Home A numerical investigation of the magnetized water-based hybrid nanofluid flow over an extending sheet with a convective condition: Active and passive controls of nanoparticles
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A numerical investigation of the magnetized water-based hybrid nanofluid flow over an extending sheet with a convective condition: Active and passive controls of nanoparticles

  • Humaira Yasmin EMAIL logo , Laila A. AL-Essa , Rawan Bossly , Hussam Alrabaiah , Showkat Ahmad Lone and Anwar Saeed EMAIL logo
Published/Copyright: May 31, 2024
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Nanotechnology Reviews
From the journal Nanotechnology Reviews Volume 13 Issue 1

Abstract

This study presents a numerical investigation of a viscous and incompressible three-dimensional flow of hybrid nanofluid composed of Ag and Al2O3 nanoparticles over a convectively heated bi-directional extending sheet with a porous medium. The main equations are converted into dimensionless form by using appropriate variables. The effects of magnetic field, porosity, Brownian motion, thermophoresis, and chemical reaction are investigated. Furthermore, the mass flux and zero-mass flux constraints are used to study heat and mass transfer rates. The obtained data show that the growing magnetic factor has reduced the velocity profiles while increasing the thermal profile. The increased porosity factor has decreased the velocity profiles. The increased thermal Biot number has increased the concentration and thermal profiles. When compared to passive control of nanoparticles, the hybrid nanofluid flow profiles are strongly influenced by the embedded factor in the active control of nanoparticles.

Keywords: nanofluid; hybrid nanofluid; MHD; porous media; Brownian motion; thermophoresis effects; convective and mass flux conditions

1 Introduction

Nanofluids, colloidal suspensions of nanoparticles in pure fluid, have gained considerable interest due to their remarkable thermal properties and potential to improve heat transfer. The addition of nanoparticles, typically ranging from a few to hundreds of nanometers in size, significantly enhances the thermal conductivity of pure fluid, leading to improved heat transfer rates as perceived initially by Sus [1]. This improvement is particularly noticeable in convective heat transfer, where nanofluids have shown higher convective heat transfer coefficients compared to conventional fluids. Shoaib et al. [2] investigated the effects of thermal transportation on Maxwell nanofluid flow on a movable surface with magnetohydrodynamic (MHD) effects and analyzed the main equations using an artificial neural network (ANN) approach. Khan et al. [3] examined thermal flow analysis for nanofluid flow between two permeable sheets using thermophoresis and Brownian motion effects. Nanofluids in microchannel heat exchangers show potential for improving heat transfer due to their improved thermal properties, making them appropriate for applications such as electronics cooling and micro reactors. Khan et al. [4] studied carbon nanotube nanofluid flow with thermally radiated performance and heat sink/source through a microchannel. With the passage of time, it was discovered that suspending two different varieties of nanoparticles in pure fluid can further broaden the thermal behavior of pure fluids [5]. Hybrid nanofluids represent a sophisticated advancement in thermal engineering, combining the benefits of traditional nanofluids with additional additives or nanostructures to further improve heat transfer processes [6]. Abas et al. [7] studied the passive control strategy for blood-based hybrid nanofluid flow over a bi-directional, convectively heated elongating surface. Mumtaz et al. [8] discussed the thermal flow behavior of trihybrid nanofluid flow on an elongating curved sheet using chemical reaction. The hybrid nature of these fluids allows for fine-tuning of properties such as viscosity, stability, and heat transfer performance, making them highly adaptable to specific application requirements [9]. In practice, hybrid nanofluids have demonstrated superior heat transfer performance in various heat exchange systems, including cooling applications in electronics, solar thermal collectors, and automotive radiators [10]. However, challenges such as optimizing the composition and dispersion of hybrid nanoparticles, as well as ensuring long-term stability, remain active research areas to fully unlock the potential of hybrid nanofluids for effective heat transfer processes.

MHD fluid flow is the study of fluid motion using the effects of magnetic fields. When a conductive fluid interacts with a magnetic field, it experiences Lorentz forces that influence its flow behavior. In terms of heat transfer phenomena, MHD fluid flow can have significant impacts due to its ability to alter fluid motion and thermal conductivity [11]. The Lorentz forces induced by the magnetic field can change the velocity profile of the fluid, leading to modifications in heat transfer rates and temperature distributions. Ullah et al. [12] used ANN approach to treat computationally squeezed fluid flow through a circular sheet and have found that fluid velocity decreased as the magnetic factor increased. Jawad et al. [13] studied three-dimensional (3D) MHD thin film nanofluid, mass, and thermal transportation on a permeable gyrating disk. MHD fluid flow has implications for various engineering applications, including nuclear reactors, metallurgical processes, and geophysical fluid dynamics [14]. MHD fluid flow is crucial for optimizing heat transfer processes in these applications, as magnetic fields can be harnessed to manipulate fluid behavior and improve thermal performance [15]. Fiza et al. [16] studied 3D gyrating Jeffery fluid flow through two sheets with effects of Hall current and concluded that the mass flux decreased at the bottom surface. Khan et al. [17] examined dual diffusive Jeffery MHD fluid flow through the cone and disk zones using the effects of different geometries' rotations.

Porous media describe the materials that contain interconnected void spaces. These voids can be filled with air, water, or other fluids, and they play a crucial role in various natural and engineered systems [18]. Examples of porous media include soil, rocks, sponges, and filters. Fluid flow through porous media is a complex process with significant implications for heat transfer velocity. Porous media, such as soil, rocks, and packed beds, contain a network of interconnected voids that alter fluid flow behavior compared to non-porous materials. When a fluid flows through porous media, it encounters resistance due to interactions with the solid matrix, leading to pressure gradients and changes in flow velocity [19]. Darcy’s law governs this flow, relating the fluid velocity to the pressure gradient. In terms of heat transfer, porous media play a crucial role in redistributing thermal energy [20]. The presence of pores provides a large surface area for heat exchange between the fluid and the solid matrix, enhancing heat transfer rates compared to non-porous materials [21]. Convection is the dominant heat transfer mechanism in fluid-saturated porous media, where the flowing fluid carries heat with it, influencing temperature distributions within the porous medium. Additionally, the thermal conductance and diffusivity of the porous medium impact heat transfer velocity, with variations in pore size, shape, and connectivity affecting thermal transport properties [22]. These interactions are essential for optimizing heat transfer processes in various applications, including geothermal energy extraction, groundwater flow, and the heat exchanger design [23]. Modeling simulation and experimental studies are key tools used to investigate fluid flow in permeable media and its impacts on heat transfer velocity, supporting the expansion of efficient and maintainable engineering solutions. Megahed et al. [24] discussed the cross flow of fluid in a permeable medium using chemical reactivity and thermally radiative effects.

Brownian motion, a fundamental phenomenon, refers to the unpredictable, haphazard movement of particles suspended in a fluid medium due to collisions with surrounding molecules. In the framework of heat and mass transfer, Brownian motion significantly influences the behavior of colloidal suspensions and nanofluids [25]. For heat transfer phenomena, Brownian motion enhances thermal mixing and increases the effective contact area between particles and the surrounding fluid, facilitating more efficient heat dissipation [26]. This effect is particularly prominent at small length scales, like in microfluidic devices or nanoporous materials, where thermal fluctuations play a significant role. In mass transfer phenomena, Brownian motion administers the dispersion and diffusion of solute particles, impacting concentration gradients and mass transfer rates [27]. Accurately modeling Brownian motion is crucial for optimizing heat and mass transfer processes in various applications, including pharmaceuticals, biotechnology, and environmental engineering. Thermophoresis, in contrast, refers to the directed motion of particles in a fluid medium in response to a temperature gradient [28]. In heat transfer phenomena, thermophoresis effects arise when temperature gradients induce particle migration, leading to concentration variations within the fluid [29]. This occurrence is mostly pertinent in systems where temperature gradients exist like in combustion processes, microfluidic devices, or industrial furnaces. Thermophoresis impacts heat transfer by influencing particle deposition, aggregation, or dispersion, which in turn can alter convective heat transfer rates and heat distribution within the system [29]. Additionally, in mass transfer processes, thermophoresis affects the transport of particles and solutes, leading to non-uniform concentration profiles and modifying overall mass transfer rates [30,31]. The relationship between thermophoresis and other transport mechanisms is essential for accurately predicting heat and mass transfer phenomena in complex systems and optimizing engineering processes for various applications [32]. Brownian motion and thermophoresis are essential for controlling mass and heat transmission in fluid flow systems [33,34]. Brownian motion enhances thermal mixing and mass dispersion, while thermophoresis influences particle migration and concentration gradients [35]. These two phenomena optimize heat and mass transfer efficiency in various applications.

In many scientific and engineering fields, the study of hybrid nanofluid flow across a three-dimensional extending sheet has great potential. Through the use of advanced control mechanisms, this study seeks to maximize fluid dynamics and heat transfer, improving energy efficiency in a variety of applications comprising renewable energy systems, cooling systems, and aeronautical engineering. The knowledge gathered from this research can also be applied to the biomedical engineering field to improve medication delivery systems and the material processing sectors to improve production processes. Furthermore, this finding has significant environmental consequences since it may lead to improvements in pollution control and wastewater treatment systems. Additionally, the investigation of nanofluid flow behavior in microfluidic devices creates new avenues for applications in medical diagnostics, lab-on-a-chip technologies, and analytical chemistry. Fundamentally, research into the active and passive regulation of nanofluid flow over expanding sheets is a multidisciplinary approach that has broad ramifications for many other sectors. This method encourages creativity and advances the development of effective and sustainable solutions. Therefore, in this work, we have scrutinized the viscous and incompressible 3D flow of a hybrid nanofluid composed of Ag and Al2O3 nanoparticles over a convectively heated bi-directional extending sheet using a porous medium. Furthermore, both the mass flux and zero-mass flux conditions are imposed to examine the heat and mass transfer rates.

2 Problem formulation

Assume the viscous and incompressible 3D flow of hybrid nanofluid composed of Ag and Al2O3 nanoparticles on a bi-directional extending sheet using a porous medium. The sheet stretches with velocities u w ( x ) = c x in the x-direction and v w ( y ) = b y in the y-direction, respectively. Here, a ( > 0 ) and b ( > 0 ) are the stretching constants (Figure 1). A magnetic field of strength B 0 is applied normally (z-direction) to the hybrid nanofluid flow. The flow is convectively heated with a hot working fluid having a heat transfer coefficient ( h f ) such that T f > T w , where T f is the hot working fluid temperature and T w is the constant temperature of the stretching sheet. Furthermore, both the mass flux and zero-mass flux conditions are imposed to examine the heat and mass transfer rates. The chemical reaction, Brownian motion, and thermophoresis impacts are considered to examine the hybrid nanofluid flow analysis. The leading equation takes the following form [36,37]:

(1) ∂ u ∂ x + ∂ v ∂ y + ∂ w ∂ z = 0 ,

(2) u ∂ u ∂ x + v ∂ u ∂ y + w ∂ u ∂ z = ν hnf ∂ 2 u ∂ z 2 − ν hnf K ⁎ u − σ hnf ρ hnf B 0 2 u ,

(3) u ∂ v ∂ x + v ∂ v ∂ y + w ∂ v ∂ z = ν hnf ∂ 2 v ∂ z 2 − ν hnf K ⁎ v − σ hnf ρ hnf B 0 2 v ,

(4) u ∂ T ∂ x + v ∂ T ∂ y + w ∂ T ∂ z = k hnf ( ρ C p ) hnf ∂ 2 T ∂ z 2 + ( ρ C p ) np ( ρ C p ) hnf D B δ ∂ T ∂ z ∂ C ∂ z + D T T ∞ ∂ T ∂ z 2 ,

(5) u ∂ C ∂ x + v ∂ C ∂ y + w ∂ C ∂ z = D B ∂ 2 C ∂ z 2 + δ D T T ∞ ∂ 2 T ∂ z 2 − K 1 ( C − C ∞ ) ,

with boundary conditions [38–40]:

(6) u w ( x ) = c x , v w ( y ) = b y , w = 0 , − k hnf ∂ T ∂ z = h f ( T f − T ) , C = C w , at z = 0 , u → 0 , v → 0 , T → T ∞ , C → C ∞ as z → ∞ . ︸ Active control of nanopartilces

(7) u w ( x ) = c x , v w ( y ) = b y , w = 0 , − k hnf ∂ T ∂ z = h f ( T f − T ) , − D B δ ∂ C ∂ z + D T T ∞ ∂ T ∂ z = 0 , at z = 0 , u → 0 , v → 0 , C → C ∞ , T → T ∞ , as z → ∞ . ︸ Pasive control of nanopartilces

Figure 1 Flow problem geometrical view.
Figure 1

Flow problem geometrical view.

The dynamic viscosity ( μ ), density ( ρ ), heat capacitance ( ρ C p ), thermal conductivity ( k ), and electrical conductivity ( σ ) are defined as [41,42]

(8) μ hnf μ f = 1 ( 1 − φ 1 − φ 2 ) , ρ hnf = ( 1 − φ 2 ) { ρ f ( 1 − φ 1 ) + ρ p 1 φ 1 } + ρ p 2 φ 2 , ( ρ C p ) hnf = ( 1 − φ 2 ) { ( ρ C p ) f ( 1 − φ 1 ) + ( ρ C p ) p 1 φ 1 } + ( ρ C p ) p 2 φ 2 , k hnf k nf = k p 2 − 2 φ 2 ( k nf − k p 2 ) + 2 k nf k p 2 + φ 2 ( k nf − k p 2 ) + 2 k nf , k nf k f = k p 1 − 2 φ 1 ( k f − k p 1 ) + 2 k f k p 1 + φ 1 ( k f − k p 1 ) + 2 k f , σ hnf σ nf = σ p 2 − 2 φ 2 ( σ nf − σ p 2 ) + 2 σ nf σ p 2 + φ 2 ( σ nf − σ p 2 ) + 2 σ nf , σ nf σ f = σ p 1 − 2 φ 1 ( σ f − σ p 1 ) + 2 σ f σ p 1 + φ 1 ( σ f − σ p 1 ) + 2 σ f .

The subscripts f, nf , hnf , p 1 , and p 2 show the fluid, nanofluid, hybrid nanofluid, first nanoparticle, and second nanoparticle, respectively. Also, φ 1 and φ 2 denote the volume fraction of these nanoparticles. Table 1 shows the thermophysical properties of the base fluid and nanoparticles.

Table 1

Thermophysical properties of H 2 O , Ag , and Al 2 O 3 [43]

Thermophysical properties Base fluid and nanoparticles
H 2 O Ag Al 2 O 3
ρ [ kg m ‒ 3 ] 997.1 10,500 3,970
k [ W K ‒ 1 m ‒ 1 ] 0.613 429 40
C p [ J K ‒ 1 kg ‒ 1 ] 4,179 235 765
σ [ S m ‒ 1 ] 5.5 × 10−6 3.6 × 107 3.5 × 107

The similarity variables are defined as [44,45]

(9) u = c f ′ ( η ) x , v = c g ′ ( η ) y , w = − ν f c { ( g ( η ) + f ( η ) ) } , θ ( η ) = T − T ∞ T f − T ∞ , ϕ ( η ) = C − C ∞ C w − C ∞ , η = z c ν f . ︸ Active control of nanopartilces

(10) u = c f ′ ( η ) x , v = c g ′ ( ξ ) y , w = − ν f c { ( g ( η ) + f ( η ) ) } , θ ( η ) = T − T ∞ T f − T ∞ , ϕ ( η ) = C − C ∞ C ∞ , η = z c ν f . ︸ Passive control of nanopartilces

By employing the above similarity variables, we obtained the following dimensionless equations:

(11) μ r ρ r f ‴ ( η ) + f ″ ( η ) ( g ( η ) + f ( η ) ) − f ′ 2 ( η ) − σ r ρ r M f ′ ( η ) − μ r ρ r γ f ′ ( η ) = 0 ,

(12) μ r ρ r g ‴ ( η ) + ( g ( η ) + f ( η ) ) g ″ ( η ) − g ′ 2 ( η ) − σ r ρ r M g ′ ( η ) − μ r ρ r γ g ′ ( η ) = 0 ,

(13) k r ( ρ C p ) r θ ″ ( η ) + Pr ( f ( η ) + g ( η ) ) θ ′ ( η ) + Pr ( ρ C p ) r ( Nb ϕ ′ ( η ) θ ′ ( η ) + θ ′ 2 ( η ) Nt ) = 0 ,

(14) ϕ ″ ( η ) + Nt Nb θ ″ ( η ) + Sc ( f ( η ) + g ( η ) ) ϕ ′ ( η ) − k Sc ϕ ( η ) = 0 ,

with boundary conditions:

(15) f ( 0 ) = 0 , f ′ ( 0 ) = 1 , g ( 0 ) = 0 , θ ′ ( 0 ) = − Bi ( ρ C p ) r ( 1 − θ ( 0 ) ) , g ′ ( 0 ) = α , ϕ ( 0 ) = 1 , f ′ ( ∞ ) = 0 , g ′ ( ∞ ) = 0 , θ ( ∞ ) = 0 , ϕ ( ∞ ) = 0 . ︸ Active control of nanoparticles ,

(16) f ( 0 ) = 0 , g ′ ( 0 ) = α , g ( 0 ) = 0 , f ′ ( 0 ) = 1 , θ ′ ( 0 ) = − Bi ( ρ C p ) r ( 1 − θ ( 0 ) ) , Nt θ ′ ( 0 ) + Nb ϕ ′ ( 0 ) = 0 , f ′ ( ∞ ) = 0 , θ ( ∞ ) = 0 , g ′ ( ∞ ) = 0 , ϕ ( ∞ ) = 0 . Passive control of nanoparticles

In equations (11)–(16), M denotes the magnetic factor, γ designates the porous factor, α signifies the ratio factor, Pr denotes the Prandtl number, Bi shows the thermal Biot number (Bi), k denotes the chemical reaction factor, Nb signifies the Brownian motion factor, and Nt shows the thermophoresis factor

(17) μ r = μ hnf μ f , ρ r = ρ hnf ρ f , σ r = σ hnf σ f , k r = k hnf k f , ( ρ C p ) r = ( ρ C p ) hnf ( ρ C p ) f , α = b c , γ = ν f c K ⁎ , M = σ f B 0 2 ρ f c , Bi = h f k f ν f c , Pr = μ f ( C p ) nf k f , k = K 1 c , Nb = ( ρ C p ) np ( ρ C p ) f δ D B C ∞ ν f ( passive control of nanoparticles ) , Nt = ( ρ C p ) np ( ρ C p ) f D T ( T f − T ∞ ) T ∞ ν f , Nb = ( ρ C p ) np ( ρ C p ) f δ D B ( C w − C ∞ ) ν f ( active control of nanoparticles ) .

The skin friction coefficients, local Nusselt number, and Sherwood number are defined as

(18) C fx = τ wx ρ f u w 2 , C fy = τ wy ρ f v w 2 ,

(19) Nu x = q w k f ( T f − T ∞ ) ,

(20) Sh x = q m D B ( C w − C ∞ ) ,

where

(21) τ wx = μ hnf ∂ u ∂ z z = 0 , τ wy = μ hnf ∂ v ∂ z z = 0 , q w = − k hnf ∂ T ∂ z z = 0 , q m = − D B ∂ C ∂ z z = 0 .

Since we have considered the mass flux and zero-mass flux conditions, for the case of zero-mass flux condition, there is no mass flow at the sheet surface (i.e., C w = 0 ), which vanishes the rate of mass transfer at the surface of the sheet. However, for the case of mass flux condition, there is no mass flow at the sheet surface (i.e., C w ≠ 0 ). Therefore, equations (18)–(20) are reduced as

(22) Re x C fx = μ r f ″ ( 0 ) , Re y C fy = μ r g ″ ( 0 ) ,

(23) Nu x Re x = − k r θ ′ ( 0 ) ,

(24) Sh x Re x = − ϕ ′ ( 0 ) , active control of nanoparticles ,

where Re x = c x 2 ν f and Re y = c y 2 ν f are the local Reynolds numbers.

3 Numerical solution

Bvp4c is a potent numerical methodology used extensively in many scientific and engineering domains. Due to its adaptability, it is extremely helpful in solving differential equations with boundary conditions, especially when obtaining analytical solutions is challenging or impossible. One well-known application is in mechanical engineering, where complicated structural systems like beams, plates, and shells are modeled using bvp4c. Engineers can improve designs, strengthen structural integrity, and guarantee safety requirements in projects ranging from bridges to aerospace components by precisely forecasting stress distributions and deformation behaviors. Bvp4c is also widely used in mathematical biology and epidemiology, where it helps with simulating ecological interactions, disease propagation, and population dynamics. The error tolerance of 10−6 is defined to show the higher accuracy of the modeled problem. To apply this method, we have converted the higher-order model equations (11)–(14) with boundary conditions (15) and (16) into first-order differential equations; we assume that

(25) f = ∐ ( 1 ) , f ′ = ∐ ( 2 ) , f ″ = ∐ ( 3 ) , f ‴ = ∐ ′ ( 3 ) , g = ∐ ( 4 ) , g ′ = ∐ ( 5 ) , g ″ = ∐ ( 6 ) , g ‴ = ∐ ′ ( 6 ) , θ = ∐ ( 7 ) , θ ′ = ∐ ( 8 ) , θ ″ = ∐ ′ ( 8 ) , ϕ = ∐ ( 9 ) , ϕ ′ = ∐ ( 10 ) , ϕ ″ = ∐ ′ ( 10 ) ,

then, the final equations are reduced as

(26) ∐ ′ ( 3 ) = − ( ∐ ( 4 ) + ∐ ( 1 ) ) ∐ ( 3 ) − ( ∐ ( 2 ) ) 2 − σ r ρ r M ∐ ( 2 ) − μ r ρ r γ ∐ ( 2 ) μ r ρ r ,

(27) ∐ ′ ( 6 ) = − ( ∐ ( 4 ) + ∐ ( 1 ) ) ∐ ( 6 ) − ( ∐ ( 5 ) ) 2 − σ r ρ r M ∐ ( 5 ) − μ r ρ r γ ∐ ( 5 ) μ r ρ r ,

(28) ∐ ′ ( 8 ) = − Pr ( ∐ ( 1 ) + ∐ ( 4 ) ) ∐ ( 8 ) + Pr ( ρ C p ) E ( Nb ∐ ( 8 ) ∐ ( 10 ) + N t ( ∐ ( 8 ) ) 2 ) k r ( ρ C p ) r ,

(29) ∐ ′ ( 10 ) = − Nt Nb ∐ ′ ( 8 ) + Sc ( ∐ ( 1 ) + ∐ ( 4 ) ) ∐ ( 10 ) − k Sc ∐ ( 9 ) ,

with boundary conditions:

(30) ∐ a ( 1 ) − 0 , ∐ a ( 4 ) − 0 , ∐ a ( 2 ) − 1 , ∐ a ( 5 ) − α , ∐ a ( 8 ) + Bi ( ρ C p ) r ( 1 − ∐ a ( 7 ) ) , ∐ a ( 9 ) − 1 , ∐ b ( 2 ) − 0 , ∐ b ( 7 ) − 0 , ∐ b ( 5 ) − 0 , ∐ b ( 9 ) − 0 . ︸ Active control of nanoparticles

(31) ∐ a ( 1 ) − 0 , ∐ a ( 4 ) − 0 , ∐ a ( 2 ) − 1 , ∐ a ( 5 ) − α , ∐ a ( 8 ) + Bi ( ρ C p ) r ( 1 − ∐ a ( 7 ) ) , Nb ∐ a ( 10 ) + Nt ∐ a ( 8 ) , ∐ ( 2 ) − 0 , ∐ ( 5 ) − 0 , ∐ ( 7 ) − 0 , ∐ ( 9 ) − 0 . Passive control of nanoparticles

4 Validation

In Table 2, the validation of present results with published results by Eswaramoorthi and Bhuvaneswari [36] and Qayyum et al. [37] for − f ″ ( 0 ) and − g ″ ( 0 ) via growing α when M = γ = φ 1 = φ 2 = 0.0 has been presented. From this table, we have determined that the present results are closely related to those published results by Eswaramoorthi and Bhuvaneswari [36] and Qayyum et al. [37] for − f ″ ( 0 ) and − g ″ ( 0 ) via increasing α when M = γ = φ 1 = φ 2 = 0.0 .

Table 2

Validation of the present results of ‒ f ″ ( η = 0 ) and ‒ g ″ ( η = 0 ) against α when M = γ = φ 1 = φ 2 = 0.0

α ‒ f ″ ( η = 0 ) ‒ g ″ ( η = 0 )
Eswaramoorthi and Bhuvaneswari [36] Qayyum et al. [37] Present results Eswaramoorthi and Bhuvaneswari [36] Qayyum et al. [37] Present results
0.0 1.00000 1.000000 1.000062 0.00000 0.000000 0.000000
0.1 1.02026 1.020259 1.020300 0.06685 0.066847 0.066859
0.2 1.03950 1.039495 1.039523 0.14874 0.148736 0.14875
0.3 1.05795 1.057954 1.057974 0.24336 0.243359 0.243372
0.4 1.07479 1.075788 1.075802 0.34921 0.349208 0.349219
0.5 1.09309 1.093095 1.093105 0.46521 0.465204 0.465213
0.6 1.10995 1.109946 1.109955 0.59063 0.590528 0.590535
0.7 1.12640 1.126397 1.126403 0.72453 0.724531 0.724537
0.8 1.14249 1.142428 1.142493 0.86668 0.866682 0.866687
0.9 1.15826 1.158253 1.158257 1.01654 1.016538 1.016542
1.0 1.17331 1.173720 1.173723 1.17371 1.173720 1.173723
1.2 — — 1.203849 — — 1.508794
1.4 — — 1.233019 — — 1.869722
1.6 — — 1.261347 — — 2.254771
1.8 — — 1.288916 — — 2.662509
2.0 — — 1.315799 — — 3.091739

5 Results and discussion

5.1 Analysis of results

5.2 Discussion of results

This section represents the debate on results obtained for viscous and incompressible 3D flow of hybrid nanofluid composed of Ag and Al2O3 nanoparticles over a bi-directional extending sheet using porous medium. The outcomes are displayed in Figures 2–16 and Tables 3 and 4. Figure 2 shows the distinction in velocity profile ( f ′ ( η ) ) due to the increasing magnetic factor ( M ). From this figure, a diminishing impact in f ′ ( η ) for upsurge in M is observed. The Lorentz force becomes active in the existence of a magnetic field, specifically for electrically conducted fluids like liquid metals or ionized gasses. This force causes fluid deflection and flow resistance because it works perpendicular to the directions of the current and the magnetic field. Consequently, when obstructions to fluid motion are encountered, the velocity profile can decrease. Furthermore, in MHD, the fluid–magnet field interaction can modify flow patterns, which in turn can modify the velocity profile. A decrease in the velocity profile may occur from eddy currents and drag forces that are caused by magnetic fields and have an impact on fluid motion. Therefore, a declining behavior in the velocity via magnetic factor is determined. A similar impact of M on g ′ ( η ) is found in Figure 3. Since the hybrid nanofluid is moving along both directions, a similar behavior is found here. Figure 4 shows the variation in temperature profile ( θ ( η ) ) due to the increasing magnetic parameter ( M ). From this figure, an increasing impact of M on θ ( η ) is observed. Through electromagnetic induction, magnetic fields may cause electrical currents to flow through conducting fluids. Because of the fluid’s electrical resistance, these currents known as eddy currents produce heat; this phenomenon is known as Joule heating. Furthermore, adiabatic compression caused by the magnetic field’s confinement and the compression of the plasma can raise the temperature profile in situations like magnetically confined plasmas. Because of energy conservation, the temperature of fluid upsurges as it compresses. Figure 5 shows the effect of porosity factor ( γ ) on f ′ ( η ) . From this figure, a decreasing impact of γ on f ′ ( η ) is observed. The porosity factor, which denotes the existence of empty spaces in a material medium, significantly reduces flow velocity for a number of important reasons. First of all, as a fluid moves through the complex web of spaces, porous materials obstruct its path and increase resistance. The porous structure’s convoluted flow route adds to this resistance and hinders fluid passage even more. Second, the fluid’s velocity is decreased as a result of frictional forces acting along the pores’ walls. Furthermore, whereas low permeability can worsen flow obstructions and amplify the effect on velocity, larger porosity indicates enhanced permeability. All things considered, porosity greatly reduces the flow velocity of the hybrid nanofluid by creating barriers and frictional resistance. A similar impact of γ on g ′ ( η ) is found in Figure 6. Since the hybrid nanofluid is moving along both directions, a similar behavior is found here. Figure 6 shows the effect of thermal Bi on θ ( η ) . From this figure, a growing impact of Bi on θ ( η ) is observed. Thermal flow features across an elongated surface are determined by the thermal Bi. Heat transport at the fluid–solid interface intensifies as the thermal Bi increases, suggesting a greater thermal opposition at the interface than within the fluid. The fluid’s temperature at the surface rises more quickly as a result of this increased heat transfer creating a greater temperature gradient close to the surface. Furthermore, changes in the temperature profile may lead to greater temperatures further away from the surface as a result of the creation of a thicker thermal boundary layer close to the surface. Thus, a growing influence is found here. It is also found that the higher Bi has strongly influenced temperature profile in the scenario of active control hybrid nanofluid flow when compared to passive of flow. Figure 7 displays the effect of thermal Bi on concentration profile ( ϕ ( η ) ). From this figure, an increasing impact of Bi on ϕ ( η ) is observed. Similar to the above, the scattering of nanoparticles in the fluid is influenced by the rise in thermal Bi in the context of concentration profile across a stretched surface. More heat transmission at the fluid–solid interface facilitates more effective dispersion and mixing of the nanoparticles. This may have an impact on the concentration profile and, in contrast to cases in which the thermal Bi is smaller, can result in a more uniform dispersion of nanoparticles within the fluid. As a result of enhanced dispersion and increased heat transmission, the concentration profile of nanoparticles in the fluid is changed, indicating the influence of the thermal Bi on concentration dynamics. When it comes to active control of nanoparticles, the thermal Bi does have a greater impact on the concentration profile than it does on passive control. This discrepancy results from the different processes and impacts of passive and active control strategies on the dispersion and distribution of nanoparticles in the hybrid nanofluid flow. Figure 9 shows the effect of Brownian motion factor ( Nb ) on θ ( η ) . From this figure, an increasing impact of Nb on θ ( η ) is observed. The Brownian motion that is present in suspensions of nanoparticles is important for heat transfer processes in fluids. Brownian motion causes nanoparticles to come into contact with adjacent fluid molecules, which enhances thermal conduction and heat dispersion in the fluid. As a result of Brownian motion’s ability to promote more effective heat transfer mechanisms, the temperature profile tends to rise. The temperature near the surface rises as a result of the faster heat loss from the surface caused by the higher thermal conduction. Furthermore, Brownian motion lowers temperature gradients and promotes thermal homogeneity by aiding in the fluid’s thermal mixing. Consequently, because Brownian motion promotes heat transmission and thermal homogenization processes in the fluid, it tends to raise the temperature profile. It is also found that the higher Nb has strongly influenced temperature profile in the case of active control hybrid nanofluid flow when compared to passive hybrid nanofluid flow. Figure 10 shows the effect of Nb on ϕ ( η ) . From this figure, a decreasing impact of Nb on ϕ ( η ) is observed. Brownian motion also has an effect on the concentration profile of the nanoparticles in the fluid. Because the nanoparticles move randomly, their movements become more uncertain, which increases the dispersion and diffusion of the nanoparticles throughout the fluid. This uniform distribution of the nanoparticles reduces the specific concentrations near the surface because the random motion of the nanoparticles keeps them from building up in particular areas, which lowers the concentration gradients. As a result, the concentration profile shows a decline in highest concentrations near the surface because of the reducing influence of Brownian motion on nanoparticle aggregation and localization. In addition, Brownian motion’s enhanced dispersion encourages a more uniform dispersion of nanoparticles within the fluid, which helps to lessen concentration gradients and smooth the concentration profile. Consequently, through improving nanoparticle dispersion and lowering localized concentrations close to the surface, Brownian motion tends to lower the concentration profile. When it comes to active control of nanoparticles, Nb does have a greater impact on the concentration profile than it does on passive control. This discrepancy results from the different processes and impacts of passive and active control strategies on the dispersion and distribution of nanoparticles in the hybrid nanofluid flow. Figure 11 shows the effect of thermophoresis factor ( Nt ) on θ ( η ) . From this figure, an increasing impact of Nt on θ ( η ) is observed. In a fluid including nanoparticles, thermophoresis, the process where particles move in accordance with temperature gradients, plays a crucial part in heat transfer mechanisms. In the presence of thermophoresis, the properties of the fluid and the nanoparticles cause the particles to move in the direction of greater or lesser temperatures. Improved heat transmission mechanisms are produced as a result of the concentration gradient created by the movement of nanoparticles throughout the temperature field. In particular, by causing localized concentrations of nanoparticles in areas with temperature gradients, thermophoresis tends to raise the temperature profile, improving its thermal conductivity and the dissipation of heat from the outermost layer. High temperatures are found close to the surface as a result of enhanced heat transmission from the surface to the surrounding fluid caused by the presence of nanoparticles aggregated in temperature gradient areas. Furthermore, thermal mixing within the fluid is facilitated by the concentration gradient created by thermophoresis, which raises heat transfer rates and encourages thermal homogenization. Consequently, because thermophoresis facilitates improved heat transmission processes and encourages thermal mixture within the fluid, it tends to raise the temperature profile. Figure 12 shows the effect of thermophoresis factor ( Nt ) on ϕ ( η ) . From this figure, an increasing impact of Nt on ϕ ( η ) is observed. The concentration profile of the nanoparticles in the fluid is similarly significantly affected by thermophoresis. Because they move in reaction to temperature gradients, nanoparticles have a tendency to gather in areas with these gradients. Peak concentrations are higher in the vicinity of temperature gradients as a result of the localized nanoparticle concentrations created by this accumulation. As a result of the nanoparticle accumulation brought on by thermophoresis, the concentration profile shows an overall rise. Additionally, thermophoresis encourages nanoparticle migration into areas with temperature gradients, which can oppose the effects of other dispersion processes like fluid flow or Brownian motion. The concentration profile is further elevated by the nanoparticles’ greater migratory propensity towards temperature gradient areas. Furthermore, greater nanoparticle dispersion and mixing within the fluid are caused by the concentration gradient created by thermophoresis, which encourages a more uniform distribution of nanoparticles. Consequently, by causing localized concentrations of nanoparticles in areas with temperature gradients and encouraging nanoparticle dispersion throughout the fluid, thermophoresis tends to improve the concentration profile. It is also found that the higher Nt has strongly influenced the concentration profile in the case of active control hybrid nanofluid flow when compared to passive hybrid nanofluid flow. Figure 13 shows the effect of chemical reaction factor ( k ) on ϕ ( η ) . From this figure, a declining impact of k on ϕ ( η ) is observed. The greater chemical reaction reduces the boundary layer thickness of a concentration profile, resulting in a decreasing concentration distribution. It is also found that the higher k has strongly influenced the concentration profile in the case of active control hybrid nanofluid flow when compared to passive hybrid nanofluid flow. Figure 14 shows the effect of Schmidt number ( Sc ) on ϕ ( η ) . From this figure, a declining impact of Sc on ϕ ( η ) is observed. A lower Sc signifies more efficient mass transfer relative to momentum transfer within the fluid. This condition promotes rapid diffusion and dispersion of nanoparticles, leading to a more uniform distribution throughout the flow domain. Consequently, the concentration profile of nanoparticles tends to decrease, with reduced localized concentrations near the extending sheet. The enhanced mass diffusivity facilitates the rapid diffusion of nanoparticles away from regions of high concentration, resulting in a smoother concentration gradient across the flow field. Overall, the lower Sc in the nanofluid flow system enhances mass transfer processes, leading to a more uniform distribution of nanoparticles and consequently reducing the concentration profile near the extending sheet. It is also found that the higher Sc has strongly influenced concentration profile in the case of active control hybrid nanofluid flow when compared to passive hybrid nanofluid flow. Figures 15 and 16 display the impacts of M and α on Re x C fx and Re y C fy . We have observed from these figures, that higher M and α increase both Re x C fx and Re y C fy . Table 3 displays the impacts of Bi , Nt and Nb on Nu x Re x . Here, we have observed that higher Bi , Nt , and Nb increase Nu x Re x for both passive and active controls. The effects of these parameters are discussed in above respective figures. It is also found that these parameters have strongly influenced Nu x Re x in the case of active control hybrid nanofluid flow when compared to passive hybrid nanofluid flow. Table 4 shows the effects Nb , Nt , Sc , and k on Sh x Re x . From this table, we have observed that the greater values Nb , Sc , and k increase Sh x Re x and the greater values of Nt reduce Sh x Re x .

Figure 2 Effect of M M on f ′ ( η ) f^{\prime} (\eta ) .
Figure 2

Effect of M on f ′ ( η ) .

Figure 3 Effect of M M on g ′ ( η ) g^{\prime} (\eta ) .
Figure 3

Effect of M on g ′ ( η ) .

Figure 4 Effect of M M on θ ( η ) \theta (\eta ) .
Figure 4

Effect of M on θ ( η ) .

Figure 5 Effect of γ \gamma on f ′ ( η ) f^{\prime} (\eta ) .
Figure 5

Effect of γ on f ′ ( η ) .

Figure 6 Effect of γ \gamma on g ′ ( η ) g^{\prime} (\eta ) .
Figure 6

Effect of γ on g ′ ( η ) .

Figure 7 Effect of Bi \text{Bi} on θ ( η ) \theta (\eta ) .
Figure 7

Effect of Bi on θ ( η ) .

Figure 8 Effect of Bi \text{Bi} on ϕ ( η ) \phi (\eta ) .
Figure 8

Effect of Bi on ϕ ( η ) .

Figure 9 Effect of Nb \text{Nb} on θ ( η ) \theta (\eta ) .
Figure 9

Effect of Nb on θ ( η ) .

Figure 10 Effect of Nb \text{Nb} on ϕ ( η ) \phi (\eta ) .
Figure 10

Effect of Nb on ϕ ( η ) .

Figure 11 Effect of Nt \text{Nt} on θ ( η ) \theta (\eta ) .
Figure 11

Effect of Nt on θ ( η ) .

Figure 12 Effect of Nt \text{Nt} on ϕ ( η ) \phi (\eta ) .
Figure 12

Effect of Nt on ϕ ( η ) .

Figure 13 Effect of k k on ϕ ( η ) \phi (\eta ) .
Figure 13

Effect of k on ϕ ( η ) .

Figure 14 Effect of Sc \text{Sc} on ϕ ( η ) \phi (\eta ) .
Figure 14

Effect of Sc on ϕ ( η ) .

Figure 15 Effects of M M and α \alpha on Re x C fx \sqrt{{\mathrm{Re}}_{\text{x}}}{C}_{\text{fx}} .
Figure 15

Effects of M and α on Re x C fx .

Figure 16 Effects of M M and α \alpha on Re y C fy \sqrt{{\mathrm{Re}}_{\text{y}}}{C}_{\text{fy}} .
Figure 16

Effects of M and α on Re y C fy .

Table 3

Effects of Bi , Nt , and Nb on Nu x Re x

Bi Nt Nb Nu x Re x
Active control Passive control
0.1 0.105476 0.104761
0.2 0.223467 0.220038
0.3 0.356961 0.347578
0.1 0.658281 0.680756
0.2 0.675423 0.689774
0.3 1.083114 0.700570
0.1 0.648743 0.689774
0.2 0.648743 0.735752
0.3 0.648743 0.806770
Table 4

Effects of Nb , Nt , Sc , and k on Sh x Re x

Nb Nt Sc k Sh x Re x
Active control
0.1 0.015993
0.2 0.365269
0.3 0.478633
0.1 0.372568
0.2 0.015993
0.3 −0.34412
1.0 0.015993
2.0 0.512234
3.0 0.891084
0.1 0.015993
0.2 0.124563
0.3 0.221747

6 Conclusions

A numerical investigation of a viscous and incompressible 3D flow of hybrid nanofluid composed of Ag and Al2O3 nanoparticles over a bi-directional extending sheet using porous medium has been presented here. The impacts of magnetic field, porosity, Brownian motion, thermophoresis, and chemical reaction are investigated. The flow is considered to be convectively heated with a hot working fluid. Furthermore, both the mass flux and zero-mass flux conditions are imposed to investigate the heat and mass transfer rates. Based on the above assumptions, the following key points are obtained:

  • The increased magnetic factor has decreased the velocity profiles while increasing the temperature profile.

  • The increased porosity factor has reduced the velocity profiles.

  • The increased thermal Bi has led to intensification in temperature and concentration distributions.

  • The increased Brownian motion factor has led to an increase in the temperature profile and a decrease in the concentration profile.

  • The increased thermophoresis factor has led to an increase in temperature and concentration profiles.

  • The increased chemical reaction factor and Sc have led to a decrease in the concentration profiles.

  • The greater magnetic and ration factors have increased the friction forces at the surface of the sheet.

  • The greater thermal Bi, thermophoresis, and Brownian motion factors have increased the heat transfer rates.

  • The hybrid nanofluid flow profiles are strongly influenced by the embedded factor in the case of active control of nanoparticles when compared to the case of passive control of nanoparticles.

  1. Funding information: This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R443), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, the Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (GrantA264).

  2. Author contributions: Conceptualization, A.S. and H.Y.; methodology, L.A.E; software, A.S.; R.B.; validation, H.A., and S.A.L.; formal analysis, H.Y.; investigation, S.A.L and A.S.; writing original draft preparation, L.A.E. and S.A.L.; writing review and editing, H.Y. R.B., and S.A.L.; visualization, H.Y. and H.A. 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: All data generated or analyzed during this study are included in this published article.

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Received: 2024-01-07
Revised: 2024-05-04
Accepted: 2024-05-07
Published Online: 2024-05-31

© 2024 the author(s), published by De Gruyter

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

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  106. Micelle-based nanoparticles with stimuli-responsive properties for drug delivery
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  110. Incorporation of organic photochromic molecules in mesoporous silica materials: Synthesis and applications
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  120. Exploiting the nanotechnological approaches for traditional Chinese medicine in childhood rhinitis: A review of future perspectives
  121. Twisto-photonics in two-dimensional materials: A comprehensive review
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  123. Recent process of using nanoparticles in the T cell-based immunometabolic therapy
  124. Future prospects of gold nanoclusters in hydrogen storage systems and sustainable environmental treatment applications
  125. Preparation, types, and applications of one- and two-dimensional nanochannels and their transport properties for water and ions
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  127. Functionalized nanostructures and targeted delivery systems with a focus on plant-derived natural agents for COVID-19 therapy: A review and outlook
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  133. Seed regeneration aided by nanomaterials in a climate change scenario: A comprehensive review
  134. Surface-engineered quantum dot nanocomposites for neurodegenerative disorder remediation and avenue for neuroimaging
  135. Graphitic carbon nitride hybrid thin films for energy conversion: A mini-review on defect activation with different materials
  136. Nanoparticles and the treatment of hepatocellular carcinoma
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  138. Highly safe lithium vanadium oxide anode for fast-charging dendrite-free lithium-ion batteries
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  140. Special Issue on Advanced Nanomaterials for Carbon Capture, Environment and Utilization for Energy Sustainability - Part II
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  148. Engineering in ceramic albite morphology by the addition of additives: Carbon nanotubes and graphene oxide for energy applications
  149. Nanoscale synergy: Optimizing energy storage with SnO2 quantum dots on ZnO hexagonal prisms for advanced supercapacitors
  150. Aging assessment of silicone rubber materials under corona discharge accompanied by humidity and UV radiation
  151. Tuning structural and electrical properties of Co-precipitated and Cu-incorporated nickel ferrite for energy applications
  152. Sodium alginate-supported AgSr nanoparticles for catalytic degradation of malachite green and methyl orange in aqueous medium
  153. An environmentally greener and reusability approach for bioenergy production using Mallotus philippensis (Kamala) seed oil feedstock via phytonanotechnology
  154. Micro-/nano-alumina trihydrate and -magnesium hydroxide fillers in RTV-SR composites under electrical and environmental stresses
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  156. Nanoscale engineering of semiconductor photocatalysts boosting charge separation for solar-driven H2 production: Recent advances and future perspective
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  161. Special Issue on Catechol Based Nano and Microstructures
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  163. In vitro anticancer activity of melanin-like nanoparticles for multimodal therapy of glioblastoma
  164. Poly-3,4-dihydroxybenzylidenhydrazine, a different analogue of polydopamine
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  166. Advancing resource sustainability with green photothermal materials: Insights from organic waste-derived and bioderived sources
  167. Bioinspired neuromelanin-like Pt(iv) polymeric nanoparticles for cancer treatment
  168. Special Issue on Implementing Nanotechnology for Smart Healthcare System
  169. Intelligent explainable optical sensing on Internet of nanorobots for disease detection
  170. Special Issue on Green Mono, Bi and Tri Metallic Nanoparticles for Biological and Environmental Applications
  171. Tracking success of interaction of green-synthesized Carbopol nanoemulgel (neomycin-decorated Ag/ZnO nanocomposite) with wound-based MDR bacteria
  172. Green synthesis of copper oxide nanoparticles using genus Inula and evaluation of biological therapeutics and environmental applications
  173. Biogenic fabrication and multifunctional therapeutic applications of silver nanoparticles synthesized from rose petal extract
  174. Metal oxides on the frontlines: Antimicrobial activity in plant-derived biometallic nanoparticles
  175. Controlling pore size during the synthesis of hydroxyapatite nanoparticles using CTAB by the sol–gel hydrothermal method and their biological activities
  176. Special Issue on State-of-Art Advanced Nanotechnology for Healthcare
  177. Applications of nanomedicine-integrated phototherapeutic agents in cancer theranostics: A comprehensive review of the current state of research
  178. Smart bionanomaterials for treatment and diagnosis of inflammatory bowel disease
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https://doi.org/10.1515/ntrev-2024-0035
Keywords for this article
nanofluid; hybrid nanofluid; MHD; porous media; Brownian motion; thermophoresis effects; convective and mass flux conditions
Creative Commons
BY 4.0

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  161. Special Issue on Catechol Based Nano and Microstructures
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