Darcy-Forchheimer hybrid nanofluids flow with quadratic convection over a stretched tube

Abstract

The study of well-known Darcian fluidic streams and hybrid nanofluids is an essential need of nowadays researchers in regard to their exceptional heat transfer rates and implementations. The significance of hybrid nanofluids in controlling heat transmission cannot be overemphasized. Therefore, this article scrutinizes the Darcy Forchheimer flow of hybrid nanofluid toward a flexible tube. The flow rates near the surface of the cylinder are investigated by applying the Darcy-Forchheimer theory while examining the heat transfer rate, nonlinear convection expressions are used. The analysis also considers thermal radiation and chemical reactions. One of sophisticated numerical approach Runge–Kutta method (RK4) is selected for the proposed problem’s solution. The main novelty of present research is to examine the characteristics of hybrid nanofluids in the context of heat transfer over an extending cylinder for the purposes of enhancement regarding thermal transference and inertial impacts. Results shows that hybrid-based nanofluid provides upsurges in solid volume fraction of nanoparticles accompanied by an enhancement in the heat transfer rate by 4.32%, 5.8%, and 14.46% individually. The outcomes witnessed that hybrid based proposed nanofluids increased thermal transportation processes more effectively than other nanofluids.

Keywords

Darcy-Forchheimer hybrid nanofluids flow extending cylinder nonlinear convection nonlinear thermal radiation Runge Kutta 4th order (RK4) method

Introduction

Nanofluids, the evolution of regular fluids, are of interest to researchers of modern time due to their industrial applications. For the very first time, the terminology was introduced by Choi and Eastman¹ for optimal heat transmissions. The fabrication of nanofluids consists of dispersing nano-sized chemicals into fundamental fluids and is based on nanotechnology to subsequently enhance heat transfer rate. These mediums have a wide variety of applications in the transmission of heat, such as engineering, manufacturing, microfluidics, microelectronics, scientific, and micro-based processors. Later, his theory was applied to the creation of hybrid nanofluids by combining different nanoparticles with different chemical properties.² Hybrid nanofluids are defined as formulations by adding different nanoparticles with different chemical properties to a base fluid which can be used to retrieve the thermological competence of nanofluids. Gul et al.³ examined the thermal performance of $Cu + F e_{3} O_{4} + H_{2} O$ from the perspective of conical alteration between a disk and a cone along with uniform magnetic field acting along axial direction. Scrutinizing an unsteady hybrid nanoliquid flow induced by a rotating disk, Waini et al.⁴ explored multiple solutions and conducted stability analysis along. Gumber et al.⁵ examined the effects of suction/injection on transporting heat in a micropolar hybrid nanoliquid flow using $CuO + Ag + H_{2} O$ , when heat is generated/absorbed and radiated in various aspects. A three-dimensional investigation by Oke et al.⁶ summarized boundary layer flow and heat transfer using $Cu + A l_{2} O_{3} + H_{2} O$ hybrid nanofluid over a rotating plate, that was exposed to an exponentially stretching and inclined magnetic field as well, where they showed that both primary and secondary velocities are diminished because the magnetic field opposes the flow. Gowda et al.⁷ looked into water-based Jeffrey nanofluid scattered with graphene nanoparticles where they successfully demonstrated molecular-level analysis of the effect of liquid–solid interface layer and morphological diameter of nanoparticles. During a study conducted by Khan et al.,⁸ an innovative concept of ternary nanofluid floating over a rotating sphere was studied, where water-based ternary hybrid nanofluids were demonstrated to be potent cooling methods. Recently, Gowda et al.⁹ demonstrated laminar flow of a magnetic liquid induced by a stretching sheet under the influence of thermal radiation and Cattaneo-Christov heat flux while comparatively examining the flow dynamics regarding propylene glycol-water fusion + paraffin wax nanofluid and $ZnO + SAE 50$ nanolubricant, as a result of which they concluded the latter showed optimal performance contrary to the earlier one. Further improvements were illustrated by Kumar et al.¹⁰ when the researchers considered water-based ternary hybrid nanofluid carrying three distinct morphological nanoadditives of $Ti$ (spherical)+CNTs (cylindrical)+ graphene (platelets) and compared it with the performance of $ZnO + SAE 50$ nanolubricant. Besides Newtonian fluids, many mathematical models describing non-Newtonian fluidic mediums have been considered as base fluids for forming hybrid nanofluids. Gul et al.¹¹ estimated the stagnation point flow of blood-based Casson hybrid nanofluid, loaded with $Ti O_{2}$ and $Ag$ , over a rotating sphere. Other interesting work associated with the hybrid nanofluid flows can be seen in Refs.^12–20

Darcy’s extensive and well-known version is the Darcian fluid stream, which is typically used for inertia impacts. With the addition of the square demonstration at the end of speed, inertia is announced and who have consented to Forchheimer’s modification. Rasool et al.²¹ studied the radiative Rosseland procedure on reactive Maxwell nanofluid flows to an isothermally heated sheet using the Darcy–Forchheimer model. In the same year, Rasool et al.²² worked on the entropy generation for Cleveland Z-Staggered Cavity utilizing Darcy-Forchheimer flow of nanofluid. The transport of the Darcy-Forchheimer convective flux using hybrid nanoparticles in force on a rotating disk was considered by Hiremath.²³ Rasool et al.²⁴ investigated the numerical significance of the Darcy-Forchheimer model for MHD nanofluid flow bounded by a stretching surface. Alshehri and Shah²⁵ investigated computational observations on the Darcy–Forchheimer nanofluid on a porous medium with a viscous dissipation effect. During their work, Rasool et al.²¹ considered Buongiorno’s model in order to investigate the effects of Darcy-Forchheimer space, radiative flux, Brownian diffusion, and thermophoresis on MHD Maxwell nanofluid flow subject to a surface being stretched. Several researchers have defined nanomaterials in order to describe their properties for engineering and industrial applications, see Refs.^26–30

In the view of demands for meteoric energy transportations, heat can be transferred at a faster rate by using thermal radiative source.³¹ These electromagnetic radiations travel with the help of wavelength and have optimal capability of penetration into media. Thus, they hold vital importance of in any thermodynamic transport system. By taking the linear Rosseland approximation into account, the impacts of thermal radiation on the flow of hybrid-based nanofluids, imposing a modified version of the Buongiorno nanofluid, has been well observed by Wakif et al.³² Khan et al.³³ Studied the radiation-convective flux of the Casson gold-bloodnanofluid of the non-linear coupled differential system on a rotating surface. The researchers of Reddy et al.³⁴ analyzed the effects of radiation and chemical reaction on Casson nanofluid on a stretching surface during MHD heat transfer. Further surveys tell us about tremendous industrial and mechanical applications that involve rapid cooling stretching bodies. An initial concept of flow analysis over a stretcher cylinder was provided by Wang³⁵ and specifically observed the influence of stretching parameter on fluids motion. Later on, the similar idea was employed by Vajravelu et al.³⁶ when they modified the Wang’s model for investigating the heat transfer rate and impact of drag force. Sarada et al.³⁷ examined the flow of a water-based ternary hybrid nanofluid containing graphene, CNTs, and silver nanomaterials, bound by a curved sheet, and discussed how stretching, activation energy, and non-Fourier heat flux affected the flow. Likewise, the fluid streams along an extending cylindrical channel were analyzed with various reverberations by many analysts from last decades, such as Refs.^38–43

Figure 1 show the preparation of hybrid nanofluids and geometry of the problem. In literature, numerous research works are available on nanofluids with different systematic methodologies and distinct variables. However, in view of the above literature, no one has taken hybrid nanofluids with this kind of flow setup. The main novelty of present research is to examine the characteristics of hybrid nanofluids in the context of heat transfer over an extending cylinder for the purposes of enhancement regarding thermal transference and inertial impacts. Mostly, the researchers strive to deal with the flow phenomena, involving external moving plates, whereas the motivation of the proposed research work is to examine the fluid flow generated by extended cylinder. Therefore, the Darcy-Forchheimer based mathematical model is used with combination of nanoparticles that is, TiO₂ and Ag. In addition, thermal radiation and chemically reacting terms are used in the proposed model. For the solution of the proposed model RK-4 methods is utilized.

Figure 1.

Contemplated hybrid nanofluid containing nanoparticles (Ag + TiO₂) and Water (H₂O) as base fluid drop showing its constituents.

Formulation

Consider incompressible, laminar and steady boundary layer flow of hybrid nanofluid accompanied by natural convection which is evolved at the external surface of an extending and radiating cylinder. The assumption of steady flow enables the definition of a streamline as the path traced by a fluid particle moving in the flow field, from which it follows that a streamline is a line in the flow that is everywhere tangent to the local velocity field. The incompressible flow means that the density is assumed to be constant which means that the equations containing derivatives of density are zero. The other major simplification is that the number of equations to be solved is reduced. If the density is constant, then there cannot be large variations in temperature, and the temperature may be assumed to be constant as well.

Water $(H_{2} O)$ is considered as base fluid dispersed with silver $(Ag)$ and titanium dioxide $(T i_{2} O_{3})$ nanoparticles. The cylinder radius is taken as $a > 0$ (constant), while assumed to be extending with a velocity $U_{w} (x) = \frac{u_{o} x}{L}$ linearly. The Darcy-Forchheimer concept of the medium is contemplated. Moreover, the nonlinear radiative flux and chemical reaction is appraised. The flow is allowed horizontally along $x -$ direction whereas the vertical domain is marked as $a \leq r \leq \infty$ (as shown in Figure 2). The equations of the flow field model are stated in accordance with existing literature.^{22,30,31,42,43}

\frac{\partial (rv)}{\partial r} + \frac{\partial (ru)}{\partial x} = 0,

(1)

\begin{matrix} [u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial r}] = μ_{hnf} \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial u}{\partial r}) - \frac{u μ_{hnf}}{k^{*}} - \frac{C_{b}}{\sqrt{k^{*}}} u^{2} \\ + g [(T - T_{\infty}) β_{hnf} + {(T - T_{\infty})}^{2} β_{hnf}^{2}] \end{matrix}

(2)

[u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial r}] = \frac{k_{hnf}}{{(ρ cp)}_{hnf}} \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial T}{\partial r}) + \frac{16}{3 {(ρ cp)}_{hnf}} (\frac{σ^{*} T_{\infty}^{3}}{k^{*}} \frac{\partial^{2} T}{\partial r^{2}}),

(3)

while the necessary Dirichlet conditions are:

\begin{matrix} at r = a, u = U_{w}, T = T_{w}, v = 0, \\ as r \to \infty, u \to 0, T \to T_{\infty}, \end{matrix}

(4)

Figure 2.

Flow geometry.

In the governing model; equation (1) is continuity, equation (2) is momentum, equation (3) is energy, and equation (4) are boundary conditions. $(u, v)$ are the velocity constituents along $(x, r)$ axes, $T$ signifies the absolute temperature, and $ρ_{hnf}$ denotes the non-varying hybrid nanofluid density. Besides, $μ_{hnf}$ , $k_{hnf}$ , $C_{p}$ , $(ρ C_{p})_{hnf}$ , and $β_{hnf}$ characterizes the hybrid nanofluid dynamic viscosity, thermal conductivity, specific heat, specific heat capacity, and thermal expansion coefficient accordingly. In all the above description, note that the subscripts “ $hnf$ ” is used for representing relative attributes of hybrid nanofluid. With the help of similarity variables,

\begin{matrix} η = (\frac{r^{2} - a^{2}}{2 a}) \sqrt{\frac{u_{0}}{υ L}}, u = \frac{u_{0} x}{L} f' (η), ψ (η) = - \frac{a}{r} \sqrt{\frac{u_{0} υ x^{2}}{L}} f (η), \\ v = - \frac{a}{r} \sqrt{\frac{u_{0} v}{L}} f (η), Θ (η) = (T - T_{\infty}) {(T_{w} - T_{\infty})}^{- 1}, \end{matrix}

(5)

In equation (5), $ψ (η)$ denote stream function while $f' (η)$ and $Θ (η)$ characterizes the dimensionless velocity and temperature, respectively.

Equations (1)–(4) transforms to dimensionless form as:

\begin{matrix} \begin{matrix} (2 γ η + 1) f ″' + 2 γ f ″ - \frac{μ_{f}}{μ_{hnf}} \frac{ρ_{hnf}}{ρ_{f}} ((1 + Fr) {f'}^{2} - ff ″) \\ + Gr Θ \frac{μ_{f}}{μ_{hnf}} \frac{β_{hnf}}{β_{f}} + G r^{*} Θ^{2} \frac{μ_{f}}{μ_{hnf}} {(\frac{β_{hnf}}{β_{f}})}^{2} - krf' = 0, \end{matrix} \end{matrix}

(6)

\begin{matrix} [\frac{k_{hnf}}{k_{bf}} + \frac{4 Rd}{3} (\frac{d}{d η} {[Θ (θ_{w} + 1) + 1]}^{3})] [((2 γ η + 1) Θ ″ + 2 γ Θ')] \\ + \frac{{(ρ cp)}_{hnf}}{{(ρ cp)}_{f}} \Pr f Θ', \end{matrix}

(7)

and

\begin{matrix} f' (η) = Θ (η) = 1, f (0) = 0, when η \to 0 \\ f' (η) = Θ (η) \to 0, when η \to \infty \end{matrix}

(8)

Further,

\begin{matrix} γ = \sqrt{\frac{υ L}{u_{0} a^{2}}}, Gr = \frac{g β T_{\infty} (θ_{w} - 1) L^{3}}{υ_{f}^{2}}, G r^{*} = \frac{g {[β T_{\infty} (θ_{w} - 1)]}^{2} L^{3}}{υ_{f}^{2}}, \\ \Pr = \frac{μ cp}{k}, Fr = \frac{C_{b} L}{\sqrt{k^{*}}}, Rd = 4 \frac{σ^{*} T_{\infty}^{3}}{k^{*} k}, θ_{w} = \frac{T_{w}}{T_{\infty}}, \end{matrix}

(9)

Here, the non-dimensional parameters obtained are $Gr, G r^{*}$ local Grashof numbers, $\Pr$ Prandtl number, $Fr$ local inertia coefficient, $Rr$ radiation parameter, and $θ_{w}$ temperature ratio parameter.

Through the assistance of Refs,^3,28,29 the related thermal and physical characteristic of hybrid nanofluids relative to base fluids are, hereby, stated in equations (10)–(14) sequentially.

\frac{μ_{hnf}}{μ_{f}} = (1 - ϕ_{Ag})^{- 5 / 2} (1 - ϕ_{Ti O_{2}})^{- 5 / 2},

(10)

\frac{ρ_{hnf}}{ρ_{f}} = (1 - ϕ_{Ag}) - (1 - ϕ_{Ag}) (ϕ_{TiO} - ϕ_{TiO} \frac{ρ_{Ti O_{2}}}{ρ_{f}}) + \frac{ρ_{Ag}}{ρ_{f}} ϕ_{Ag},

(11)

\begin{matrix} \frac{{(ρ C_{p})}_{hnf}}{{(ρ C_{p})}_{f}} = (1 - ϕ_{Ag}) - (1 - ϕ_{Ag}) (ϕ_{Ti O_{2}} - ϕ_{Ti O_{2}} \frac{{(ρ C_{p})}_{Ti O_{2}}}{{(ρ C_{p})}_{f}}) \\ + \frac{{(ρ C_{p})}_{Ag}}{{(ρ C_{p})}_{f}} ϕ_{Ag}, \end{matrix}

(12)

\begin{matrix} \frac{{(ρ β)}_{hnf}}{{(ρ β)}_{f}} = (1 - ϕ_{Ag}) - (1 - ϕ_{Ag}) (ϕ_{Ti O_{2}} - ϕ_{Ti O_{2}} \frac{{(ρ β)}_{Ti O_{2}}}{{(ρ β)}_{f}}) \\ + \frac{{(ρ β)}_{Ag}}{{(ρ β)}_{f}} ϕ_{Ag}, \end{matrix}

(13)

\begin{matrix} \frac{k_{hnf}}{k_{nf}} = (\frac{k_{Ti O_{2}} - 2 (k_{nf} - k_{Ti O_{2}}) ϕ_{A l_{2} O_{3}} + 2 k_{nf}}{k_{Ti O_{2}} + ϕ_{Ti O_{2}} (k_{nf} - k_{Ti O_{2}}) + 2 k_{nf}}), \frac{k_{nf}}{k_{f}} \\ = (\frac{k_{Ag} - 2 (k_{f} - k_{Ag}) ϕ_{Ag} + 2 k_{f}}{k_{Ag} + (k_{f} - k_{Ag}) ϕ_{Ag} + 2 k_{f}}), \end{matrix}

(14)

The physical quantities of prime concern are skin friction coefficient $C_{f}$ and Nusselt number $N u_{x}$ that are accordingly defined so (see Prasannakumara and Gowda³¹ and Zeeshan et al.⁴⁰):

C_{f} = \frac{τ_{w}}{ρ_{f} {U_{\infty}}^{2}}, N u_{r} = \frac{q_{w} - q_{r}}{k_{f} T_{\infty} (θ_{w} - 1)},

(15)

where $τ_{w} = μ_{hnf} {(\frac{\partial u}{\partial r})}_{r = a}$ , $q_{w} = - k_{hnf} {(\frac{\partial T}{\partial r})}_{r = a}$ , and $q_{r} = \frac{4}{3} Rd (\frac{d}{d η} {[Θ (θ_{w} + 1) + 1]}^{3})$ are the surface shear stress, heat flux, and thermal radiative flux, sequentially. Using equation (5), equation (3) revises so (see Prasannakumara and Gowda³¹ and Zeeshan et al.⁴⁰):

\begin{matrix} {Re}^{0.5} C_{f} = \frac{μ_{hnf}}{μ_{f}} f ″ (0), {Re}^{- 0.5} N u_{r} \\ = - [\frac{k_{hnf}}{k_{f}} + \frac{4}{3} Rd (\frac{d}{d η} {[Θ (θ_{w} + 1) + 1]}^{3})] Θ' (0), \end{matrix}

(16)

where ${Re}_{r} = \frac{U_{w} x}{υ_{f}}$ is the local Reynolds number.

Solution methodology

In order to better understand the flow phenomena, the non-linear model, consisting of coupled equations (6)–(8) and subjected to substantial constraints in equation (8), is solved numerically. For this purpose, the fourth order Runge–Kutta method (RK4) technique combined with the shooting method is adopted. RK4 is a family of classical Runge–Kutta approaches which are among the most effective and widely used iterative methods (implicit and explicit) for solving the initial value problem (IVP) of ordinary differential equations. One of the most significant advantages of Runge-Kutta formulae is that it requires the function’s values at some specified points. For the ambient boundary, it is necessary to select a suitable finite value such as $η < \infty$ representing $η \to \infty$ and a precise step-size $h$ . It is noteworthy that RK4 method is only applicable to first order ODEs. The fourth-order formula is:

h_{1} = k f (x_{n}, y_{n}),

(17)

h_{2} = k f (x_{n} + \frac{1}{2} k, y_{n} + \frac{1}{2} h_{1}),

(18)

h_{3} = k f (x_{n} + \frac{1}{2} k, y_{n} + \frac{1}{2} h_{2}),

(19)

h_{4} = k f (x_{n} + k, y_{n} + h_{3}),

(20)

Equations (17)–(20) are coded in MATLAB in the form of a matrix to solve numerically. Also, a conversion is made for $C_{f}$ and $N u_{r}$ accordingly. At the end, a specified level of accuracy is achieved with RK4 repeated iteratively.

Discussion

This section reflects the influences regarding fluxes in dimensionless prospective of velocity $(f' (η))$ , temperature $(Θ (η))$ , skin friction coefficient $({Re}^{0.5} C_{f})$ , and Nusselt number $({Re}^{- 0.5} N u_{r})$ , which are of substantial interest as opposed to variations in significant parameters. Besides, the efficiency of base fluid is checked for mono and bi-hybrid nanoparticles as well. The performances of $Ti O_{2}$ and $Ti O_{2} + Ag$ are comparatively observed in regard to the velocity and temperature. On the other hand, a thorough discussion of skin friction and Nusselt number corresponding to sundry flow parameters has been conducted for $Ag$ and $Ag + Ti O_{2}$ . All the relative outcomes are encapsulated in graphs and tables.

Depiction of velocity profiles

Figures 3 to 7 represent the visible upshots associated with velocity field $(f' (η))$ corresponding to distinct flow parameters for the present model. The performance of flow rate against the increasing values of Farchemmier’s parameter is depicted in Figure 3. It is noticed that for higher $Fr$ , the fluid flow movement takes a downward trend because of the high range of mass transfer. Figure 4 gives the variation in velocity profile under the impact of increasing measures of $kr$ , where decreasing patterns in velocity are observed. Physically, this happens since kinematic viscosity intensifies for larger $kr$ . Figure 5 provides an overview of the favorable results upon $f' (η)$ against the rising Grashof number $Gr$ . These effects can be explained by the interaction between buoyancy force and resistance force which occurs as a result of heat and mass transfer on a solid surface, as result recommending enhanced convection and improved velocity profiles. Besides, from Figure 6, certain increments in flow field are depicted based on increasing order of $G r^{*}$ . Due to the closed relationship between the nonlinear Grashof number and the free surface, this effect is more reliable. Moreover, the ramifications associated with $f' (η)$ due to enlarging measures of $β$ is presented in Figure 7. In physics, the fluid flow velocity is highly influenced by the role of $β$ . In fact, the increase in the amount of the non-Newtonian effect decreases the motion of the fluid because of the heightened viscosity effect. Lastly, it is the most crucial and paramount observations from Figures 3 to 7 that all the above-mentioned impacts caused by $kr$ , $Gr$ , $G r^{*}$ , and $β$ are a bit more intense for the case of $Ti O_{2} + Ag + H_{2} O$ rather than $Ti O_{2}$ contrary to the influence of $Fr$ , where the adverse upshots on $Ti O_{2} + H_{2} O$ are comparatively more significant. However, the results because of parameters $kr$ , $Gr$ , $G r^{*}$ , and $β$ are not so influential as a whole.

Figure 3.

Significant impressions of local inertia coefficient $Fr$ on velocity profile $f' (η)$ .

Figure 4.

Significant impressions of porosity factor $kr$ on velocity profile $f' (η)$ .

Figure 5.

Significant impressions of Grashof number $Gr$ on velocity profile $f' (η)$ .

Figure 6.

Significant impressions of Grashof number $G r^{*}$ on velocity profile $f' (η)$ .

Figure 7.

Significant impressions of thermos expansion coefficient $β$ on velocity profile $f' (η)$ .

Depiction of temperature profiles

Figures 8 and 9 are the performance of distinguishing fluid parameters on a temperature profile for the suggested flow model. The direct relation of radiation Rd with temperature profile is shown in Figure 8. It has been examined that for larger radiation parameter, temperature of the proposed fluid is raised due to increase in heat of fluid. The solid volume fraction of the aforementioned nanomaterials $ϕ_{1}$ , $ϕ_{2}$ is illustrated in Figure 9. Graph indicated that for increased amount of $ϕ_{1}$ , $ϕ_{2}$ , behavior of temperature profile is enhanced. It has been found that the flexible quantity of volume fraction can also improve the thermal characteristics of the traditional liquid and thus improve the temperature. Despite all the-shown impacts, as a final note, it is highlighted in Figures 8 and 9 that the impacts caused by $ϕ_{1}$ , $ϕ_{2}$ seem to be more intense over the influence of $Rd$ .

Figure 8.

Significant impressions of Radian parameter $Rd$ on temperature profile $Θ (η)$ .

Figure 9.

Significant impressions of solid volume fraction of nanoparticles $ϕ_{1}, ϕ_{2}$ on temperature profile $Θ (η)$ .

Depiction of skin friction coefficient and Nusselt number

Figures 10 and 11 show skin friction rate in correspondance with different values of thermo-Grashof numbers $Gr$ and $G r^{*}$ , whereas Figures 12 and 13 present the essential roles of $θ_{w}$ and $Rd$ upon heat transfer rate, accordingly. From Figures 10 and 11, it is clear that at high ranges of considered parameters, the skin friction coefficient is simultaneously decreasing for either of $Ag + H_{2} O$ and $Ag + Ti O_{2} + H_{2} O$ nanoliquids by the virtue of the relative ratio between the buoyancy force and the viscous force appearing on a liquid flow. In spite of that, Figures 12 and 13 imply that there is an increasing relationship between Nusselt number (or heat transfer rate) and pondered parameters ( $θ_{w}$ and $Rd$ ) and for both types of nanoliquids. However, $Ag + Ti O_{2} + H_{2} O$ is showing higher and improved thermal conductance than $Ag + H_{2} O$ . This statement means that hybrid nanocomposites escalates the heat transfer phenomenon in comparison to mono nanomaterials in any fluid system, thus can be emphasized for rapid heat transfer and cooling processes. It is noteworthy that an increment in $ϕ_{1}$ , $ϕ_{2}$ leads to improving conduct in rate of heat transfer. This fact is illustrated through Figure 14 that shows the percentage-based enhancement in ${Re}^{- 0.5} N u_{r}$ versus distinct values of nanomaterial’s solid volume fraction $ϕ_{1}$ and $ϕ_{2}$ . In case of $Ag$ , the data framed in Figure 14 states refinements about 3.71%, 5.87%, and 11.27% relative to $ϕ_{1}, ϕ_{2} = 0.01$ , $0.02$ , and $0.03$ , accordingly. Comparatively, regarding the same preceding values of $ϕ_{1}$ and $ϕ_{2}$ , further augmentations were reported in percent-wise recordings for the situation of $Ag + Ti O_{2}$ , which are listed as 4.32%, 7.62%, and 14.46%, respectively. Moreover, all the graphs in Figures 3 to 13 are existing in perfect agreement. Furthermore, the obtained results are validated through comparing the reduced skin friction $(f ″ (0))$ and Nusselt number $(- Θ (0))$ with the published work⁴¹ and displayed in Tables 1 and 2.

Figure 10.

Significant impressions of Grashof number $Gr$ on skin friction rate.

Figure 11.

Significant impressions of Grashof number $G r^{*}$ on skin friction rate.

Figure 12.

Significant impressions of temperature ratio parameter $θ_{w}$ on heat transfer rate.

Figure 13.

Significant impressions of Radiation parameter $Rd$ on heat transfer rate.

Figure 14.

Percentage-wise variations in heat transfer rate against solid volume fraction of nanoparticles $ϕ_{1}, ϕ_{2}$ .

Table 1.

The validation of reduced skin friction $f ″ (0)$ against $Gr$ and $G r^{*}$ while the rest of parameters are kept common.

		Zeeshan et al.⁴⁰	[Present]
		$f ″ (0)$	$f ″ (0)$
0.1	0.1	0.817049	0.813017
0.2		0.818277	0.814150
	0.2	0.818395	0.815940
	0.3	0.819914	0.817830

Table 2.

The validation of reduced Nusselt number $- Θ (0)$ against $Rd$ while the rest of parameters are kept common.

	Zeeshan et al.⁴⁰	[Present]
	$- Θ (0)$	$- Θ (0)$
0.1	3.608447	3.60636
0.3	3.625493	3.62461
0.5	3.64108	3.640445
0.7	3.66637	3.66255

Concluding remarks

An area of current research focuses on the Darcy-Forchheimer flow model for impingement normally oriented toward porous extendable tubes with silver and titanium dioxide. On the behalf of above-mentioned discussions and investigations, the conclusions end with the following key considerations:

A negative impact upon the velocity field is recorded due to increased values of local inertia coefficient, porosity parameter, and thermal expansion coefficient. However, the positive effect is achieved for velocity due to the higher number of thermo Grashof numbers, respectively.

The skin friction coefficient boosts up with increasing values of the porosity parameter and local inertia coefficient.

Contrarily, the surface frictional sources are reduced by Grashof numbers while affecting the fluids motion productively.

According to the percentage-wise enhancement in heat transfer rate, hybrid nanofluids are more effective at improving heat transfer mechanisms.

In view of all the discussions, hybrid nanofluids can generate required cooling efficiency of moving pipe conduits in industrial and manufacturing factories.

Moreover, hybrid nanofluid is hoped to have productive selection cooling-efficiency and cost-friendly, and thus can be considered as a best anti-freeze agent for automobiles in the future.

Footnotes

Handling Editor: Chenhui Liang

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Taza Gul

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