Thermal analysis of radiative and electromagnetic flowing of hybridity nanofluid via Darcy–Forchheimer porous material with slippage constraints

Abstract

The goal of this research is to address the electrical magnetohydrodynamic flow of hybridity nanoparticles through a stretched surface with the impact of the Darcy–Forchheimer permeable medium. By identifying appropriate similarity factors, the partial differential equation is converted into a set of ordinary differential equation with the help of R-K fourth-order approach based on shooting technique. The velocity, thermal, friction factor coefficient, and heat transport are investigated comprehensively for the influence of numerous blossoming parameters refer to in the present study by plots graphically. It is accomplished that due to the influence of the electric field associated with first- and second-order velocity slip impacts, the motion of liquid slows down. Moreover, the stream velocity grows for the boosting scales of mixed convection, which leads to eliminating sticking impact. The variation of important flowing variables on heat transport and the drag coefficient is also investigated. Here augmenting valuation of mixed convection parameter corresponds to boost up velocity curve. Furthermore, the influence of thermal field and heat transport are similar in view of the Biot number.

Keywords

Hybrid nanofluid Darcy–Forchheimer thermal radiation porous medium uniform heat source/sink unsteady flow

Introduction

A nanofluid is a liquid that contains particles the size of a nanometer. Colloidal suspensions in this fluid are made up of nanomolecules in a conventional liquid. The most frequent materials used to create nanomolecules for nanofluids include metals, oxides, carbides, or carbon nanotubes. Different standard fluids have been used as reactive liquids to convey heat to various operations. Water is a regularly used active liquid because of its ubiquitous availability, however, it is not considered an efficient heat transporter due to its weak thermal conductivity. Engine oil, ethylene glycol, and other nonexchange fluids are also used in a variety of implementations, although utilization in heat transference processes is limited due to high viscosities and dangerous surroundings. The hybrid nanoparticle is a type of nanofluid in which two or several distinct nanomaterial components are described in a flowing liquid in various patterns. The nanoparticle combinations were chosen to include the positive impact of both nanoparticles into a uniform stably homogenous system. Hybridity nanofluids are a step forward in thermal conductance obtained by the mixing of different nanoparticles. Silver-silicon, silver-magnesium oxide, copper-alumina, copper-titanium dioxide, and silicon carbide-titanium dioxide are two examples of nanoparticle mixtures in a conventional base liquid (H₂O). There are numerous approaches for hybridizing nanoparticle mixtures, including a two-step process, ball milling, in-situ, thermal decomposition, vapor deposition, mechanically alloying, wet chemically, and thermally solved. He noted that the two-step approach is mostly used by researchers due to its low cost and ability to manufacture hybridity nanofluids on a large scale. According to their ideal proportional rate and thermal mechanism, the goal of employing a hybrid nanofluid is to further improve the thermal interchange between the qualities and disadvantages of separate suspensions. Hybrid nanoliquids are a fast-expanding topic. Hybrid nanoscales have several applications, which add advanced automated processes, biomedicine production, hybrid electric mechanisms, solar heat, automobile thermal dissipation, and residential refrigeration. Choi¹ presented a study on the temperature characteristic of nanofluid, which compelled investigators to pay interest to the subject. Due to specific innovative research in this field, we simply mention some achievements. Scholars are attentive to hybrid nanoscale due to their improving requirement in the heat transport procedure. Literature^2–8 highlight a notable improvement in hybrid nanofluid flow. The effect of a nonuniform heat source on Burger's nanoparticle flowing with thermal variable conductivity was viewed by Khan et al.⁹ Thermodiffusion impact on hybridity nanoparticle flowing toward inclined oscillatory permeable surface with radiative heat transport was illustrated by Chu et al.¹⁰ Influence of viscous dissipation on hybridity nanoparticle with the characteristic of suction/injection by stability analysis were reported by Dero et al.¹¹ In this respect, a few of the most important and remarkable studies are illustrated in the literature.^12–18 Impact of Darcy–Forchheimer on hybridity nanoparticle flowing in an electrohydrodynamic by machine learning algorithm were explored by Shafiq et al.¹⁹ Theoretical investigation of the chemically reactive flowing of Ree–Eyring fluid with Darcy–Forchheimer by utilizing artificial neural networks was reported by Shafiq et al.²⁰ Food preparation, electronics chillers, nuclear collecting, photovoltaic storage, room climate control, fluid crystal growth, biomedicine, heat exchangers, liquid crystal development, and many more industries are examples. Most available kinds of literature claim that injecting nanomaterials into conventional fluids increases the rate of heat transfer dramatically. Because nanofluids have better thermal conductance, than conventional fluids, we may boost their thermal conductivity by suspending gold, silver, aluminum, nanotubes, graphite, silica, zinc, and many other nanostructures in conventional fluids.^21–24

Thermal radiative fluxing may have a significant impact on heat transference rate and temperature patterns at high operational temperatures. As a result, the amount of research conducted on thermal radiation impacts on boundary layer issues is growing. Thermal radiation has assisted in the development of better energy conversion equipment that can function at high temperatures. Heat transfer is a natural phenomenon that takes place when there is a temperature differential between two things or inside the same item, resulting in the phenomena of heat transition.²⁵ Furthermore, heat transmission properties in fluid flow have several uses in industry and manufacturing. A great deal of theoretical and experimental research has been performed recently on the heat transfer features of fluid motion. As a result, much effort has been expended in forecasting heat transfer behavior. The most significant of these are in the oilfield, hydropower, and a variety of other fields.^26–29 Assessment of the stream and heat transport characteristics of nanoliquid flow toward a stretched surface has been associated with the importance of several scientists across the universe because of its numerous engineering and manufacturing implementation, including spinning of metals, glass blowing, wire drawing by hot rolling, and extrusion of polymers, etc. The cooling of the stretchy surface is necessary to create higher quality things and maintain thermal control, however, a knowledge of energy transport and flow in such structures is important. Crane³⁰ was the first to analyze the boundary layer flowing toward a linearly movable stretched surface. Time-dependent second-grade nanoparticles flowing along vertical stretching surface with mixed convective and radiative heat transport were examined by Abbas et al.³¹ Seth et al.³² investigated the striation point stream of nanoscale with the Hartmann field and explored the velocity and thermal transport properties of nanofluid stream generated by shrink/stretch surface with temperature source-sink and permeability impacts. Assessment of the Lie group was used by Layek et al.³³ to explore the influence of heat radiative on the striation point stream of liquid flowing through a stretched/shrunk surface. Chemically reactive effects on thermal and solutal transport in radiative unsteady flowing with viscous dissipative and ohmic heating were reported by Li et al.³⁴ The unsteady, striation point stream of nanoscale across a stretched surface in the influence of thermophoretic and Hartman fields was studied by Zaib et al.,³⁵ which discovered a dual solution to the governing equations. They noticed that the temperature transport rates slow down with improving nanoparticle friction. Impacts of distinct slip mechanisms and energy constraints on bioconvectional rate type nanoparticles were viewed by Liu et al.³⁶ Above examination^37–39 are not consider the thermal radiation impact.

In consideration of the above analysis, the aim of this research is the numeric explore the combining impacts of heat transport and temperature radiative on the flow of hybrid nanofluid via variable features of permeable space. Even such concepts in electrical MHD streams are not widely explored. We investigate electrical MHD streams with such approaches. The condition of velocity slips is investigated flow is produced by a stretchy sheet. The shooting approach is utilized to determine the solution to the governmental nonlinear problem. The result of several emergent flow factors is depicted in graphs. We believe that electrical magnetohydrodynamic toward heat radiative in the theory of Darcy–Forchheimer stream of hybrid nanomaterials makes a new improvement.

Mathematical formulation

In this analysis, we have assumed the unsteady, 2D, incompressible, and electrical MHD flowing of hybridity nanomaterials toward a stretchable surface with Darcy–Forchheimer and radiative heat transfer effects. The velocity of the stretchable sheet is determined as $u_{w} (x, t)$ . The influence of the Hartman and electrical fields are applied to the liquid flowing in a perpendicular direction. Hartmann's number is considered extremely low, as an outcome, hall current and Hartman inducement are insignificant. In this study, $T_{w} (x, t)$ represent surface thermal is higher than the $T_{\infty}$ named as ambient thermal as indicated in Figure 1.

Figure 1.

Flow geometry of the model.

According to all these ideas, the governmental stream formulation of the current scheme is as followed by.^40,41

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

(1)

\begin{matrix} \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = v_{h n f} (\frac{\partial^{2} u}{\partial y^{2}} - \frac{u}{K *}) + \frac{σ_{h n f}}{ρ_{h n f}} (E B - B^{2} u) \\ + \frac{{(ρ β_{T})}_{h n f}}{ρ_{h n f}} (T - T_{\infty}) g - F_{c} u^{2} \end{matrix}}

(2)

\begin{matrix} \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = α_{h n f} \frac{\partial^{2} T}{\partial y^{2}} + \frac{1}{{(ρ C_{p})}_{h n f}} \frac{16 σ * T_{\infty}^{3}}{3 k *} \frac{\partial^{2} T}{\partial y^{2}} \\ + \frac{μ_{h n f}}{{(ρ C_{p})}_{h n f}} [{(\frac{\partial u}{\partial y})}^{2} + \frac{u^{2}}{K *}] + \frac{Q_{0}}{{(ρ C_{p})}_{h n f}} (T - T_{\infty}) \end{matrix}}

(3)

With

\begin{matrix} \begin{matrix} u = u_{w} = \frac{a x}{\sqrt{1 - a t}} + U_{s l i p}, \\ v = v_{w} = \frac{- v_{0}}{\sqrt{1 - a t}}, \\ - k_{h n f} \frac{\partial T}{\partial y} = h_{f} (T_{f} - T), \end{matrix}} a t y = 0, \\ u \to 0, T \to T_{\infty} a s y \to \infty \end{matrix}}

(4)

where

u, v, K *, σ_{h n f}, ρ_{h n f}, F_{c}, α_{h n f}, (ρ C_{p})_{h n f}, β_{T}, v_{h n f}, σ *, g, k *, μ_{h n f}, U_{s l i p} = α \frac{\partial u_{1}}{\partial y} + β \frac{\partial^{2} u_{2}}{\partial y^{2}}

and

h_{f}

represent the velocity components, porous media, electric conductivity, density, nonuniform inertia coefficient, thermal diffusivity, specific heat capacity, thermal expansion coefficient, kinematic viscosity, Stefan–Boltzman constant, gravitational acceleration, mean absorption coefficient, dynamic viscosity, slip velocity and heat exchange coefficient respectively.

Further, a hybrid nanoparticle made up of a based and Cu and TiO₂ nanomaterials was employed in this study. The volume fractions of Cu nanoparticle and TiO₂ nanoparticle are denoted as $ϕ_{1}$ and $ϕ_{2}$ , respectively, with $ϕ_{1}$ and $ϕ_{2}$ varying from 1% to 8%. Table 1^40–43 illustrates the correlations utilized for hybrid nanofluid, while Table 2^{40, 42, 44} summarizes the numerical valuation for the characteristic of the nanomaterials (Cu, TiO₂) and base liquid.

Table 1.

Thermophysical characteristics of hybrid nanofluid.

Properties	Hybrid nanofluid
Dynamic viscosity	$μ_{h n f} = \frac{μ_{f}}{{(1 - ϕ_{1})}^{2.5} {(1 - ϕ_{2})}^{2.5}}$
Thermal conductivity	$\frac{k_{h n f}}{k_{f}} = [\frac{2 k_{f} + (\frac{ϕ_{1} k_{s 1} + ϕ_{2} k_{s 2}}{ϕ_{h n f}}) + 2 (ϕ_{1} k_{s 1} + ϕ_{2} k_{s 2}) - 2 ϕ_{h n f} k_{f}}{2 k_{f} - (ϕ_{1} k_{s 1} + ϕ_{2} k_{s 2}) + (\frac{ϕ_{1} k_{s 1} + ϕ_{2} k_{s 2}}{ϕ_{h n f}}) + ϕ_{h n f} k_{f}}]$
Density	$ρ_{h n f} = [ϕ_{2} ρ_{s 2} + ρ_{f} (1 - ϕ_{2}) ((1 - ϕ_{1}) + ϕ_{1} \frac{ρ_{s 1}}{ρ_{f}})]$
Thermal expansion coefficient	$(ρ β_{T})_{h n f} = (1 - ϕ_{2}) [(1 - ϕ_{1}) {(ρ β_{T})}_{f} + ϕ_{1} {(ρ β_{T})}_{s 1}] + ϕ_{2} (ρ β_{T})_{s 2}$
Electric conductivity	$\frac{σ_{h n f}}{σ_{f}} = [\frac{2 σ_{f} + (\frac{ϕ_{1} σ_{s 1} + ϕ_{2} σ_{s 2}}{ϕ_{h n f}}) + 2 (ϕ_{1} σ_{s 1} + ϕ_{2} σ_{s 2}) - 2 ϕ_{h n f} σ_{f}}{2 σ_{f} - (ϕ_{1} σ_{s 1} + ϕ_{2} σ_{s 2}) + (\frac{ϕ_{1} σ_{s 1} + ϕ_{2} σ_{s 2}}{ϕ_{h n f}}) + ϕ_{h n f} σ_{f}}]$
Heat capacitance	$(ρ C_{p})_{h n f} = (1 - ϕ_{2}) [(1 - ϕ_{1}) {(ρ C_{p})}_{f} + ϕ_{1} {(ρ C_{p})}_{s 1}] + ϕ_{2} (ρ C_{p})_{s 2}$

Table 2.

Thermophysical characteristic amounts for water toward various types of nanoscales.

Properties	Titanium Oxide (TiO₂)	Pure water	Copper (Cu)
$ρ (k g m^{- 3})$	4250	997.1	8933
$k (W m^{- 1} K^{- 1})$	8.9538	0.613	401
$σ (S m^{- 1})$	2.38 $\times 10^{6}$	0.05	59.6 $\times 10^{6}$
$C_{p} (J k g^{- 1} K^{- 1})$	686.2	4179	385
$B_{T} (K^{- 1}) \times 10^{- 5}$	0.9	21	1.67

Now, apply the similarity transformation described below:

\begin{matrix} ψ = \sqrt{\frac{b v_{f}}{1 - a t}} x f (η), η = y \sqrt{\frac{b}{v_{f} (1 - a t)}}, θ (η) = \frac{T - T_{\infty}}{T_{f} - T_{\infty}}, \\ u (x, t) = \frac{b x}{1 - a t} f^{'} (η), v (x, t) = \sqrt{\frac{b v_{f}}{1 - a t}} f (η) \end{matrix}}

(5)

Equation (1) is fulfilled, and equations (2)–(3) are simplified to the nondimensional form shown below:

\begin{matrix} \frac{μ_{h n f}}{μ_{f}} [f^{″'} - K f^{'}] + \frac{ρ_{h n f}}{ρ_{f}} [f f^{″} - D_{f} {(f^{'})}^{2} - A (f^{'} + \frac{η}{2} f^{″})] \\ + \frac{σ_{h n f}}{σ_{f}} M (E - f^{'}) + \frac{{(ρ β_{T})}_{h n f}}{{(ρ β_{T})}_{f}} λ θ = 0 \end{matrix}}

(6)

\begin{matrix} {\frac{k_{h n f}}{k_{f}} + \frac{4}{3} R d {[1 + (θ_{w} - 1) θ]}^{3}} θ^{″} + \frac{μ_{h n f}}{μ_{f}} Pr E c [{(f^{″})}^{2} - K {(f^{'})}^{2}] + Pr Q θ \\ + 3 R d (θ_{w} - 1) {(θ^{'})}^{2} {(1 + (θ_{w} - 1) θ)}^{2} \frac{{(ρ C_{p})}_{h n f}}{{(ρ C_{p})}_{f}} Pr (f θ^{'} - \frac{A}{2} η θ^{'}) = 0 \end{matrix}}

(7)

with

\begin{matrix} \begin{matrix} f (0) = f_{w}, \\ f^{'} (0) = 1 + δ_{1} f^{″} (0) + δ_{2} f^{″'} (0), \\ θ^{'} (0) = - \frac{k_{f}}{k_{h n f}} B i (1 - θ (0)), \end{matrix}} a t η = 0, \\ f^{'} (\infty) \to 0, θ (\infty) \to 0 a s η \to \infty \end{matrix}}

(8)

With dimensionless variables:

\begin{matrix} Re = \frac{b x^{2}}{v_{f} (1 - a t)}, A = \frac{a}{b}, M = \frac{σ_{f} B_{0}^{2}}{b ρ_{f}}, G r = \frac{g {(β_{T})}_{f} (T_{f} - T_{\infty}) x^{3}}{{(v_{f})}^{2}}, θ_{w} = \frac{T_{w}}{T_{\infty}}, \\ λ = \frac{G r}{R e^{2}}, f_{w} = \frac{v_{0}}{\sqrt{v_{f}} b}, E = \frac{E_{0}}{U_{w} B_{0}}, Pr = \frac{v_{f}}{α}, K = \frac{v_{f}}{b k *}, D_{f} = \frac{C_{b}}{\sqrt{k *}}, δ_{1} = \sqrt{\frac{b}{v_{f}}} \\ Q = \frac{Q_{0}}{b {(ρ C_{p})}_{f}}, E c = \frac{u_{w}^{2}}{{(C_{p})}_{f} (T_{w} - T_{\infty})}, B i = \frac{h}{k_{f}} \sqrt{\frac{v_{f}}{b}}, δ_{2} = β \sqrt{\frac{b}{v_{f}}}, R d = \frac{4 σ * T_{\infty}^{3}}{k_{f} k *}, \end{matrix}}

(9)

The formulation for the drag coefficient

(C_{f})

and Nusselt number (

N u_{x}

) are given as follows:

\begin{matrix} C_{f} = \frac{τ_{w}}{ρ_{h n f} u_{w}^{2}}, \\ N u_{x} = \frac{x q_{w}}{(T_{w} - T_{\infty}) k_{h n f}} \end{matrix}}

(10)

where (

τ_{w}

) represent shear stress and

(q_{w})

heat flux is given by:

\begin{matrix} τ_{w} = μ_{h n f} {(μ \frac{\partial u}{\partial y})}_{y = 0} \\ q_{w} = {(- k_{h n f} \frac{\partial T}{\partial y} + q_{r})}_{y = 0} \end{matrix}}

(11)

In dimensionless form

\begin{matrix} {Re}_{x}^{\frac{1}{2}} C_{f} = \frac{μ_{h n f}}{μ_{f}} f^{″} (0), \\ {Re}_{x}^{\frac{- 1}{2}} N u_{x} = - {\frac{4}{3} R d {[1 + (θ_{w} - 1) (θ (0))]}^{3} + \frac{k_{h n f}}{k_{f}}} θ^{'} (0), \end{matrix}}

(12)

R e_{x} = \frac{b x^{2}}{v_{f} (1 - a t)}

describes as Reynolds number.

Numerical procedure

In this section, the impact of Darcy–Forchheimer on unsteady hybridity nanoparticle along porous stretching surfaces with radiative heat and uniform heat source/sink are considered. The dimensionless equations (6) and (7) with boundary condition (8) are solved numerically by utilizing the shooting technique. The procedure is performed as selected by the initial guesses and required the linear operator for motion and thermal field.^{45, 46}

h_{1} = f, h_{2} = f^{'}, h_{3} = f^{″}, h_{4} = θ, h_{5} = θ^{'}

(13)

\begin{matrix} h_{1}^{'} = h_{2}, \\ h_{2}^{'} = h_{3}, \\ \begin{matrix} h_{3}^{'} = K h_{2} - \frac{1}{\frac{μ_{h n f}}{μ_{f}}} [\begin{matrix} \frac{ρ_{h n f}}{ρ_{f}} [h_{1} h_{3} - D_{f} {(h_{2})}^{2} - A (h_{2} + \frac{η}{2} h_{3})] + \frac{{(ρ β_{T})}_{h n f}}{{(ρ β_{T})}_{f}} λ h_{4} \\ + \frac{σ_{h n f}}{σ_{f}} M (E - h_{2}) \end{matrix}], \\ h_{4}^{'} = h_{5}, \\ h_{5}^{'} = - \frac{[\begin{matrix} 3 R d (θ_{w} - 1) {(h_{5})}^{2} {(1 + (θ_{w} - 1) h_{4})}^{2} \frac{{(ρ C_{p})}_{h n f}}{{(ρ C_{p})}_{f}} Pr (h_{1} h_{5} - \frac{η}{2} A h_{5}) \\ + \frac{μ_{h n f}}{μ_{f}} Pr E c ({(h_{3})}^{2} - K {(h_{2})}^{2}) + Pr Q h_{4} \end{matrix}]}{(\frac{k_{h n f}}{k_{f}} + \frac{4}{3} R d {(1 + (θ_{w} - 1) h_{4})}^{3})}, \end{matrix} \end{matrix}}

(14)

with

\begin{matrix} \begin{matrix} h_{1} (0) = f_{w}, \\ h_{2} (0) = 1 + δ_{1} h_{3} + δ_{2} h_{3}^{'}, \\ h_{5} (0) = \frac{k_{f}}{k_{h n f}} B i (1 - h_{5}), \end{matrix}}, a t η \to 0 \\ h_{2} (\infty) \to 0, h_{4} (\infty) \to 0}, a s η \to \infty \end{matrix}}

(15)

The presence of a continuous consequence promotes the grid selection and error measure in this evolution. The margin is still set at $10^{- 4}$ . The assessment of $η \to \infty$ means that any number respond achieves asymptotic features completely using this technique. The detailed flow chart has also been provided for a clearer and better comprehension of the existing shooting technique (see Figure 2).

Figure 2.

The flow chart of the numerical procedure.

Code validity

A comprehensive investigation of the drag friction factor coefficient with the previous work is described in Table 3. It is evaluated that the present outcome is in good agreement with a preceding solution in a limiting sense.

Table 3.

Evaluation of skin friction coefficient $f^{″} (η)$ with Mumriaz et al.⁴⁰ and Gayatri et al.⁴⁷ for various valuations of Prandtl number M and suction parameter $(f_{w})$ while the other parameter is static.

$M$	$f_{w}$	Ref.⁴⁰	Ref.⁴⁷	Present study
0.0	0.5	1.28082	1.280778	1.280781
0.5	−	−	1.500000	1.500001
1.0	−	−	1.686141	1.686137
1.5	−	−	1.850781	1.850778
2.0	−	−	2.000000	2.000001
1.0	0.0	−	1.414214	1.414209
−	0.2	−	1.517745	1.517747
−	0.7	−	1.806880	1.806879
−	1.0	2.00001	2.000000	2.000000

Outcomes and discussion

In this analysis, the important aim is to exhibit the numeric results of velocity $f^{'} (η)$ , thermal $θ (η)$ , friction factors $f^{″} (0)$ , and Nusselt number $- θ^{'} (0)$ . To change the important variable and other variables are fixed. The characteristics of the associated variables along $f^{'} (η)$ , $θ (η)$ , $f^{″} (0)$ , and $- θ^{'} (0)$ of species are depicted in Figures 3 to 7.

Figure 3.

Impact of $δ_{1}, δ_{2}, f_{w}$ , and K on $f^{'} (η)$ .

Figure 4.

Impact of $ϕ_{1}, ϕ_{2}, λ$ , and $B i$ on $f^{'} (η)$ and $θ (η)$ .

Figure 5.

Effect of $E c, ϕ_{1}, ϕ_{2}$ , and $θ_{w}$ on $θ (η)$ .

Figure 6.

Effect of $R d, ϕ_{1}, f_{w}$ , and $λ$ on $θ (η), f^{″} (0)$ and $- θ^{'} (0)$ .

Figure 7.

Impact of $B i$ and $R d$ on $- θ^{'} (0)$ .

The influence of the first- and second-order velocity slip $(δ_{1} = δ_{2} = 0.0, 0.5, 1.0, 1.5)$ on the $f^{'} (η)$ curve is highlighted in Figure 3(a)–(b). The larger velocity slips scale climbs the wall share stress which restricts the liquid stream. Furthermore, escalating valuations of $δ_{1}, δ_{2}$ the momentum boundary layer are considerably reduced. The variation of $(f_{w} = 1, 2, 3, 4)$ on $f^{'} (η)$ curve is depicted in Figure 3(c). We discovered that the $f^{'} (η)$ estimation is decay with higher $f_{w}$ . The diminishing tendency in Figure 3(c) can be decreased by $f_{w}$ seeming to slow the liquid molecules along the permeable media and thus reducing the thicknesses of the boundary layer. Figure 3(d) depicts the role of $(K = 1, 2, 3, 4)$ on $f^{'} (η)$ curve. It is revealed that a greater association between decline $f^{'} (η)$ distribution for hybrid nanoparticles. The change in $f^{'} (η)$ estimation with respect to $(ϕ_{1} = 0.0, 0.1, 0.15, 0.2)$ and $(ϕ_{2} = 0.1, 0.4, 0.5, 0.6)$ are reported in Figure 4(a)–(b). In these figures, the $f^{'} (η)$ curve appears to enhance as the numeric values of the volume fraction (Cu) and TiO₂ nanostructures improve. Physically, the incrementation in the value of $(ϕ_{1}, ϕ_{2})$ reflects a growth in the number of nanoscales scattered in the base liquid. The outcome indicated hybrid nanofluid is enhanced by the minimal volume fraction. Figure 4(c) displays the role of $(λ = 0.5, 1.0, 1.5, 2.0)$ on $f^{'} (η)$ curve. An improvement in the velocity of the liquid is noted for higher levels of $λ$ . The buoyant forces are augmented due to $λ$ which boosts $f^{'} (η)$ the curve. Similarly, the interrelated intimacy of the boundary layer also reduces for a raising valuation of $λ$ .

The effect of $(B i = 0.1, 0.2, 0.3, 0.4)$ on the thermal $θ (η)$ curve is noticed in Figure 4(d). When $B i$ raises, it means that the plate's inner heat transport is higher than its outward heat transport. As an outcome, thermal boosts with growing $B i$ scales. This improvement in thermal is more prominent in the occurrence of TiO₂/H₂O. Figure 5(a) depicts the thermal $θ (η)$ distribution varies as a function of $(E c = 0.0, 0.5, 1.0, 1.5)$ . Furthermore, $E c$ is utilized to highlight the impact of liquid self-heated due to dissipating impacts. In this figure, it is assumed that the thermal of the flowing is augmented by inner resistance between the greater velocity liquid layers. It is considered that the thermal distribution improves with growth in $E c$ . Figure 5(b)–(c) exhibits the graphs of the flowing liquid of thermal $θ (η)$ distribution versus the volume fraction $(ϕ_{1} = 0.0, 0.1, 0.15, 0.2)$ and $(ϕ_{2} = 0.1, 0.4, 0.5, 0.6)$ of the copper (Cu) and titanium oxide TiO₂. These figures, clearly reveal that the thermal curve improves with the augmentation of nanostructures volume fraction $(ϕ_{1}, ϕ_{2})$ of (Cu, TiO₂) in base liquid (H₂O). Figure 5(d) indicates the tendency of the thermal ratio $(θ_{w} = 0.0, 1.0, 1.6, 2.0)$ variable along the thermal $θ (η)$ curve. $θ_{w}$ is improved to climb the $θ (η)$ and thicknesses of the thermal boundary layer. The variation in thermal $θ (η)$ distribution with growing values of $(R d = 0.0, 1.0, 2.0, 3.0)$ is illustrated in Figure 6(a). It is assumed that thermal growths with an improvement in the $R d$ . Prominently, $R d$ quantifies the comparability input of heat transmission via conductivity to transport via $R d$ . It is supposed that a boost in $R d$ leads to enhancing thermal.

Figure 6(b) demonstrates the variation of $ϕ_{1}$ and $λ$ on the drag coefficient $C_{f}$ . It is noticeable that the intensity of $(C_{f})$ is decay for augmenting the valuation of $ϕ_{1}$ and $λ$ variables. Figure 6(c) depicts the results of the drag coefficient $C_{f}$ for various scales of $f_{w}$ and $λ$ variables. It is apparent that lesser is attained for greater scales of $f_{w}$ , whereas an opposing tendency is gained for strong $λ$ . Figure 6(d) displays the role of $ϕ_{1}$ and $R d$ on $N u$ the Nusselt number. This graph noticed that the reduction in $N u$ is indicated for augmenting values of $ϕ_{1}$ and $R d$ variables. Figure 7 presents the prominent characteristics of $B i$ and $R d$ . Furthermore, the cohesiveness of the layer is also augmented by a raised valuation of $B i$ and $R d$ .

Concluding remarks

In this research, the velocity slip effects are utilized to explore hybridity nanoparticles (Cu-TiO₂/water) flow through a stretchable sheet. The hybrid nanofluid stream is modeled using association based on the volume fraction of nanomaterials (Cu and TiO₂). The following are key observations.

➢ For higher valuations of first and second-order velocity slip parameter and porosity parameter, the hybridity nanoparticle velocity $f^{'} (η)$ break down.

➢ Thermal $θ (η)$ distribution grows up with improvement in Biot number $(B i)$ and Ecker number $(E c)$ .

➢ The opposite behavior is observed for Skin friction coefficient $f^{″} (0)$ and velocity $f^{'} (η)$ profile with boosting values of the nanoparticle volume friction parameter.

➢ Reverse pattern of heat transport $- θ^{'} (0)$ is observed in view of the nanoparticle volume friction parameter and Biot number $(B i)$ .

➢ Thermal $θ (η)$ field and velocity $f^{'} (η)$ curve boost up via nanoparticle volume friction parameter.

In the future, the existing method might be used for a number of physical and technical obstacles.^48–56

Footnotes

Availability of data and materials

All data generated or analyzed during this study are included in this published article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Acknowledgement

Researchers supporting project number (RSPD2023R535), King Saud University, Riyadh, Saudi Arabia.

ORCID iDs

M. Israr Ur Rehman

Wasim Jamshed

Correction (September 2023):

The article has been updated with the author's correct affiliations.

Nomenclature

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