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Role of localized magnetic field in vortex generation in tri-hybrid nanofluid flow: A numerical approach

  • Shabbir Ahmad EMAIL logo , Hidemasa Takana , Kashif Ali , Yasmeen Akhtar , Ahmed M. Hassan and Adham E. Ragab
Published/Copyright: July 11, 2023
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Nanotechnology Reviews
From the journal Nanotechnology Reviews Volume 12 Issue 1

Abstract

Tri-hybrid nanofluid (THNF) can achieve a higher heat transfer rate than conventional hybrid nanofluid by combining three different nanoparticles with synergistic effects. It can have more diverse physical and thermal properties by choosing different combinations of nanoparticles. That is why it has more potential applications in various fields such as solar thermal, biomedical, and industrial processes. On the other hand, vortices are circular motions of liquid or gas that occur when there is a velocity difference. They are important for understanding how fluids mix and transport mass. They can be found in nature, such as in tornadoes and hurricanes. The aim of the current study is to mainly investigate the complex interaction of Lorentz force with the tri-hybrid nanoparticles inside a lid-driven square cavity. It can be seen that the magnetic field has caused the evolution of new vortices (which are very important while analyzing any flow model due to their importance in interpreting fluid mixing and mass transport phenomena) in the flow field, thus adding much more significance to our work. Most of the scientific literature is enriched with investigations dealing with the problems assuming a uniform magnetic field occupying the flow field, but in this research, a vertical strip of magnetism within the flow field will be introduced. It may be the first effort to interpret the role of the applied magnetic field in the formation of the new vortices in the flow field. A single-phase model is utilized to describe THNF whereas a numerical solution to the governing differential equations has been obtained by employing an algorithm based on the central difference discretization and the alternating direction implicit method. The analysis reveals that the magnetic field intensity may result in up to 13 and 119% increase in the skin friction and Nusselt number, respectively. Similarly, a remarkable change in the Nusselt number and the skin friction is also observed by raising the Reynolds number Re. Moreover, the localization or confinement of the magnetic field does not always increase or decrease the Nusselt number. Thus, it is concluded that there will be a certain width of the magnetic corridor for which the Nusselt number would be optimal. Further, the THNF containing Al2O3, Ag, and TiO2 outperforms in terms of enhancing the average Nusselt number, compared to the simple nanofluid containing the abovementioned particles.

Keywords: Reynolds number; single-phase model; tri-hybrid nanofluids; localized magnetic field

Nomenclature

Ec

Eckert number

H ˜ ( x , y )

magnetic field intensity of a dipole

K

pyro magnetic factor

M ˜

magnetization property

Mn

magnetic number

Pr

Prandtl number

Re

Reynolds number

T ¯ c

Curie temperature

V 0

constant velocity

∂ H ˜ / ∂ x

magnetic force components along the X-axis

∂ H ˜ / ∂ y

magnetic force components along the Y-axis

Greek symbols

γ

magnetic field strength

μ hnf

viscosity of hybrid nanofluid

σ hnf

electrical conductivity of hybrid nanofluid

φ ⌢ 1

volume fraction of iron oxide

ρ ⌢ nf

density of nanofluid

φ ⌢

nanoparticle volume fraction

ε

dimensionless number

μ ¯ 0

dynamic viscosity

k hnf

thermal conductivity

φ ⌢ 2

volume fraction of copper

ρ ⌢ hnf

density of nanofluid of hybrid nanofluid

1 Introduction

The study of vortex generation in lid-driven cavities by localized magnetic field is useful for understanding the fluid dynamics of biological fluids under the influence of high magnetic fields and its possible medical applications, such as cell separation, magnetic tracers, and magnetic drug targeting [1]. It is also useful for exploring the size-dependent effects of fluid flows at small scales, which can be relevant for liquid crystals, lubrication, and microelectromechanical devices [2]. Moreover, it can provide insights into the topology and stability of corner vortices in the lid-driven cavity flow, which is a benchmark problem for testing numerical methods and studying fundamental aspects of incompressible flows in confined volumes [3]. A localized magnetic field is a vector field that varies in space due to the presence of magnetic materials or electric currents. Some real-world applications of localized magnetic fields are indoor localization, magnetic anomaly detection, magnetoencephalography, and quantum physics. Magnetic fields can be used to estimate the position and orientation of a device inside a building by measuring the distortions of the earth’s magnetic field caused by ferromagnetic objects [4,5].

Amalgamation of the nanoparticles that are exhibiting magnetic features may produce magnetic nanofluids which are rather different from ordinary nanofluids. Since ferromagnetic materials such as ferrites, mercury, and nickel are likely more striking towards the influence of the external magnetic fields, these nanomaterials disseminate the fluid throughout the enclosure by altering the flow regime. In a magnetic field setup, the flow of viscous and conductive fluids (e.g., Brine, nano lubricants, plasma, electrolytes, liquid metals, etc.) has a variety of applications in the manufacturing of medical apparatuses, control networks in industries, propulsion systems in submarine and spacecraft, wastewater treatment, liquid metal fast reactor, power generation, etc. In recent times, many researchers have focused on magneto-hydrodynamic (MHD) flows through different geometries. Recently, the heat and mass transfer of a micro-rotating fluid between two plates was explored by Jalili et al. [6] by using both electric and magnetic fields. They computed the nonlinear and coupled equations and evaluated how the Nusselt number, Sherwood number, and skin friction influence the fluid’s temperature, concentration, and velocity. MHD convection and the development of entropy of Cu–Al2O3/water nanofluid in a porous square cavity are numerically investigated with sinusoidal boundary temperatures by Abdel-Nour et al. [7]. The isotherms vary with Rayleigh number and Hartmann number (Ha). A high Ha increases conduction transfer. MHD mixed convection in Al2O3–Cu nanostructures in the water nanofluid in an L-shaped cavity with two heat sources at bottom corners and cold ends is proposed by Armaghani et al. [8]. The size of the enclosure and heat source location affect the heat transfer and flow. The numerical study of unsteady hybrid nanofluid (Fe3O4 and carbon nanotubes in water) flow with heat source/sink effects between two sheets with variable thermal conductivity and magnetic field has been conducted by Dinarvand et al. [9]. The heat transfer decreases with an oblique magnetic field and also temperature-dependent thermal conductivity. The spherical particles have a particularly high temperature with horizontal magnetic field.

Convective flows through enclosures and cavities have captivated the attention of scientists due to their increasingly versatile applications in several industries which include chemical and food processing industries, paints and molten polymer processing units, microelectronics, HVAC systems of buildings and mechanical complexes, nuclear power stations, solar collectors, etc. [10,11,12]. Natural convection flow problems are mainly concerned with the heat transfer performance of the working fluids. Fluid flow and heat transfer in a square cavity with varying wall temperatures and Cu–Al2O3water nanofluid is numerically investigated by Gorla et al. [13]. A higher Ha lowers fluid velocity and a higher nanoparticle volume fraction boosts the heat transfer rate. The heat transfer of a hybrid nanofluid in a triangular cavity with a magnetic field and a heat source at the bottom is studied by Rashad et al. [14]. The slanted side is cold and the other sides are insulated. The heat transfer is affected by the nanofluid volume fraction when convection is very low. A hybrid nanofluid with an equal nanostructure of copper (Cu) and aluminum oxide (Al2O3) in water does not improve the mean Nusselt number more than a regular nanofluid. Excellent research on the thermophysical characteristics of various nanofluids in fascinating problems was also conducted by Ahmad et al. [15,16,17,18,19,20,21,22]. Liu et al. [23] examined the magnetic flow of Fe3O4 nanoparticles inside a porous cavity with the latest numerical technique taking impacts of various combinations of Rayleigh numbers, the volume fraction of nanocomposites, porosity parameter, and magnetic parameters on the thermal distributions of the nano liquid. Nepal [24] calculated a numerical solution of MHD flow through an enclosure with numerous heat sources of variable breadth at the bottom most layer of the enclosure. The author employed a powerful numerical technique, the successive over-relaxation method, entrenched with relevant finite differences of differential equations.

The advantage of tri-hybrid nanofluids (THNFs) over hybrid nanofluids is that they have higher thermal conductivity and heat transfer properties than hybrid nanofluids and simple nanofluids [25,26]. This means that THNFs can enhance the cooling performance and energy efficiency of various applications, such as microfluidic devices, electric ovens, nuclear reactors, etc. However, the effectiveness of THNFs depends on several factors, such as the types, sizes, shapes, and mixture ratios of nanoparticles, as well as the base fluids [27]. Jakeer et al. [28] studied the magneto copper based nanofluid flow in a non-Darcy porous square enclosure by using the Cattaneo–Christov heat flux model. The heat transfer and flow depend on the heated obstacle. Higher Ha reduces the local Nusselt number, whereas hybrid nanofluid enhances heat transfer more than nanofluids. The thermal diffusivity profile in oblique stenosis artery with Al2O3–Cu based hybrid nanofluid was designed by Jalili et al. [29] by taking care of several numerical methods (AGM, FEM, and Runge–Kutta). They scrutinized the influence of heat source parameters and volume fraction on the temperature evolution and reported that increasing the nanoparticles volume fraction decreases the maximum temperature profile and rising the heat source increases the maximum heat temperature. They also discovered that Al2O3 has more impact on reducing the temperature profile than Cu. The best cooling of the fluid was accomplished by Abdollahi et al. [30] by perusing the thermal and fluid parameters of copper/water hybrid solution, such as Darcy number and Nusselt number. They also fitted a piece of fin on the wall of the heat sink to surge the coefficient of displacement heat transfer and conductive heat transfer with the surrounding air-fluid and the effectiveness of the system. Tawade et al. [31] examined the steady laminar flow on a two-dimensional boundary layer with heat transfer of Casson over a linearly stretching sheet. The governing partial differential equations (PDEs) were changed into nonlinear ordinary differential equations (ODEs) by means of similarity transformation. For more related information, please refer to refs [32,33]. Hameed et al. [34] studied the two-dimension flow of Casson hybrid nanofluid on a nonlinear extending surface with a magnetic field, absorption, heat generation, and viscous dissipation. The main aim is to enhance the heat transfer relationship between the engineering industries and manufacturing.

In most recent times, various nanoparticles such as alumina, titanium, spinel ferrites, nickel, copper, silver, graphite, etc., are used in the working host fluids to produce thermally vigorous fluids usually termed nanofluids [35,36,37,38]. The study of a micro-polar nanofluid in a rotating system with electric and magnetic fields between two plates was conducted by Jalili et al. [39]. They considered a steady-state fluid flow and integrated the coupled and nonlinear equations with appropriate similar variables, and employed new semi-analytical and numerical methods to define the accuracy of the results. They reported that concentration value decays by increasing the thermophoretic parameter and Reynolds number. Heat transfer and entropy generation in a square enclosure with aluminum and copper based nanofluid at different wall temperatures associated with imposed magnetic field are studied by Mansour et al. [40]. A new semi-analytical technique was used by Jalili et al. [41] to crack the heat and mass transfer problem. They considered a viscous, laminar axisymmetric, and incompressible flow of a micropolar fluid under the effect of a magnetic field between two stretchable disks. They presented the impact of different parameters such as radiation parameter, Eckert number, magnetic field, Reynolds number, Prandtl number, and Schmidt number on the profiles of microrotation, temperature, velocity, and concentration with the corresponding graphs. Saeed et al. [42] explored the complex dynamics of nanofluid flow between two parallel plates that squeeze and rotate with various physical properties. They used the Navier–Stokes equation, energy equation, and concentration equation to model the unsteady behavior of the fluid and its thermal and mass transport characteristics. The problem of variable viscosity and inclined Lorentz force influences on Williamson nanofluid on a stretching plate was unraveled by Jalili et al. [43] by considering a semi-analytical approach. The variable viscosity was chosen to change with temperature as a linear function. The system of PDEs was transformed into ODEs using suitable transformations. The outcomes demonstrated that the heat transfer rate reduced by increasing the Prandtl number and the temperature decreased by rising the thermophoresis parameter.

One of the imperative problems of heat transfer through bounded surfaces is natural convection, in which the governing equations are highly nonlinear and coupled in nature, and require advanced computational fluid dynamics tools to obtain their numerical solutions. Yuan et al. [44] numerically solved the hydromagnetic flow of nano liquid inside a U-shaped cavity. In order to prevent the bafflement on the fluid movement inside the enclosure, clockwise and anti-clockwise vortices are observed at greater scales of Rayleigh number. The convection heat transfer in a turbulent flow in a countercurrent double-tube heat exchanger with various fins was investigated by Jalili et al. [45]. The double-pipe heat exchanger was studied for the real cooling or heating process of fluids. Water–titanium dioxide and water–aluminum oxide nanofluids at quatern concentrations (0.4, 2, 4, and 6%) were considered as the cold fluid in the innermost tube, and water as the hot fluid in the annular space. The findings indicated that the Al2O3 + H2O nanofluid had an improved convection heat transfer coefficient than TiO2 + H2O and pure water. A numerical model of a shell and tube heat exchanger associated with non-continuous helical baffles (NCHB-STHX) was investigated by Jalili et al. [46]. This heat exchanger has miscellaneous applications in different arenas of science and industry. They subjugated the ratio of heat transfer coefficient to pressure drop (HTC/ΔP) from various angles and intensities. Dharmaiah et al. [47] analyzed the behavior of nanofluid flows through a sinusoidally varying circular cylinder numerically by applying the similarity transformations which reduce the governing equations to nonlinear ODEs. Sherwood number, Skin friction, and Nusselt number are also evaluated with the use of bvp4c. Dharmaiah et al. [48] considered an incompressible non-Newtonian nanofluid in a two-dimensional transitory boundary layer through a cone. The radiation absorption and Arrhenius activation energy are taken into account by the non-Newtonian nanofluid model. The nonlinear equations are numerically solved using ODE45 MATLAB bvp4c and the Runge–Kutta integration method. Skin friction is reduced when the activation energy parameter is higher, while Nusselt and Sherwood values are increased.

The novelty or originality of the study lies in the combination of lid-driven cavity flow, localized magnetic field, and THNFs, which introduce additional boundary conditions. A vertical strip of magnetism will be introduced within the flow field and its impact on the flow characteristics will be observed. The work under consideration may be the first effort to interpret the role of the applied magnetic field in the formation of the new vortices in the flow field. The work mainly aims to investigate the complex interaction of Lorentz force with the tri-hybrid nanoparticles inside a lid-driven square cavity. Lorentz force causes the spinning of the nano-sized hybrid particles and thus generates a complicated structure of vortices.

2 Problem description

The schematic representation of the two-dimensional square enclosure taken into consideration is depicted in Figure 1. The physical domain of a square with side L. A localized magnetic field is constantly embedded in the lower horizontal wall. A single-phase model is being used to build the governing equations for a two-dimensional enclosure.

Figure 1 Schematic illustration of the problem.
Figure 1

Schematic illustration of the problem.

2.1 Basic assumptions

  • A magnetic source creates the magnetic field with intensity H, as given by

    H ˜ ( x , y ) = H 0 { tanh A ′ 1 ( x − x 1 ) − tanh A ′ 2 ( x − x 2 ) } , in the strip defined by x 1 ≤ x ≤ x 2 , and 0 ≤ y ≤ L .

  • Both vertical walls are insulated, whereas the enclosure’s top and bottom horizontal walls are kept at a constant temperature.

  • Lower and upper horizontal walls slide uniformly from left to rightward with a constant velocity V 0 .

  • Newtonian, incompressible, and laminar nanofluids are assumed in the current analysis.

  • The solid nanoparticles (silver; Ag, aluminum oxide; Al2O3, and titanium dioxide; TiO2), and the base fluid (water) are in thermal equilibrium.

  • The thermo-physical parameters of the nanofluid are taken to be constant.

3 Governing equations and dimensionless variables

The basic laws, assumptions, and derivation of the Navier–Stokes equations are based on the following principles:

  • The fluid is a continuum, i.e., it is treated as a continuous substance rather than discrete particles.

  • The fluid is incompressible, i.e., its density is constant and independent of pressure [49].

  • The fluid is Newtonian, i.e., its stress is proportional to its strain rate [49].

  • The fluid obeys the conservation of mass, i.e., the mass of a fluid element does not change as it moves.

  • The fluid obeys the conservation of momentum, i.e., the net force acting on a fluid element is equal to its rate of change of momentum.

  • The fluid obeys the conservation of energy, i.e., the net work done on a fluid element is equal to its rate of change of energy

The dimensional form of these equations is as follows [50]:

Continuity equation

(1) ∂ U ˜ ∂ x + ∂ V ˜ ∂ y = 0 ,

Momentum equation

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

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

Energy equation

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

It is to point out that the energy equation is based on the following energy balance:

The time rate of change of energy in a system at time t = [The net rate of heat transfer into a system at time t] – [The net rate of work out of a system at time t].

In the above equations,

μ ¯ 0 M ˜ ∂ H ˜ ∂ x Magnetic force component along the X-axis μ ¯ 0 M ˜ ∂ H ˜ ∂ y Magnetic force component along the Y-axis
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 [51]
∂ 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

It is important to keep in mind that the subscript (hnf) is used to indicate the physical properties of a THNF.

It is important to keep in mind that convection terms refer to the terms in the equations on the left side of the momentum equations. Convection is a physical phenomenon that takes place in a gas flow in which the ordered movement of the flow transports some property. The diffusion terms are those on the right-hand side of the momentum equations which are multiplied by the inverse Reynolds number. Diffusion is a physical process that occurs in a flow of gas and involves the random motion of the gas molecules transporting some property. The stress tensor and the viscosity of the gas are related to diffusion. Diffusion in the flow causes turbulence and the formation of boundary layers.

Eliminating the pressure term results in

(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 current problem’s dimensional boundary conditions are interpreted as follows:

Left and right walls (Adiabatic)

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

Top wall

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

Bottom wall

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

3.2 Thermophysical characteristics of THNFs

To look into the features of THNFs’ heat transport, a particular combination of thermophysical parameters for Al2O3–Ag–TiO2 tri-hybrid nanoparticles is taken into account. These thermophysical characteristics were derived and validated from the published literature [52]. Below is a description of each symbol that appears in Table 1.

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

Thermophysical characteristics of THNF and pure nanofluids

Properties 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

The formulas suggested in the literature can be used to analyze the thermophysical characteristics of a THNF.

The following set of dimensionless variables should be used right now:

(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 ′ .

Now equations (4) and (5) imply that

(8) ∂ J ∂ t + u ∂ J ∂ ξ + v ∂ J ∂ η = ( 1 − φ ⌢ 3 ) ( 1 − φ ⌢ 1 ) ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 ρ ⌢ f × ( 1 − φ ⌢ 1 ) 2.5 ( 1 − φ ⌢ 2 ) 2.5 ( 1 − φ ⌢ 3 ) 2.5 1 Re ∇ 2 J + Mn ( 1 − φ ⌢ 3 ) ( 1 − φ ⌢ 1 ) ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 ρ ⌢ f H ∂ H ∂ η . ∂ θ ⌢ ∂ ξ − ∂ H ∂ ξ . ∂ θ ⌢ ∂ η ,

(9a) ∇ 2 θ ⌢ = Pr ⁎ ( 1 − φ ⌢ 3 ) ( 1 − φ ⌢ 1 ) ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 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 S1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 c p S2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 c p S3 ρ ⌢ s3 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 ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 ρ ⌢ f ∂ θ ⌢ ∂ ξ ∂ ψ ˜ ∂ η − ∂ θ ⌢ ∂ η ∂ ψ ˜ ∂ ξ + Pr ⁎ ( 1 − φ ⌢ 3 ) ( 1 − φ ⌢ 1 ) ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 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 S1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 c p S2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 c p S3 ρ ⌢ s3 ρ ⌢ f ⁎ Mn ( 1 − φ ⌢ 3 ) ( 1 − φ ⌢ 1 ) ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 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 ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 ρ ⌢ f EcH ( ε − ψ ˜ ) ∂ H ∂ ξ ∂ ψ ˜ ∂ η − ∂ H ∂ η ∂ ψ ˜ ∂ ξ + Pr ⁎ ( 1 − φ ⌢ 3 ) ( 1 − φ ⌢ 1 ) ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 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 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 ρ ⌢ f Ec ∂ 2 ψ ˜ ∂ η 2 − ∂ 2 ψ ˜ ∂ ξ 2 2 + 4 ∂ 2 ψ ˜ ∂ ξ ∂ η 2 ,

where

(9b) H ( ξ , η ) = H 0 { tanh A ′ 1 ( ξ − ξ 1 ) − tanh A ′ 2 ( ξ − ξ 2 ) } ,

in the strips defined by ξ 1 ≤ ξ ≤ ξ 2 , 0 ≤ η ≤ 1 .

The aforementioned equations represent the stream-vorticity formulation, which is a modified form of equations (1)–(4) with

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

In a similar manner, the dimensionless boundary conditions are interpreted as follows:

Left and right walls (Adiabatic)

(11a) ∂ θ ∂ ξ ξ = 0 = 0 , ∂ θ ∂ ξ ξ = 1 = 0 , u ⌢ ( 0 , η ) = v ⌢ ( 0 , η ) = u ⌢ ( 1 , η ) = v ⌢ ( 1 , η ) = 0 , 0 < η < 1

Top wall

(11b) u ⌢ ( ξ , 1 ) = 1 , v ⌢ ( ξ , 1 ) = 0 , θ ⌢ ( ξ , 1 ) = 1 , 0 < ξ < 1

Bottom wall

(11c) u ⌢ ( ξ , 0 ) = 1 , v ⌢ ( ξ , 0 ) = 0 , θ ⌢ ( ξ , 0 ) = 1 , 0 < ξ < 1 .

Now the Nusselt number and the skin friction are the basic quantities of our interests, which are given by

N u = q ⌢ L k thnf Δ T and C f = 2 τ ρ ⌢ thnf v 0 2 ,

where

q ⌢ = − k thnf ∂ T ∂ η η = 0 , L Heat flux τ = μ ˜ thnf ∂ U ⌢ ∂ η η = 0 , L Shear stress

Using the dimensionless variables gives

C f Re = 2 ( 306 φ ⌢ 2 − 0.19 φ ⌢ + 1 ) 1 − φ ⌢ + φ ⌢ ρ ⌢ s ρ ⌢ f ∂ u ⌢ ∂ y , and Nu = ∂ θ ⌢ ∂ y .

4 Computational methodology

The combined system of non-dimensional Navier–Stokes, and energy equations (4)–(6) may be numerically solved using an Alternating Direction Implicit (ADI) methodology combined with central difference approximations for the derivatives. A finite difference method is generally the most straightforward numerical technique to perceive and apply when handling differential equations that have structured discretization assumptions. Finite differences can also be helpful in other areas when someone needs to approximate derivatives. In a nutshell, the method is easy to achieve high-order approximations and needs a structured grid.

The details of the scheme in moving from n to (n + 1) time level are given below:

(12) 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 ,

(13) 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 ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 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 S1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 c p S2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 c p S3 ρ ⌢ s3 ρ ⌢ f ⁎ Mn ( 1 − φ ⌢ 3 ) ( 1 − φ ⌢ 1 ) ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 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 ⌢ s2 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 ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 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 S1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 c p S2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 c p S3 ρ ⌢ s3 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 ,

(14) 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 ) ,

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

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

(17) 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 ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 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 + M n ( 1 − φ ⌢ 3 ) ( 1 − φ ⌢ 1 ) ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 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 ,

(18) 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 ⌢ s3 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 S1 ρ ⌢ s1 ρ ⌢ f + φ ⌢ 2 c p S2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 c p S3 ρ ⌢ s3 ρ ⌢ f ⁎ Mn ( 1 − φ ⌢ 3 ) ( 1 − φ ⌢ 1 ) ( 1 − φ ⌢ 2 ) ρ ⌢ f + φ ⌢ 1 ρ ⌢ s 1 ρ ⌢ f + φ ⌢ 2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 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 ⌢ s2 ρ ⌢ f + φ ⌢ 3 ρ ⌢ s3 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 ⌢ s2 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 S1 ρ ⌢ s1 ρ ⌢ f + φ ⌢ 2 c p S2 ρ ⌢ s2 ρ ⌢ f + φ ⌢ 3 c p S3 ρ ⌢ s3 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 .

The iterative process is finally stopped once the 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

is reached, which identifies that the steady-state solution has arrived. For the purpose of this study, the present problem assumes TOL < 10 − 6 . Figure 2 depicts the flow diagram for the pseudo-transient technique.

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

A flow chart of pseudo-transient approach.

5 Validation of the numerical technique

In order to confirm the accuracy of our numerical procedure, the numerical results for the horizontal velocity profiles (for the limiting case when φ ⌢ 1 = φ ⌢ 2 = 0 , Mn = 0 , and the flow is driven by only one moving lid), along the three different horizontal lines (y = 0.25, 0.5, 0.75), are compared with ones presented by Yasmin et al. [53] (Figure 3).

Figure 3 Numerical results in regard with Shih and Tan’s analytical solution [54].
Figure 3

Numerical results in regard with Shih and Tan’s analytical solution [54].

To validate the code further, the well-known problem of convection-driven flow in a cavity studied by Chen et al. [55] and Davis [56] is used as a benchmark. The Lattice Boltzmann method and the finite difference method are employed by these studies, respectively.

Table 2 shows that the computational approach used in this study, which yields a Nusselt number that is very consistent with the values reported by the abovementioned studies.

Table 2

Comparison of obtained results with literature

Re Average Nusselt number along the heated wall
Lattice Boltzmann method [55] Finite difference method (Davis [56]) Our approach
10 3 1.1192 1.1181 1.1182
10 4 2.2531 2.2432 2.2481

We will explore how governing parameters such as the nanoparticle volume fraction ( 0 ≤ φ ⌢ 1 , φ ⌢ 2 , φ ⌢ 3 ≤ 0.20 ) , magnetic field localized in the strip ( 0.4 ≤ ξ ≤ 0.6 , and 0 ≤ η ≤ 1 ) , Reynolds number ( 1 ≤ Re ≤ 200 ) , and magnetic number ( 0 ≤ Mn ≤ 200 ) affect the cavity flow. Water is regarded as the base fluid whereas Ag, Al2O3, and TiO2 (as a mixture) are used as the nanoparticles. Additionally, ε = 0 . 02 in our simulations, and the actual properties of water correspond to Pr = 6.2 . Due to the comparably lower Reynolds number, it is assumed that the Eckert number is very small. (e.g., 10−5). Further, the physical properties of the nanoparticles and the base fluid used in the present work can be seen in Table 3.

Table 3

Thermophysical characteristics of water and nanoparticles (Ag–Al2O3–TiO2)

C p ( J kg ‒ 1 K ‒ 1 ) βK‒1 ρ ⌢ ( k gm ‒ 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

On other hand, the convergence of our numerical results with the step size may be seen in Figure 4 which not only shows the stability of our numerical scheme but also depicts the independence of the results from the grid size.

Figure 4 Grid independence analysis.
Figure 4

Grid independence analysis.

6 Results and discussion

This section is devoted to the understanding of results presented in the form of different figures (which include the streamlines, and isotherm) and tables (showing the skin friction and Nusselt number along the lower and upper horizontal walls of the cavity) for the case when a flow is mainly driven by both horizontal lids of the square cavity moving along the direction of the +ve x-axis. Further, the localized magnetic field is also affecting the flow field in the form of a single vertical strip of width 2 units. Finally, the impact of the tri-hybrid nature of the fluid, as well as the intensity of the confined magnetic fields on the Nusselt number and the skin friction factor, will also be discussed.

It is to note that because of having two simultaneous forces, acting in the flow field, such as (i) the inertial force developed by the mechanical effect of the moving lid and (ii) secondly the Lorentz force due to the presence of a magnetic source just outside the cavity near the middle of the left vertical wall, the fluid motion becomes more complex and interesting. Further, the low-density fluid generated due to thermal buoyancy force tries to move from heated walls towards the upward direction whereas such movement is opposed by the high-density fluid originating from the strong magnetic force. Our objective, in this section, is to thoroughly investigate how the two forces affect the flow and heat transfer characteristics of the problem.

6.1 Impact of the magnetic parameter

The impact of the intensity of the magnetic field (characterized as a single vertical strip defined in the regions ( 0.4 < ξ < 0.6 and 0 < η < 1 ) on the flow can be seen in Figure 5. In the absence of a magnetic field, two almost symmetric vortices are observed in the flow field. However, the intensity of the magnetic sources elongates the existing vortices which eventually break down to create two new smaller and weaker vortices rotating in clockwise or counter-clockwise directions. It is noted that the direction of the rotation of vortices is determined by the direction in which the top and bottom lids of the cavity are moving due to some externally imposed mechanical arrangement. In addition, the localized magnetic fields generate a set of counter-rotating vortices (shown in the lighter blue or reddish colors in the streamlines) in the flow which has been created as a result of the break-up of primary vortices. Finally, the elongated vortices in the flow field tend to occupy a major portion of the cavity.

Figure 5 Streamlines representing the flow for various magnetic numbers (Mn).
Figure 5

Streamlines representing the flow for various magnetic numbers (Mn).

Now, Figure 6 shows how the parameter (Mn) influences the thermal signature across the flow field. In the absence of any magnetic force, it can be seen that the major part of the isotherms is red-colored in the flow field, which predicts a relatively higher temperature. Further, a concentration of isotherms is also noted near the top wall of the cavity. The magnetic force, on the other hand, eliminates any linear temperature distribution in any part of the cavity. Additionally, it is possible to detect a decrease in the thermal gradient near the lower horizontal wall of the enclosure.

Figure 6 Isotherms for the flow against various magnetic numbers (Mn).
Figure 6

Isotherms for the flow against various magnetic numbers (Mn).

6.2 Impact of the Reynolds number

The effect of the Reynolds number on the flow may be seen in Figure 7. It is to note that if the dimension of the cavity and the properties of the fluid are constant, an increase in the Reynolds number would simply mean an enhancement in the velocity with which the lids are moving. So, in this way, the Reynolds number may be connected with the lid velocity. At smaller Re, an increase in the parameter results in the elongation of two primary vortices, thus giving rise to new vortices occupying the flow field.

Figure 7 Streamlines representing the flow for various Reynolds numbers (Re).
Figure 7

Streamlines representing the flow for various Reynolds numbers (Re).

The effect of the Reynolds number on the thermal distribution can be seen in Figure 8. As discussed earlier, an increase in the Reynolds number translates into a rise in the velocity of the moving lids, subject to the condition that the fluid properties and the size of the cavity are fined. The faster moving lids will cause a more rigorous mixing of the layer of fluid at different temperatures. This will obviously distort any uniformity in the pattern of the isotherms. Finally, similar to the magnetic parameter, a concentration of isotherms near the upper horizontal wall of the enclosure has been identified.

Figure 8 Isotherms for the flow against various Reynolds numbers (Re).
Figure 8

Isotherms for the flow against various Reynolds numbers (Re).

6.3 Impact of localization

An obvious point to ponder may be the role of the width of the magnetic field in the present problem. To understand this, a single strip defined by ( 0.4 − L ) < ξ < ( 0.6 + L ) , and 0 < η < 1 ) , within the flow field has been considered. It is clear from Figure 9 that the parameter L determines the width of the strip, obviously, L = 0.4, corresponds to the case when the magnetic field is occupying the whole cavity whereas L = 0.0 is the case when the magnetic field is confined to the strip ( 0.4 < ξ < 0.6 , and 0 < η < 1 ) . It is to point out that the confinement of the magnetic field intensity gives rise to new vortices while elongating the primary vortices. The isotherms, on the other hand, are squeezed towards the upper horizontal lid thus giving rise to higher temperature gradients in this region (Figure 10).

Figure 9 Streamlines for various values of the magnetic strip length (L).
Figure 9

Streamlines for various values of the magnetic strip length (L).

Figure 10 Isotherms for various values of the magnetic strip length (L).
Figure 10

Isotherms for various values of the magnetic strip length (L).

6.4 Nusselt number and skin friction dependence on different parameters

Let us now go over how the Nusselt number (Nu) and the skin friction factor (CfRe) vary with the governing parameter Mn and Re for the same range of two parameters. The Nu and CfRe profiles as shown in Figures 11 and 12 are drawn. It is clear that the magnetic parameter significantly enhances heat transfer. On the other hand, the Reynolds number causes a significant rise in the Nusselt number. However, this rise is much more in magnitude than for the magnetic number (for the same range of the two parameters). Finally, as for the skin friction factor CfRe is concerned, it shows that the two parameters influence the CfRe in the same qualitative sense. The quantitative impact may, however, be a little bit different as expected, the particle volume fraction φ 1 , φ 2 and φ 3 cause a notable rise in the Nusselt number with a comparatively lower impact on the skin friction (CfRe). However, compared to φ 2 and φ 3 , the Nusselt number depends more strongly on φ 1 due to the inherent physical properties of the nanoparticles. It has also been noted that the localization or confinement of the magnetic field does not always increase or decrease the Nusselt number. Thus, it is concluded that there will be a certain width of the magnetic corridor for which the Nusselt number would be optimal. However, this may be the topic of a subsequent study.

Figure 11 Dependence of skin friction on different parameters.
Figure 11

Dependence of skin friction on different parameters.

Figure 12 Dependence of Nusselt number on different parameters.
Figure 12

Dependence of Nusselt number on different parameters.

6.5 Impact of the solid volume-fraction (Ag, Al2O3, and TiO2)

The following analysis focuses on how a magnetic field, Reynolds Number, the nanoparticle volume fractions, Nusselt number, and skin friction interact with each other. Unless otherwise stated, we have fixed Mn = 5, Re = 5, φ ⌢ 1 = 0.05 , φ ⌢ 2 = 0.02 , and φ ⌢ 3 = 0.02 . Table 4 shows that the magnetic field intensity may result in up to 13 and 119% increase in the skin friction and Nusselt number, respectively. On the other hand, a remarkable change in the Nusselt number and the skin friction is also observed by raising the Reynolds number Re (Table 5).

Table 4

Variation in Nu and C f with Mn

Mn ∣ Nu ∣ ∣ C f Re ∣
00 0.7699 46.7320
50 0.7319 48.8759
100 1.0854 50.4625
150 1.4127 51.7376
200 1.6864 52.8780
Table 5

Variation in Nu and C f with Re

Re ∣ Nu ∣ ∣ C f Re ∣
01 0.6724 46.6926
50 3.0374 56.1526
100 6.1516 64.0634
150 8.7981 69.1599
200 10.8247 73.0435

It is evident from Tables 6–8 that the Nusselt number (Nu) increased by 62% for φ ⌢ 1 , while decreased by almost 25% for φ ⌢ 2 and φ ⌢ 3 . Moreover, there is a significant decrease (113%) in skin friction for φ ⌢ 1 , while, on the other hand, there is a similar amount of (but reverse) change for φ ⌢ 2 and φ ⌢ 3 . This means that φ ⌢ 1 is more impactful on skin friction in comparison to φ ⌢ 2 and φ ⌢ 3 when there is 20% rise for nanostructures in the base fluid.

Table 6

Variation in Nu and C f with ( φ ⌢ 1 ) for Re = 20 and Mn = 20

φ ⌢ 1 ∣ Nu ∣ ∣ C f Re ∣
0.00 1.8494 46.8612
0.05 2.1034 55.9203
0.10 2.3784 67.2506
0.15 2.6792 81.5945
0.20 3.0115 99.9932
Table 7

Variation in Nu and C f with ( φ ⌢ 2 )

φ ⌢ 2 ∣ Nu ∣ ∣ C f Re ∣
0.00 2.2016 56.8119
0.05 1.9618 55.0545
0.10 1.7590 54.7303
0.15 1.6392 55.6909
0.20 1.6473 57.9029
Table 8

Variation in Nu and C f with ( φ ⌢ 3 )

φ ⌢ 3 ∣ Nu ∣ ∣ C f Re ∣
0.00 2.2153 57.1067
0.05 1.9449 54.6858
0.10 1.7212 53.8729
0.15 1.5858 54.4275
0.20 1.5780 56.2618

With the understanding that the parameter L actually determines one of the geometric dimensions of the rectangle in which the magnetic field is localized, it can be concluded (from Table 9) that the occupation of the magnetic field in a wider part of the domain results in a serious decline in both the Nusselt number and the skin friction factor.

Table 9

Variation in Nu and C f with L

L ∣ Nu ∣ C f
00 2.1034 55.9203
0.10 1.9053 57.0784
0.20 1.2740 53.4481
0.30 1.2361 46.2474
0.35 1.6694 46.7971

6.6 Comparison of thermal performance of the nanoparticles

There are two things that may be inferred from Figure 13: one there is almost a linear relationship between the average Nusselt number (along the vertical wall adjacent to the magnetic source) and the nanoparticle volume fraction, second, the THNF containing Ag, Al2O3, and TiO2 outperforms in terms of enhancing the average Nusselt number, compared to the simple nanofluid with Ag, Al2O3, or TiO2 particles.

Figure 13 Comparison of concerned nanoparticles with THNF.
Figure 13

Comparison of concerned nanoparticles with THNF.

6.7 Impact of magnetic field intensity

It is to point that ξ 1 , ξ 2 , ξ 3 , and ξ 4 appearing in equation (9b) define the two magnetic fields, confined in the form of two vertical strips, as ξ 1 ≤ ξ ≤ ξ 2 , ξ 3 ≤ ξ ≤ ξ 4 and 0 ≤ η ≤ 1 . Further, the parameter A 1 and A 2 determine the amplitude of the magnetic field. Now the impact of the parameter A 1 and A 2 on the average Nusselt number is shown in Figure 14. It is obvious that the Nusselt number depends more on A 1 and A 2 at their smaller values.

Figure 14 Impact of the parameter A 1 and A 2 on average Nusselt number.
Figure 14

Impact of the parameter A 1 and A 2 on average Nusselt number.

6.8 Impact of the localization of magnetic field

The magnetic strip is defined as follows: 0.4 − L ≤ x ≤ 0.6 + L , 0 ≤ η ≤ 1 . This means that the case L = 0 indicates that the strip has a fixed width of 0.2 units. As L is increased (that is, when the strip is widened), it is noted that the Nusselt number first decreases and then increases. Further, the Nusselt number is more sensitive to the nanoparticle concentration at the larger value of φ ⌢ 1 , φ ⌢ 2 , and φ ⌢ 3 (Figure 15).

Figure 15 Impact of the magnetic strip width on the average Nusselt number.
Figure 15

Impact of the magnetic strip width on the average Nusselt number.

7 Concluding remarks

The novelty or originality of the study lies in the combination of lid-driven cavity flow, localized magnetic field, and THNFs, which introduce additional boundary conditions. Most of the scientific literature is enriched with investigations dealing with problems assuming a uniform magnetic field occupying the flow field. But in this investigation, a vertical strip of magnetism within the flow field has been introduced. It may be the first effort to interpret the role of the applied magnetic field in the formation of the new vortices in the flow field. The following conclusions may be drawn.

Without any magnetic field, the flow field shows a couple of primary vortices. But when the magnetic sources are turned on, they stretched and tore apart the vortices, creating two new weaker ones that spin in opposite directions. The magnetic fields have a significant impact on the flow field, as it breaks up the primary vortices into a collection of counter-rotating vortices. This results in a complex pattern of many vortices that have different sizes and shapes. One notable feature of this pattern is that the elongated vortices tend to occupy a large portion of the cavity, making them more visible and dominant than the others. The isotherms have a relatively higher temperature without the magnetic force. The magnetic field also makes the isotherms more concentrated near the upper wall of the enclosure, where the temperature gradient is higher. The magnetic force also affects the temperature distribution in the enclosure, as it disrupts the uniform temperature distribution in the cavity, creating a more varied pattern. It also lowers the temperature difference near the bottom wall of the cavity, making it less hot than before. When the fluid properties and cavity size are carefully tuned, the velocity of the moving lids increases with the Reynolds number. The faster moving lids mixed the fluid layers at different temperatures, disrupting the uniform pattern of the isotherms. Moreover, like a magnetic parameter, the isotherms clustering near the upper horizontal wall of the enclosure have also been observed. The primary vortices stretch and split into new ones when the magnetic field intensity is confined. Meanwhile, the isotherms squeeze towards the top horizontal lid, creating higher temperature gradients in that area. The magnetic field and Reynolds number affect the skin friction factor in the same way, but not by the same amount. The particle volume fraction boosts the Nusselt number more than the skin fraction. The Nusselt number also depends more on aluminum oxide, because of its physical properties. The Nusselt number can go up or down depending on how the magnetic field is confined or localized. This means that there is an optimal magnetic corridor width for the Nusselt number. The magnetic field intensity has a strong influence on the Nusselt number and skin friction, as it can make them increase by up to 119 and 13%, respectively. Similarly, the Reynolds number also has a remarkable impact on the Nusselt number and skin friction, as it changes the velocity of the moving lids. The average Nusselt number increases almost linearly with the nanoparticle volume fraction. Moreover, the THNF with Ag, Al2O3, and TiO2 boosts the average Nusselt number more than the conventional nanofluids such as Ag, Al2O3, or TiO2. The Nusselt number drops first and then rises. It is more sensitive to the high nanoparticle concentration, as it affects the heat transfer rate.

An interesting future direction emerging from the current study may be the consideration of multiple magnetic sources producing fields of different intensities within the flow regime in the form of localized strips, which may result in multiple vortices of different strengths.

Acknowledgments

The authors extend their appreciation to King Saud University for funding this work through Researchers Supporting Project number (RSPD2023R711), King Saud University, Riyadh, Saudi Arabia.

  1. Funding information: The Researchers Supporting Project number (RSPD2023R711), King Saud 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 during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2023-03-19
Revised: 2023-05-12
Accepted: 2023-05-19
Published Online: 2023-07-11

© 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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  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
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  97. Critical review on experimental and theoretical studies of elastic properties of wurtzite-structured ZnO nanowires
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  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
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  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
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  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
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  133. Decellularized extracellular matrix as a promising biomaterial for musculoskeletal tissue regeneration
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  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
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  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
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  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-2022-0561
Keywords for this article
Reynolds number; single-phase model; tri-hybrid nanofluids; localized magnetic field
Creative Commons
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  111. Recent research progress on the stimuli-responsive smart membrane: A review
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  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
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  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
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  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
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  140. Research on AFM tip-related nanofabrication of two-dimensional materials
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  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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