Numerical study of non-Darcy hybrid nanofluid flow with the effect of heat source and hall current over a slender extending sheet

Abstract

The current evaluation described the flow features of Darcy Forchhemier hybrid nanoliquid across a slender permeable stretching surface. The consequences of magnetic fields, second order exothermic reaction, Hall current and heat absorption and generation are all accounted to the fluid flow. In the working fluid, silicon dioxide (SiO₂) and titanium dioxide (TiO₂) nano particulates are dispersed to prepare the hybrid nanoliquid. TiO₂ and SiO₂ NPs are used for around 100 years in a vast number of diverse products. The modeled has been designed as a nonlinear set of PDEs, Which are degraded to the dimensionless system of ODEs by using the similarity transformation. The reduced set of nonlinear ODEs has been numerically estimated through bvp4c package. The outcomes are tested for validity and consistency purpose with the published report and the ND solve technique. It has been noted that the energy curve lessens with the influence of thermodiffusion, Brownian motion and rising number of nanoparticles, while boosts with the result of magnetic field. Furthermore, the concentration outline of hybrid nanoliquid improves with the upshot of chemical reaction.

Introduction

The analysis of fluid flow over a slendering surface has frequent implementations in various fields, containing manufacture of glass, aerodynamic, polymer industry, firmness of plastic slips and metal tubular^1–3. Gul et al.⁴ examined and evaluated the proficiency of a hybrid nanofluid along an increasing sheet. It was discovered that the magnetism influence altered the instability of liquid. Bilal et al.⁵ employed the PCM methodology to imitate the movement of nanoliquids through a stretchable material with the effects of suction and injection. The physical and chemical properties of nanofluid flow passing through permeable stretching was documented by Safwa et al.⁶. Moreover, Hussain et al.⁷ reported the energy conversions of MHD nanoliquid flow along an elongating surface. Shuaib et al.⁸ described the ferrofluid flow along with the characteristics of energy conveyance through spinning sheet. Hussain et al.⁹ assessed the energy transport through nanoliquid flow over an extending cylinder. Uddin et al.¹⁰ analysed the energy transmission through water-based nanoliquid across an expanding surface. Rasool et al.¹¹ documented the nanoliquid flow across a contracting surface. Ahmad et al.¹² assessed nanoliquid fluid across a slender stretching sheet.

Hybrid nanofluid has greater thermal efficiency and mostly utilized in industry for cooling purposes¹³. Hybrid nanofluid work in solar energy, energy transition, air conditioners, generators, the vehicle sector, radioactive systems, electrical coolers, ships, biotechnology and transmitters^14–16. TiO₂ and SiO₂ have non-toxic, non-reactive characteristics and absorb UV rays used for skin cancer, drug delivery, recording devices and solar cells¹⁷. Traciak et al.¹⁸ conducted an experimentally assessed the density, optical characteristics and surface tension of SiO₂-containing nanoliquids based on ethylene glycol. Using the bvp4c software, Bhatti et al.¹⁹ provided a detailed discussion of SiO2 and carbon nanocrystals over an elastic substrate. Ahmed et al.²⁰ scrutinized the nanoliquid flow and energy conveyance through Al₂O₃ and TiO₂ nps based nanoliquid, to augments the thermal efficiency of base solvent, such as thermal diffusivity and heat transport coefficient. Khashi'ie et al.^21,22 highlighted the comportment of Al₂O₃-Cu based hybrid nanoliquid flow and its thermal properties as they were driven by an elongating Riga plate. Alwawi et al.²³ addressed the impact of magnetism on nanofluid streaming in the scenario of coupled convection across a circular cylinder. The findings show that increasing the coupled convection factor's value improves the Nusselt number, velocity, skin friction and rotational velocity while reducing the thermal contour's trends. Abbasi et al.²⁴ comparatively reported the thermal assessment of three distinct sorts of nano particulates, including TiO₂, SiO₂ and aluminum oxide through curved sheet. Khashi’ie et al.²⁵ used Cu-Al₂O₃ hybrid nanoparticles to study the Blasius flow across a rotating plate. De²⁶ and Mondal et al.²⁷ investigated the combined influence of Soret-Dufour interactions in a nanoliquid flow. Recently, a number of investigators have described on the evaluation of hybrid nanoliquid flow over distinct configuration^28–32.

Hall current can be detected if the fluid density is small, or the magnetic flux amplitude is strong. In many practical operations that call for an intense electric affect and smaller atomic concentration, hall effects should not be undervalued. Electron transport, where electrons move more quickly than ions, is what results in isotropic conductivity. Ohm's law needs to be revised for the purposes to consider the Hall effect. It has several applications in Hall activators, circuits, pumps, electric inverters, turbines and other equipment, Nanoliquid flow with the upshot of Hall current and magnetic effect has drawn the attention of scientists^33,34. Using an extended sheet, Khan and Nadeem³⁵ examined a spinning Maxwell nanoliquid flow with a magnetism, Hall current and kinetic energy. An asymmetrical reactive nanoliquid flow induced by a magnetization revolving plate and the Hall impact is described by Acharya et al.³⁶ along with the flow dynamics and energy variations. They found that the energy transference was improved by 84.61% by nanocomposites. The Hall effect in nanofluid flow has recently been the subject of numerous investigations^37–40.

The purpose of the current assessment is to study the flow features of Darcy Forchhemier hybrid nanoliquid across a slender permeable stretching surface. The consequences of magnetic fields, second order exothermic reaction, Hall current and heat absorption and generation are all accounted to the fluid flow. In the working fluid, SiO₂ and TiO₂ nano particulates are dispersed to prepare the hybrid nanoliquid. The modeled has been designed as a nonlinear set of PDEs. Which are transmute to the dimensionless system of ODEs by using the similarity replacement. The reduced set of nonlinear ODEs has been numerically estimated through bvp4c package.

Mathematical framework

We assumed a steady 2D MHD hybrid nanoliquid flow through impermeable slendering substrate. The surface is stretching with velocity $U_w\left(x\right)={\left(x+b\right)}^n{U}_0,$ as described in Fig. 1, where n is the power index. The sheet irregularity is assumed as $y=A{\left(x+b\right)}^{\frac{1-n}{2}},$ (A is the stretching constant). The Hall and magnetic effect are employed for flow motion in y-direction. Heat source, Brownian motion, thermo-diffusion and chemical reactions are all observed in current analysis.

Figure 1.

The fluid flow across a slandering expanding cylinder.

The basic equations responsible for the fluid flow are characterized as⁴¹:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$ 1

$${\rho }_{\mathit{hnf}}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)={\mu }_{\mathit{hnf}}\frac{{\partial }^{2}u}{\partial y^2}-\frac{{\sigma }_{\mathit{hnf}}}{1+m^2}{B}^2\left(x\right)\left(u+mw\right)-\frac{{\upsilon }_{\mathit{hnf}}}{K^{\ast }}u-\frac{1}{{\rho }_{\mathit{hnf}}}F{u}^2,$$ 2

$${\rho }_{\mathit{hnf}}\left(u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}\right)={\mu }_{\mathit{hnf}}\frac{{\partial }^{2}w}{\partial y^2}-\frac{{\sigma }_{\mathit{hnf}}}{1+m^2}{B}^2\left(x\right)\left(mu-w\right)-\frac{{\upsilon }_{\mathit{hnf}}}{K^{\ast }}w-\frac{1}{{\rho }_{\mathit{hnf}}}F{w}^2,$$ 3

$$\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=\frac{k_{\mathit{hnf}}}{{\left(\rho C_p\right)}_{\mathit{hnf}}}\left(\frac{{\partial }^{2}T}{\partial y^2}\right)+\left(D_B\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2\right)+\frac{Q_0\left(T-T_{\infty }\right)}{\rho C_p},$$ 4

$$\left(u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}\right)=D_B\left(\frac{{\partial }^{2}C}{\partial y^2}\right)+\frac{D_T}{T_{\infty }}\frac{{\partial }^{2}T}{\partial y^2}-K{c}^2\left(C-C_{\infty }\right),$$ 5

here, $m={\tau }_e{w}_e$ is the Hall current, Kc², Q₀, K^* and $F=C_b/r{K}^{\ast 1/2}$ are the chemical reaction rate, heat source, permeability factor and non-uniform inertia factor respectively.

The initial and boundary conditions are:

$$\left(\begin{array}{cc}& u=U_w\left(x\right)=U_0{\left(x+b\right)}^n,v=0,w=0,D_B\frac{\partial C}{\partial y}+\frac{D_T}{T_{\infty }}\frac{\partial T}{\partial y}=0,T=T_w\text{at}y=A{\left(x+b\right)}^{\frac{1-n}{2}}\hfill \\ \hfill & u\to 0,T\to T_{\infty },w\to 0,C\to C_{\infty }asy\to \infty .\hfill \\ \hfill \end{array}\right\}$$ 6

The transformation variables are:

$$\begin{array}{cc}& \eta =y\sqrt{\frac{n+1}{2}\frac{U_0}{{\nu }_f}{\left(x+b\right)}^{m-1}},\psi =\sqrt{\frac{2}{n+1}{\nu }_f{U}_0{\left(x+b\right)}^{m+1}}f\left(\eta \right),\phi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\hfill \\ \hfill & w=U_0{\left(x+b\right)}^{n}h\left(\eta \right),\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }}.\hfill \\ \hfill \end{array}$$ 7

By merging Eq. (7) in Eqs. (1)–(6), we get:

$$f^{″\prime }+\frac{{\vartheta }_1}{{\vartheta }_2}\left(\left(f{f}^{″}-\frac{2n}{n+1}Fr{f}^{\prime 2}\right)-\frac{{\vartheta }_3}{{\vartheta }_1}\left(\frac{2M}{\left(n+1\right)\left(1+m^2\right)}\right)f^{\prime }+\lambda mg\right)=0,$$ 8

$$g^{″}+\frac{{\vartheta }_1}{{\vartheta }_2}\left(\left(f{g}^{\prime }-\frac{2n}{n+1}Frg{f}^{\prime }\right)-\frac{{\vartheta }_3}{{\vartheta }_1}\left(\frac{2M}{\left(n+1\right)\left(1+m^2\right)}\right)m{f}^{\prime }-\lambda g\right)=0,$$ 9

$${\theta }^{″}+Pr\frac{{\vartheta }_4}{{\vartheta }_5}\left(f{\theta }^{\prime }+Nb{\phi }^{\prime }{\theta }^{\prime }+Nt{\theta }^{\prime 2}\right)+Q_1\theta =0,$$ 10

$${\phi }^{″}+\frac{\mathit{Nt}}{\mathit{Nb}}{\theta }^{″}+Lef{\phi }^{\prime }-Kr\phi =0.$$ 11

here, ${\vartheta }_1=\frac{{\rho }_{\mathit{hnf}}}{{\rho }_f},{\vartheta }_2=\frac{{\mu }_{\mathit{hnf}}}{{\mu }_f},{\vartheta }_3=\frac{{\sigma }_{\mathit{hnf}}}{{\sigma }_f},{\vartheta }_4=\frac{{\left(\rho C_p\right)}_{\mathit{hnf}}}{{\left(\rho C_p\right)}_f},{\vartheta }_5=\frac{k_{\mathit{hnf}}}{k_f}.$

The conditions for system of ODEs are:

$$\left(\begin{array}{cc}& f\left(\eta \right)=\eta \left(\frac{1-n}{1+n}\right),Nb{\phi }^{\prime }\left(\eta \right)+Nt{\theta }^{\prime }\left(\eta \right)=0,f^{\prime }\left(\eta \right)=1,g\left(\eta \right)=0,\theta \left(\eta \right)=1\hfill \\ \hfill & f^{\prime }\to 0,g\to 0,\theta \to 0,\phi \to 0as\eta \to \infty \hfill \\ \hfill \end{array}\right\}$$ 12

here, the M, $\lambda$, Pr, Nt, Nb, Fr, $Q_1$, Le, Gr, $\delta$,, Gc and Kr is mathematically expressed as:

$$\begin{array}{cc}& M=\frac{B_0^2{\sigma }_f}{{\rho }_f{T}_{\infty }},Pr=\frac{{\mu }_f{\left(\rho C_p\right)}_f}{{\rho }_f{k}_f},\lambda =\frac{\nu }{k^{\ast }b},Nt=\frac{\tau D_T\left(T_w-T_{\infty }\right)}{{\nu }_f{T}_{\infty }},Fr=\frac{C_b}{K^{\ast 1/2}},Nb=\frac{\tau D_B{C}_{\infty }}{{\nu }_f},Q_1=\frac{x{Q}_0}{\rho C_p},\hfill \\ \hfill & Le=\frac{{\nu }_f}{D_B},\delta =A\sqrt{\frac{n+1}{2}\frac{U_0}{{\nu }_f}},Gr=\frac{g{\beta }_{T_f}\left(T_w-T_{\infty }\right)n}{U_w^2},Gc=\frac{g{\beta }_{C_f}\left(C_w-C_{\infty }\right)n}{U_w^2},Kr=\frac{K{c}^2}{b}.\hfill \\ \hfill \end{array}$$ 13

The physical interest quantities are:

$$C_{f_x}=\frac{2{\tau }_{w_1}}{U_w^2{\rho }_f},C_{f_z}=\frac{{\tau }_{w_2}}{U_w^2{\rho }_f},Nu=\frac{q_w\left(x+b\right)}{\left(T_w-T_{\infty }\right)k_f},Sh=\frac{j_w\left(x+b\right)}{\left(C_w-C_{\infty }\right)D_B}.$$ 14

where,

$$\left(\begin{array}{cc}& {\tau }_{w_1}={\mu }_{\mathit{hnf}}{\left(\frac{\partial u}{\partial y}\right)}_{y=A{\left(x+b\right)}^{\frac{1-n}{2}}},{\tau }_{w_2}={\mu }_{\mathit{hnf}}{\left(\frac{\partial v}{\partial y}\right)}_{y=A{\left(x+b\right)}^{\frac{1-n}{2}}},\hfill \\ \hfill & q_w=-k_{\mathit{hnf}}{\left(\frac{\partial T}{\partial y}\right)}_{y=A{\left(x+b\right)}^{\frac{1-n}{2}}},j_w=-D_B{\left(\frac{\partial C}{\partial z}\right)}_{y=A{\left(x+b\right)}^{\frac{1-n}{2}}}.\hfill \\ \hfill \end{array}\right\}$$ 15

The dimensionless structure of Eq. (14) is:

$$\left(\begin{array}{cc}& C_{f{r}_x}=\sqrt{R{e}_x}{C}_{\mathit{fx}}={\left(1-{\varphi }_1\right)}^{-2.5}{\left(1-{\varphi }_2\right)}^{-2.5}\sqrt{2\left(n+1\right)}{f}^{″}\left(0\right),\hfill \\ \hfill & C_{f{r}_z}=\sqrt{R{e}_x}{C}_{\mathit{fz}}={\left(1-{\varphi }_1\right)}^{-2.5}{\left(1-{\varphi }_2\right)}^{-2.5}\sqrt{2\left(n+1\right)}{g}^{\prime }\left(0\right),\hfill \\ \hfill & N{u}_r=\frac{\mathit{Nu}}{\sqrt{R{e}_x}}=-\frac{k_{\mathit{hnf}}}{k_f}\sqrt{\frac{n+1}{2}}{\theta }^{\prime }(0),S{h}_r=\frac{\mathit{Sh}}{\sqrt{R{e}_x}}=-\sqrt{\frac{n+1}{2}}{\phi }^{\prime }(0).\hfill \\ \hfill \end{array}\right\}$$ 16

Numerical methodology

The system of Eqs. (8)–(12) are simplified to 1st order set of ODEs and solved through bvp4c package as^42,43

$$\left(\begin{array}{cc}& {\xi }_1=f(\eta ),{\xi }_3=f^{″}(\eta ),{\xi }_5=g^{\prime }(\eta ),{\xi }_7={\theta }^{\prime }(\eta ),{\xi }_9={\phi }^{\prime }(\eta ),\hfill \\ \hfill & {\xi }_2=f^{\prime }(\eta ),{\xi }_4=g(\eta ),{\xi }_6(\eta )=\theta (\eta ),{\xi }_8=\phi (\eta ).\hfill \\ \hfill \end{array}\right\}$$ 17

By putting (17) in (8–12), we get:

$${\xi }_3^{\prime }+\frac{{\vartheta }_1}{{\vartheta }_2}\left(\left({\xi }_1{\xi }_3-\frac{2n}{n+1}Fr{\xi }_2^2\right)-\frac{{\vartheta }_3}{{\vartheta }_1}\left(\frac{2M}{\left(n+1\right)\left(1+m^2\right)}\right){\xi }_2+\lambda m{\xi }_4\right)=0,$$ 18

$${\xi }_5^{\prime }+\frac{{\vartheta }_1}{{\vartheta }_2}\left(\left({\xi }_1{\xi }_5-\frac{2n}{n+1}Fr{\xi }_4{\xi }_2\right)-\frac{{\vartheta }_3}{{\vartheta }_1}\left(\frac{2M}{\left(n+1\right)\left(1+m^2\right)}\right)m{\xi }_2-\lambda {\xi }_4\right)=0,$$ 19

$${\xi }_7+Pr\frac{{\vartheta }_4}{{\vartheta }_5}\left({\xi }_1{\xi }_7+Nb{\xi }_9{\xi }_7+Nt{\xi }_7^2\right)+Q_1{\xi }_6=0,$$ 20

$${\xi }_9^{\prime }+\frac{\mathit{Nt}}{\mathit{Nb}}{\xi }_7^{\prime }+Le{\xi }_1{\xi }_9-K{r}^2{\xi }_8=0.$$ 21

the transform conditions are:

$$\left(\begin{array}{cc}& {\xi }_1\left(\eta \right)=\eta \left(\frac{1-n}{1+n}\right),Nb{\xi }_9\left(\eta \right)+Nt{\xi }_7\left(\eta \right)=0,{\xi }_2\left(\eta \right)=1,{\xi }_4\left(\eta \right)=0,{\xi }_6=1\hfill \\ \hfill & {\xi }_2\to 0,{\xi }_4\to 0,{\xi }_6\to 0,{\xi }_8\to 0as\eta \to \infty \hfill \\ \hfill \end{array}\right\}$$ 22

Result and discussion

This segment estimates the exhibition of velocity, energy and concentration outlines versus interest constraints and explain the physics behind each table and figures. The dimensionless set of ODEs (Eqs. (18)–(22)) are solved through bvp4c package.

Velocity curve $f^{\prime }\left(\eta \right)$

Figure 2a–d communicates the demonstration of velocity $f^{\prime }\left(\eta \right)$ curve versus m, $\delta$, n, ${\varphi }_1,{\varphi }_1$, Gc and Gr respectively. Figure 2a,b revealed that flow velocity amplifies with the outcome of m and diminishes with the impact of $\delta$. Figure 2c,d exhibits that flow velocity augments with the influence of n and lessen with the impact of ${\varphi }_1,{\varphi }_2$ respectively. The result of n decreases the shear stress of surface, as a result the fluid velocity improves with action of n. The developing amount of TiO₂ + SiO₂ nps grows the fluid viscosity, which triggers the retardation effect. Figure 2e,f emphasized that velocity outline boosts with the increment of thermal and mass Grashop number. The extending velocity of surface drops with the upshot of Gc and Gr which triggers the rises in the velocity outline.

Figure 2.

The exposition of velocity $f^{\prime }\left(\eta \right)$ curve versus constraints m, $\delta$, n, ${\varphi }_1,{\varphi }_2$, Gc and Gr respectively.

Figure 3a–d demonstrated the comportment of $g\left(\eta \right)$ outline verssu parameter m, n, $\delta$, Fr and M. Figure 3a–b revealed that the flow velocity considerably upsurges with the change of m and n. While declines with the addition of $\delta$ and M. The magnetic upshot causes Lorentz effect, which prevents the flow moment, so the velocity profile drops. Figure 3e presents that the consequences of Fr pointedly de accelerates the velocity field in the radial direction.

Figure 3.

The exposition of velocity $g\left(\eta \right)$ curve versus the m, n, M, $\delta$ and Fr respectively.

Energy curve $\theta \left(\eta \right)$

Figure 4a–d demonstrates the arrangement of temperature $\theta \left(\eta \right)$ curve against $\delta$, m, n and Q. Figure 4a,b describes that the energy $\theta \left(\eta \right)$ outline enlarged with the action of m and reduces under the upshot of $\delta$. Hall current result also creates confrontation, which uplifts the energy contour as perceived in Fig. 4a. Figure 4c,d represent the significances of n and Q, that their effects augment the energy profile of SiO₄₂ + TiO₂/C₂H₆O₂–H₂O hybrid nanoliquid. The consequence of Q term operational as a energy mediator for the nanoliquid, which directly effects the temperature outline $\theta \left(\eta \right)$.

Figure 4.

The exposition of energy $\theta \left(\eta \right)$ curve versus the m, $\delta$, n and Q respectively.

Figure 5a–d emphasized the appearance of heat $\theta \left(\eta \right)$ contour relative to Nb, Nt, ${\varphi }_1,{\varphi }_1$ and M. Figure 5a–c designated that fluid energy curve drops with the effect of Nb, Nt and ${\varphi }_1,{\varphi }_1$, while enhances with the influence of magnetic field. The mounting number of nano particulates intensifies the flow velocity as well as the heat capacity of the ordinary fluid, which fallouts such scenario. As earlier deliberated that the repellant strength created by the magnetic field, absolutely effects the energy curve $\theta \left(\eta \right)$.

Figure 5.

The exposition of energy $\theta \left(\eta \right)$ curve verus the Nb, Nt, ${\varphi }_1,{\varphi }_2$ and M respectively.

Mass profile $\phi \left(\eta \right)$

Figure 6a–c defined the exhibition of concentration $\phi \left(\eta \right)$ contur versus m, $\delta$, n and Kr. The concentration conversion of hybrid nanoliquid intensify with the upshot of m and declines with the impact of $\delta$ as exhibited in Fig. 6a,b. Figure 6c,d described that the upshot of Kr and n both augments the mass transport. The factor Kr boosts the kinetic force within the nanofluid, which results in the quick communication of concentration $\phi \left(\eta \right)$.

Figure 6.

The concentration $\phi \left(\eta \right)$ outline versus m, $\delta$, n and Kr respectively.

Figure 7 emphasized the relative examination of nanofluid (SiO₂ + TiO₂) and hybrid nanoliquid (SiO₂ + TiO₂/C₂H₆O₂–H₂O) for the energy and the velocity outline. Tables 1 and 2 represent the tentative values and mathematical model for SiO₂, TiO₂ and base fluid. Table 3 described the numerical calculation of the present outcomes with the ND solver approach, to approve the authenticity of the results. Table 4 discovered the arithmetic valuations of SiO₂ + TiO₂/C₂H₆O₂–H₂O hybrid nanoliquid for $C_{f_x},C_{f_z}$, $N{u}_r$ and $S{h}_r$. It is identified that the upshot of m augments the energy interaction rate and drag force.

Figure 7.

The comparation between nanoliquid and hybrid nanoliquid.

Table 1.

The tentative values of TiO₂, TiO₂ and C₂H₆O₂–H₂O⁴⁴.

Nano particulates and base fluid	$\rho (kg/m^3)$	$k(W/mK)$	$Cp(j/kgK)$	$\sigma (S/m)$
C2H6O2–H2O	1063.8	0.387	3630	0.00509
SiO2	2270	1.4013	3630	$3.5\times {10}^6$
TiO2	4250	8.953	686.2	$2.38\times {10}^6$

Table 2.

The mathematical model of the hybrid nanoliquid $\left({\varphi }_1={\varphi }_{Si{O}_2},{\varphi }_2={\varphi }_{Ti{O}_2}\right)$⁴⁴.

Models
$\frac{{\mu }_{\mathit{hnf}}}{{\mu }_{\mathit{bf}}}=\frac{1}{{\left(1-{\varphi }_{Si{O}_2}-{\varphi }_{Ti{O}_2}\right)}^2}$
$\frac{{\rho }_{\mathit{hnf}}}{{\rho }_{\mathit{bf}}}={\varphi }_{Si{O}_2}\left(\frac{{\rho }_{Si{O}_2}}{{\rho }_{\mathit{bf}}}\right)+{\varphi }_{Ti{O}_2}\left(\frac{{\rho }_{Ti{O}_2}}{{\rho }_{\mathit{bf}}}\right)+\left(1-{\varphi }_{Si{O}_2}-{\varphi }_{Ti{O}_2}\right)$
$\frac{{(\rho C_p)}_{\mathit{hnf}}}{{(\rho C_p)}_{\mathit{bf}}}={\varphi }_{Si{O}_2}\left(\frac{{(\rho C_p)}_{Si{O}_2}}{{(\rho C_p)}_{\mathit{bf}}}\right)+{\varphi }_{Ti{O}_2}\left(\frac{{(\rho C_p)}_{Ti{O}_2}}{{(\rho C_p)}_{\mathit{bf}}}\right)+\left(1-{\varphi }_{Si{O}_2}-{\varphi }_{Ti{O}_2}\right)$
$\frac{{(\rho {\beta }_T)}_{\mathit{hnf}}}{{(\rho {\beta }_T)}_{\mathit{bf}}}={\varphi }_{Si{O}_2}\left(\frac{{(\rho {\beta }_T)}_{Si{O}_2}}{{(\rho {\beta }_T)}_{\mathit{bf}}}\right)+{\varphi }_{Ti{O}_2}\left(\frac{{(\rho {\beta }_T)}_{Ti{O}_2}}{{(\rho {\beta }_T)}_{\mathit{bf}}}\right)+\left(1-{\varphi }_{Si{O}_2}-{\varphi }_{Ti{O}_2}\right)$
$\frac{k_{\mathit{hnf}}}{k_{\mathit{bf}}}=\left[\frac{\left(\frac{{\varphi }_{Si{O}_2}{k}_{Si{O}_2}+{\varphi }_{Ti{O}_2}{k}_{Ti{O}_2}}{{\varphi }_{Si{O}_2}+{\varphi }_{Ti{O}_2}}\right)+2k_{\mathit{bf}}+2\left({\varphi }_{Si{O}_2}{k}_{Si{O}_2}+{\varphi }_{Ti{O}_2}{k}_{Ti{O}_2}\right)-2\left({\varphi }_{Si{O}_2}+{\varphi }_{Ti{O}_2}\right)k_{\mathit{bf}}}{\left(\frac{{\varphi }_{Si{O}_2}{k}_{Si{O}_2}+{\varphi }_{Ti{O}_2}{k}_{Ti{O}_2}}{{\varphi }_{Si{O}_2}+{\varphi }_{Ti{O}_2}}\right)+2k_{\mathit{bf}}-2\left(k_{Si{O}_2}{\varphi }_{Si{O}_2}+k_{Ti{O}_2}{\varphi }_{Ti{O}_2}\right)+2\left({\varphi }_{Si{O}_2}+{\varphi }_{Ti{O}_2}\right)k_{\mathit{bf}}}\right]$
$\frac{{\sigma }_{\mathit{hnf}}}{{\sigma }_{\mathit{bf}}}=\left[\frac{\left(\frac{{\sigma }_{Si{O}_2}{\varphi }_{Si{O}_2}+{\varphi }_{Ti{O}_2}{\sigma }_{Ti{O}_2}}{{\varphi }_{Ti{O}_2}+{\varphi }_{Si{O}_2}}\right)+2{\sigma }_{\mathit{bf}}+2\left({\sigma }_{Si{O}_2}{\varphi }_{Si{O}_2}+{\varphi }_{Ti{O}_2}{\sigma }_{Ti{O}_2}\right)-2\left({\varphi }_{Si{O}_2}+{\varphi }_{Ti{O}_2}\right){\sigma }_{\mathit{bf}}}{\left(\frac{{\varphi }_{Si{O}_2}{\sigma }_{Si{O}_2}+{\varphi }_{Ti{O}_2}{\sigma }_{Ti{O}_2}}{{\varphi }_{Si{O}_2}+{\varphi }_{Ti{O}_2}}\right)+2{\sigma }_{\mathit{bf}}-\left({\sigma }_{Si{O}_2}{\varphi }_{Si{O}_2}+{\varphi }_{Ti{O}_2}{\sigma }_{Ti{O}_2}\right)+\left({\varphi }_{Si{O}_2}+{\varphi }_{Ti{O}_2}\right){\sigma }_{\mathit{bf}}}\right]$

Table 3.

The numerical outcomes for skin friction $-f^{″}(0)$.

n	Ref.41 $-f^{″}(0)$	Ref.45 $-f^{″}(0)$	bvp4c	ND-solve
			Current outcomes $-f^{″}(0)$	Current outcomes $-f^{″}(0)$
1.0	1.000001	1.0000	1.000000	1.000000
2.0	1.023410	1.0234	1.023511	1.023500
3.0	1.035871	1.0359	1.035970	1.035961
5.0	1.048615	1.0486	1.048514	1.048503
7.0	1.055049	1.0550	1.055248	1.055227
9.0	1.058920	1.0589	1.058941	1.058901
10	1.060329	1.0603	1.060428	1.060407

Table 4.

The numerical results for $\left(C_{f_x},C_{f_z}\right)$,$N{u}_r$ and $S{h}_r$.

m	$\delta$	N	$C_{f_x}$	$C_{f_z}$	$N{u}_r$	$S{h}_r$
0.2			− 1.689775	0.418367	1.225167	3.371834
0.4			− 1.286627	0.685572	1.259011	3.234460
0.6			− 1.865864	0.726487	1.289550	3.104671
0.8			− 0.1646348	0.765092	1.412453	3.006296
1.0			− 1.496956	0.791047	1.429020	2.935582
	0.1		− 1.509798	0.420456	1.18262	1.723911
	0.2		− 1.689775	0.418367	1.625167	2.371834
	0.3		− 1.877942	0.414772	1.701782	6.058711
	0.4		− 1.074212	0.409747	2.303375	8.311254
	0.5		− 1.278457	0.403394	2.723852	10.47904
		0.2	− 1.689775	0.418367	1.325167	5.371834
		0.3	− 2.214652	0.497395	1.253300	3.381260
		0.4	− 2.686005	0.567704	1.265827	2.474134
		0.5	− 3.116150	0.631884	1.222736	0.677470
		0.6	− 3.513876	0.691371	1.206127	0.572587

Conclusion

We have examined the flow features of hybrid nanoliquid through a slender stretching surface. The consequences of second order exothermic reaction, heat source, Hall current and magnetic fields are all also described. The modeled equations are assessed by using the numerical approach bvp4c package. The important conclusions are:

• The $f^{\prime }\left(\eta \right)$ outline augments with the outcome of n and m and while reduces with the rising quanitity of nano particulates ${\varphi }_1,{\varphi }_2$ and parameter $\delta .$

• The velocity $g\left(\eta \right)$ curve substantially upsurges with effect of n and m. While decresaes with the effect of $\delta$ and M.

• The energy $\theta \left(\eta \right)$ curve is enhances with the variation of m and diminishes with the $\delta$.

• The energy $\theta \left(\eta \right)$ contour diminish with the influence of Nb, Nt and ${\varphi }_1,{\varphi }_1$, while boosts with the outcome of magnetic effect.

• The concentration $\phi \left(\eta \right)$ outline of hybrid nanoliquid improves with the upshot of Kr and n.

• The current model may be expanded to other type of fluid and can be used different chemical composition nanoparticles in the base fluid for desire output. Furthermore, different numerical, analytical and fractional methods can also be used to solve such problems.

Abbreviations

n

Power index

U₀

Reference velocity

$m$

Hall current

K^*

Permeability factor

2D

Two-dimension

$F$

Non-uniform inertia factor

T

Temperature

Nt

Thermodiffusion

Pr

Prandtl number

${\mu }_{\mathit{hnf}}$

Viscosity

${(\rho C_p)}_{\mathit{hnf}}$

Thermal capacity

$k_{\mathit{hnf}}$

Thermal conductivity

${\varphi }_1={\varphi }_{Si{O}_2}$

Nanoparticles volume friction

C₂H₆O₂

Ethyne glycol

$\left(u,v,w\right)$

Velocity component

A

Stretching constant

$U_w$

Stretching velocity

Kc²

Chemical reaction rate

Q₀

Heat source

MHD

Magnetohydrodynamics

C

Concentration

$\lambda$

Porosity factor

Nb

Brownian motion

Gr

Grashof number

${\rho }_{\mathit{hnf}}$

Density

${(\rho {\beta }_T)}_{\mathit{hnf}}$

Thermal expansion

${\sigma }_{\mathit{hnf}}$

Electrical conductivity

${\varphi }_2={\varphi }_{Ti{O}_2}$

Nanoparticles volume friction

TiO₂

Titanium dioxide

SiO₂

Silicon dioxide

Author contributions

All authors equally contributed.

Data availability

All data used in this manuscript have been presented within the article.

Competing interests

The authors declare no competing interests.