0.2, and the lowest ameliorating percentage (2.40%) occurred in the viscous nanoliquid when modifying ${\mathsf{\Gamma }}_1$ from 0 to 0.05. In the case of the $A{A}^{*}$ parameter, the highest ameliorating percentage (19.78%) was exhibited in the $Ag-$based nanoliquid when modifying $A{A}^{*}$ from 0.6 to 0.8, and the lowest ameliorating percentage (5.21%) occurred in the viscous nanoliquid when modifying $A{A}^{*}$ from 0 to 0.2. In the case of the $B{B}^{*}$ parameter, the highest ameliorating percentage (18.11%) was exhibited in the $Ag-$based nanoliquid when modifying $B{B}^{*}$ from 0.6 to 0.8, and the lowest ameliorating percentage (2.89%) occurred in the viscous nanoliquid when modifying $B{B}^{*}$ from 0 to 0.2.

5. Conclusions

The current exploration scrutinizes the consequences of a magnetic field with a Darcy–Forchheimer flow of a $Cu/Ag-$$H_{2}O$-based nanoliquid over a heated stretchy sheet with the CCHF theory, radiation, and viscous dissipation. Consider water as a base liquid and $Cu/Ag-$ as a nanoliquid. First, the energy equation is framed by the heat sink/source and Cattaneo–Christov’s theory. Next, suitable symmetry variables reform the governing PDE expressions into ODE expressions. Finally, the obtained ODEs are resolved by using the BVP4c scheme. Our study’s key findings are as follows:

• The nanoliquid speed reduces when strengthening the porosity parameter, Forchheimer number, magnetic parameter, and injection/suction parameter.
• The nanoliquid warmth rises when enhancing the radiation, heat sink/source, viscous dissipation, and magnetic field parameter.
• The nanoliquid temperature declines when raising the heat relaxation time and injection/suction parameter, and it exalts when strengthening the Biot number and nanoparticle volume fraction.
• The skin friction depresses for the higher quantity of the Forchheimer number, porosity, and magnetic field parameter.
• The Nusselt number slumps when mounting the values of the Eckert number and magnetic field parameter. It strengthens the radiation and injection/suction parameters.
• In the future, we will expand this flow model to include the Riga plate and the convective heating scenario.