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Abstract

Fluid flow and heat transfer inside a rectangular framework have numerous engineering applications, including energy generation, power systems, cooling technologies, and nuclear reactors. Researchers are currently focusing on improving heat transfer rates through the use of nanofluids. With this motive, the current study investigates the constant, incompressible flow of nanofluids containing copper nanoparticles suspended in ethylene glycol and water, within a rectangular framework. The study investigates the non-Newtonian Ree-Eyring nanofluid flow with impact of Hall Effects. Furthermore, the impacts of Joule dissipation, thermal radiation, and viscous dissipation are examined. Additionally, the physical quantities skin friction coefficient and Nusselt number are investigated. The mathematical modeling is examined, followed by the application of similarity transformation to the nonlinear governing partial differential equations (PDEs) with particular boundary conditions. This procedure simplifies the derivation of dimensionless ordinary differential equations. The dimensionless scheme of equations is then solved numerically using the bvp4c numerical technique, which is integrated into the MATLAB software. Graphical and tubular representations are used to study the logical impact of fascinating aspects on subjective dimensions. According to the findings, greater magnetic parameters reduce velocity components, whereas larger Ree-Eyring fluid values enhance them. The temperature profile improves when magnetic parameters increase, but declines as Ree-Eyring fluid parameters rise. As suction parameters increase, the skin friction coefficient drops. The Nusselt number drops with a rising Biot number. This study have examined the first time to the best of authors knowledge.

Keywords

Ree-Eyring fluid nanofluid Hall Effect joule heating viscous dissipation BVP4C

Introduction

A nanofluid is made up of nanoparticles, which are miniscule particles on a nano scale that improve the fluid’s viscosity and heat conductivity. To ensure their uniform distribution, these nanoparticles are added to the solution via dispersion procedures including chemical stabilization or ultrasonic. By raising the effective surface area for better heat dissipation, this improves heat transfer efficiency. Because they are so good at improving energy absorption, nanofluids are used in a wide range of heat transfer systems, such as solar energy systems, electrical devices, and automobile engines. The issue of nanoparticles and the environment is extensive, and a thorough study cannot be provided here. With increasing financing, research and development initiatives have expanded, resulting in a large increase in the number of publications. Covering all of these articles is beyond the scope of this article. Instead, a succinct review of the myriad scientific and technological problems and potential is presented, emphasizing the importance of collaborative efforts to increase understanding. Choi and Eastman¹ introduced the term “nanofluid” during his key paper. Nanofluid is a liquid that contains a dispersion of solid particles (nanoparticles) ranging from 1 to 100 nm in size. The field of nanotechnology has advanced significantly in the last few decades. Shah et al.² examined the rotating flow of a micro polar nanofluid including CNT nanoparticles in different base fluids between parallel and horizontal plates in a porous media defined by Darcy-Forchheimer’s law. Sharma et al.³ investigated the effect of heat radiation on grapheme Maxwell nanofluid flow across a linearly extended sheet. Ge-Jile et al.⁴ investigated the heat transfer properties of a nanofluid smoothly through a vertical cone in the occurrence of thermal radiation. They investigated a water-based nanofluid with ferrous nanoparticles. Mishra and Kumar⁵ investigated the effects of radiation on the heat transfer efficiency of a nanofluid flow across a stretched sheet. Rashed et al.⁶ investigated nanofluid behavior near a sliding vertical plate, specifically how nanoparticle dispersion within the boundary layer influences fluid characteristics. Their study shows that nanofluids can greatly improve heat transfer efficiency, which is consistent with previous findings. Rashed et al.⁷ explored the impact of a variable magnetic source on a ferro-nanofluid containing CoFe₂O₄ nanoparticles and water as the base fluid. Their findings revealed that nanofluid velocity and shear stress rose with higher nanoparticle volume fraction and magnetic field strength, but reduced with increasing Pr. Rashed et al.⁸ explored MHD fluid flow in both stretching and shrinking contexts. Their theoretical analysis showed that greater magnetic field parameter values led to a notable increase in fluid heat transfer, with the effect being more pronounced during stretching than in shrinking. The Hall Effect is important because it affects magnetic force and changes the current density’s direction and magnitude. The effects of Hall current on velocity and temperature fields in the occurrence of unstable Hartmann numbers were examined by Attia.⁹ Three-dimensional Coquette flow with transpiration cooling was examined by Singh¹⁰ in the existence of a static magnetic field perpendicular to the flow. Singh¹¹ investigated an additional aspect of this flow, concentrating on the case in which the moving plate stayed motionless inside the magnetic field. The discussion in this field was aided by Prasada Rao and Krishna¹² who examined the Stokes and Eckert problems in magneto hydrodynamics with Hall effects. The conductivity perpendicular to the magnetic field decreases in an ionized gas with low density and/or large magnetic fields because of the free rise motion of electrons and ions along the magnetic field lines before encountering collisions. Furthermore, a perpendicular current is brought to the magnetic and electric fields. Turkyilmazoglu¹³ investigated the flow over a porous disk in the existence of a Hall current, emphasizing its effects and practical applications.

The Hall Effect is a well-established phenomenon that has received extensive attention in the literature. Magneto hydrodynamic flows with Hall currents are of great interest in technical applications, especially in the domains of flight magneto hydrodynamics, Hall accelerators, and magneto hydrodynamic generators. Due to its frequent occurrence in a variety of industrial and technological applications, fluid flow within a circular duct or pipe is of significant importance within the technical realm. Although a great deal of study has been done in this area by scholars like Szymanski.^14–21

Non-Newtonian fluids are remarkable for their capacity to defy the simplicity of Newton’s viscosity law, showing changing viscosity under diverse shear rates. Ree-Eyring is a popular model for complex designs that are necessary to capture their complex activities. This model is intended specifically for fluids that behave as if they are shear-thinning, meaning that as shear rates increase, viscosity falls. Researchers are interested in Ree-Erying fluid because its wide series applications in biological and industrial processes. Al-Mdallal et al.²² investigated radioactive Ree-Eyring nanofluid flow between two stretching rotating disks and solve the obtained equations numerically. Alzahrani and Khan²³ investigated entropy formation of reactive Ree-Eyring nanofluid flow between two rotating disks with the influence of joule heating. The outcome indicates that rising Wiesenberger number, boost the velocity profile. Shoaib et al.²⁴ investigated the irreversibility of Ree-Eyring Nanofluid flow among two spinning discs, considering the Arrhenius chemical reaction, viscous dissolution, and nonlinear thermal radiation. They addressed the problem using a mix of the Bayesian Regularization approach and the Homotopy analysis method. Their findings show that as the Wiesenberger number is adjusted, both the velocity components rise, while the axial velocity profile decreases. Rooman et al.²⁵ studied the Hall effects and entropic effect on Ree-Eyring nanoliquid flow over leaky rotating discs and discovered that the Sherwood number is reduced due to Brownian diffusivity. Shah et al.²⁶ investigated the flow of Ree-Eyring non-Newtonian nanofluids with AA7072 and AA7075 nanoparticles over a stretching sheet, taking into account magnetic dipoles and using the Koo-Kleinstreuer and Cattaneo-Christov models for heat transfer. The results showed that AA7075 nanoparticles outperformed AA7072 in terms of thermal efficiency improved as volume fraction and magnetic interactions increased.

The diverse applications include physics and technology, particularly in material and equipment construction, aerospace engineering, and gas generators, thermal radiation’s impact on natural convection phenomena is becoming increasingly important. England and Emery²⁷ studied the effects of heat radiation on a lamina, focusing on mixed convection boundary layers descending a vertical shallow for both absorbing and non-absorbing gases. Furthermore, Mahanthesh et al.²⁸ investigated the numerical properties of boundary layer two-phase nanofluid flow containing particles such as $A l_{2} O_{3}$ and $H_{2} O_{2}$ across a perpendicular plane surface in the presence of quadratic thermo-convection and quadratic heat radiation. Khader and Sharma²⁹ explored the effects of non-uniform heat source/sink and thermal radiation on unsteady MHD micropolar nanofluid flow on a stretching/shrinking surface. Rashed et al.³⁰ investigated the steady-state flow of electrically conducting Newtonian fluids near a heated vertical plate, influenced by a magnetic field and a heat source/sink. The key findings demonstrated that raising the plate’s permeability coefficient greatly increased flow velocity and somewhat increased heat flux, but shear stress and temperature distribution reduced. Khan et al.³¹ considered the impression of nonlinear thermal radiation and Arrhenius activation energy on the flow of a improved second-grade nanofluid. Nadeem et al.³² investigated the flow of reactive chemical nanofluid over a rotatable disc enclosed with non-Darcy porous media while accounting for heat radiation. Elsaid and Abedel-Aal³³ used numerical analysis to examine the effect of thermal radiation on nanofluid flow through a porous sheet. Hussain et al.³⁴ investigated the Hall effect-induced flow of hybrid nanofluids. Bouslimi et al.³⁵ examined heat transmission in a solar collector.

The main purpose of this research is to appearance at how the Hall Effect the flow of a Ree-Eyring nanofluid based on ethylene glycol water over a rectangular frame. The investigation contains the effects of suction and stretching, as well as viscous dissipation, Joule dissipation, and thermal radiation effects. The work presents findings for nanofluids comprising copper nanoparticles.

Dispersed ethylene glycol water, subject to certain limitations. To explain the findings, visual representations of numerical analyses are presented in the form of tables and graphs. The primary objective of this study was to answer the following research questions:

How does the Hall Effect the temperature and velocity fields?

How does the nanofluid effect the rate of heat transfer?

How does altering the volume percentage of nanoparticles effect velocity and temperature fields?

What is the impact of Ree-Eyring parameter on temperature and velocity profiles?

Mathematical formulation

A magnetic field of strength $B_{0}$ and an electric field of strength $E_{0}$ act in the $y$ -direction. The model of micropolar nanofluids also takes into account the impact of Hall current. These fluids are electrically conductive, and as the magnetic field increases, Hall current emerges, influencing the behavior of the micropolar nanofluids. This phenomenon produces a force in the z-direction, which causes a cross-flow in the same way. As a result, the flow of micropolar nanofluid is deflected in three dimensions. The generalized form of Ohm’s law, which includes the Hall current, is

\tilde{j} + \frac{ω_{e} t_{e}}{B_{0}} \times (\tilde{j} \times \tilde{B}) - σ_{nf} (\tilde{E} + \tilde{V} \times \tilde{B}) - \frac{σ_{nf} P_{e}}{e n_{e}} = 0,

(1)

Here $\tilde{j} = (J_{x}, J_{y}, J_{z})$ represents current density and $\tilde{J} = σ_{nf} (\tilde{E} + \tilde{V} \times \tilde{B}),$

$\tilde{B} = (0, B_{0}, 0)$ represent magnetic field, $\tilde{E}$ represent an intensity of electric field, $\tilde{E} = (0, E_{0}, 0), \tilde{V} = (\tilde{u}, \tilde{v}, \tilde{w})$ represent velocity components, $ω_{e}$ represent oscillating frequency of the electron, $t_{e}$ denotes the time of electron collision, $σ_{nf}$ denotes electrical conductivity, $e$ is the charge of electron, $n_{e}$ is number density of electron, and $p_{e}$ is the pressure of electron. For weakly ionized molecules, the generalized Law of Ohm yields the flow field ( $J_{y} = 0$ ). Using these assumptions, obtained $J_{x}$ and $J_{y}$ as³⁶

J_{x} = \frac{σ_{nf} \tilde{B_{0}^{2}}}{1 = m^{2}} (m \tilde{u} - \tilde{w})

(2)

J_{z} = \frac{σ_{nf} \tilde{B_{0}^{2}}}{1 = m^{2}} (\tilde{u} - m \tilde{w})

(3)

Here $m = ω_{e} t_{e}$ is the Hall parameter.

This study investigates the constant, incompressible three-dimensional flow of nanofluids made up of copper nanoparticles dispersed in a base fluid of ethylene glycol and water. The analysis takes into account the effect of magnetic fields and heat radiation. The velocity components ( $u, v, w$ ) correspond to the $x, y$ , and $z$ directions of the rectangular frame, respectively. The velocities are shown as horizontal surfaces along the $x$ and $y$ axes, with $u_{w} = ax, ν_{w} = by$ representing the velocities caused by dual linear stretchable boundary layer flows, as proposed by Prandtl³⁷ and Sakiadis (Figure 1).³⁸

Figure 1.

Fluid flow geometry.

The Ree-Eyring hybrid nanofluid flow’s continuity, momentum, and energy equations can be stated as follows using the usual boundary layer approximation with the viscous dissipation term:

\nabla . V = 0

(4)

ρ_{n f} (\frac{\partial V}{\partial t} + V . \nabla V) = \nabla . τ

(5)

{(ρ c_{p})}_{nf} (\frac{\partial T}{\partial t} + V . \nabla T) = k_{nf} \nabla^{2} T + Φ + \frac{16 σ^{*} T_{\infty}}{3 k^{*}} \nabla^{2} T

(6)

Where $τ$ represent the extra stress tensor for Ree-Eyring nanofluid and is defined as³⁹:

τ = (μ_{nf} + \frac{1}{β_{1} λ_{2}} \sin h^{- 1} (\frac{λ_{2}}{c_{1}})) A_{1}

(7)

Where

λ_{2} = \sqrt{(\frac{1}{2}) tr A_{1}^{2}}

(8)

Since

\sin h^{- 1} (\frac{λ_{2}}{c_{1}}) \approx \frac{λ_{2}}{c_{1}} for \frac{λ_{2}}{c_{1}} \leq 1

(9)

Thus

τ = (μ_{nf} + \frac{1}{β_{1} λ_{2}}) A_{1}

(10)

Here $A_{1}$ is the first Rivilin Ericksin tensor, which is showed as³⁹

A_{1} = \nabla V + (\nabla V)^{t}

(11)

Based on what has been said above, the Ree-Eyring boundary layer equations as follows^26,39

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

(12)

\begin{matrix} ρ_{nf} (u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}) = (μ_{nf} + \frac{1}{β c}) \frac{\partial^{2} u}{\partial z^{2}} \\ + \frac{μ_{nf} β_{0}^{2}}{1 + m^{2}} (v - mu) \end{matrix}

(13)

\begin{matrix} ρ_{nf} (u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z}) = (μ_{nf} + \frac{1}{β c}) \frac{\partial^{2} v}{\partial z^{2}} \\ - \frac{σ_{nf} β_{0}^{2}}{1 + m^{2}} (u - mv) \end{matrix}

(14)

\begin{matrix} (u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z}) = \\ \begin{matrix} (\frac{k_{nf}}{{(ρ c_{p})}_{nf}}) \frac{\partial^{2} T}{\partial z^{2}} + \frac{σ_{nf} β_{0}^{2}}{({(ρ C_{p})}_{nf}) (1 + m^{2})} (u^{2} - v^{2})) + \\ \frac{1}{{(ρ C_{p})}_{nf}} (μ_{nf} + \frac{1}{β c}) [{(\frac{\partial u}{\partial z})}^{2} + {(\frac{\partial u}{\partial z})}^{2}] - \frac{1}{{(ρ C_{p})}_{nf}} (\frac{\partial q_{r}}{\partial y}) \end{matrix} \end{matrix}

(15)

Boundary condition⁴⁰

\begin{matrix} u = u_{w} = ax, w = w_{w} (= - \sqrt{a ϑ_{nf}}) \\ \int \frac{v}{ay} d ξ + z_{w}) at z = 0 and y = 0 \\ U \to 0, as z \to \infty and y = 0 \end{matrix}

\begin{matrix} V = v_{w} (x) = \frac{b^{2} y}{a}, w = w_{w} (= - \sqrt{a ϑ_{nf}}) \\ \int \frac{u}{dx} d ξ + z_{w}), at z = 0 and x = 0 \\ v \to 0, as z \to \infty and x = 0 \end{matrix}

\begin{matrix} - k_{o} [\frac{\partial T}{\partial y}] = h_{f} (T_{w} - T) at z = 0 as (x, y) \to \infty \\ T \to T_{\infty} as z \to \infty, and (x, y) \to \infty \end{matrix}

(16)

Rosseland approximation

Using Rosseland’s approximation for radiative heat flux, the expression for radiative heat flow can be written as follows:

q_{r} = - \frac{4 σ^{*}}{3 K^{*}} \frac{\partial T^{2}}{\partial y} = - \frac{4 σ^{*} T^{3}}{3 K^{*}} \frac{\partial T}{\partial y}

(17)

In the above expression, the symbol “ $σ^{*}$ ” represents the Stefan-Boltzmann constant, and “ $K^{*}$ ” signifies the mean immersion coefficient (Table 1 shows the thermosphysical characteristics of the basefluid and nanoparticles).

Table 1.

Thermo physical characteristics of the basefluid and nanoparticles.

Property	Ethylene glycol water based nanofluid
Dynamic viscosity	$μ_{nf} = μ_{f} {(1 - \frac{φ_{a}}{φ_{m}})}^{[ℵ] φ_{m}}$
Density	$ρ_{nf} = (1 - φ_{a}) ρ_{f} + φ_{a} ρ_{p}$
Heat capacity	${(ρ C_{p})}_{nf} = (1 - φ_{a}) {(ρ C_{p})}_{f} + φ_{a} {(ρ C_{p})}_{p}$
Thermal conductivity	$K_{nf} = K_{f} [\frac{K_{a} + 2 K_{f} - 2 φ_{a} (K_{f} - K_{a})}{K_{a} + 2 K_{f} + φ_{a} (K_{f} - K_{a})}]$
Electrical conductivity	$\begin{array}{l} σ_{n f} / σ_{f} = \\ (1 + \frac{3 (σ_{p} / σ_{f} - 1) φ_{a}}{φ_{p} / σ_{f} + 2} - (σ_{p} / σ_{f} - 1) φ_{a}) \end{array}$

The abbreviation “nf” in the above properties refers to the thermo physical properties of nanofluids. In this nomenclature, $p$ properties represents solid nanoparticles, while $f$ represents the base fluid. The symbol “ $φ$ ” represents the volumetric concentration of nanoparticles.

Similarity transformation

Similarity transformation is utilized to find non-dimensional equations, which are listed below.⁴⁰

\begin{matrix} u = ax f' (ξ), v = by g' (ξ) w = - \sqrt{a ϑ_{f}} [f (ξ) + dg (ξ)] \\ ξ = z \sqrt{\frac{a}{ϑ_{nf}}}, θ = \frac{T - T_{\infty}}{T_{w} - T_{\infty}} \end{matrix}

(18)

Using similarities between dimensional expressions, constructed the dimensionless equations:

\begin{matrix} (\frac{μ_{nf}}{μ_{f}} + α_{1}) f ″' - \frac{ρ_{nf}}{ρ_{f}} ({f'}^{2} - (f + dg) f^{″}) \\ - \frac{σ_{nf}}{σ_{f}} (\frac{M}{1 + m^{2}}) (dL g' - m f') = 0 \end{matrix}

(19)

\begin{matrix} (\frac{μ_{nf}}{μ_{f}} + α_{1}) g ″' - \frac{ρ_{nf}}{ρ_{f}} (d {g'}^{2} - (f + dg) g^{″}) \\ - \frac{σ_{nf}}{σ_{f}} (\frac{M}{1 + m^{2}}) (f' - mdL g') = 0 \end{matrix}

(20)

\begin{matrix} (\frac{k_{nf}}{k_{f}} + Rd) θ ″' + \frac{M P_{r} E_{c}}{1 + m^{2}} (f'^{2} + d^{2} L^{2} g'^{2}) \\ + (\frac{μ_{nf}}{μ_{f}} + α_{1}) P_{r} E_{c} (f''^{2} + d^{2} L^{2} g''^{2}) + \frac{{(ρ C_{p})}_{nf}}{{(ρ C_{p})}_{f}} P_{r} (f + dg) θ' = 0 \end{matrix}

(21)

Boundary condition

\begin{matrix} f' = 1, f = f_{w}, g' = \frac{f_{w}}{d}, θ' (0) = - γ (1 - θ (0)) at ξ = 0 \\ f' \to 0, g' \to 0, θ \to 0, ξ \to \infty \end{matrix}

(22)

Here

\begin{matrix} \frac{ρ_{nf}}{ρ_{f}} = 1 - φ_{a} + φ_{a} \frac{ρ_{p}}{ρ_{f}}, \frac{μ_{nf}}{μ_{f}} = (1 - \frac{φ_{a}}{φ_{m}}) \\ \frac{{(ρ C_{p})}_{nf}}{{(ρ C_{p})}_{f}} = 1 - φ_{a} + φ_{a} \frac{{(ρ C_{p})}_{p}}{{(ρ C_{p})}_{f}}, \frac{k_{nf}}{k_{f}} = \frac{k_{a} + 2 k - 2 φ_{a} (k_{f} - k_{a})}{k_{a} + 2 k_{f} + 2 φ_{a} (k_{f} - k_{a})} \end{matrix}

(23)

Thermal conductance aggregation was produced by adapting Maxwell’s model to create the Bargeman model. This method was chosen for its precision, combined with the intrinsic aggregation properties of nanofluids. The thermal conductivity of the collective model is calculated as follows.

\frac{k_{a}}{k_{f}} = \frac{1}{4} [\begin{matrix} [(3 φ_{m} - 1) \frac{K_{p}}{K_{f}} + [3 (1 - φ_{m}) - 1] + \\ {[{((3 φ_{m} - 1) \frac{K_{p}}{K_{f}} + [3 (1 - φ_{m}) - 1])}^{2} + \frac{8 k_{p}}{k_{f}}]}^{1 / 2} \end{matrix}]

(24)

The collective volume fraction is the volume occupied by aggregates divided by the largest aggregate’s maximum packing fraction.

φ_{a} = φ {(\frac{r_{a}}{r_{p}})}^{3 - D}

(25)

Let $φ$ be the volume fraction of small nanoparticles, and $φ_{m}$ represent the greatest volume fraction attainable for the tiniest nanoparticles [ $ℵ$ ]. Also, indicate the Einstein coefficient as D, denoting the fractal indices $r_{a}$ and $r_{p}$ , which correspond to the radii of sums and nanoparticles, respectively. In addition, include non-dimensional factors such as the suction velocity, thermal radiation, magnetic, and stretching ratio parameter, as well as the Biot and Prandtl numbers (the range of these non-dimensional parameters are shown in Table 2).

\begin{matrix} d = \frac{b}{a}, M = \frac{σ_{f} β_{0}^{2}}{C ρ_{f}}, Rd = \frac{16 σ^{*} T_{\infty}^{3}}{3 K^{*} k_{f}}, γ = \frac{h_{f}}{K_{0}} \sqrt{\frac{v_{f} (1 - ξ t)}{b}} \\ \Pr = \frac{{(ρ C_{p})}_{f} ϑ_{f}}{k_{f}}, Ec = \frac{a^{2} x^{2}}{C_{p} (T_{w} - T_{\infty})}, α_{1} = \frac{1}{β_{1} C_{1} μ_{1}}, L = \frac{y}{x} \end{matrix}

(26)

Table 2.

Range of non-dimensional parameters.

Name of the parameters	Mathematical form	Consider range
Suction velocity parameter	$f_{w} = {\frac{- Z_{w}}{\sqrt{a} ϑ}}_{f}$	$[0.1 0.7]$
Stretching ratio parameter	$d = \frac{b}{a}$	[1.0 4.0]
Magnetic parameter	$M = \frac{σ_{f} β_{0}^{2}}{C ρ_{f}}$	[0.1 1.3]
Biot number	$γ = \frac{h_{f}}{K_{0}} \sqrt{\frac{v_{f} (1 - ξ t)}{b}}$	[0.1 1.0]
ReeEyring fluid parameter	$α_{1} = \frac{1}{β_{1} C_{1} μ_{1}}$	[1.0 7.0]

Physical quantities of engineering interest

In this portion, the native skin friction coefficient $C_{f_{x}}$ and Nusselt number $N_{u_{x}}$ are mentioned as

C_{f_{x}} = \frac{(μ_{nf} + \frac{1}{β_{1} C_{1}})}{ρ {fu}_{W}^{2}} {(\frac{\partial u}{\partial z})}_{z = 0}, C_{f_{y}} = \frac{(μ_{nf} + \frac{1}{β_{1} C_{1}})}{ρ {fv}_{W}^{2} (x)} {(\frac{\partial v}{\partial z})}_{z = 0}

N u_{x} = - \frac{x}{k_{f} (T_{w} - T_{\infty})} (k_{nf} + \frac{16 σ^{*} T_{\infty}^{3}}{3 k^{*}}) {(\frac{\partial T}{\partial z})}_{z = 0}

(27)

\begin{matrix} C_{fx} \sqrt{R e_{x}} = (\frac{μ_{nf}}{μ_{f}} + α_{1}) f^{″} (0) \\ C_{f_{y}} \sqrt{R e_{y}} = (\frac{μ_{nf}}{μ_{f}} + α_{1}) g^{″} (0) \\ \frac{N u_{x}}{\sqrt{R e_{x}}} = - \frac{k_{nf}}{k_{f}} θ' (0) \end{matrix}

(28)

The Reynolds number $\sqrt{R e_{y}} = \frac{a^{\frac{1}{2}} y}{ϑ_{bf}^{2}}, \sqrt{R e_{x}} = \frac{a^{\frac{1}{2}} x}{ϑ_{bf}^{2}}$ ,

Numerical scheme

In this portion, the ordinary differential equations and their boundary conditions equation (15) precisely presented. The shooting algorithm, implemented through the bvp4c in MATLAB, is used to find the flow model’s numerical solution. In this technique, the higher order is first transformed (Figure 2).

\begin{matrix} f = y_{1}, f' = y_{2}, f^{″} = y_{3}, f ″' = y_{3}', \\ g = y_{4}, g' = y_{5}, g^{″} = y_{6}, g ″' = y_{6}' \\ θ = y_{7}, θ' = y_{8}, θ^{″} = y_{8}' \end{matrix}

(29)

\begin{matrix} y_{3}' = \frac{1}{(\frac{μ_{nf}}{μ_{f}} + α_{1})} \\ [\frac{ρ_{nf}}{ρ_{f}} (y_{1}^{2} - (y_{1} + d y_{4}) y_{3}) + \frac{σ_{nf}}{σ_{f}} (\frac{M}{1 + m^{2}}) (dL y_{5} - m y_{2})] \end{matrix}

(30)

\begin{matrix} y_{6}' = \frac{1}{(\frac{μ_{nf}}{μ_{f}} + α_{1})} \\ [\frac{ρ_{nf}}{ρ_{f}} ({dy}_{5}^{2} - (y_{1} + d y_{4}) y_{6}) + \frac{σ_{nf}}{σ_{f}} (\frac{M}{1 + m^{2}}) (y_{2} - mdL y_{5})] \end{matrix}

(31)

\begin{matrix} y_{8}' = \frac{- 1}{(\frac{μ_{nf}}{μ_{f}} + Rd)} [\begin{matrix} \frac{MPrEc}{1 + m^{2}} (y_{2}^{2} + d^{2} L^{2} y_{5}^{2}) + A_{3} \Pr (y_{1} + d y_{4}) y_{8} \\ + (A_{1} + α_{1}) PrEc (y_{3}^{2} + d^{2} L^{2} y_{6}^{2}) \end{matrix}] \end{matrix}

(32)

\begin{matrix} y_{2} (0) = 1, y_{1} (0) = f_{w}, y_{5} (0) = d, y_{4} (0) = \frac{f_{w}}{d}, \\ y_{8} (0) = - γ (1 - y_{7} (0)) \\ y_{2} (0) \to 0, y_{5} (0) = 0, y_{7} (0) = 0 as ξ \to \infty \end{matrix}

(33)

Figure 2.

(bvp4c) Flow chart.

Result and discussion

This section discusses in depth the graphical and tabular results generated from a numerical solution using the bvp4c method. Tables and graphs are used to investigate the impact of many governing factors, including fluid, magnetic, stretching ratio, Biot number, Ree Eyring fluid parameter, suction velocity, and volume fraction parameters, on velocity profiles $(f^{'} (η), g^{'} (η)),$ and temperature (θ(η)), Nusselt number, and skin friction.

Velocity profile

Figures 3 and 4 represent the variation of velocity profile along $x$ and $y$ direction for several values of fluid parameter $α_{1}$ , respectively. The velocity profile tends to grow in both the $x$ and $y$ directions when the Ree-Eyring fluid parameter is raised in a nanofluid that contains copper nanoparticles mixed with a base fluid like ethylene glycol water, because the fluid’s rheological behavior is influenced by the Ree-Eyring fluid parameter. Higher viscosities are usually out come of increasing this parameter. The viscosity of nanofluids can be further altered by the addition of nanoparticles. Higher velocities are required to maintain the same flow rate as a result of the increased resistance to flow.

Figure 3.

Impact of Ree Eyring fluid parameter $α_{1}$ on velocity profile $f' (η)$ .

Figure 4.

Impact of Ree Eyring fluid parameter $α_{1}$ on velocity profile $g' (η)$ .

Figures 5 and 6 show the behavior of velocity profile in x and y path respectively for distinct values of suction velocity parameter $f_{w}$ . When the suction velocity parameter is increased, both the $x$ and $y$ directions velocity profiles tend to decrease. Increasing the suction velocity parameter implies that fluid is being evacuated from the system at a faster pace. This significantly lowers the motive force for fluid flow. As a result, the fluid’s velocity drops, resulting in reduced speeds in both the $x$ and $y$ axes.

Figure 5.

Impact of Suction velocity parameter $f_{w}$ on velocity profile $f' (η)$ .

Figure 6.

Impact of Suction velocity parameter $f_{w}$ on velocity profile $g' (η)$ .

Figures 7 and 8 examine the variations in the velocity fields $f' (η)$ and $g' (η)$ as influenced by different values of the magnetic parameter $M$ for nanofluids comprising an ethylene glycol-water base fluid with copper nanoparticles. The figures show that for increasing $M$ , $f^{'} (η)$ declined, while $g^{'} (η)$ enhanced. Because the magnetic parameter controls the strength of the magnetic field supplied to the nanofluid. The magnetic field strengthens as the magnetic parameter growths. In MHD flow, the Lorentz force generated by the magnetic field works perpendicular to both the magnetic field direction and the fluid velocity. As a result, it can restrict fluid travel in the direction parallel to the magnetic field $(x),$ causing velocity to drop. In contrast, the direction is vertical to both the magnetic field and the flow direction $(y),$ the Lorentz force adds to fluid motion, resulting in an increase in velocity. Figures 9 and 10 indicate the behavior of velocity profile in x and $y$ direction respectively for distinct values of volume fraction parameter $ϕ$ .The inclusion of nanoparticles increases drag force within the nanofluid. As the volume fraction parameter grows, more nanoparticles are spread throughout the fluid volume, resulting in greater drag forces on the fluid. These drag forces slow the fluid velocity in both the $x$ and $y$ directions. Figures 11 and 12 represent the variation of velocity profiles $f^{'} (η)$ and $g^{'} (η)$ , separately, for different values of stretching ratio parameter $d$ . When the stretching ratio parameter $d$ rises the surface stretches more quickly in the $d$ direction of flow. This additional stretching can reduce velocity in the $x$ -direction. One way to think about it is that when the surface expands more, the fluid particles in touch with it are dragged along more powerfully, resulting in a slower velocity at the surface in the $x$ -direction. Stretching the surface in the $y$ direction (perpendicular to the flow) can provide a “pumping” effect. As the surface stretches, it can draw fluid in the $y$ -direction with it, boosting velocity in that direction significantly. The boost in velocity in the $y$ direction may be caused by the stretching process imparting momentum to the fluid in that direction. Figures 13 and 14 indicates the variation of velocity profiles $f^{'} (η)$ and $g^{'} (η)$ , respectively, for different values of Hall effect parameter $m$ . When the Hall effect parameter is increased, both the $x$ and $y$ directions velocity profiles tend to increase because The Hall effect parameter governs how charged particles behave in a fluid exposed to a magnetic field. When this effect is applied to a base fluid, such as an ethylene glycol-water combination containing copper nanoparticles, it alters the fluid’s velocity profile in both the x and y directions.

Figure 7.

Impact of Magnetic parameter $M$ on velocity profile $f' (η)$ .

Figure 8.

Impact of Magnetic parameter $M$ on velocity profile $g' (η$ ).

Figure 9.

Impact of Volume fraction parameter $ϕ$ on velocity profile $f' (η)$ .

Figure 10.

Impact of Volume fraction parameter $ϕ$ on velocity profile $g' (η)$ .

Figure 11.

Impact of Stretching ratio parameter $d$ on velocity profile $f' (η)$ .

Figure 12.

Effect of Stretching ratio parameter $d$ on velocity profile $g' (η)$ .

Figure 13.

Impact of Hall effect parameter $m$ on velocity profile $f' (η)$ .

Figure 14.

Impact of Hall effect parameter $m$ on velocity profile $g' (η) f' (η)$ .

Temperature profile

Figure 15 illustrates the temperature profile for different values of Ree-Eyring fluid parameter $α_{1}$ .The temperature distribution is improved with a greater $α_{1}$ . Nanoparticles, such as copper have much better heat conductivity than the base fluid. When nanoparticles are inserted to nanofluids, they increase the fluid’s total thermal conductivity. As the Ree-Eyring fluid parameter rises, this can effect the concentration or effectiveness of these nanoparticles in the fluid. The nanofluid can transmit heat more efficiently to its higher thermal conductivity. This improved heat transmission capabilities may result in an increase in the temperature profile.

Figure 15.

Effect of Ree Eyring fluid parameter $α_{1}$ on temperature profile $θ (η)$ .

Here in Figure 16 it is analyzed that temperature drops as the Biot number grows. As the Biot number grows, it suggests that convective heat transmission at the solid-fluid interface is more efficient than internal heat conduction within the nanoparticles. In other words, the amount of heat transfer among the nanoparticles and the surrounding fluid (ethylene glycol-water mixture) exceeds the rate of heat transmission within the nanoparticles. This efficient convective heat transmission results in a more consistent temperature distribution throughout the nanofluid. As a result, the temperature profile of nanofluid reduces.

Figure 16.

Influence of Biot number $γ$ on temperature profile $θ (η)$ .

Figure 17 illustrates the temperature profile for different values of magnetic parameter $M$ . The temperature distribution is improved with a greater magnetic parameter. As the heat transfer rate rises due to MHD convection, the temperature gradient within the nanofluid steepens.

Figure 17.

Influence of Magnetic parameter $M$ on temperature profile $θ (η)$ .

The increased heat transfer rate from the heated surface to the adjacent nanofluid causes a temperature increase in the nanofluid. As the magnetic parameter increases and MHD convection becomes more evident, the temperature profile within the nanofluid rises, resulting in greater temperatures across the nanofluid. Figure 18 refers the temperature profile in $x$ direction decreases when increase the volume fraction Copper nanoparticles have better thermal conductivity compared to the base fluid, ethylene glycol. As a result, as the solid volume percentage of nanoparticles grows, so does the nanofluids thermal conductivity. Higher thermal conductivity improves heat conduction inside the nanofluid, allowing heat to be distributed more effectively across the fluid. As a result of higher thermal conductivity, the temperature differential within the nanofluid decreases, resulting in a lower temperature profile.

Figure 18.

Effect of volume fraction parameter $ϕ$ on temperature profile $θ (η)$ .

Figure 19 illustrates the temperature profile in x direction decreases when increase the Hall effect parameter. As the Hall effect parameter rises in a fluid, such as an ethylene glycol-water combination containing copper nanoparticles, the magnetic force (Lorentz force) deflects charged particles and heat-conducting nanoparticles from the x-direction to the y-direction. This diversion limits heat transport along the x-axis, resulting in a lower temperature profile in that direction. Additionally, a smaller thermal boundary layer might reduce the efficiency of heat transport along the x-axis.

Figure 19.

Effect of Hall effect parameter $m$ on temperature profile $θ (η)$ .

Table 3 indicates the skin friction coefficient for magnetic parameter M, volume fraction parameter $ϕ$ , stretching ratio parameter $d$ , and suction velocity parameter $f_{w}$ . It is observed that for increasing $M$ , the skin friction in $x$ direction increases and in $y$ direction decreases, because the magnetic field causes the fluid flow to align in the same direction as the field. In MHD flows, the Lorentz force generated by the interaction of the magnetic field and the current created in the conducting fluid tends to resist flow perpendicular to the magnetic field. As a result, if the $x$ -direction coincides with the magnetic field’s direction, the flow will encounter more resistance, resulting in greater skin friction in that direction. The Lorentz force reduces flow resistance in the $y$ -direction (assuming it is perpendicular to the magnetic field direction). This decrease in skin friction can occur because the magnetic field prevents the creation of turbulent eddies or vortices, which would otherwise increase skin friction in that direction. From Table 2 it is also detected that increasing $ϕ$ the skin friction in $x$ direction increases and $y$ direction decreases because when the volume fraction parameter is increased, skin friction increases in the direction of flow due to increased particle-fluid interactions, but decreases perpendicular to the flow due to particles’ disruptive effect on flow patterns. These effects are important in a wide range of commercial operations that involve suspensions or colloidal systems, including paint and food processing, as well as sediment movement in environmental engineering. The skin friction in both $x$ and $y$ directions falls for greater values of $d$ , because Increased skin friction in the $x$ -direction. This occurs because expanding the flow domain in the $x$ -direction increases the surface area with which the fluid interacts. More specifically, in a flow with a higher stretching ratio in the $x$ -direction, the fluid has a larger surface area to interact with solid surfaces or other fluid layers, resulting in more skin friction due to increased frictional forces acting on the flow.

Table 3.

Deviation in Skin Friction coefficient with varying of different parameter.

$M$	$ϕ$	$d$	$f_{w}$	$C_{fx} \sqrt{R e_{x}}$	$C_{f_{y}} \sqrt{R e_{y}}$
0.1				−1.839012	−0.1016444
0.2				−1.829325	−0.1045622
0.3				−1.819716	−0.1073031
	0.1			−1.839012	−0.1016444
	0.2			−2.516461	−0.1382931
	0.3			−3.373611	−0.1833424
		0.1		−1.839012	−0.1016444
		0.2		−1.849868	−0.2099861
		0.3		−1.867448	−0.3364231
			0.1	−1.839012	−0.1016444
			0.2	−2.23832	−0.1212187
			0.3	−2.687257	−0.1429897

Increase in skin friction in the $y$ -direction. Similarly, increasing the stretching ratio parameter in the $y$ -direction causes an increase in skin friction in that direction. The skin friction decreases in $x$ and $y$ direction, for large values of $f_{w}$ . This happens because the flow domain’s elongation in the $y$ -direction increases the surface area. Increasing the suction velocity parameter effectively pulls the fluid in the negative $x$ -direction (opposite to the natural flow direction). This reduces skin friction in the $x$ direction by counteracting the fluid’s natural flow resistance. Essentially, the suction helps the fluid overcome the resistance it would normally encounter, resulting in less skin friction along the $x$ -axis. Similarly, increasing suction velocity can cause a decrease in skin friction in the $y$ direction. While the major effect is in the $x$ -direction since the suction is delivered in that direction, secondary effects can also occur in the $y$ -direction.

Table 4 represent the deviation of Nussle number for different values of magnetic parameter $M$ , Eckert number $Ec$ , Stretching ratio parameter $d$ , and Biot number $γ$ . Table 4 show that the large values of $M$ , declined the Nusselt number, because it suppresses fluid motion perpendicular to the magnetic field, modifies boundary layer dynamics, and reduces turbulent mixing. This behavior has ramifications for a variety of technical applications, including the cooling of liquid metal blankets in fusion reactors and the thermal control of conducting fluids in industrial processes. Copper nanoparticles often have higher thermal conductivity than the base fluid. When mixed with ethylene glycol-water, they increase the fluid’s total thermal conductivity. This can lead to better results in heat transfer and higher Nussle values. The data also indicates that higher $d$ , rise the Nusselt number. This increase in stretching ratio can have a range of effects on flow and heat transmission, most notably a rise in the Nussle number, which evaluates the efficiency of convective heat transfer. Greater Biot number indicates that the convective heat transfer resistance at the solid-fluid interface is lower than the internal conductive heat transfer resistance within the solid. This means that heat transmission at the solid-fluid interface dominates heat transfer within the solid. As a result, the convective heat transfer coefficient rises, leading to faster convective heat transfer rates. Table 5 show comparison with previous work, an excellent agreement is found between both present and previous results.

Table 4.

Variation in Nussle number with varying of different parameter.

$M$	$Ec$	$d$	$γ$	$ϕ$	$\frac{N u_{x}}{\sqrt{R e_{x}}}$
0.1	0.1	0.1	0.1	0.1	0.08301156
0.2					0.08293458
0.3					0.08285409
	0.1				0.08301156
	0.2				0.07277185
	0.3				0.06253213
		0.1			0.08301156
		0.2			0.08349259
		0.3			0.08420377
			0.1		0.08301156
			0.2		0.1449972
			0.3		0.1930474
				0.1	0.08301156
				0.2	0.09291623
				0.3	0.1002625

Table 5.

Comparison results with previous literature.⁴¹

$\Pr$	Current study	Gorla and Sidwai	Error%
2.0	0.9109495	0.9114	0.049454
7.0	1.896485	1.8954	0.057211103
20.0	3.350709	3.3538	0.09225
70.0	6.460099	6.4621	0.031114

Conclusion

This study investigates the non-Newtonian Ree-Eyring nanofluid flow with the influence of Hall effects within rectangular framework. Furthermore, the impacts of Joule dissipation heating, viscous dissipation, and thermal radiation are also considered. Additionally, the physical quantities are investigated through tables. In this study the nanofluids containing copper nanoparticles suspended in ethylene glycol-water. The non-linear governing PDEs are altered to non-linear system of ODEs utilizing similarity transformation. The obtained non-linear system is solved by numerical technique (bvp4c). The key finding of this investigation is summarized as follows:

The velocity profile in both $x$ and $y$ direction increases for high values of Ree-Eyring fluid parameter.

For high values of suction velocity parameter and volume fraction, the velocity profile in both $x$ and $y$ directions declined.

The velocity profile in $x$ direction declined for higher values Magnetic parameter of stretching ratio parameter, while contrary behavior is detected for velocity profile in the $y$ direction.

The temperature profile enhanced for large values of Magnetic parameter and Biot number, while opposite comportment is observed for large values of volume fraction and Ree-Eyring fluid parameter.

The skin friction declined in both $x$ and $y$ direction for greater values of volume fraction, stretching ratio parameter, suction velocity parameters.

For large values of Magnetic parameter, the skin friction in $x$ direction increases, while in $y$ direction decreases.

The Nusselt number enhanced for higher values of stretching ratio parameter and Biot number, while declined for large number of Magnetic parameter and Eckert numbers.

Footnotes

Appendix

Notation

u , v , w	Velocity components ( m s − 1 )	M	Magnetic parameter
x , y , z	Coordinates axis ( m )	β o	Magnetic field strength ( Nm A − 1 )
d	Stretching ratio parameter	Pr	Prandtl number
ρ nf	Density of nanofluid	Rd	Radiation parameter
a , b	Stretching rates	u w	Horizontal velocity in x direction
f w	Suction velocity parameter	μ nf	Dynamic viscosity
T	Temperature of fluid ( K )	K nf	Thermal conductivity [ ML / T 3 K ]
ν w	Horizontal velocity in y direction	σ nf	Electrical conductivity
σ *	Stefan Boltzmann constant	C p	Specific heat
h f	Heat transfer coefficient ( W ( m − 2 K − 1 ) )	T w	Temperature of fluid at the wall ( K )
q r	Thermal radiative heat flux ( W m − 2 )	K *	Mean obserption coefficient
( ρ C p ) nf	Heat capacitance of nanofluid ( Jk g − 1 K − 1 )	( ρ C p ) sp	Heat capacitance of nanoparticles
ϕ	Volume fraction	nf	Nanofluid
f	Fluid	sp	Solid particles
γ	Biot number	α 1	Ree-Eyring fluid parameter
L	Dimensional less quantity	Er	Eckerd number
m	Hall effect parameter

Handling Editor: Chenhui Liang

Author contributions

All authors participated equally in the conceptualization, investigation, analysis, original draught writing, review, and editing of the work.

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 project was supported by Researchers Supporting Project number (RSP2024R411), King Saud University, Riyadh, Saudi Arabia.” “Project financed by Lucian Blaga University of Sibiu through research grant LBUS - IRG - 2023 - 09.”

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.

ORCID iDs

Zahir Shah

Mansoor H Alshehri

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Impact of Hall current on radiative Ree-Eyring nanofluid flow across rectangular frame with Joule and viscous dissipation effects

Abstract

Keywords

Introduction

Mathematical formulation

Rosseland approximation

Similarity transformation

Physical quantities of engineering interest

Numerical scheme

Result and discussion

Velocity profile

Temperature profile

Conclusion

Footnotes

Appendix

Author contributions

Declaration of conflicting interests

Funding

Data availability

ORCID iDs

References