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Abstract

This article explores heat transfer characteristics of magnetohydrodynamics water-base silver (Ag) and iron oxide (Fe₃O₄) hybrid nanomaterials flow in a Darcy-Forchheimer porous medium induced by a stretching/shrinking surface with impacts of heat sink/source. Moreover, thermal radiation effects and the slip boundary conditions are also incorporated in the given problem. Governing partial differential equations (PDEs) are first altered into the ordinary differential equations (ODEs) using suitable similarity transformations. These achieved ODEs are solved by the well-known shooting technique in Maple software to get the required numerical solutions for the variation in different physical parameters. Here, the numerical findings show duality in solutions in case of stretching/shrinking parameter over different ranges of the comprised distinguished parameters. In this regard, the stability analysis is done and the first solution is found stable and physically acceptable, while the second one unstable and physically infeasible. Besides, the skin friction increases for the case of shrinking but it decreases for case of stretching parameter due to the greater impacts of the mass transfer parameter while the heat transfer phenomenon upsurges for the case of shrinking parameter. Moreover, the skin friction, and the heat transfer rise with variation of the suction parameters when the quantity of solid nanoparticles volume fraction is increased.

Keywords

Magnetohydrodynamics (MHD)hybrid nanofluid porous medium Darcy-Forchheimer slip effects

Introduction

Over the last few decades, the nanoparticles (NPs) have garnered a lot of attention owing to their superparamagnetic properties, biocompatibility, and lack of toxicity. Recent developments in the biological generation of iron oxide NPs using environmentally friendly techniques have greatly enhanced both their quality and biological applications. On the other hand, the recent changes in nano-technology are also motivating the technical communities toward new innovative directions in all sectors of engineering and modern industries. Similarly, heat removal from the surfaces plays an important role of the several processes in industries such as heat transfer management in the pharmaceutical industries, cooling of different devices in the electromechanical sectors and treating the temperature in power plants. The thermal conductivity of the matter is one of the compelling factor for achieving anticipated heat removals rate. Traditional fluids, such as oils, glycine, water and common polymeric fluids possess poor thermal conductivity compared to solid particles. However, a homogeneous mixture of nanoparticles contains two liquids known as nanofluids. Nanoparticles operating in nanofluids are composed of carbon nanotubes, oxides, metals, and carbides. The base fluid is made up of liquid oil, ethylene glycol, and water. The nanofluid was first proposed by Choi and Eastman.¹ Nanofluids have applications in microelectronics, microfluidics, transportation, biomedical, X-rays, material processing, and scientific measurement. Buongiorno proposed an analytical model for convective transport in nanofluids, which incorporate the effects of Brownian diffusion and thermophoresis. These nanoparticles also hold a lot of significance in the areas of biological and medical applications. Some nanoparticles can bind many drugs, proteins, and target cancer cells. Since many nanoparticles have high atomic numbers that can produce heat, they lead to the treatment of tumor-selective photo thermal therapy. Most of the nanoparticles can cure and help in targeting the deadly cancer cells. Flow through porous medium with nanoparticles has significant applications in biomedical science (such as drug delivery and cancer treatment to treat radiotherapy) and chemical engineering.² Many researchers have worked on nanofluids from different points of views on different surfaces.^3–5

Hybrid Nanofluids (HNF) are the most updated version of the nanofluids, where more than one types of nano-sized particles are dispersed into the pure working base fluid. Sulochana and Aparna⁶ have explored how unsteady flow of hybrid liquids on an enlarging surface is affected by Brownian motion and thermophoresis. They concluded that the heat transfer rate decreases due to the influence of Brownian motion and thermophoresis. Devi and Devi⁷ examined the Cu-Al₂O₃/H₂O HNF flow over a permeable stretching surface. That investigation indicates that the volumetric fractions of nano-sized particles are essential to achieve the desired capability in the heat exchange rates. The thermo-physical features of EO-TC4/Ni mixtures along with the radiative effects have been analyzed.⁸ Magnetohydrodynamics is an important field of the fluid mechanics, which deals with the magnetic features of electrical conduction in fluids. Aladdin et al.⁹ scrutinize the magnetic field and suction influence on a stretching surface along with the aluminum-copper nanoparticles. In this study, duality in solution is observed owing to the presence of suction parameter, and better enhancement of transfer of heat is observed.

The magnetohydrodynamics (MHD) is concerned with physical and mathematical scaffold, which shows magnetic dynamics in the electrical conducting fluid. The applications of magnetohydrodynamics are especially useful in the modern industrial and engineering areas, such as the drawing of the plastic wires and films, polymer extrusion in the melt spinning process, crystal growth, paper production, glass fiber manufacturing, fluid film condensation processes, food production, electronic chips, electrochemical processes, flow through the filtering devices, and thermal energy storage. The incorporation of nanoparticles in MHD fluids has a wide range of applications in medical sciences,¹⁰ materials processing,¹¹ and various other industries. The magnetic and electrical fields are very essential for controlling fluid flow, which is the most important need in nanofluids flow problems. Anandakumar and Umamaheswari¹² conducted a study on laminar forced flow of Fe₃O₄ with magnetic effects, demonstrating that the magnetic field enhances heat transfer rates in fluids. Krishna et al.,¹³ investigated radiative MHD flow of Casson hybrid nanofluid over an infinite exponentially accelerated vertical porous surface. Many other researchers have also explored MHD flow in this context. Moreover, the influence of the magnetic parameter on nanofluid flows has been a subject of observation by many researchers.^13–17

The Darcy law, which establishes a proportional relationship between pressure gradients and velocity, may not be suitable for problems involving fluids with high-velocity flow. However, when dealing with models featuring higher velocities, several advanced fluid flow issues emerge. Consequently, Forchheimer’s nonlinear terms can be introduced to investigate fluid flow, particularly in permeable media along with higher Reynolds numbers. A study conducted by Ghadikolaei et al.¹⁸ explored the flow of HNF over a curved, stretched surface, taking into account radiation and the influence of a constant temperature source. The findings revealed that the blade-shaped nanoparticles show greater enhancement in temperature as compared to brick-shaped nanoparticles. Another investigation, carried out by Shaiq et al.,¹⁹ delved into three-dimensional flow over a vertically stretching surface using a 50-50 mixture of ethylene glycol and water. Their results indicated that the shape factor and radiation enhance the heat transfer rate, with a more pronounced improvement observed in the case of hybrid nanofluids. Ghadikolaei and Gholinia²⁰ considered the temperature base viscosity model to examine the heat enactment of SiO₂ and MoS₂ nanoparticle along the base fluid ethylene glycol. In the realm of non-Darcy flow, Chamkha,²¹ investigated hydromagnetic free convection from a cone and a wedge in porous media, while Chamkha,²² delved into non-Darcy fully developed mixed convection in a porous medium channel with heat generation/absorption and hydromagnetic effects.

Motivated by the above literature, the current examination explores model development and magnetohydrodynamics radiative flow in the case of Darcy-Forchheimer porous medium in the presence of hybrid nanofluid past a stretching/shrinking surface with thermal radiation. In addition, heat absorption/generation and the slip conditions at the boundary are also incorporated into the study. Iron oxide (Fe₃O₄) and silver (Ag) nanoparticles combine with the base (water) fluid to form the proposed hybrid nanofluid. Dual (first and second) branch outcomes are also calculated against the specific range of suction as well as the stretching/shrinking parameters. The stability analysis is also conducted to forecast stable solutions as time evolves. According to the authors’ knowledge, no one has yet investigated the aforementioned problem.

Mathematical formulation

A two-dimensional (2D) incompressible steady boundary layer flow and heat transfer characteristic of $Ag + F e_{3} O_{4}$ /water based HNF past on stretching/shrinking surface is investigated. The flow model is portrayed in Figure 1.

Figure 1.

The flow model and coordinate system: (a) stretching surface (λ > 0) and (b) shrinking surface (λ < 0).

Moreover, the porous space of Darcy-Forchheimer is assumed. Thermal radiation, thermal slip and velocity slip, heat sink/source parameters are also considered in energy equation. A uniform magnetic field with $B_{0}$ strength is used to transverse to linearly stretched surface, invoking a Lorentz magneting body force with plane surface. Furthermore, $T_{w}$ and $T_{\infty}$ represent wall’s temperature and the free stream, respectively. Using the above assumptions and Alzahrani et al.,²³ the proposed problem can written as:

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

(1)

u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \frac{μ_{hnf}}{ρ_{hnf}} \frac{\partial^{2} u}{\partial y^{2}} - \frac{1}{ρ_{hnf}} (\frac{μ_{hnf}}{K^{*}} + σ_{hnf} B_{0}^{2}) u - \frac{F}{ρ_{hnf}} u^{2},

(2)

\begin{matrix} u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \frac{k_{hnf}}{{(ρ c_{p})}_{hnf}} \frac{\partial^{2} T}{\partial y^{2}} + \frac{16 σ^{*} T_{\infty}^{3}}{3 {(ρ c_{p})}_{hnf} k^{*}} \frac{\partial^{2} u}{\partial y^{2}} \\ + \frac{Q_{0}}{{(ρ c_{p})}_{hnf}} (T - T_{\infty}) . \end{matrix}

(3)

The boundary conditions are

{\begin{matrix} v = v_{w} (x), u = λ u_{w} (x) + A \frac{\partial u}{\partial y}, T = T_{w} + B \frac{\partial T}{\partial y} at y = 0 \\ u \to 0, T \to T_{\infty}, as y \to \infty . \end{matrix}

(4)

Where, $u$ , $v$ , are the respective components of velocity in $x$ and $y$ directions. $K$ is permeability of porous medium, $F = C_{b} / x K^{1 / 2}$ is variable coefficient of the inertia, $B_{0}$ is the magnetic field’s strength, $Q_{0}$ is heat rate for source/sink. Further, $σ_{hnf}$ is the electrical conductivity of HNF, while $P$ represents the improved pressure accounting for the centrifugal force term. Furthermore, $μ_{hnf}, k_{hnf}, {(ρ c_{p})}_{hnf}$ and $, ρ_{hnf}$ are the corresponding dynamic viscosity, thermal conductivity, heat capacity, and density of HNF. Here, the parameter $λ$ represents the type of surface, where the stretching surface is represented by $λ > 0$ , shrinking by $λ < 0$ , and static by $λ = 0$ . The permeability in porous medium is denoted by $K^{*}$ . In addition, the subscript $hnf$ denotes the thermophilic characteristics of HNF. Thermophysical characteristics of HNF comprising heat capacity, density, thermal conductivity, and dynamic viscosity are reported in Zainal et al.²⁴

{\begin{matrix} μ_{h n f} = μ_{f} {(1 - ϕ_{2})}^{- 2.5} {(1 - ϕ_{1})}^{- 2.5}, ρ_{h n f} = [(1 - ϕ_{1}) ρ_{f} + ϕ_{1} ρ_{1}] (1 - ϕ_{2}) + ϕ_{2} ρ_{2} \\ k_{h n f} = \frac{2 k_{n f} - 2 ϕ_{2} (k_{n f} - k_{2}) + k_{2}}{2 k_{n f} + ϕ_{2} (k_{n f} - k_{2}) + k_{2}} \times (k_{n f}), where k_{n f} = \frac{2 k_{f} - 2 ϕ_{1} (k_{f} - k_{1}) + k_{1}}{2 k_{f} + ϕ_{1} (k_{f} - k_{1}) + k_{1}} \times (k_{f}) \\ {(ρ c_{p})}_{h n f} = [(1 - ϕ_{1}) {(ρ c_{p})}_{f} + ϕ_{1} {(ρ c_{p})}_{1}] (1 - ϕ_{2}) + ϕ_{2} {(ρ c_{p})}_{2} \\ σ_{h n f} = \frac{σ_{2} + 2 σ_{n f} - 2 ϕ_{2} (σ_{n f} - σ_{2})}{σ_{2} + 2 σ_{n f} + ϕ_{2} (σ_{n f} - σ_{2})} \times (σ_{n f}), w h e r e, σ_{n f} = \frac{σ_{1} + 2 σ_{f} - 2 ϕ_{1} (σ_{f} - σ_{1})}{σ_{1} + 2 σ_{f} + ϕ_{1} (σ_{f} - σ_{1})} \times (σ_{f}) . \end{matrix}

Here, $φ_{1}$ corresponds to volume fraction for nanomaterials while Ag and Fe₃O₄ nanoparticles are represented by $φ_{2}$ in the above expression.

To convert the equations (2)–(4) following similarity transformations²¹ are considered here.

{\begin{matrix} u = cxf' (η), v = - \sqrt{c ϑ} f (η), η = y \sqrt{\frac{c}{ϑ_{f}}} \\ θ (η) = (T - T_{\infty}) / (T_{w} - T_{\infty}) . \end{matrix}

(5)

By utilizing equation (5) in equations (2)–(4), it is obtained:

\begin{matrix} \frac{μ_{hnf} / μ_{f}}{ρ_{hnf} / ρ_{f}} f^{″'} - \frac{μ_{hnf} / μ_{f}}{ρ_{hnf} / ρ_{f}} K f^{'} + [f f^{″} - (1 + Fr) f^{' 2}] - \frac{σ_{hnf} / σ_{f}}{ρ_{hnf} / ρ_{f}} \\ M f^{'} = 0, \end{matrix}

(6)

\begin{matrix} \frac{1}{\Pr {(ρ c_{p})}_{hnf} / {(ρ c_{p})}_{f}} (k_{hnf} / k_{f} + \frac{4}{3} R) θ ″ + θ' f \\ + \frac{Q}{{(ρ c_{p})}_{hnf} / {(ρ c_{p})}_{f}} θ^{'} = 0 . \end{matrix}

(7)

The boundary condition are:

{\begin{matrix} f (0) = S, f' (0) = δ f'' (0) + λ, θ (0) = δ_{T} θ' (0) + 1 \\ f' (η) \to 0, θ (η) \to 0, as η \to \infty . \end{matrix}

(8)

Here, $F r = \frac{C_{b}}{K^{1 / 2}}$ is Forchheimer number, $Rd = \frac{4 σ^{*} T_{\infty}^{3}}{3 k k^{*}}$ is radiation factor, $M = \frac{σ B_{0}^{2}}{c ρ_{f}}$ is magnetic parameter, $\Pr = \frac{ϑ}{α}$ is Prandtl number, $Q = \frac{Q_{0}}{c (ρ C_{P})}$ is heat source/sink factor. Moreover, $S = - \frac{v_{0}}{\sqrt{c ϑ}}$ is a parameter of suction ( $S > 0$ ), $δ = A \sqrt{\frac{c}{ϑ}}$ is velocity slip, and $δ_{T} = B \sqrt{\frac{c}{ϑ}}$ is the thermal slip. The velocity sheet in $x$ direction is $u_{w} (x) = cx$ . Mass flux of velocity is $v_{w} (x) = - S \sqrt{c ϑ_{f}}$ .

Key engineering quantities of interest

The physical quantities of significant importance are skin friction coefficient $(C f_{x})$ and Nusselt number $(N u_{x})$ that are written as:

C f_{x} = \frac{τ_{w}}{\frac{1}{2} ρ_{f} {(u_{w})}^{2}} and N u_{x} = \frac{x q_{w}}{k_{f} (T_{w} - T_{\infty})} .

(9)

Here $τ_{w}$ and $q_{w}$ are respectively shear stress, and heat exchange (flux) at surface. Using equation (5) in equation (9), it is obtained:

\begin{matrix} C f_{x} {(R e_{x})}^{0.5} = \frac{2}{{(1 - ϕ_{2})}^{2.5} {(1 - ϕ_{1})}^{2.5}} f'' (0) and N u_{x} {(R e_{x})}^{- 0.5} \\ = - \frac{k_{nf}}{k_{f}} θ' (0) \end{matrix}

(10)

Stability analysis

In case of existing multiple solutions, it is important to assess stability of achieved solutions to examine their feasibility. The stability of solutions here is assessed using a procedure suggested by Weidman et al.²⁵ and Merkin.²⁶ According to this procedure, the governing equations (2) and (4) are converted into unsteady form as:

\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \frac{μ_{hnf}}{ρ_{hnf}} \frac{\partial^{2} u}{\partial y^{2}} - \frac{1}{ρ_{hnf}} (\frac{μ_{hnf}}{K^{*}} + σ_{hnf} B_{0}^{2}) u - \frac{F}{ρ_{hnf}} u^{2},

(11)

\begin{matrix} \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \frac{k_{hnf}}{{(ρ c_{p})}_{hnf}} \frac{\partial^{2} T}{\partial y^{2}} + \frac{16 σ^{*} T_{\infty}^{3}}{3 {(ρ c_{p})}_{hnf} k^{*}} \frac{\partial^{2} u}{\partial y^{2}} \\ + \frac{Q_{0}}{{(ρ c_{p})}_{hnf}} (T - T_{\infty}) . \end{matrix}

(12)

Here, $t$ denotes time. By introducing non-dimensional time dependent variable τ, the similarity transformation (5) for unsteady-state equations (11) and (12) is written as:

{\begin{matrix} u = ax f' (η, τ), v = - \sqrt{a ϑ} f (η, τ), η = y \sqrt{\frac{a}{ϑ}}, \\ θ (η, τ) = (T - T_{\infty}) / (T_{w} - T_{\infty}), τ = at . \end{matrix}

(13)

By using equation (13) in equations (11) and (12), it is obtained:

\begin{matrix} \frac{μ_{hnf} / μ_{f}}{ρ_{hnf} / ρ_{f}} \frac{\partial^{3} f (η, τ)}{\partial η^{3}} - \frac{μ_{hnf} / μ_{f}}{ρ_{hnf} / ρ_{f}} K \frac{\partial f (η, τ)}{\partial η} + f (η, τ) \frac{\partial^{2} f (η, τ)}{\partial η^{2}} \\ - (1 + Fr) {(\frac{\partial f (η, τ)}{\partial η})}^{2} - \frac{σ_{hnf} / σ_{f}}{ρ_{hnf} / ρ_{f}} M \frac{\partial f (η, τ)}{\partial η} - \frac{\partial^{2} f (η, τ)}{\partial τ \partial η} = 0 \end{matrix}

(14)

\begin{matrix} \frac{{(ρ c_{p})}_{f}}{\Pr {(ρ c_{p})}_{hnf}} (k_{hnf} / k_{f} + \frac{4}{3} Rd) \frac{\partial^{2} θ (η, τ)}{\partial η^{2}} + f (η, τ) \frac{\partial θ (η, τ)}{\partial η} \\ + \frac{Q {(ρ c_{p})}_{f}}{{(ρ c_{p})}_{hnf}} \frac{\partial θ (η, τ)}{\partial η} - \frac{\partial θ (η, τ)}{\partial τ} = 0, \end{matrix}

(15)

with boundary conditions:

{\begin{matrix} f (0, τ) = S, \frac{\partial f (0, τ)}{\partial η} = λ + δ \frac{\partial^{2} f (0, τ)}{\partial η^{2}}, θ (0) = 1 + δ_{T} \frac{\partial θ (0, τ)}{\partial η} \\ \frac{\partial f (η, τ)}{\partial η} \to 0, θ (η, τ) \to 0, as η \to \infty . \end{matrix}

(16)

The technique proposed in Weidman et al.²⁵ can be employed to find the stability analysis owing to dual solutions. Thus, the following functions are introduced.

{\begin{matrix} f (η, τ) = f_{0} (η) + e^{- χ τ} F (η, τ) \\ θ (η, τ) = θ_{0} (η) + e^{- χ τ} G (η, τ) . \end{matrix}

(17)

Here $F (η, τ),$ and $G (η, τ)$ respectively are smaller compared to $f_{0} (η)$ and $θ_{0} (η)$ . Here, $χ$ indicates the set of unknown eigenvalues by showing $χ_{1} < χ_{2} < χ_{2} . . . .$ . By utilizing equation (17) in equations (14)–(16), it is obtained:

\begin{matrix} \frac{μ_{hnf} / μ_{f}}{ρ_{hnf} / ρ_{f}} (F_{0} ″' - {KF}_{0}') + f_{0} F_{0} ″ + F_{0} f_{0} ″ - 2 ((1 + Fr)) f_{0}' F_{0}' \\ - \frac{σ_{hnf} / σ_{f}}{ρ_{hnf} / ρ_{f}} {MF}_{0}' + χ F_{0}' = 0 \end{matrix}

(18)

\begin{matrix} \frac{{(ρ c_{p})}_{f}}{\Pr {(ρ c_{p})}_{hnf}} (k_{hnf} / k_{f} + \frac{4}{3} Rd) G_{0} ″ + f_{0} G_{0}' + F_{0} θ_{0}' \\ + \frac{Q {(ρ c_{p})}_{f}}{{(ρ c_{p})}_{hnf}} G_{0} + χ G_{0} = 0 . \end{matrix}

(19)

Subjected to the given constraints:

{\begin{matrix} F_{0} (0) = 0, F_{0}^{' (0)} = δ F_{0}^{'' (0)}, G_{0} (0) = δ_{T} G_{0}' (0) \\ F_{0}' (\infty) \to 0, G_{0} (\infty) \to 0 . \end{matrix}

(20)

In view of Harris et al.,²⁷ possible ranges for eigenvalues might be found by relaxing boundary condition of $F_{0} (η)$ or $G_{0} (η)$ . This study sets $F''_{0} (0) = 1$ and solves (18) and (19) subjected to boundary conditions (20) in order to obtain the acquired eigenvalues.

Numerical methodology

Maple software is used to get the solution of equations (6) and (7) with specified boundary constraints (8) through shooting technique.²⁸ This technique converts BVPs (boundary value problems) into IVPs (initial value problems). The specified problem can be written as:

\begin{matrix} f' = F_{P}, f^{''} = F_{PP} \frac{μ_{hnf} / μ_{f}}{ρ_{hnf} / ρ_{f}} F_{PP}' - \frac{μ_{hnf} / μ_{f}}{ρ_{hnf} / ρ_{f}} K F_{P} \\ + [F F_{PP} - (1 + Fr) {(F_{P})}^{2}] - \frac{σ_{hnf} / σ_{f}}{ρ_{hnf} / ρ_{f}} M F_{P} = 0 \end{matrix}

(21)

\begin{matrix} θ' = θ_{P} \frac{1}{\Pr {(ρ c_{p})}_{hnf} / {(ρ c_{p})}_{f}} (k_{hnf} / k_{f} + \frac{4}{3} R) θ_{P}' + F θ_{P} \\ + \frac{Q}{{(ρ c_{p})}_{hnf} / {(ρ c_{p})}_{f}} θ_{P} = 0, \end{matrix}

(22)

with specified boundary constraints:

{\begin{matrix} F (0) = S, F_{P} (0) = λ + δ α_{2}, F_{PP} (0) = α_{1} \\ θ_{P} = 1 + δ_{T} α_{2}, θ_{P} (0) = α_{2}, \end{matrix}

(23)

where, $α_{1}$ and $α_{2}$ stand for missing initial conditions whose value would be taken carefully that must satisfy the boundary conditions that are mentioned in equation (8). In Maple software by aiding shootlib functions, the results are obtained by shooting techniques as suggested by Meade et al.²⁸

Results and discussion

The shooting technique in the Maple Software is utilized in order to numerically solve the system of equations (6) and (7) along with boundary conditions (8). Jaluria and Torrance²⁹ explain this technique. This technique is broadly used by several researchers to resolve all kinds of linear as well as non-linear equations related to fluid flow, heat transfer, and nanoparticles concentrations problems.^30–32 Duality in solutions have been found for two distinct initial guesses for un-known values of $f^{''} (0)$ and $- θ' (0)$ for similar set of parameters, providing two distinct profiles defining velocity and temperature. Both the profiles completely satisfy the boundary conditions (8). To test the feasibility of achieved solutions, stability analysis is conducted using 3-stage 3a Lobatto formula developed in bvp4c in Matlab software, as done by Refs.^33–35 The stability analysis revels that the first solution is feasible and stable, as indicated by positive eigenvalues, indicating continuity in the solution loop. On the other hand, the second solution is unstable and infeasible, with negative eigenvalues indicating discontinuity in the solutions loop. Therefore, it appears the second solution is extraneous solution, arising due to high nonlinearity in the equations. Consequently, this article only discusses the results of the first solution, while excluding those of the second solution. The Table 1 shows the result of stability. Moreover, to validate the accuracy of obtained results, they are compared with already obtained results^36,37 showing good resemblance between the two, as displayed in Table 2. The thermos-physical values concerned to nanoparticles with base fluid used in this study are kept in Table 3.

Table 1.

The values of $χ$ for achieved dual solutions with change in $S$ and $λ$ .

$S$	$λ$	First solution	Second solution
1.6	0.5	1.6124	−0.1048
1.8	0.5	1.8531	−0.6133
2	0.2	1.3534	−0.7429
2	0.5	1.4657	−0.6405

Table 2.

The relative findings of first solution for the local Nusselt number $θ' (0)$ for various values of $\Pr$ at $ϕ = Rd = Q = δ_{T} = δ = 0$ and $λ = 1$ .

$\Pr$	Khan and Pop³⁶	Afify³⁷	Present
0.07	0.0663	0.0662	0.0663
0.70	0.4539	0.4538	0.4537
2.00	0.9113	0.9113	0.9113
7.00	1.8954	1.8954	1.8954

Table 3.

Thermo-physical properties of Fe₃O₄, Ag nanoparticles, and base fluid $[38]$ .³⁸

	$ρ (kg m^{- 3})$	$C_{p} (k g^{- 1} k^{- 1})$	$k (W m^{- 1} k^{- 1})$	$σ (s m^{- 1})$
Water	997.1	4179	0.6130	5.5 × 10⁻⁶
Fe₃O₄	$5180$	670	80.4	1.12 × 10⁵
Ag	10,500	902	429	6.3 × 10⁷
$\Pr$	6.2

Furthermore, the graphical results indicating the effects of different parameters are presented and discussed below. The effect of parameter $S$ on $f^{''} (0)$ and the Nusselt number $θ' (0)$ , along with the variation of the stretching/shrinking parameter $λ$ , is demonstrated in Figures 2 and 3, respectively. The variation of $λ$ for four distinct values of $S$ shows duality in the solution at different ranges of $λ$ . Both solutions merge at $λ_{c_{0}}, λ_{c_{1}}, λ_{c_{2}}$ , and $λ_{c_{3}}$ for the values of $S = 3, 3.2, 3.4, 3.6$ respectively. Figure 2 suggests that an increment in S increases the skin friction rate for $λ < 0$ (shrinking case) and decreases for $λ > 0$ (stretching case). Meanwhile, increasing suction increases the Nusselt number throughout the flow, whether surface is shrinking or stretching.

Figure 2.

Variation in $f ″$ (0) against λ for varying $S$ .

Figure 3.

Variation in $- θ' (0)$ against λ for varying $S$ .

Similarly, the influence of nanoparticles’ volumetric fractions $ϕ_{hnf}$ $(ϕ_{hnf} = ϕ_{1} + ϕ_{2})$ on $f^{''} (0)$ and $θ' (0)$ , along with the variation of $S$ , is shown in Figures 4 and 5, respectively. The variation of $S$ for four different values of $ϕ_{hnf}$ shows duality in the solution at different ranges of $S$ . Both solutions merge at $S_{c_{0}}, S_{c_{1}}, S_{c_{2}}$ , and $S_{c_{3}}$ for increasing values of $ϕ_{1} = ϕ_{2} = 0, 0.05, 0.1, 0.15,$ respectively. Figures 4 and 5 show that boosting nanoparticles’ volume fraction increases the rate of skin friction and the Nusselt number for both cases of surface, whether the surface is shrinking or stretching throughout the flow. Likewise, the magnetic parameter augments the skin friction and Nusselt number with variations in $S$ , as shown in Figures 6 and 7. Moreover, $S_{c_{0}}, S_{c_{1}}, S_{c_{2}}$ , and $S_{c_{3}}$ represent critical points for $M = 0.5, 1, 1.5, 2$ , respectively, where two solution merge into one another.

Figure 4.

Variation in $f ″$ (0) against $S$ for varying $ϕ_{1} = ϕ_{2}$ .

Figure 5.

Variation in $- θ' (0)$ against S for varying $ϕ_{1} = ϕ_{2}$ .

Figure 6.

Variation in $f ″$ (0) against S for varying M.

Figure 7.

Variation in $- θ' (0)$ against S for varying M.

Figures 8 and 9 are presented to illustrate the comparative results of the skin friction coefficient and Nusselt number as they vary with the volumetric fractions of nanoparticles in a water-based fluid under three different situations. In Figure 8, it is evident that $Ag$ nanoparticles in water exhibit a higher skin friction rate compared to $F e_{3} O_{4}$ in water, as well as a combination of $Ag$ and $F e_{3} O_{4}$ in water. However, $Ag + F e_{3} O_{4}$ nanoparticles in water show the greater rate of skin friction than $F e_{3} O_{4}$ in water. Figure 9 illustrates that $F e_{3} O_{4}$ nanoparticles in water have a higher rate of Nusselt number compared to Ag in water, as well as combination of $Ag$ and $F e_{3} O_{4}$ in water. However, $Ag$ in water shows a greater rate of Nusselt number than $Ag + F e_{3} O_{4}$ in water.

Figure 8.

Variation in $f ″$ (0) against $φ$ for the use of specific nanoparticles.

Figure 9.

Variation in $- θ$ ′(0) against $φ$ for the use of specific nanoparticles.

The behavior of different parameters on velocity profiles $f' (η)$ and temperature profiles $θ (η)$ for the stable solution is shown in Figures 10 to 22. The impact of $S$ on velocity profile $f' (η)$ and the temperature $θ (η)$ is delineated in Figures 10 and 11, respectively. In the feasible solution of this study, the velocity profile increases while the temperature $θ (η)$ decreases with an increase in suction $S$ throughout the flow.

Figure 10.

Variation in $f' (η)$ for varying S.

Figure 11.

Variation in $θ (η)$ for varying S.

Figure 12.

Variation in $f' (η)$ for varying M.

Figure 13.

Variation in $θ (η)$ for varying M.

Figure 14.

Variation in $f' (η)$ for varying K.

Figure 15.

Variation in $θ (η)$ for varying K.

Figure 16.

Variation in $f' (η)$ for varying Fr.

Figure 17.

Variation in $θ (η)$ for varying Fr.

Figure 18.

Variation in $θ (η)$ for varying Rd.

Figure 19.

Variation in $θ (η)$ for varying Q.

Figure 20.

Variation in $θ (η)$ for varying $δ_{T}$ .

Figure 21.

Variation in $f' (η)$ for varying $δ$ .

Figure 22.

Variation in $θ (η)$ for varying $δ$ .

Physically, a substantial amount of the fluid flows to the outside of the surface due to the presence of pores in the suction process, which decelerates the flow of the fluid. Therefore, fluid velocity as well as temperature, decreases. On the other hand, the increasing rate of magnetic parameter $M$ decreases velocity $f' (η)$ and the temperature $θ (η)$ profiles, as demonstrated in Figures 12 and 13, respectively. In reality, an increase in the magnetic field increases Lorentz forces, which in turn, develop resistance in flow.

Figures 14 and 15 respectively, are drawn to show the effects of parameter $K$ on profiles of velocity $f' (η)$ and temperature $θ (η)$ . The Figures suggest that the velocity profile $f' (η)$ and the temperature profile $θ (η)$ decline with an increase in $K$ . Meanwhile, an increase in $Fr$ increases velocity and temperature profiles with their boundary layer thicknesses throughout the flow, as shown in Figures 16 and 17.

Figure 18 shows the impact of the parameter $Rd$ on temperature profile $θ (η) .$ The thermal boundary layer thickness and temperature are seen increasing with an increase in the thermal radiation parameter. Physically, it occurs due to the fact that the greater value of $Rd$ reduces the absorbing factor $k^{*}$ , which grows the divergence in radiating thermal exchange. Therefore, the radiative parameter discharges heat energy in the flow regime. Hence, the rate of radiating heat transfer increases to the hybrid nanofluid and ultimately, the temperature of hybrid nanofluid is enlarged.

On the other hand, the thermal boundary layer and temperature are seen decreasing with an increase in heat source/sink parameters $Q$ , as shown in Figure 19. Similarly, Figure 20 also shows that a rise in slip parameter $δ_{T}$ decreases the temperature profile and the thickness of thermal boundary layers throughout the flow in the stable solution. Originally, an increase in $δ_{T}$ results a minor quantity of heat energy being transferred to the hybrid nanofluids, that reduces its temperature.

Figures 21 and 22 are drawn to show the impact of slip parameter $δ$ on velocity and the temperature profiles. These figures show that the increasing velocity slip parameter $δ$ increases the profiles of velocity and temperature along their boundary layer thicknesses throughout flow in first stable and feasible solution of the present study.

Conclusions

In this investigation, two-dimensional incompressible steady Darcy-Forchheimer boundary layer flow, and the heat transfer characteristics of Ag-Fe₂O₃/water-based hybrid nanofluid induced by a shrinking/stretching surface along radiation and magnetic impacts are considered. In addition, the magnetohydrodynamics, radiation, heat sink/source, and slip effects were also taken in the given examination. The numerical dual solutions were achieved using shooting technique in Maple software. Important results of present study are as follows:

The skin friction $f ″$ (0) and Nusselt number $- θ$ ′(0) increase for $< 0$ , and skin friction decreases for $λ > 0$ with increase in suction parameter.

Skin friction and the Nusselt number increase along the variation of suction when nanoparticles volumetric fraction is increased.

Skin friction and the Nusselt number rise with rise in magnetic parameter and as well as the suction parameter.

The Ag-water-based hybrid nanofluid has a greater the rate of the skin friction compared to Ag- $F e_{3} O_{4} /$ water and $F e_{3} O_{4}$ -water-based hybrid nanofluid.

The $F e_{3} O_{4}$ -water-based hybrid nanofluid has a greater the rate of the Nusselt number as compare to Ag- $F e_{3} O_{4} /$ water and $Ag$ -water-based hybrid nanofluid.

An increase in suction, magnetic, $K$ and velocity slip parameters decreases the velocity profiles, and opposite to it, velocity profile increases with an increase in $Fr .$

An increase in suction, magnetic, porous permeability parameter, heat source/sink, thermal, and velocity slips parameters decreases the temperature profiles while it is heightened owing to the superior impacts of thermal radiation.

The present study will prove beneficial regarding hybrid nanofluid and for those interested in working on multiple solutions such as dual and triple branch solutions. This work can be extended to three-dimensional flow problems with concentration equation, along with the momentum and heat transfer equations, where chemical reaction parameters can also be considered. In addition, the waste discharge pollutant concentration can also be taken in the extension of this problem.

Footnotes

Appendix

Acknowledgements

The authors are thankful for the support of Researchers Supporting Project number (RSP2023R33), King Saud University, Riyadh, Saudi Arabia.

Handling Editor: Chenhui Liang

Author contributions

Conceptualization, M.A.M., K.J., and U.K.; methodology, M.A.M., K.J., and U.K.; software, M.A.M., K.J., and U.K.; validation, M.A.M., K.J., and U.K.; formal analysis, M.A.M., K.J., and U.K.; investigation, H.B.L. and I.P.; resources, I.P.; data curation, H.B.L.; writing—original draft preparation, H.B.L., I.P., and El-S. M. S.; writing—review and editing, H.B.L., I.P., and El-S. M. S.; visualization, El-S. M. S.; supervision, El-S. M. S.; project administration, El-S. M. S.; funding acquisition, El-S. M. S. All authors have read and agreed to the published version of the manuscript.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the Researchers Supporting Project number (RSP2023R33), King Saud University, Riyadh, Saudi Arabia.

ORCID iD

Umair Khan

El-Sayed M Sherif

Data availability statement

Data will be available on request.

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Thermal analysis for hydromagnetic flow of Darcy-Forchheimer hybrid nanofluid with velocity and temperature slip effects: Scrutinization of stability and dual solutions

Abstract

Keywords

Introduction

Mathematical formulation

Key engineering quantities of interest

Stability analysis

Numerical methodology

Results and discussion

Conclusions

Footnotes

Appendix

Acknowledgements

Author contributions

Declaration of conflicting interests

Funding

ORCID iD

Data availability statement

References