Startseite A novel exploration of how localized magnetic field affects vortex generation of trihybrid nanofluids
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A novel exploration of how localized magnetic field affects vortex generation of trihybrid nanofluids

  • Shabbir Ahmad EMAIL logo , Kashif Ali , Fareeha Khalid , John Joseph McKeon , Tmader Alballa , Hamiden Abd El-Wahed Khalifa und Jianchao Cai EMAIL logo
Veröffentlicht/Copyright: 20. November 2023
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Nanotechnology Reviews
Aus der Zeitschrift Nanotechnology Reviews Band 12 Heft 1

Abstract

Nanofluidics have better thermal properties than regular fluids, which makes them useful for heat transfer applications. This research investigated the complex dynamics of confined magnetic forces that influence the rotation of nanostructures and vortex formation in a tri-hybrid nanofluid (Ag, Al2O3, TiO2) flow regime. The study shows that the magnetic field can change the flow and heat transfer of nanofluidic, depending on its direction and strength. The study also provides insights into the complex physics of nanofluid flow and heat transfer, which can help design devices that use nanofluids more efficiently for cooling electronics, harvesting solar energy, and generating power from fuel cells. We used a single-phase model to model the nanofluids while the governing partial differential equations were solved numerically. An alternating-direction implicit approach has been employed to analyze the impact of confined magnetic fields on the nanofluid flow and thermal properties. Unlike previous studies that assumed uniform magnetic fields, we introduced multiple confined magnetic fields in the form of horizontal and vertical strips. Using our custom MATLAB codes, we systematically examined various parameters, including the magnetic field strength, number of strips and their position, and nanoparticle volume fraction, to assess their effects on nanofluid flow and thermal characteristics. Our findings revealed that the confined Lorentz force induced the spinning of tri-hybrid nanoparticles, resulting in a complicated vortex structure within the flow regime. In the absence of a magnetic field, a single symmetric vortex can be seen in the flow field. However, the introduction of magnetic sources stretches this vortex until it splits into two smaller, weaker vortices in the lower cavity, rotating clockwise or counterclockwise. Furthermore, the magnetic field strength significantly reduces both skin friction and the Nusselt number, while Reynolds numbers mainly affect the Nusselt number.

Keywords: nanofluidics; alternating direction implicit approach; efficiency; localized magnetic field

Nomenclature

Re

Reynolds number

M ˜

magnetization property ( A m − 1 )

H ˜ ( x , y )

magnetic field intensity ( T )

∂ H ˜ / ∂ x

magnetic force components along the X-axis ( T )

Mn

magnetic number

Pr

Prandtl number

K

pyro magnetic factor ( Cm 2 K )

T ¯ c

Curie temperature (K)

V 0

constant velocity ( m s − 1 )

∂ H ˜ / ∂ y

magnetic force components along the Y-axis ( T )

Ec

Eckert number

Greek symbols

γ

magnetic field strength ( T )

μ thnf

viscosity of the tri-hybrid nanofluid ( Pa s )

σ thnf

electrical conductivity of the tri-hybrid nanofluid (S m−1)

φ ⌢ 1

volume fraction of silver

φ ⌢ 3

volume fraction of titanium dioxide

ρ ⌢ nf

density of the nanofluid ( kg m − 3 )

ε

dimensionless number

μ ¯ 0

magnetic permeability ( H m − 1 )

k thnf

thermal conductivity of the tri-hybrid nanofluid ( W m − 1 K − 1 )

φ ⌢ 2

volume fraction of aluminum oxide

ρ ⌢ thnf

density of the tri-hybrid nanofluid ( kg m − 3 )

1 Introduction

Localized magnetic fields are fascinating phenomena that occur at the intersection of fundamental physics and practical innovation. These fields vary in space due to the presence of magnetic materials or electric currents and have a wide range of applications in real-world settings, such as indoor localization, magnetic anomaly detection, magnetoencephalography, and quantum physics [1,2].

Indoor localization is one area where localized magnetic fields are particularly useful. By measuring and analyzing changes in the Earth’s magnetic field, which are caused by ferromagnetic materials, researchers and engineers can estimate the orientation and precise position of devices within a building. This has led to the development of solutions that improve navigation, security, and efficiency in indoor spaces.

Here are some specific examples of how localized magnetic fields are used in indoor localization:

  • Magnetic fingerprinting: This technique involves creating a map of the magnetic field at different locations in a building. This map can then be used to determine the location of a device by comparing its measured magnetic field to the map.

  • Trilateration: This technique uses the strength of the magnetic field at three or more known locations to determine the location of a device.

  • Motion tracking: This technique uses the changes in the magnetic field over time to track the movement of a device.

Localized magnetic fields are a promising technology for indoor localization, and their use is likely to grow in the future. This is because magnetic fields are ubiquitous, infrastructure-free, and energy-efficient.

Nanofluids are liquid solutions containing suspended nanoparticles, which can significantly enhance their thermal characteristics even at low nanoparticle concentrations. Numerous studies are devoted to unraveling the behavior of nanofluids, especially their ability to improve direct heat transfer. This improvement is of paramount importance across various industrial sectors, such as nuclear reactors, food production, electronics, transportation, and biomedical applications. The superior thermophysical properties of nanofluids are due to the presence of nanoparticles, which leverage their nanoscale structural features to augment the fluid’s effective thermal conductivity. For example, in a study conducted by Gürdal et al. [3], iron oxide (Fe3O4) and copper (Cu) nanoparticles were used in a hybrid nanofluid study. They observed that increasing the nanoparticle volume fraction and magnetic interaction parameter led to higher Nusselt number (Nu) and friction factors. Xian et al. [4] chose a base fluid consisting of ethylene glycol and distilled water to create hybrid nanofluids enriched with graphene nanoplatelets and titanium dioxide (TiO2). Meanwhile, Agrawal and Kaswan [5] explored the flow of hybrid nanoparticles (Ag–Fe3O4) within an ethylene glycol and water-based medium, considering Darcy–Forchheimer effects and entropy generation. Ahmad et al. [6,7,8,9,10,11,12,13] have made notable contributions to the investigation of the thermal properties of various nanofluids in intriguing problem domains. Finally, Urmi et al. [14] performed a comprehensive thermophysical analysis of hybrid nanofluids (TiO2–Al2O3) in an ethylene glycol-based suspension, where the ethylene glycol concentration was 40%.

Trihybrid nanofluids offer remarkable advantages over both hybrid and simple nanofluids, exhibiting enhanced heat transfer capabilities and improved thermal conductivity [15,16]. This leads to higher cooling efficiency and greater energy conservation potential across a wide range of applications, including nuclear reactors, electric ovens, and microfluidic devices. However, the efficacy of trihybrid nanofluids depends on several critical factors, such as composition ratios, nanoparticle sizes, types, and shapes [16,17]. In a study by Jakeer et al. [18], they investigated the flow of magneto copper-based nanofluid within a non-Darcy porous square cavity, using the Cattaneo–Christov heat flux model. They found that the heat transfer and fluid flow dynamics were influenced by the presence of the heated obstacle. A higher Hartmann number (Ha) was observed to reduce the Nu, while hybrid nanofluids outperformed conventional nanofluids in terms of heat transport. In a study by Hameed et al. [19], they examined two-dimensional Casson flow saturated with a hybrid nanostructure on a nonlinear extending surface exposed to a magnetic field, absorption, heat generation, and viscous dissipation. Their main objective was to enhance the heat transport relations within engineering industries and manufacturing. Jalili et al. [20] addressed the thermal diffusivity gradient in stenosis arteries using Al2O3–Cu-based nanofluid and various numerical methods (AGM, Runge–Kutta, finite element method). Their results indicated that increasing nanoparticle volume fraction decreased the temperature evolution while increasing the heat source had the opposite effect. Moreover, aluminum oxide (Al2O3) had a more pronounced impact on lowering the temperature profile compared to copper (Cu). Sajid et al. [21] analyzed the fluid flow between spinning discs in a porous medium using two nanofluid models (Yamada-Ota and Xue) containing tri-hybrid nanoparticles and diathermic oil. They systematically explored how these characteristics affected the nanofluid behavior.

Natural convection flow phenomena primarily revolve around the heat transport capability of the working fluid. Early in the twenty-first century, scientists embarked on a quest to enhance fluid thermal conductivities by incorporating minuscule nanocrystalline particles into carrier fluids [22,23,24]. The impact of nanofluids on the thermo-hydraulic efficiency of heat transfer devices was elucidated by Awais et al. [25]. Meanwhile, Zhang et al. [26] delved into the realm of a hybrid nanofluid (CuO–Al2O3/water), investigating its flow efficiency and turbulent thermal behavior within a round tube. Khan [27] conducted a study involving a novel nanocomposite fluid, TiO2–Ag/water, designed to augment thermal conductivity. A comparison was drawn between this nanocomposite and a monolithic TiO2/water nanofluid. The research conducted by Ahmad et al. [28,29,30,31] focused on elucidating the heat and mass transport characteristics of semi-hybrid nanofluids, incorporating various nanoparticles such as iron oxide (Fe3O4), graphene oxide (GO), manganese zinc ferrite (MnZnFe2O4), and silver (Ag). They also examined the influence of physical parameters on friction factors, heat transfer rates, temperature profiles, and fluid velocities. Significantly enhanced heat transport rates were observed in the case of hybrid nanofluids when compared to conventional nanofluids. Lone et al. [32] investigated the flow of microconstituents over a flat surface, incorporating hybrid nanoparticles under the influence of mixed convection and magnetohydrodynamics. For a comprehensive assessment of effective nanoparticle selection and implementation in various thermal engineering equipment, Tahmooressi et al. [33] scrutinized the effects of nanoparticles on nanofluid thermal conductivity using the lattice Boltzmann method (LBM), employing mesoscale numerical simulations.

The detailed examination of delaminated single-layer and multilayer nanofluids, utilizing ethylene glycol (EG) as the base fluid, was conducted by Bao et al. [34] to explore their overall performance characteristics. The significance of enhanced thermal performance is evident in various systems, where it leads to cost reduction, increased effectiveness, and better environmental effects. Such advancements are particularly crucial in both cooling and heating power systems. Cavity flows play a pivotal role in numerous applications, such as solar collectors, boilers, subsurface water flow, and nuclear reactors, among others. Researchers have extensively explored the dynamics of flows within cavities. As highlighted by Zumbrunnen et al. [35], cavity flows find applications in drag-reducing riblets, where fine polymer composites are employed, as well as in mixing cavities, which offer insights into the eddy structures within these flows. Additional insights into the utilization of cavity flows are provided by Shankar and Deshpande [36]. Khan et al. [37] investigated the non-Newtonian liquid behavior within a square cavity subjected to external and internal forces. Their study involved constant slip velocity on the upper lid and a constant temperature on the lower boundary, disrupting thermal equilibrium, while the remaining walls remained cold. Zhang et al. [38] employed a finite volume approach to model laminar mixed convection in a water/Ag nanofluid with variable Grashof (Gr) and Richardson (Ri) numbers. They observed that the Nu increased as Ri decreased and Gr increased.

This study explores a novel synergy of localized magnetic fields, tri-hybrid nanofluids, and lid-driven square cavity flow, introducing additional boundary conditions to the flow dynamics. Within the flow field, a vertical magnetic strip is introduced, and its influence on flow characteristics is examined. These findings could mark the pioneering insights into how the imposed Lorentz force contributes to the emergence of distinct vortices. The main objective of this investigation is to explore the intricate interaction between tri-hybrid nanofluids and magnetic forces within a square enclosure. The magnetic force induces rotation in the hybrid nanostructures, leading to the complex formation of vortices.

2 Problem description

Figure 1 shows a complex diagram with a two-dimensional square cavity, having a length of side “L” where a powerful magnetic field in the shape of strips is embedded into the walls of the cavity. To understand the amazing phenomena inside the cavity, we use the single-phase model, a brilliant tool that applies the science of fluid dynamics and thermodynamics. The stream vorticity formulation solves the basic equations of mass, momentum, and energy conservation. These equations are expressed in a two-dimensional Cartesian coordinate system, revealing the intricate patterns of fluid flow and heat transfer inside an enclosure.

Figure 1 Illustration of the portion of the magnetic field in a diagram using straight arrows.
Figure 1

Illustration of the portion of the magnetic field in a diagram using straight arrows.

2.1 Basic assumptions

The following assumptions are the basis of this study:

  • A magnetic source produces a magnetic field with intensity H, given by the following equations:

H ˜ 1 ( x , y ) = H 0 { tanh A ′ 1 ( x − x 1 ) − tanh A ′ 2 ( x − x 2 ) } ,

H ˜ 2 ( x , y ) = H 0 { tanh A ′ 1 ( x − x 3 ) − tanh A ′ 2 ( x − x 4 ) } ,

H ˜ 3 ( x , y ) = H 0 { tanh A ′ 1 ( y − y 1 ) − tanh A ′ 2 ( y − y 2 ) } ,

H ˜ 4 ( x , y ) = H 0 { tanh A ′ 1 ( y − y 3 ) − tanh A ′ 2 ( y − y 4 ) } .

  • A magnetic field is confined to a strip defined by x 1 ≤ x ≤ x 2 , x 3 ≤ x ≤ x 4 ; 0 ≤ y ≤ L , and y 1 ≤ y ≤ y 2 , y 3 ≤ y ≤ y 4 ; 0 ≤ x ≤ L , respectively.

  • A square cavity with insulated vertical walls and non-insulated horizontal walls that have different temperatures and move in opposite directions with the same velocity.

  • The tri-hybrid nanofluid has Newtonian, laminar, and incompressible properties. It is created by the interaction of solid nanostructures containing Ag, Al2O3, and TiO2 with water, which is the base fluid.

  • The thermophysical characteristics of the nanofluid, including parameters like thermal conductivity, density, viscosity, and specific heat, remain consistent.

3 Mathematical formulation

These equations can be expressed in a dimensional form as follows [39]:

Continuity equation

(1) ∂ U ˜ ∂ x + ∂ V ˜ ∂ y = 0 ,

Momentum equation

(2) ∂ U ˜ ∂ t ′ + V ˜ ∂ U ˜ ∂ y + U ˜ ∂ U ˜ ∂ x = − 1 ρ ⌢ hnf ∂ P ∂ x + υ ⌢ hnf ∂ 2 U ˜ ∂ y 2 + ∂ 2 U ˜ ∂ x 2 + μ ¯ o M ˜ ρ ⌢ hnf ∂ H ˜ ∂ x ,

(3) ∂ V ˜ ∂ t ′ + U ˜ ∂ V ˜ ∂ x + V ˜ ∂ V ˜ ∂ y = − 1 ρ ⌢ hnf ∂ P ∂ y + υ ⌢ hnf ∂ 2 V ˜ ∂ x 2 + ∂ 2 V ˜ ∂ y 2 + μ ¯ o M ˜ ρ ⌢ hnf ∂ H ˜ ∂ y .

Energy equation

(4) ( ρ ⌢ c p ) hnf k hnf U ˜ ∂ T ∂ x + V ˜ ∂ T ∂ y + μ ¯ o k hnf T ∂ M ˜ ∂ T V ˜ ∂ H ˜ ∂ y + U ˜ ∂ H ˜ ∂ x = ∇ 2 T + μ ˜ hnf k hnf 2 ∂ U ˜ ∂ x 2 + ∂ V ˜ ∂ x + ∂ U ˜ ∂ y 2 + 2 ∂ V ˜ ∂ y 2 ,

where

μ ¯ 0 M ˜ ∂ H ˜ ∂ x The X-axis magnetic force components μ ¯ 0 M ˜ ∂ H ˜ ∂ y The Y-axis magnetic force components
μ ¯ 0 T ∂ M ˜ ∂ T U ˜ ∂ H ˜ ∂ x + V ˜ ∂ H ˜ ∂ y Magneto-caloric phenomenon γ Magnetic field strength
M ˜ = K H ˜ ( T ¯ c − T ) . Magnetization property T ¯ c Curie temperature [40]
∂ U ∼ ∂ t ′ + V ∼ ∂ U ∼ ∂ y + U ∼ ∂ U ∼ ∂ x Convection terms ∂ 2 U ∼ ∂ y 2 + ∂ 2 U ∼ ∂ x 2 Diffusion terms
∂ V ∼ ∂ t ′ + U ∼ ∂ V ∼ ∂ x + V ∼ ∂ V ∼ ∂ y Convection terms ∂ 2 U ∼ ∂ y 2 + ∂ 2 U ∼ ∂ x 2 Diffusion terms

By removing the pressure component, we obtain the following equation:

(5) ∂ ∂ t ′ ∂ U ˜ ∂ y − ∂ V ˜ ∂ x + V ⌢ ∂ ∂ y ∂ U ˜ ∂ y − ∂ V ˜ ∂ x + U ˜ ∂ ∂ x ∂ U ˜ ∂ y − ∂ V ˜ ∂ x = υ ⌢ thnf ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ∂ U ˜ ∂ y − ∂ V ˜ ∂ x + ∂ μ ¯ 0 M ˜ ρ ⌢ thnf ∂ H ˜ ∂ x ∂ y − ∂ μ ¯ 0 M ˜ ρ ⌢ thnf ∂ H ˜ ∂ y ∂ x .

3.1 Boundary conditions

The dimensional boundary conditions that apply to the current problem are as follows:

Left and right vertical walls (adiabatic). No heat flux occurs across these walls, that is,

(6a) U ˜ ( 0 , y ) = U ˜ ( L , y ) = 0 , ∂ T ∂ x x = 0 = ∂ T ∂ x x = L = 0 , V ˜ ( 0 , y ) = V ˜ ( L , y ) = 0 ; 0 < y < L .

Upper horizontal wall

(6b) U ˜ ( x , L ) = − V 0 , T ( x , L ) = T c , V ˜ ( x , L ) = 0 ; 0 < x < L .

Lower horizontal wall

(6c) U ˜ ( x , 0 ) = V 0 , T ( x , 0 ) = T h , V ˜ ( x , 0 ) = 0 ; 0 < x < L .

3.2 Features of tri-hybrid nanofluids comprising Ag, Al2O3, and TiO2

This section presents the thermophysical properties of the tri-hybrid nanofluids that are used in this analysis. These properties are essential for the investigation of the heat transport problem. We consider a specific combination of thermophysical properties for the tri-hybrid nanofluids composed of Ag–Al2O3–TiO2 nanoparticles. These properties (such as density, specific heat, thermal conductivity, and viscosity) are confirmed from the existing literature [41,42,43,44,45]. The thermophysical properties of tri-hybrid nanofluids versus conventional nanofluids are explained in Table 1.

For tri-hybrid nanofluids For hybrid nanofluids For nanofluids For base fluids (water) For solid
Silver ( s 1 ) Aluminum oxide ( s 2 ) Titanium dioxide( s 3 )
Viscosity μ thnf μ hnf μ nf μ f
Density ρ ⌢ thnf ρ ⌢ f ρ ⌢ s 1 ρ ⌢ s2 ρ ⌢ s3
Electrical conductivity σ thnf σ hnf σ nf σ f σ s 1 σ s2 σ s 3
Thermal conductivity k thnf k hnf k nf k f k s 1 k s2 k s 3
Nanoparticles volume fraction φ ⌢ 1 φ ⌢ 2 φ ⌢ 3

Table 1

Tri-hybrid nanofluids versus conventional nanofluids: a thermophysical analysis

Properties Tri-hybrid nanofluid (thnf)
Density ρ ⌢ thnf = ( 1 ‒ φ ⌢ 3 ) [ ( 1 ‒ φ ⌢ 1 ) { ( 1 ‒ φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 } + φ ⌢ 2 ρ ⌢ s 2 ] + φ ⌢ 3 ρ ⌢ s 3
Heat capacity ( ρ ⌢ cp ) thnf = ( 1 ‒ φ ⌢ 3 ) [ ( 1 ‒ φ ⌢ 2 ) { ( 1 ‒ φ ⌢ 1 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 } + φ ⌢ 2 ρ ⌢ s 2 ] + φ ⌢ 3 ρ ⌢ s 3
Viscosity μ thnf = μ f ( 1 ‒ φ ⌢ 1 ) 2.5 ( 1 ‒ φ ⌢ 2 ) 2.5 ( 1 ‒ φ ⌢ 3 ) 2.5
Thermal conductivity k thnf k hnf = k s 3 + ( n ‒ 1 ) ( k hnf ‒ ( n ‒ 1 ) φ ⌢ s 3 ) ( k hnf ‒ k s 3 ) k s 3 + ( n ‒ 1 ) k hnf + φ ⌢ s 3 ( k hnf ‒ k s 3 )
where k hnf k nf = k s 2 ‒ ( n ‒ 1 ) φ ⌢ 2 ( k nf ‒ k s 3 ) + ( n ‒ 1 ) k nf k s 2 + φ ⌢ s 2 ( k nf ‒ k s 3 ) + ( n ‒ 1 ) k nf
and k nf k f = k s 1 ‒ ( n ‒ 1 ) φ ⌢ 1 ( k f ‒ k s 1 ) + ( n ‒ 1 ) k f k s 1 + φ ⌢ 1 ( k f ‒ k s 1 ) + ( n ‒ 1 ) k f
Electric conductivity σ thnf σ nf = σ s 3 ‒ 2 φ ⌢ 3 ( σ nf ‒ σ s 3 ) + 2 σ nf σ s 3 + φ ⌢ 3 ( σ nf ‒ σ s 3 ) + 2 σ nf
where σ hnf σ nf = σ s 2 ‒ 2 φ ⌢ 2 ( σ nf ‒ σ s 2 ) + 2 σ nf σ s 2 + φ ⌢ 2 ( σ nf ‒ σ s 2 ) + 2 σ nf
and σ nf σ f = σ s 1 ‒ 2 φ ⌢ 1 ( σ f ‒ σ s 1 ) + 2 σ f σ s 1 + φ ⌢ 1 ( σ f ‒ σ s 1 ) + 2 σ f

To perform the analysis, the following dimensionless variables are used:

(7) ξ = x L ⌢ , y = y L ⌢ , u = U ˜ V 0 , v = V ˜ V 0 , θ ⌢ = T − T c Δ T , H = H ˜ H 0 , t = V 0 L ⌢ t ′ .

We can infer from equations (4) and (5) that

(8) ∂ J ∂ t + u ∂ J ∂ ξ + v ∂ J ∂ η = ( 1 − φ ⌢ 1 ) ( 1 − φ ⌢ 2 + φ ⌢ 3 ( ρ ⌢ s 1 / ρ ⌢ f ) + φ ⌢ 2 ( ρ ⌢ s 2 / ρ ⌢ f ) ) ( 1 − φ ⌢ 2 ) 2.5 ( 1 − φ ⌢ 3 ) 2.5 1 Re ∇ 2 J + Mn ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 1 + φ ⌢ 2 ( ρ ⌢ s 1 / ρ ⌢ f ) + φ ⌢ 3 ( ρ ⌢ s 2 / ρ ⌢ f ) ) H ∂ H ∂ η . ∂ θ ⌢ ∂ ξ − ∂ H ∂ ξ . ∂ θ ⌢ ∂ η ,

(9) ∇ 2 θ ⌢ = Pr × ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 1 + φ ⌢ 2 ( ρ ⌢ s 1 / ρ ⌢ f ) + φ ⌢ 3 ( ρ ⌢ s 2 / ρ ⌢ f ) ) ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 2 ) − 2.5 × ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 1 + φ ⌢ 2 ( ρ ⌢ s 1 × c p ρ ⌢ s 1 / ρ ⌢ f × c p f ) + φ ⌢ 3 ( ρ ⌢ s 3 × c p ρ ⌢ s 3 / ρ ⌢ f × c p f ) ) Re × ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 2 ) − 2.5 ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 1 + φ ⌢ 2 ( ρ ⌢ s 1 / ρ ⌢ f ) + φ ⌢ 2 ( ρ ⌢ s 3 / ρ ⌢ f ) ) ∂ θ ⌢ ∂ ξ ∂ ψ ˜ ∂ η − ∂ θ ⌢ ∂ η ∂ ψ ˜ ∂ ξ + Pr × ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 1 + φ ⌢ 2 ( ρ ⌢ s 1 / ρ ⌢ f ) + φ ⌢ 3 ( ρ ⌢ s 2 / ρ ⌢ f ) ) ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 2 ) − 2.5 × ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 1 + φ ⌢ 1 ( ρ ⌢ s 1 × c p ρ ⌢ s 1 / ρ ⌢ f × c p f ) + φ ⌢ 2 ( ρ ⌢ s 2 × c p ρ ⌢ s 2 / ρ ⌢ f × c p f ) ) × Mn ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 1 + φ ⌢ 1 ( ρ ⌢ s 1 / ρ ⌢ f ) + φ ⌢ 2 ( ρ ⌢ s 2 / ρ ⌢ f ) ) Re × ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 3 ) − 2.5 ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 1 + φ ⌢ 1 ( ρ ⌢ s 1 / ρ ⌢ f ) + φ ⌢ 3 ( ρ ⌢ s 3 / ρ ⌢ f ) ) Ec H ( ε − ψ ˜ ) ∂ H ∂ ξ ∂ ψ ˜ ∂ η − ∂ H ∂ η ∂ ψ ˜ ∂ ξ + Pr × ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 1 + φ ⌢ 1 ( ρ ⌢ s 1 / ρ ⌢ f ) + φ ⌢ 3 ( ρ ⌢ s 2 / ρ ⌢ f ) ) ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 2 ) − 2.5 × ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 1 + φ ⌢ 1 ( ρ ⌢ s 1 × c p ρ ⌢ s 1 / ρ ⌢ f × c p f ) + φ ⌢ 2 ( ρ ⌢ s 2 × c p ρ ⌢ s 2 / ρ ⌢ f × c p f ) ) Ec ∂ 2 ψ ˜ ∂ η 2 − ∂ 2 ψ ˜ ∂ ξ 2 2 + 4 ∂ 2 ψ ˜ ∂ ξ ∂ η 2 ,

where

(10) H 1 ( ξ , η ) = H 0 { tanh A ′ 1 ( ξ − ξ 1 ) − tanh A ′ 2 ( ξ − ξ 2 ) } H 2 ( ξ , η ) = H 0 { tanh A 1 ( ξ − ξ 3 ) − tanh A 2 ( ξ − ξ 4 ) } H 3 ( ξ , η ) = H 0 { tanh A ′ 1 ( η − η 1 ) − tanh A ′ 2 ( η − η 2 ) } H 4 ( ξ , η ) = H 0 { tanh A 1 ( η − η 3 ) − tanh A 2 ( η − η 4 ) } ,

in the strips defined by ξ 1 ≤ ξ ≤ ξ 2 , ξ 3 ≤ ξ ≤ ξ 4 ; 0 ≤ η ≤ 1 and η 1 ≤ η ≤ η 2 , η 3 ≤ η ≤ η 4 ; 0 ≤ ξ ≤ 1 , respectively.

Finally,

(11) H ( ξ , η ) = H 1 ( ξ , η ) + H 2 ( ξ , η ) + H 3 ( ξ , η ) + H 4 ( ξ , η )

We use the stream–vorticity formulation, which modifies equations (1)–(4) with

(12) u ⌢ = ∂ ψ ˜ ∂ η , v ⌢ = ∂ ψ ˜ ∂ ξ and ∂ u ⌢ ∂ η − ∂ v ⌢ ∂ ξ = − ω ⌢ or ∂ 2 ψ ˜ ∂ ξ 2 + ∂ 2 ψ ˜ ∂ η 2 = − ω ⌢ .

The dimensionless boundary conditions are the following:

Left and right vertical walls (adiabatic)

(13a) u ⌢ ( 0 , η ) = u ⌢ ( 1 , η ) = 0 , ∂ θ ∂ ξ ξ = 0 = 0 , ∂ θ ∂ ξ ξ = 1 = 0 , v ⌢ ( 0 , η ) = v ⌢ ( 1 , η ) = 0 , 0 < η < 1 .

Upper horizontal wall

(13b) u ⌢ ( ξ , 1 ) = − 1 , θ ⌢ ( ξ , 1 ) = 0 , v ⌢ ( ξ , 1 ) = 0 ; 0 < ξ < 1 .

Lower horizontal wall

(13c) u ⌢ ( ξ , 0 ) = 1 , θ ⌢ ( ξ , 0 ) = 1 , v ⌢ ( ξ , 0 ) = 0 ; 0 < ξ < 1 .

3.3 Physical quantities of interest

Fluid dynamics, the study of how fluids (liquids and gases) behave and interact under various conditions, involves the analysis of a wide range of physical quantities to understand the behavior of fluids. In this work, we are interested in studying the heat transfer (Nu) and the wall shear stress (CfRe).

Nusselt number: Nu is a dimensionless quantity used in the field of heat transfer. It relates the convective heat transfer rate in a fluid to the conductive heat transfer rate through a stationary medium. In other words, it helps us understand how effectively heat is being transferred from a solid surface to a fluid that flows over or around it. The Nu takes into account the effect of convection, which enhances heat transfer due to the fluid motion.

Mathematically, the Nu is defined as

Nu = q ⌢ L k hnf Δ T ,

where q ⌢ = − k hnf ( ∂ T ∂ η ) η = 0 , L is the heat flux, ΔT is the temperature difference between both horizontal plates, L is a characteristic length related to the geometry of the system, and k hnf is the thermal conductivity of the tri-hybrid nanofluid.

Using the dimensionless variables, we obtain

Nu = k thnf k nf ∂ θ ⌢ ∂ y .

A higher Nu indicates more efficient heat transfer. It is often used to correlate and predict heat transfer coefficients in various situations, such as the fluid flow over a solid surface, heat exchangers, and other convective heat transfer scenarios.

Skin friction: Skin friction is a term used in fluid dynamics to describe the frictional force per unit area acting on a fluid as it flows over a solid surface. It is a crucial concept in understanding the resistance a fluid encounters when moving across a surface. Skin friction is primarily responsible for the drag force experienced by objects moving through a fluid, like an aircraft through air or a ship through water.

Mathematically, skin friction is related to the shear stress ( τ ) at the solid–fluid interface:

CfRe = 2 τ ρ ⌢ hnf v 0 2 ,

where τ = μ ˜ hnf ( ∂ U ⌢ ∂ η ) η = 0 , L .

Using the dimensionless variables, we have

CfRe = 2 [ ( 1 − φ ⌢ 1 ) ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 3 ) ] 2.5 ( 1 − φ ⌢ 3 ) ( 1 − φ ⌢ 2 ) 1 − φ ⌢ 1 + φ ⌢ 1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 ρ ⌢ s 2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s 3 ρ ⌢ f ∂ u ⌢ ∂ y .

Higher skin friction values indicate greater resistance to fluid flow and higher drag forces. Reducing skin friction is an important consideration in the design of streamlined surfaces to improve the efficiency of vehicles and reduce energy consumption.

In short, the Nu quantifies convective heat transfer efficiency, while skin friction measures the drag force and resistance encountered by a fluid as it flows over a solid surface. The Nu and skin friction are indispensable tools for engineers and researchers in a wide range of fields from heat exchangers to aerodynamics. They provide insights into heat transfer efficiency, fluid dynamics, and the design of energy-efficient systems. By leveraging these concepts, engineers can optimize designs, improve energy efficiency, and enhance the performance of various engineering applications.

4 Numerical techniques

A numerical method has been used to solve equations (4)–(6) that describe the fluid flow and heat transfer in a cavity. The method is called ADI, which means alternating direction implicit. It solves the equations in one direction at a time, while keeping the other direction fixed. This makes the solution faster and more stable. The method also uses central difference, which means using the average values of the nearby points to estimate the change at a point. This is a good and accurate way to change the equations into algebraic equations on a grid. The method we use is a type of finite difference method, which is a general way to solve governing equations. Finite differences can also help us find changes from data that are not continuous. One of the benefits of the finite difference method is that it is easy to do and can be very accurate by using more points in the stencil, which is a group of points that we use to calculate the change at a point.

The steps for solving the equations are the following:

  1. Solve the fluid flow equation in the horizontal direction, while keeping the vertical direction fixed.

  2. Solve the heat transfer equation in the horizontal direction, while keeping the vertical direction fixed.

  3. Solve the fluid flow equation in the vertical direction, using the solution from step 1.

  4. Solve the heat transfer equation in the vertical direction, using the solution from step 2.

  5. We repeat these steps until we get the final solution.

(14) w ⌢ i , j ( n + 1 / 2 ) − w ⌢ i , j ( n ) δ t / 2 = Re × ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 2 ) − 2.5 ( 1 − φ ⌢ 3 ) − 2.5 ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 ρ ⌢ s 3 / ρ ⌢ f ) × w ⌢ i − 1 , j ( n + 1 / 2 ) − 2 w ⌢ i , j ( n + 1 / 2 ) + w ⌢ i + 1 , j ( n + 1 / 2 ) h 2 + w ⌢ i , j − 1 ( n ) − 2 w ⌢ i , j ( n ) + w ⌢ i , j + 1 ( n ) k 2 + Mn ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 ρ ⌢ s 3 / ρ ⌢ f ) × H i , j H i , j + 1 − H i , j − 1 2 k θ ⌢ i + 1 , j ( n ) − θ ⌢ i − 1 , j ( n ) 2 h − H i + 1 , j − H i − 1 , j 2 h θ ⌢ i , j + 1 ( n ) − θ ⌢ i , j − 1 ( n ) 2 k − u ⌢ i , j ( n + 1 / 2 ) w ⌢ i + 1 , j ( n + 1 ) − w ⌢ i − 1 , j ( n + 1 ) 2 h − v ⌢ i , j ( n + 1 / 2 ) w ⌢ i , j + 1 ( n ) − w ⌢ i , j − 1 ( n ) 2 k ,

(15) θ ⌢ i , j ( n + 1 / 2 ) − θ ⌢ i , j ( n ) δ t / 2 = θ ⌢ i − 1 , j ( n + 1 / 2 ) − 2 θ ⌢ i , j ( n + 1 / 2 ) + θ ⌢ i + 1 , j ( n + 1 / 2 ) h 2 + θ ⌢ i , j − 1 ( n ) + θ ⌢ i , j + 1 ( n ) − 2 θ ⌢ i , j ( n ) k 2 + Pr × ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 ρ ⌢ s 3 / ρ ⌢ f ) ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 2 ) − 2.5 ( 1 − φ ⌢ 3 ) − 2.5 × ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 c p s 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 c p S 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 c p S 3 ρ ⌢ s 3 / ρ ⌢ f ) × Mn ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 ρ ⌢ s 3 / ρ ⌢ f ) × Re × ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 2 ) − 2.5 ( 1 − φ ⌢ 3 ) − 2.5 ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 1 + φ ⌢ 1 ( ρ ⌢ s 1 / ρ ⌢ f ) + φ ⌢ 2 ( ρ ⌢ s 2 / ρ ⌢ f ) ) H i , j ( ψ ˜ i , j − ε ) u ⌢ i , j ( n + 1 / 2 ) H i , j + 1 − H i , j − 1 2 k + v ⌢ i , j ( n + 1 / 2 ) H i + 1 , j − H i − 1 , j 2 h − u ⌢ i , j ( n + 1 / 2 ) θ ⌢ i + 1 , j ( n + 1 / 2 ) − θ ⌢ i − 1 , j ( n + 1 / 2 ) 2 h − v ⌢ i , j ( n + 1 / 2 ) θ ⌢ i , j + 1 ( n ) − θ ⌢ i , j − 1 ( n ) 2 k − Pr × ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 ρ ⌢ s 3 / ρ ⌢ f ) ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 2 ) − 2.5 × ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 c p S 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 c p S 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 c p S 3 ρ ⌢ s 3 / ρ ⌢ f ) × Ec u ⌢ i , j + 1 ( n + 1 / 2 ) − u ⌢ i , j − 1 ( n + 1 / 2 ) 2 k + v ⌢ i + 1 , j ( n + 1 / 2 ) − v ⌢ i − 1 , j ( n + 1 / 2 ) 2 h 2 + 4 u ⌢ i + 1 , j ( n + 1 / 2 ) − u ⌢ i − 1 , j ( n + 1 / 2 ) 2 h 2 ,

(16) ψ ˜ i − 1 , j ( n + 1 ) + ψ ˜ i + 1 , j ( n + 1 ) − 2 ψ ˜ i , j ( n + 1 ) h 2 + ψ ˜ i , j − 1 ( n + 1 ) + ψ ˜ i , j + 1 ( n + 1 ) − 2 ψ ˜ i , j ( n + 1 ) k 2 = − w ⌢ i , j ( n + 1 / 2 ) ,

(17) u ⌢ i , j ( n + 1 ) = − ψ ˜ i , j − 1 ( n + 1 ) + ψ ˜ i , j + 1 ( n + 1 ) 2 k ,

(18) v ⌢ i , j ( n + 1 ) = − ψ ˜ i + 1 , j ( n + 1 ) − ψ ˜ i − 1 , j ( n + 1 ) 2 h ,

(19) w ⌢ i , j ( n + 1 ) − w ⌢ i , j ( n + 1 / 2 ) δ t / 2 = Re × ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 2 ) − 2.5 ( 1 − φ ⌢ 3 ) − 2.5 ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 ρ ⌢ s 3 / ρ ⌢ f ) × w ⌢ i − 1 , j ( n + 1 / 2 ) − 2 w ⌢ i , j ( n + 1 / 2 ) + w ⌢ i + 1 , j ( n + 1 / 2 ) h 2 + w ⌢ i , j − 1 ( n + 1 ) − 2 w ⌢ i , j ( n + 1 ) + w ⌢ i , j + 1 ( n + 1 ) k 2 + Mn ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 ρ ⌢ s 3 / ρ ⌢ f ) × H i , j H i , j + 1 − H i , j − 1 2 k θ ⌢ i + 1 , j ( n + 1 ) − θ ⌢ i − 1 , j ( n + 1 ) 2 h − H i + 1 , j − H i − 1 , j 2 h θ ⌢ i , j + 1 ( n + 1 ) − θ ⌢ i , j − 1 ( n + 1 ) 2 k − u ⌢ i , j ( n + 1 / 2 ) w ⌢ i + 1 , j ( n + 1 ) − w ⌢ i − 1 , j ( n + 1 ) 2 h − v ⌢ i , j ( n + 1 / 2 ) w ⌢ i , j + 1 ( n + 1 ) − w ⌢ i , j − 1 ( n + 1 ) 2 k ,

(20) θ ⌢ i , j ( n + 1 ) − θ ⌢ i , j ( ( n + 1 / 2 ) ) δ t / 2 = θ ⌢ i − 1 , j ( n + 1 / 2 ) − 2 θ ⌢ i , j ( n + 1 / 2 ) + θ ⌢ i + 1 , j ( n + 1 / 2 ) h 2 + θ ⌢ i , j − 1 ( n + 1 ) + θ ⌢ i , j + 1 ( n + 1 ) − 2 θ ⌢ i , j ( n + 1 ) k 2 + Pr × ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 ρ ⌢ s 3 / ρ ⌢ f ) ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 2 ) − 2.5 ( 1 − φ ⌢ 3 ) − 2.5 × ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 c p S 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 c p S 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 c p S 3 ρ ⌢ s 3 / ρ ⌢ f ) × Mn ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 ρ ⌢ s 3 / ρ ⌢ f ) × Re × ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 2 ) − 2.5 ( 1 − φ ⌢ 3 ) − 2.5 ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 ρ ⌢ s 3 / ρ ⌢ f ) × H i , j ( ψ ˜ i , j − ε ) u ⌢ i , j ( n + 1 / 2 ) H i , j + 1 − H i , j − 1 2 k + v ⌢ i , j ( n + 1 / 2 ) H i + 1 , j − H i − 1 , j 2 h − u ⌢ i , j ( n + 1 / 2 ) θ ⌢ i + 1 , j ( n + 1 / 2 ) − θ ⌢ i − 1 , j ( n + 1 / 2 ) 2 h − v ⌢ i , j ( n + 1 / 2 ) θ ⌢ i , j + 1 ( n + 1 ) − θ ⌢ i , j − 1 ( n + 1 ) 2 k − Pr × ( 1 − φ ⌢ 2 ) ( 1 − φ ⌢ 1 + φ ⌢ 1 ( ρ ⌢ s 1 / ρ ⌢ f ) + φ ⌢ 2 ( ρ ⌢ s 2 / ρ ⌢ f ) ) ( 1 − φ ⌢ 1 ) − 2.5 ( 1 − φ ⌢ 2 ) − 2.5 × ( ( 1 − φ ⌢ 3 ) [ ( 1 − φ ⌢ 1 ) { ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 c p S 1 ρ ⌢ s 1 / ρ ⌢ f } + φ ⌢ 2 c p S 2 ρ ⌢ s 2 / ρ ⌢ f ] + φ ⌢ 3 c p S 3 ρ ⌢ s 3 / ρ ⌢ f ) × Ec u ⌢ i , j + 1 ( n + 1 / 2 ) − u ⌢ i , j − 1 ( n + 1 / 2 ) 2 k + v ⌢ i + 1 , j ( n + 1 / 2 ) − v ⌢ i − 1 , j ( n + 1 / 2 ) 2 h 2 + 4 u ⌢ i + 1 , j ( n + 1 / 2 ) − u ⌢ i − 1 , j ( n + 1 / 2 ) 2 h 2 .

We stop the iteration scheme when we reach the following criterion:

max { abs ( ψ ˜ i , j ( n + 1 ) − ψ ˜ i , j ( n ) ) , abs ( w i , j ( n + 1 ) − w i , j ( n ) ) , abs ( θ i , j ( n + 1 ) − θ i , j ( n ) ) } < TOL,

which means that we have found a steady-state solution. We set TOL < 10−6 for this study.

4.1 Illustration of our numerical technique

Figure 2 shows the computational model for the pseudo-transient technique.

Figure 2 A flow chart of the pseudo-transient approach.
Figure 2

A flow chart of the pseudo-transient approach.

4.2 Benchmarking the numerical scheme

To verify the reliability and accuracy of our numerical approach, we conduct a rigorous validation process. Our benchmark for comparison is the reputable work of Yasmin et al. [46], whose findings provide a solid reference. We focus our attention on the horizontal velocity profiles, which clearly show the oscillations in the horizontal component of fluid velocity along a vertical axis. In our investigation, we examine a remarkable limiting scenario where several influential factors are set to zero, i.e., ( φ ⌢ 1 = φ ⌢ 2 = 0, Mn = 0). In this fascinating domain, the fluid’s motion is driven only by the motion of one of the cavity’s lids. Our analysis takes place along three different horizontal lines, strategically located at y = 0.25, 0.50, and 0.75. Here, “y” denotes the vertical coordinates, carefully normalized by the height of the lid-driven cavity, as shown in Figure 3a. The horizontal velocity along the cross-section defined by x = 0.25, 0.50, and 0.75 is given in Figure 3b, by using the present approach as well as Yasmin et al. [46]. It is obvious to note that exact comparison validates our technique.

Figure 3 Illustration of the Numerical Simulation and the Analytical Solution by Yasmin et al. [46] for the Horizontal Velocity Components along (a) the y-axis and (b) the x-axis.
Figure 3

Illustration of the Numerical Simulation and the Analytical Solution by Yasmin et al. [46] for the Horizontal Velocity Components along (a) the y-axis and (b) the x-axis.

To validate our code, we compared it with the well-known problem of natural convection within a cavity. This problem has been previously studied by Chen et al. [47] and Davis [48], who used the LBM and finite difference methods, respectively. Table 2 shows that the Nu values obtained from our numerical analysis are in good agreement with those reported in previous studies.

Table 2

Comparing our findings with previous studies: A quality assessment

Re Average Nu for the hot wall
LBM approach [47] Finite difference approach [48] Our approach
10 3 1.1192 1.1181 1.1182
10 4 2.2531 2.2432 2.2481

In this investigation, we use water as the basic base fluid, enriched with a mixture of nanoparticles including Ag, Al2O3, and TiO2. These nanoparticles are cleverly integrated into the base fluid to increase thermal conductivity and enable efficient heat transfer. In our simulations, we choose ε = 0.02 to represent the thermophysical characteristics of this remarkable nanofluid. The properties of water underlying our simulations are based on Pr = 6.2.

To provide a context for our work, we introduce two important dimensionless parameters:

  • The Reynolds number (Re) is a key indicator measuring the balance between inertial and viscous forces in fluid flow.

  • The Eckert number is another vital dimensionless parameter that describes the relationship between the kinetic energy and enthalpy change in fluid flow.

When the Re is very small, it results in a very low Eckert number (e.g., 10−5), indicating the dominance of viscous forces while kinetic energy becomes negligible. For a comprehensive understanding, Table 3 presents a detailed overview of the fundamental properties of both nanomaterials and the base fluid used in our research. These properties include density, particle size, specific heat, thermal conductivity, and viscosity, highlighting the foundation upon which our exploration rests.

Table 3

Thermal and physical features of water and nanomaterials (Ag–Al2O3–TiO2)

Thermophysical properties C p ( J kg ‒ 1 K ‒ 1 ) β ( K ‒ 1 ) ρ ⌢ ( kg m ‒ 3 ) σ ( S m ‒ 1 ) k ( W m ‒ 1 K ‒ 1 )
Water 4,179 21 × 10 ‒ 5 997.1 0.05 0.613
Silver (Ag) 235 1.89 × 10 ‒ 5 8,933 3.6 × 10 7 429
Aluminum oxide (Al2O3) 765 0.85 × 10 ‒ 5 3,970 1 × 10 ‒ 1 40
Titanium dioxide (TiO2) 686 0.90 × 10 ‒ 5 5,200 1 × 10 ‒ 12 8.95

Alternatively, Figure 4 shows the convergence of our numerical results with respect to the step size, which is the distance between adjacent grid points. As the step size decreases, our numerical findings converge smoothly. This convergence confirms the stability of our numerical technique, ensuring that it does not generate spurious oscillations or divergent results. Moreover, this convergence indicates that our numerical findings are robust to changes in the grid size – the total number of grid points used to discretize the domain. In other words, it implies that beyond a certain point, increasing the grid size does not significantly affect our numerical results. This reveals the reliability of our approach.

Figure 4 Assessment of grid independence for normal velocity distribution along the line y = 0.5.
Figure 4

Assessment of grid independence for normal velocity distribution along the line y = 0.5.

5 Results and discussion

In this section, we present and discuss the results, using figures and tables. These figures show the fluid movement and temperature inside the enclosure with different curves and colors. The tables give numbers for important things, such as the Nu and CfRe (skin friction coefficient Re) values, which measure the fluid–wall interaction and the heat transfer rate. These are essential for our analysis. We mainly look at a situation where both lids along the horizontal walls of the cavity move in opposite directions, causing the fluid to move inside the enclosure. We also consider some basic factors, such as the nanoparticle volume fraction ( 0 ≤ φ ⌢ 1 , φ ⌢ 2 , φ ⌢ 3 ≤ 0.20 ) , the Re ( 1 ≤ Re ≤ 200 ) , and the magnetic number ( 0 ≤ Mn ≤ 200 ) . We keep some values constant unless stated otherwise: Mn = 5, Re = 5, φ ⌢ 1 = 0.05, φ ⌢ 2 = 0.02, φ ⌢ 3 = 0.02, and ε = 0.02. We make our research more interesting by applying a magnetic field that has four stripes – two horizontals ( 0.2 ≤ ξ ≤ 0.3 , 0.7 ≤ ξ ≤ 0.8 , and 0 ≤ η ≤ 1 ) and two verticals ( 0.2 ≤ η ≤ 0.3 , 0.7 ≤ η ≤ 0.8 , and 0 ≤ ξ ≤ 1 ) , each with a width of 1 unit. These magnetic stripes divide the cavity into nine different parts, each with its own features. Our research also explores the tri-hybrid nanofluid, which means how Ag, Al2O3, and TiO2 work together in the fluid. From this analysis, we learn how the Nu and CfRe values change because of this tri-hybrid mixture.

5.1 Outcomes of the magnetizing force

As shown in Figure 5, the magnetic field intensity affects the flow, which is expressed by the horizontal and vertical strips in the regions ( 0.2 < ξ < 0.3 , 0.7 < ξ < 0.8 , 0 < η < 1 and 0.2 < η < 0.3 , 0.7 < η < 0.8 , 0 < ξ < 1 ) . When there is no magnetic field, there is a single vortex that is almost symmetric about the line η = 0.5 . However, as the magnetic field intensity increases, the vortex elongates and eventually breaks down into two smaller and weaker vortices. These vortices rotate in clockwise or counter-clockwise directions, depending on the direction of motion of the top and bottom lids of the cavity. The magnetic field creates a force on the fluid, which interacts with the force applied by the lids to create the vortices. The orientation of the net force determines the direction of rotation of the vortices. At higher intensities of the magnetic field, the number of vortices increases. This is because the magnetic field creates a stronger force on the fluid, which can break down the larger vortices into smaller ones. The formation of vortices can be seen from the stream surface. The stream surface is a visualization of the fluid flow, and it shows how the fluid is flowing in different directions. The vortices appear as regions where the fluid is rotating. The physical mechanism behind the formation of vortices is complex, but it can be understood in terms of the forces acting on the fluid. The magnetic field creates a force on the fluid, and the lids also create a force on the fluid. The net force on the fluid determines the direction of rotation of the vortices.

Figure 5 Visualization of streamlines and stream surfaces at various Mn.
Figure 5

Visualization of streamlines and stream surfaces at various Mn.

Figure 6 shows how the parameter Mn affects the temperature in the flow field. Without any magnetic force, most of the isotherms in the flow field are red, meaning higher temperatures. Also, the isotherms are close together near the upper and lower walls of the cavity. With magnetic force, the temperature changes considerably. The isotherms are not linear anymore in any part of the cavity, which is the cause of new vortices in the flow. These vortices mix fluid masses with different temperatures, making the temperature vary throughout the enclosure. This shows how the magnetic force makes the temperature gradients interact, causing turbulence in the temperature profile across the enclosure. This shows the complex relationship between magnetic forces, fluid dynamics, and temperature distribution.

Figure 6 Visualization of temperature fields and isotherms at various Mn.
Figure 6

Visualization of temperature fields and isotherms at various Mn.

5.2 Impact of the Re

Figure 7 shows how the Re affects the flow dynamics in a fixed cavity size and with constant fluid properties. A higher Re corresponds to faster motion of the cavity’s lids. As the Re increases, the main vortex in the flow field is affected, and new vortices appear. This makes the flow field more complex and interesting. It is important to note that the equations governing the flow field have a term that depends on both the magnetic parameters and the Re. Therefore, a higher Re also increases the effect of this term in the model. This means that the Re and the magnetic parameter have similar effects on the flow field, and they both are connected to how the system behaves.

Figure 7 Visualization of streamlines and stream surfaces at various Re.
Figure 7

Visualization of streamlines and stream surfaces at various Re.

Figure 8 shows how the Re affects the flow characteristics in a cavity with constant dimensions and fluid properties. As Re increases, the velocity of the fluid increases, especially when the upper and lower lids are moving in opposite directions. A careful examination of the streamlines shows that the localized magnetic field has a greater impact on the flow pattern as Re increases. This is because the product of Re and the Mn appears in the governing equations for the problem. This observation highlights the strong interaction between Re and Mn, and their combined influence on the flow behavior. This is a significant finding in our study.

Figure 8 Visualization of temperature fields and isotherms at various Re.
Figure 8

Visualization of temperature fields and isotherms at various Re.

5.3 Impact of localization

The width of the magnetic field plays an interesting role in this problem. The strips that encompass the flow field are defined by ( 0.2 − L < ξ < 0.3 + L , 0.7 − L < ξ < 0.8 + L , 0 < η < 1 ) and ( 0.2 − L < η < 0.3 + L , 0.7 − L < η < 0.8 + L , 0 < ξ < 1 ) . The parameter L controls the width of these strips, where L = 0.2 means that the magnetic field engulfs the entire cavity, and L = 0.0 means that the magnetic field is confined to the designated strips 0.2 < ξ < 0.3 , 0.7 < ξ < 0.8 , 0 < η < 1 and 0.2 < η < 0.3 , 0.7 < η < 0.8 , 0 < ξ < 1 . The flow and temperature distributions are minimally affected when the magnetic field extends throughout the entire flow field. However, Figure 9 shows that confining the magnetic field to specific strips can cause significant changes. This confinement leads to the formation of new vortices, which break up the primary vortex. Additionally, the isotherms (lines of constant temperature) take on a zigzag trajectory due to the vigorous mixing of fluid layers at different temperatures, as shown in Figure 10. These observations shed light on the complex relationship between magnetic field width and its effects on flow dynamics and thermal behavior, which enriches our understanding of the problem.

Figure 9 Streamlines for different values of the magnetic strip lengths (L) for Mn = 150, Re = 5.
Figure 9

Streamlines for different values of the magnetic strip lengths (L) for Mn = 150, Re = 5.

Figure 10 Isotherms for different values of the magnetic strip lengths (L) for Mn = 150, Re = 5.
Figure 10

Isotherms for different values of the magnetic strip lengths (L) for Mn = 150, Re = 5.

5.4 Variations in Nu and CfRe with different parameters

Now, we will investigate the effects of the governing parameters, namely, the Re and the Mn, on the skin friction factor (CfRe) and the Nu. As shown in Figures 11 and 12, the magnetic field significantly reduced heat transfer in almost half of the domain. On the other hand, the Re had a significant positive effect on the Nu, more than the Mn. Both parameters had a similar qualitative effect on the skin friction factor but the quantitative effects were slightly different. The nanoparticle composition of the fluid had a negligible effect on the skin friction factor. As expected, the nanoparticle volume fractions φ ⌢ 1 , φ ⌢ 2 , φ ⌢ 3 had a significant effect on the Nu. Increasing ϕ increased the thermal conductivity of the nanofluid, which led to higher heat transfer coefficients and Nu. However, this also increased the fluid’s viscosity, which led to higher frictional resistance and skin friction factors. It is worth noting that the effect of viscosity on skin friction is less pronounced than the effect of thermal conductivity on Nu. The skin friction factor is affected by several factors, including the Re, enclosure geometry, and flow regime. Therefore, the nanoparticle volume fraction has a greater effect on Nu than on CfRe. An interesting finding is that confining the magnetic field does not always increase or decrease the Nu. This suggests that there is an optimal width for the magnetic corridor, which could be explored in future studies.

Figure 11 Variation of CfRe with different parameters.
Figure 11

Variation of CfRe with different parameters.

Figure 12 Variation of Nu with different parameters.
Figure 12

Variation of Nu with different parameters.

5.5 Effect of solid volume concentration (Ag, Al2O3, TiO2)

In this section, we investigate the effects of different factors on the Nu and skin friction factor (CfRe). Table 4 shows that increasing the magnetic field intensity (Mn) reduces both Nu and CfRe. The maximum reduction is 6% for Nu and 46% for CfRe. Table 5 shows that changing the Re has a significant effect on Nu but a small effect on CfRe. As Re increases, Nu increases. The reason for the different effects of Mn and Re on Nu and CfRe is that Mn affects the fluid flow, while Re affects the heat transfer. Mn increases the drag force on the fluid, which reduces the flow rate and hence the heat transfer. Re increases the velocity of the fluid, which increases the heat transfer. The results in Tables 4 and 5 show that the magnetic field and the Re are important factors that affect the heat transfer and flow in a cavity.

Table 4

Relationship between the Mns and physical parameters for Re = 10; Mn = 20

Mn ∣ Nu ∣ ∣ CfRe ∣
00 5.0499 45.2392
10 4.6416 44.8844
20 3.8578 45.5835
50 3.0039 44.8076
100 2.6776 42.4513
Table 5

Relationship between the Re and physical parameter for Re = 10; Mn = 20

R e ∣ Nu ∣ ∣ CfRe ∣
01 0.9426 45.1797
05 2.5808 45.0577
10 3.8578 45.5835
15 4.6566 45.9362
20 5.2141 45.7236

The results in Tables 6, 7 8 show that the type of nanoparticle has a significant impact on the heat transfer rate of the nanofluid. Both the Nu and the skin friction factor (CfRe) increased when nanoparticles were added, but the extent of the increase varied depending on the type of nanoparticle. The most pronounced increase was observed for silver (Ag) nanoparticles, with Nu and CfRe increasing by 67 and 108%, respectively. This suggests that there is a strong interplay between the thermal and fluid properties of the Ag nanofluid. For alumina (Al2O3) and titanium dioxide (TiO2) nanoparticles, Nu increased by 40 and 52%, respectively, while CfRe increased by 14 and 10%, respectively. This suggests that the relationship between the thermal and fluid properties of these nanofluids is weaker, or that there may be a trade-off between these two properties. The results of this study show that the type of nanoparticle is an important factor to consider when designing nanofluids for heat transfer applications.

Table 6

Relationship between the nanostructure of silver and physical parameters for Re = 20; Mn = 20

φ ⌢ 1 (silver) ∣ Nu ∣ ∣ CfRe ∣
0.00 4.6548 38.6143
0.05 5.2141 45.7236
0.10 5.8965 54.6559
0.15 6.7353 66.0160
0.20 7.7783 80.6676
Table 7

Relationship between the nanostructure of alumina and physical parameters for Re = 20; Mn = 20

φ ⌢ 2 (alumina) ∣ Nu ∣ ∣ CfRe ∣
0.00 5.0330 45.6211
0.05 5.5358 46.1127
0.10 6.1974 47.3485
0.15 7.0186 49.3335
0.20 8.0571 52.1149
Table 8

Relationship between the nanostructure of titanium dioxide and physical parameters for Re = 20; Mn = 20

φ ⌢ 3 (titanium dioxide) ∣ Nu ∣ ∣ CfRe ∣
0.00 5.0640 45.8578
0.05 5.4894 45.8004
0.10 6.0709 46.5932
0.15 6.8042 48.1869
0.20 7.7399 50.5897

The data in Table 9 show that increasing the width of the magnetic field (parameter L) leads to an increase in the Nu and a decrease in the skin friction factor (CfRe). This is because the magnetic field enhances the mixing of the fluid, which improves the heat transfer and reduces the drag force on the fluid. The increase in Nu is because the magnetic field creates a force that deflects the fluid flow, which causes the fluid to mix more effectively. This mixing brings hot and cold fluid parcels closer together, which enhances the heat transfer. The decrease in CfRe is because the magnetic field also creates a force that opposes the motion of the fluid. This force reduces the drag force on the fluid, which allows the fluid to flow more smoothly. This smoother flow reduces the friction between the fluid and the walls of the cavity, which reduces CfRe. The results in Table 9 suggest that it is beneficial to extend the coverage of the magnetic field to improve the heat transfer and reduce the drag force in a cavity.

Table 9

How Nu and CfRe change with the localized magnetic field strips in the flow

L ∣ Nu ∣ ∣ Cf Re ∣
00 5.2141 45.7236
0.10 4.1241 44.8319
0.20 4.0466 45.4963
0.30 4.2506 45.9400
0.35 8.4566 38.7864

5.6 Thermal characteristics of the nanostructures: A comparison

Figure 13 shows that the average Nu increases almost linearly with the nanoparticle volume fractions φ ⌢ 1 , φ ⌢ 2 , φ ⌢ 3 . This means that increasing the nanoparticle concentration enhances the heat transfer rate. The tri-hybrid nanofluid (comprising Ag, Al2O3, and TiO2) outperforms the conventional fluid in terms of increasing the average Nu. The increase in Nu with φ ⌢ 1 , φ ⌢ 2 , φ ⌢ 3 is because nanoparticles increase the thermal conductivity of the fluid. This means that the nanoparticles can carry more heat, which enhances the heat transfer rate. The tri-hybrid nanofluid outperforms the conventional fluid because the three types of nanoparticles work together to enhance the thermal conductivity of the fluid. Ag is a good conductor of heat, while Al2O3 and TiO2 are good at absorbing heat. The combination of these three types of nanoparticles results in a fluid that has superior heat transfer properties.

Figure 13 A comparative study of bi-hybrid and tri-hybrid nanofluids.
Figure 13

A comparative study of bi-hybrid and tri-hybrid nanofluids.

Tri-hybrid nanofluids can be used to improve heat transfer in heat exchangers, solar collectors, electronic cooling, medical applications, and other applications where heat transfer is important. The specific application of tri-hybrid nanofluids will depend on the specific requirements of the application. However, as shown in Figure 13, tri-hybrid nanofluids have the potential to improve the heat transfer rate in a variety of applications.

5.7 Impact of the magnetic field intensity on the flow and heat transfer

Figure 14 shows how the average Nu is affected by the parameters A1 and A2. These parameters control the strength of the magnetic field. The Nu is more sensitive to changes in A 1 and A 2 when these values are smaller. We found that there was little difference in our numerical results when A1 = A 2 = 50 and A 1 = A 2 = 100. Therefore, we have used A 1 = A 2 = 50 throughout this study, unless otherwise specified.

Figure 14 Impact of the parameters A 1 and A 2 on average Nu.
Figure 14

Impact of the parameters A 1 and A 2 on average Nu.

5.8 Impact of the magnetic field localization

The magnetic strips are defined as follows: ( 0.2 − L < ξ < 0.3 + L , 0.7 − L < ξ < 0.8 + L , 0 < η < 1 ) and ( 0.2 − L < η < 0.3 + L , 0.7 − L < η < 0.8 + L , 0 < ξ < 1 ) . They are two vertical strips of width 0.2 units that are located on the left and right sides of the cavity. When L = 0, it shows that the strips have a constant width size of 0.2 units. As L increases, the width of the strips increases. Figure 15 shows that the Nu initially decreases and then increases as L increases. This is because the magnetic field created by the strips initially inhibits the flow of the fluid, but as the width of the strips increases, the magnetic field becomes weaker.

Figure 15 The impact of the magnetized strip size on the average Nu.
Figure 15

The impact of the magnetized strip size on the average Nu.

6 Concluding remarks

This research investigated the numerical simulation of tri-hybrid nanofluid flow within a lid-driven cavity under the influence of a confined magnetic field. The tri-hybrid nanofluid comprises silver (Ag), aluminum oxide (Al2O3), and titanium dioxide (TiO2). The main findings are as follows:

  • A single symmetric vortex forms around the line in the flow field without a magnetic field. The magnetic field stretches this vortex, causing it to split into weaker vortices in the lower half of the cavity, rotating in different directions.

  • Magnetic fields from vertical and horizontal strips exert different forces on the flow, leading to the creation of multiple vortices at higher magnetic field intensities.

  • In the absence of a magnetic force, the flow exhibits predominantly red isotherms, indicating higher temperatures. The magnetic force disrupts this linear temperature distribution, causing new vortices that mix fluids with varying temperatures.

  • Lower Res distort the primary vortex and generate new vortices, similar to the effect of the magnetic parameter, implying their similar impact on the flow field.

  • Magnetic parameters reduce heat transfer by up to 46% across almost half of the domain, whereas Re increases the Nu more significantly than the Mn.

  • Skin friction (CfRe) is influenced similarly by two parameters but with slight quantitative differences. Nanoparticle properties have a limited impact on skin friction, while they significantly affect Nu.

  • Different types of nanoparticles affect Nu and CfRE differently. Silver (Ag) shows the highest increase in both (67 and 108%), indicating a strong link between nanofluid properties. Alumina Al2O3 and titanium dioxide (TiO2) exhibit lower increases, suggesting a weaker connection or trade-off.

  • Nu increases and CfRe decreases as the magnetic field covers more of the domain.

  • Tri-hybrid nanofluid (Ag, Al2O3, and TiO2) enhances average Nu more than simple nanofluids with individual particles.

Acknowledgments

The authors acknowledge Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R404), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

  1. Funding information: The research is funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R404), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2023-06-30
Revised: 2023-09-13
Accepted: 2023-10-11
Published Online: 2023-11-20

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  70. Comparative investigations of Ag/H2O nanofluid and Ag-CuO/H2O hybrid nanofluid with Darcy-Forchheimer flow over a curved surface
  71. Study on deformation characteristics of multi-pass continuous drawing of micro copper wire based on crystal plasticity finite element method
  72. Properties of ultra-high-performance self-compacting fiber-reinforced concrete modified with nanomaterials
  73. Prediction of lap shear strength of GNP and TiO2/epoxy nanocomposite adhesives
  74. A novel exploration of how localized magnetic field affects vortex generation of trihybrid nanofluids
  75. Fabrication and physicochemical characterization of copper oxide–pyrrhotite nanocomposites for the cytotoxic effects on HepG2 cells and the mechanism
  76. Thermal radiative flow of cross nanofluid due to a stretched cylinder containing microorganisms
  77. In vitro study of the biphasic calcium phosphate/chitosan hybrid biomaterial scaffold fabricated via solvent casting and evaporation technique for bone regeneration
  78. Insights into the thermal characteristics and dynamics of stagnant blood conveying titanium oxide, alumina, and silver nanoparticles subject to Lorentz force and internal heating over a curved surface
  79. Effects of nano-SiO2 additives on carbon fiber-reinforced fly ash–slag geopolymer composites performance: Workability, mechanical properties, and microstructure
  80. Energy bandgap and thermal characteristics of non-Darcian MHD rotating hybridity nanofluid thin film flow: Nanotechnology application
  81. Green synthesis and characterization of ginger-extract-based oxali-palladium nanoparticles for colorectal cancer: Downregulation of REG4 and apoptosis induction
  82. Abnormal evolution of resistivity and microstructure of annealed Ag nanoparticles/Ag–Mo films
  83. Preparation of water-based dextran-coated Fe3O4 magnetic fluid for magnetic hyperthermia
  84. Statistical investigations and morphological aspects of cross-rheological material suspended in transportation of alumina, silica, titanium, and ethylene glycol via the Galerkin algorithm
  85. Effect of CNT film interleaves on the flexural properties and strength after impact of CFRP composites
  86. Self-assembled nanoscale entities: Preparative process optimization, payload release, and enhanced bioavailability of thymoquinone natural product
  87. Structure–mechanical property relationships of 3D-printed porous polydimethylsiloxane films
  88. Nonlinear thermal radiation and the slip effect on a 3D bioconvection flow of the Casson nanofluid in a rotating frame via a homotopy analysis mechanism
  89. Residual mechanical properties of concrete incorporated with nano supplementary cementitious materials exposed to elevated temperature
  90. Time-independent three-dimensional flow of a water-based hybrid nanofluid past a Riga plate with slips and convective conditions: A homotopic solution
  91. Lightweight and high-strength polyarylene ether nitrile-based composites for efficient electromagnetic interference shielding
  92. Review Articles
  93. Recycling waste sources into nanocomposites of graphene materials: Overview from an energy-focused perspective
  94. Hybrid nanofiller reinforcement in thermoset and biothermoset applications: A review
  95. Current state-of-the-art review of nanotechnology-based therapeutics for viral pandemics: Special attention to COVID-19
  96. Solid lipid nanoparticles for targeted natural and synthetic drugs delivery in high-incidence cancers, and other diseases: Roles of preparation methods, lipid composition, transitional stability, and release profiles in nanocarriers’ development
  97. Critical review on experimental and theoretical studies of elastic properties of wurtzite-structured ZnO nanowires
  98. Polyurea micro-/nano-capsule applications in construction industry: A review
  99. A comprehensive review and clinical guide to molecular and serological diagnostic tests and future development: In vitro diagnostic testing for COVID-19
  100. Recent advances in electrocatalytic oxidation of 5-hydroxymethylfurfural to 2,5-furandicarboxylic acid: Mechanism, catalyst, coupling system
  101. Research progress and prospect of silica-based polymer nanofluids in enhanced oil recovery
  102. Review of the pharmacokinetics of nanodrugs
  103. Engineered nanoflowers, nanotrees, nanostars, nanodendrites, and nanoleaves for biomedical applications
  104. Research progress of biopolymers combined with stem cells in the repair of intrauterine adhesions
  105. Progress in FEM modeling on mechanical and electromechanical properties of carbon nanotube cement-based composites
  106. Antifouling induced by surface wettability of poly(dimethyl siloxane) and its nanocomposites
  107. TiO2 aerogel composite high-efficiency photocatalysts for environmental treatment and hydrogen energy production
  108. Structural properties of alumina surfaces and their roles in the synthesis of environmentally persistent free radicals (EPFRs)
  109. Nanoparticles for the potential treatment of Alzheimer’s disease: A physiopathological approach
  110. Current status of synthesis and consolidation strategies for thermo-resistant nanoalloys and their general applications
  111. Recent research progress on the stimuli-responsive smart membrane: A review
  112. Dispersion of carbon nanotubes in aqueous cementitious materials: A review
  113. Applications of DNA tetrahedron nanostructure in cancer diagnosis and anticancer drugs delivery
  114. Magnetic nanoparticles in 3D-printed scaffolds for biomedical applications
  115. An overview of the synthesis of silicon carbide–boron carbide composite powders
  116. Organolead halide perovskites: Synthetic routes, structural features, and their potential in the development of photovoltaic
  117. Recent advancements in nanotechnology application on wood and bamboo materials: A review
  118. Application of aptamer-functionalized nanomaterials in molecular imaging of tumors
  119. Recent progress on corrosion mechanisms of graphene-reinforced metal matrix composites
  120. Research progress on preparation, modification, and application of phenolic aerogel
  121. Application of nanomaterials in early diagnosis of cancer
  122. Plant mediated-green synthesis of zinc oxide nanoparticles: An insight into biomedical applications
  123. Recent developments in terahertz quantum cascade lasers for practical applications
  124. Recent progress in dielectric/metal/dielectric electrodes for foldable light-emitting devices
  125. Nanocoatings for ballistic applications: A review
  126. A mini-review on MoS2 membrane for water desalination: Recent development and challenges
  127. Recent updates in nanotechnological advances for wound healing: A narrative review
  128. Recent advances in DNA nanomaterials for cancer diagnosis and treatment
  129. Electrochemical micro- and nanobiosensors for in vivo reactive oxygen/nitrogen species measurement in the brain
  130. Advances in organic–inorganic nanocomposites for cancer imaging and therapy
  131. Advancements in aluminum matrix composites reinforced with carbides and graphene: A comprehensive review
  132. Modification effects of nanosilica on asphalt binders: A review
  133. Decellularized extracellular matrix as a promising biomaterial for musculoskeletal tissue regeneration
  134. Review of the sol–gel method in preparing nano TiO2 for advanced oxidation process
  135. Micro/nano manufacturing aircraft surface with anti-icing and deicing performances: An overview
  136. Cell type-targeting nanoparticles in treating central nervous system diseases: Challenges and hopes
  137. An overview of hydrogen production from Al-based materials
  138. A review of application, modification, and prospect of melamine foam
  139. A review of the performance of fibre-reinforced composite laminates with carbon nanotubes
  140. Research on AFM tip-related nanofabrication of two-dimensional materials
  141. Advances in phase change building materials: An overview
  142. Development of graphene and graphene quantum dots toward biomedical engineering applications: A review
  143. Nanoremediation approaches for the mitigation of heavy metal contamination in vegetables: An overview
  144. Photodynamic therapy empowered by nanotechnology for oral and dental science: Progress and perspectives
  145. Biosynthesis of metal nanoparticles: Bioreduction and biomineralization
  146. Current diagnostic and therapeutic approaches for severe acute respiratory syndrome coronavirus-2 (SARS-COV-2) and the role of nanomaterial-based theragnosis in combating the pandemic
  147. Application of two-dimensional black phosphorus material in wound healing
  148. Special Issue on Advanced Nanomaterials and Composites for Energy Conversion and Storage - Part I
  149. Helical fluorinated carbon nanotubes/iron(iii) fluoride hybrid with multilevel transportation channels and rich active sites for lithium/fluorinated carbon primary battery
  150. The progress of cathode materials in aqueous zinc-ion batteries
  151. Special Issue on Advanced Nanomaterials for Carbon Capture, Environment and Utilization for Energy Sustainability - Part I
  152. Effect of polypropylene fiber and nano-silica on the compressive strength and frost resistance of recycled brick aggregate concrete
  153. Mechanochemical design of nanomaterials for catalytic applications with a benign-by-design focus
https://doi.org/10.1515/ntrev-2023-0146
Schlagwörter für diesen Artikel
nanofluidics; alternating direction implicit approach; efficiency; localized magnetic field
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Research Articles
  2. Preparation of CdS–Ag2S nanocomposites by ultrasound-assisted UV photolysis treatment and its visible light photocatalysis activity
  3. Significance of nanoparticle radius and inter-particle spacing toward the radiative water-based alumina nanofluid flow over a rotating disk
  4. Aptamer-based detection of serotonin based on the rapid in situ synthesis of colorimetric gold nanoparticles
  5. Investigation of the nucleation and growth behavior of Ti2AlC and Ti3AlC nano-precipitates in TiAl alloys
  6. Dynamic recrystallization behavior and nucleation mechanism of dual-scale SiCp/A356 composites processed by P/M method
  7. High mechanical performance of 3-aminopropyl triethoxy silane/epoxy cured in a sandwich construction of 3D carbon felts foam and woven basalt fibers
  8. Applying solution of spray polyurea elastomer in asphalt binder: Feasibility analysis and DSR study based on the MSCR and LAS tests
  9. Study on the chronic toxicity and carcinogenicity of iron-based bioabsorbable stents
  10. Influence of microalloying with B on the microstructure and properties of brazed joints with Ag–Cu–Zn–Sn filler metal
  11. Thermohydraulic performance of thermal system integrated with twisted turbulator inserts using ternary hybrid nanofluids
  12. Study of mechanical properties of epoxy/graphene and epoxy/halloysite nanocomposites
  13. Effects of CaO addition on the CuW composite containing micro- and nano-sized tungsten particles synthesized via aluminothermic coupling with silicothermic reduction
  14. Cu and Al2O3-based hybrid nanofluid flow through a porous cavity
  15. Design of functional vancomycin-embedded bio-derived extracellular matrix hydrogels for repairing infectious bone defects
  16. Study on nanocrystalline coating prepared by electro-spraying 316L metal wire and its corrosion performance
  17. Axial compression performance of CFST columns reinforced by ultra-high-performance nano-concrete under long-term loading
  18. Tungsten trioxide nanocomposite for conventional soliton and noise-like pulse generation in anomalous dispersion laser cavity
  19. Microstructure and electrical contact behavior of the nano-yttria-modified Cu-Al2O3/30Mo/3SiC composite
  20. Melting rheology in thermally stratified graphene-mineral oil reservoir (third-grade nanofluid) with slip condition
  21. Re-examination of nonlinear vibration and nonlinear bending of porous sandwich cylindrical panels reinforced by graphene platelets
  22. Parametric simulation of hybrid nanofluid flow consisting of cobalt ferrite nanoparticles with second-order slip and variable viscosity over an extending surface
  23. Chitosan-capped silver nanoparticles with potent and selective intrinsic activity against the breast cancer cells
  24. Multi-core/shell SiO2@Al2O3 nanostructures deposited on Ti3AlC2 to enhance high-temperature stability and microwave absorption properties
  25. Solution-processed Bi2S3/BiVO4/TiO2 ternary heterojunction photoanode with enhanced photoelectrochemical performance
  26. Electroporation effect of ZnO nanoarrays under low voltage for water disinfection
  27. NIR-II window absorbing graphene oxide-coated gold nanorods and graphene quantum dot-coupled gold nanorods for photothermal cancer therapy
  28. Nonlinear three-dimensional stability characteristics of geometrically imperfect nanoshells under axial compression and surface residual stress
  29. Investigation of different nanoparticles properties on the thermal conductivity and viscosity of nanofluids by molecular dynamics simulation
  30. Optimized Cu2O-{100} facet for generation of different reactive oxidative species via peroxymonosulfate activation at specific pH values to efficient acetaminophen removal
  31. Brownian and thermal diffusivity impact due to the Maxwell nanofluid (graphene/engine oil) flow with motile microorganisms and Joule heating
  32. Appraising the dielectric properties and the effectiveness of electromagnetic shielding of graphene reinforced silicone rubber nanocomposite
  33. Synthesis of Ag and Cu nanoparticles by plasma discharge in inorganic salt solutions
  34. Low-cost and large-scale preparation of ultrafine TiO2@C hybrids for high-performance degradation of methyl orange and formaldehyde under visible light
  35. Utilization of waste glass with natural pozzolan in the production of self-glazed glass-ceramic materials
  36. Mechanical performance of date palm fiber-reinforced concrete modified with nano-activated carbon
  37. Melting point of dried gold nanoparticles prepared with ultrasonic spray pyrolysis and lyophilisation
  38. Graphene nanofibers: A modern approach towards tailored gypsum composites
  39. Role of localized magnetic field in vortex generation in tri-hybrid nanofluid flow: A numerical approach
  40. Intelligent computing for the double-diffusive peristaltic rheology of magneto couple stress nanomaterials
  41. Bioconvection transport of upper convected Maxwell nanoliquid with gyrotactic microorganism, nonlinear thermal radiation, and chemical reaction
  42. 3D printing of porous Ti6Al4V bone tissue engineering scaffold and surface anodization preparation of nanotubes to enhance its biological property
  43. Bioinspired ferromagnetic CoFe2O4 nanoparticles: Potential pharmaceutical and medical applications
  44. Significance of gyrotactic microorganisms on the MHD tangent hyperbolic nanofluid flow across an elastic slender surface: Numerical analysis
  45. Performance of polycarboxylate superplasticisers in seawater-blended cement: Effect from chemical structure and nano modification
  46. Entropy minimization of GO–Ag/KO cross-hybrid nanofluid over a convectively heated surface
  47. Oxygen plasma assisted room temperature bonding for manufacturing SU-8 polymer micro/nanoscale nozzle
  48. Performance and mechanism of CO2 reduction by DBD-coupled mesoporous SiO2
  49. Polyarylene ether nitrile dielectric films modified by HNTs@PDA hybrids for high-temperature resistant organic electronics field
  50. Exploration of generalized two-phase free convection magnetohydrodynamic flow of dusty tetra-hybrid Casson nanofluid between parallel microplates
  51. Hygrothermal bending analysis of sandwich nanoplates with FG porous core and piezomagnetic faces via nonlocal strain gradient theory
  52. Design and optimization of a TiO2/RGO-supported epoxy multilayer microwave absorber by the modified local best particle swarm optimization algorithm
  53. Mechanical properties and frost resistance of recycled brick aggregate concrete modified by nano-SiO2
  54. Self-template synthesis of hollow flower-like NiCo2O4 nanoparticles as an efficient bifunctional catalyst for oxygen reduction and oxygen evolution in alkaline media
  55. High-performance wearable flexible strain sensors based on an AgNWs/rGO/TPU electrospun nanofiber film for monitoring human activities
  56. High-performance lithium–selenium batteries enabled by nitrogen-doped porous carbon from peanut meal
  57. Investigating effects of Lorentz forces and convective heating on ternary hybrid nanofluid flow over a curved surface using homotopy analysis method
  58. Exploring the potential of biogenic magnesium oxide nanoparticles for cytotoxicity: In vitro and in silico studies on HCT116 and HT29 cells and DPPH radical scavenging
  59. Enhanced visible-light-driven photocatalytic degradation of azo dyes by heteroatom-doped nickel tungstate nanoparticles
  60. A facile method to synthesize nZVI-doped polypyrrole-based carbon nanotube for Ag(i) removal
  61. Improved osseointegration of dental titanium implants by TiO2 nanotube arrays with self-assembled recombinant IGF-1 in type 2 diabetes mellitus rat model
  62. Functionalized SWCNTs@Ag–TiO2 nanocomposites induce ROS-mediated apoptosis and autophagy in liver cancer cells
  63. Triboelectric nanogenerator based on a water droplet spring with a concave spherical surface for harvesting wave energy and detecting pressure
  64. A mathematical approach for modeling the blood flow containing nanoparticles by employing the Buongiorno’s model
  65. Molecular dynamics study on dynamic interlayer friction of graphene and its strain effect
  66. Induction of apoptosis and autophagy via regulation of AKT and JNK mitogen-activated protein kinase pathways in breast cancer cell lines exposed to gold nanoparticles loaded with TNF-α and combined with doxorubicin
  67. Effect of PVA fibers on durability of nano-SiO2-reinforced cement-based composites subjected to wet-thermal and chloride salt-coupled environment
  68. Effect of polyvinyl alcohol fibers on mechanical properties of nano-SiO2-reinforced geopolymer composites under a complex environment
  69. In vitro studies of titanium dioxide nanoparticles modified with glutathione as a potential drug delivery system
  70. Comparative investigations of Ag/H2O nanofluid and Ag-CuO/H2O hybrid nanofluid with Darcy-Forchheimer flow over a curved surface
  71. Study on deformation characteristics of multi-pass continuous drawing of micro copper wire based on crystal plasticity finite element method
  72. Properties of ultra-high-performance self-compacting fiber-reinforced concrete modified with nanomaterials
  73. Prediction of lap shear strength of GNP and TiO2/epoxy nanocomposite adhesives
  74. A novel exploration of how localized magnetic field affects vortex generation of trihybrid nanofluids
  75. Fabrication and physicochemical characterization of copper oxide–pyrrhotite nanocomposites for the cytotoxic effects on HepG2 cells and the mechanism
  76. Thermal radiative flow of cross nanofluid due to a stretched cylinder containing microorganisms
  77. In vitro study of the biphasic calcium phosphate/chitosan hybrid biomaterial scaffold fabricated via solvent casting and evaporation technique for bone regeneration
  78. Insights into the thermal characteristics and dynamics of stagnant blood conveying titanium oxide, alumina, and silver nanoparticles subject to Lorentz force and internal heating over a curved surface
  79. Effects of nano-SiO2 additives on carbon fiber-reinforced fly ash–slag geopolymer composites performance: Workability, mechanical properties, and microstructure
  80. Energy bandgap and thermal characteristics of non-Darcian MHD rotating hybridity nanofluid thin film flow: Nanotechnology application
  81. Green synthesis and characterization of ginger-extract-based oxali-palladium nanoparticles for colorectal cancer: Downregulation of REG4 and apoptosis induction
  82. Abnormal evolution of resistivity and microstructure of annealed Ag nanoparticles/Ag–Mo films
  83. Preparation of water-based dextran-coated Fe3O4 magnetic fluid for magnetic hyperthermia
  84. Statistical investigations and morphological aspects of cross-rheological material suspended in transportation of alumina, silica, titanium, and ethylene glycol via the Galerkin algorithm
  85. Effect of CNT film interleaves on the flexural properties and strength after impact of CFRP composites
  86. Self-assembled nanoscale entities: Preparative process optimization, payload release, and enhanced bioavailability of thymoquinone natural product
  87. Structure–mechanical property relationships of 3D-printed porous polydimethylsiloxane films
  88. Nonlinear thermal radiation and the slip effect on a 3D bioconvection flow of the Casson nanofluid in a rotating frame via a homotopy analysis mechanism
  89. Residual mechanical properties of concrete incorporated with nano supplementary cementitious materials exposed to elevated temperature
  90. Time-independent three-dimensional flow of a water-based hybrid nanofluid past a Riga plate with slips and convective conditions: A homotopic solution
  91. Lightweight and high-strength polyarylene ether nitrile-based composites for efficient electromagnetic interference shielding
  92. Review Articles
  93. Recycling waste sources into nanocomposites of graphene materials: Overview from an energy-focused perspective
  94. Hybrid nanofiller reinforcement in thermoset and biothermoset applications: A review
  95. Current state-of-the-art review of nanotechnology-based therapeutics for viral pandemics: Special attention to COVID-19
  96. Solid lipid nanoparticles for targeted natural and synthetic drugs delivery in high-incidence cancers, and other diseases: Roles of preparation methods, lipid composition, transitional stability, and release profiles in nanocarriers’ development
  97. Critical review on experimental and theoretical studies of elastic properties of wurtzite-structured ZnO nanowires
  98. Polyurea micro-/nano-capsule applications in construction industry: A review
  99. A comprehensive review and clinical guide to molecular and serological diagnostic tests and future development: In vitro diagnostic testing for COVID-19
  100. Recent advances in electrocatalytic oxidation of 5-hydroxymethylfurfural to 2,5-furandicarboxylic acid: Mechanism, catalyst, coupling system
  101. Research progress and prospect of silica-based polymer nanofluids in enhanced oil recovery
  102. Review of the pharmacokinetics of nanodrugs
  103. Engineered nanoflowers, nanotrees, nanostars, nanodendrites, and nanoleaves for biomedical applications
  104. Research progress of biopolymers combined with stem cells in the repair of intrauterine adhesions
  105. Progress in FEM modeling on mechanical and electromechanical properties of carbon nanotube cement-based composites
  106. Antifouling induced by surface wettability of poly(dimethyl siloxane) and its nanocomposites
  107. TiO2 aerogel composite high-efficiency photocatalysts for environmental treatment and hydrogen energy production
  108. Structural properties of alumina surfaces and their roles in the synthesis of environmentally persistent free radicals (EPFRs)
  109. Nanoparticles for the potential treatment of Alzheimer’s disease: A physiopathological approach
  110. Current status of synthesis and consolidation strategies for thermo-resistant nanoalloys and their general applications
  111. Recent research progress on the stimuli-responsive smart membrane: A review
  112. Dispersion of carbon nanotubes in aqueous cementitious materials: A review
  113. Applications of DNA tetrahedron nanostructure in cancer diagnosis and anticancer drugs delivery
  114. Magnetic nanoparticles in 3D-printed scaffolds for biomedical applications
  115. An overview of the synthesis of silicon carbide–boron carbide composite powders
  116. Organolead halide perovskites: Synthetic routes, structural features, and their potential in the development of photovoltaic
  117. Recent advancements in nanotechnology application on wood and bamboo materials: A review
  118. Application of aptamer-functionalized nanomaterials in molecular imaging of tumors
  119. Recent progress on corrosion mechanisms of graphene-reinforced metal matrix composites
  120. Research progress on preparation, modification, and application of phenolic aerogel
  121. Application of nanomaterials in early diagnosis of cancer
  122. Plant mediated-green synthesis of zinc oxide nanoparticles: An insight into biomedical applications
  123. Recent developments in terahertz quantum cascade lasers for practical applications
  124. Recent progress in dielectric/metal/dielectric electrodes for foldable light-emitting devices
  125. Nanocoatings for ballistic applications: A review
  126. A mini-review on MoS2 membrane for water desalination: Recent development and challenges
  127. Recent updates in nanotechnological advances for wound healing: A narrative review
  128. Recent advances in DNA nanomaterials for cancer diagnosis and treatment
  129. Electrochemical micro- and nanobiosensors for in vivo reactive oxygen/nitrogen species measurement in the brain
  130. Advances in organic–inorganic nanocomposites for cancer imaging and therapy
  131. Advancements in aluminum matrix composites reinforced with carbides and graphene: A comprehensive review
  132. Modification effects of nanosilica on asphalt binders: A review
  133. Decellularized extracellular matrix as a promising biomaterial for musculoskeletal tissue regeneration
  134. Review of the sol–gel method in preparing nano TiO2 for advanced oxidation process
  135. Micro/nano manufacturing aircraft surface with anti-icing and deicing performances: An overview
  136. Cell type-targeting nanoparticles in treating central nervous system diseases: Challenges and hopes
  137. An overview of hydrogen production from Al-based materials
  138. A review of application, modification, and prospect of melamine foam
  139. A review of the performance of fibre-reinforced composite laminates with carbon nanotubes
  140. Research on AFM tip-related nanofabrication of two-dimensional materials
  141. Advances in phase change building materials: An overview
  142. Development of graphene and graphene quantum dots toward biomedical engineering applications: A review
  143. Nanoremediation approaches for the mitigation of heavy metal contamination in vegetables: An overview
  144. Photodynamic therapy empowered by nanotechnology for oral and dental science: Progress and perspectives
  145. Biosynthesis of metal nanoparticles: Bioreduction and biomineralization
  146. Current diagnostic and therapeutic approaches for severe acute respiratory syndrome coronavirus-2 (SARS-COV-2) and the role of nanomaterial-based theragnosis in combating the pandemic
  147. Application of two-dimensional black phosphorus material in wound healing
  148. Special Issue on Advanced Nanomaterials and Composites for Energy Conversion and Storage - Part I
  149. Helical fluorinated carbon nanotubes/iron(iii) fluoride hybrid with multilevel transportation channels and rich active sites for lithium/fluorinated carbon primary battery
  150. The progress of cathode materials in aqueous zinc-ion batteries
  151. Special Issue on Advanced Nanomaterials for Carbon Capture, Environment and Utilization for Energy Sustainability - Part I
  152. Effect of polypropylene fiber and nano-silica on the compressive strength and frost resistance of recycled brick aggregate concrete
  153. Mechanochemical design of nanomaterials for catalytic applications with a benign-by-design focus
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