Startseite Naturwissenschaften A homotopic analysis of the blood-based bioconvection Carreau–Yasuda hybrid nanofluid flow over a stretching sheet with convective conditions
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A homotopic analysis of the blood-based bioconvection Carreau–Yasuda hybrid nanofluid flow over a stretching sheet with convective conditions

  • Humaira Yasmin EMAIL logo , Laila A. AL-Essa , Rawan Bossly , Hussam Alrabaiah , Showkat Ahmad Lone und Anwar Saeed EMAIL logo
Veröffentlicht/Copyright: 23. Mai 2024
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Nanotechnology Reviews
Aus der Zeitschrift Nanotechnology Reviews Band 13 Heft 1

Abstract

The time-independent and incompressible blood-based hybrid nanofluid flow, including Au and Cu nanoparticles across an expanding sheet, has been studied. To illustrate the non-Newtonian performance of the blood-based hybrid nanofluid flow, a non-Newtonian model known as the Carreau–Yasuda model is used. The hybrid nanofluid flow is studied under the influence of magnetic effects, thermal radiation, Brownian motion, thermophoresis, and chemical reactivity. Homotopy analysis method (HAM) is employed to evaluate the modeled equations. A study is conducted on the convergence analysis of HAM, and the HAM and numerical analyses are compared. From the present analysis, the velocity profile increases with an increase in Weissenberg number and decreases with increasing magnetic factor. The temperature, concentration, and microorganisms profiles increase in tandem with the higher thermal Biot, concentration Biot, and microorganism Biot numbers. The thermal and concentration profiles, respectively, have decreased due to the larger thermal and concentration relaxation time factors. The microorganism profiles have decreased due to the increased bioconvection of Lewis and Peclet populations. The modeled equations can be solved by both the HAM and the numerical approaches, validating both approaches to solution.

Keywords: non-Newtonian fluid; nanofluid; hybrid nanofluid; MHD; chemical reaction; motile microorganisms; convective conditions; Cattaneo-Christov heat and mass flux model

Nomenclature

Symbol

u w ( x ) ( m s − 1 )

stretching velocity

T ( K )

temperature

C ( mol m − 3 )

nanoparticle concentration

N

microorganisms concentration

μ 0

zero share stress viscosity

μ ∞

infinite share stress viscosity

Γ , D

Carreau–Yasuda liquid factors

A 1

Rivlin–Ericksen tensor

τ

shear stress

u , v ( m s − 1 )

velocity components

ν ( m − 2 s − 1 )

kinematic viscosity

σ ( S m − 1 )

electrical conductivity

ρ ( kg m − 3 )

density

B 0 ( kg s − 2 A − 1 )

magnetic field strength

k ( W m − 1 K − 1 )

thermal conductivity

C p ( J K − 1 kg − 1 )

specific heat

σ ⁎ ( W m − 2 K − 4 )

Stefan–Boltzmann constant

k ⁎

absorption coefficient

D B ( m − 2 s − 1 ) , D T ( m − 2 s − 1 )

Brownian and thermophoresis diffusion coefficients

n

power index

k c ( s − 1 )

reaction coefficient

ƛ

volume fraction of the solid nanoparticle

We

Weissenberg number

M

magnetic factor

Rd

radiation factor

Nb

Brownian motion factor

Nt

thermophoresis factor

Ec

Eckert number

Sc

Schmidt number

δ

chemical reaction factor

Pr

Prandtl number

δ T

thermal relaxation time factor

δ C

concentration relaxation factor

Lb , Pe

bioconvection Lewis and Peclet numbers

Ω

microorganism difference factor

Bi T

thermal Biot number

Bi C

concentration Biot number

Bi N

microorganism Biot number

Subscripts

hnf

hybrid nanofluid

nf

nanofluid

f

fluid

1

first nanoparticle

2

second nanoparticle

1 Introduction

Non-Newtonian fluid, unlike their Newtonian counterparts, exhibits varying viscosities under different flow circumstances, thereby influencing heat transfer processes in diverse ways. In laminar flow zones, where the viscous nature of non-Newtonian fluids can be shear-rate-dependent, the rate of heat transference is affected by the fluid’s flow behavior [1]. For instance, in shear-thinning fluids like certain polymer solutions or molten plastics, viscosity decreases with increasing shear rate, promoting enhanced mixing and convective heat transfer. Conversely, shear-thickening fluids such as concentrated suspensions or certain biological fluids exhibit increased viscosity under shear, which can hinder flow and heat transfer in some scenarios [2]. Asghar et al. [3] studied non-Newtonian flow of the fluid on a permeable sheet with magnetic effects. Moreover, non-Newtonian fluids can also form complex flow patterns, such as vortex structures and boundary layers, which alter the distribution of heat within a system [4]. Punith Gowda et al. [5] inspected non-Newtonian fluid flow on an elongating surface with influences of magnetic field and chemical reactivity. Nazeer et al. [6] studied computationally the non-Newtonian multiphase fluid flow on an inclined conduit. In practical applications, such as chemical processing, food industry, and polymer extrusion, understanding the rheological behavior of non-Newtonian fluids is crucial for optimizing heat transfer processes, designing efficient equipment, and ensuring product quality and consistency. Additionally, in thermal management systems like microfluidic devices or the cooling of electronic components, the unique rheological properties of non-Newtonian fluids can be leveraged to tailor heat transfer characteristics for specific requirements, offering opportunities for improved energy efficiency and performance. Ragupathi and Prakash [7] studied the effects of linear and nonlinear thermal radiations on a fluid flow on gyrating permeable disks using heat sinks/sources. Thus, the study and application of non-Newtonian fluids in heat transfer processes encompass a multifaceted realm where fluid dynamics, rheology, and thermal sciences intersect, offering both challenges and opportunities for innovation and optimization. Biswal et al. [8] studied magneto and radiative non-Newtonian fluid flow on a nonlinearly elongated sheet.

Nanofluids are engineered colloidal suspensions consisting of nanometer-sized particles dispersed within a pure fluid. Nanofluids exhibit unique thermal properties due to the high thermal conductance of mixed nanoparticles in these fluids, as suggested first by Choi and Eastman [9]. Nanofluids, comprising nanometer-sized particles, disperse. Yasmin et al. [10] have discussed the peristaltic mechanism in Sisko fluid flow with double-diffusive convection and induced magnetic field. It is projected that the results of this study will help in the creation of intelligent magneto-peristaltic pumps, especially for use in drug delivery and thermal applications. Bhatti et al. [11] deliberated on the performance of solar energy using nanofluid flow. This increased thermal conductance facilitates more efficient conduction and convective heat transfer processes, making nanofluids promising candidates for enhancing heat exchangers, electronics cooling systems, and thermal energy storage devices. Abdelsalam and Zaher [12] studied the role of zirconium nanoparticle flow on a viscous slim. Furthermore, nanofluids exhibit tunable rheological properties, enabling tailored flow behavior to optimize heat transfer characteristics in specific applications. Acharya et al. [13] established that growth in a nanolayer within the nanofluid flow increased the thermal flow features to 84.61% in comparison to the diameter of the nanoparticles. Researchers have observed that as time progresses, the addition of an increased quantity of nanoparticles of distinct types into traditional fluids can enhance their thermal flow behavior, leading to the emergence of a new class of fluids identified as hybrid nanofluids. Abdelsalam et al. [14] studied nanofluid flow through a dual-shape annulus placed vertically with elector-osmosis. This phenomenon has sparked considerable interest and investigation within the scientific community. Hybrid nanofluids combine the advantageous properties of different types of nanoparticles with those of pure fluid, thereby potentially achieving superior thermal conductivity and heat transfer performance compared to conventional nanofluids. The impacts of the porous medium and temperature-dependent variable viscosity are considered in analyzing the magnetohydrodynamic (MHD) hybrid nanofluid flow with slip phenomenon by Khalil et al. [15]. It is depicted that the thermal conductivity of the base fluid was greatly enhanced by the addition of nanoparticles. This enhancement has a lot of potential applications needing efficient heat transfer, particularly in engineering systems where heat dissipation is essential. By carefully selecting and blending different nanoparticles, researchers aim to engineer hybrid nanofluids with tailored thermal properties optimized for specific applications, ranging from heat exchangers and cooling systems to thermal energy storage and renewable energy technologies. Abdelsalam and Bhatti [16] studied hybrid nanofluid flow through narrowing veins using three different configurations. The synergistic effects resulting from the combination of nanoparticles offer opportunities for enhancing heat transfer efficiency, improving system performance, and addressing challenges associated with conventional heat transfer fluids. Alshehry et al. [17] investigated the MHD multiphase hybrid nanofluid flow over a stretching sheet with heat transfer. To enable meaningful comparisons, quantitative parametric research is conducted to examine the impact of various physical characteristics. The findings show that, even in the presence of a magnetic field, the hybrid nanofluid displays an increased rate of heat transfer than the conventional fluid. Furthermore, the development of hybrid nanofluids opens avenues for exploring innovative approaches to thermal management and advancing the efficiency and sustainability of various industrial processes and technologies [18].

MHD is a multidisciplinary field that studies the performance of electrically conducted fluids, like ionized gases and plasmas, using magnetic effects. In MHD flows, the interaction between the fluid motion, electromagnetic fields, and thermal effects can significantly impact heat transfer processes [19]. The use of a magnetic field alters the fluid flow patterns and induces electric currents, leading to changes in the temperature distribution and heat transfer rates within the system. Prakash et al. [20] considered the impacts of MHD on the nanofluid flow with the effects of Ohmic heating on an elongated surface immersed in a porous medium. In certain configurations, MHD can lead to increased heat transfer rates through enhanced mixing and turbulence suppression, making it beneficial for applications like nuclear fusion reactors, liquid metal cooling systems, and industrial processes involving high-temperature fluids [21,22]. However, MHD can also introduce challenges such as flow instabilities, pressure drops, and increased electrical resistance, which may need to be addressed in the design and operation of MHD systems for optimal heat transfer performance. The interaction between MHD effects and heat transfer is crucial for harnessing the potential benefits of MHD flows while mitigating associated challenges in various engineering applications. Mahesh et al. [23] debated the influence of radiation on couple stress MHD fluid flow on a permeable surface with dissipative effects. Jawad [24] studied the MHD fluid flow numerically on an elongating sheet that is positioned vertically with impacts of thermal radiations. Lund et al. [25] studied thermally dissipative MHD fluid flow in the form of uniform streams and noticed that the thermal behavior of flow amplified with progression in the concentration of nanoparticles.

Microorganisms are tiny living organisms that are imperceptible, like bacteria and fungi. Despite their small size, microorganisms play a vital role in various flow systems and industrial applications. Fluid flow with microorganisms, also known as microbial fluid dynamics, encompasses a fascinating array of phenomena occurring in diverse environments ranging from soil and marine ecosystems to human bodies [26]. In these environments, microorganisms interact with fluid flow in complex ways, influencing nutrient transport, biofilm formation, and microbial dispersion. Kada et al. [27] studied the significance of microorganisms on the Williamson fluid flow. The behavior of microorganisms in fluid flow is governed by a complex relationship of physical forces, biological processes, and environmental factors [28]. Hydrodynamic forces, such as shear stress and turbulence, affect the motility, orientation, and aggregation of microorganisms, impacting their ability to colonize surfaces or disperse within the fluid. Moreover, the rheological properties of microbial suspensions, including cell density, size distribution, and extracellular polymeric substances, can alter fluid flow patterns and lead to emergent collective behaviors, like bacterial swarming and bioconvection. Doh et al. [29] studied mass and heat transference for fluid flow through a channel placed vertically with constraints at the boundary of the third type. The dynamics of fluid flow with microorganisms is crucial for various applications, including bioremediation, wastewater treatment, and the design of biomedical devices [30]. Advances in experimental techniques, computational modeling, and interdisciplinary research have provided insights into the complex dynamics of microbial fluid interactions, shedding light on fundamental principles governing microbial ecology and evolution in dynamic fluid environments. Waqas et al. [31] studied bio-convective nanofluid flow on the movable surface with impacts of microorganisms and magnetic effects. Moreover, harnessing the unique capabilities of microorganisms in fluid flow holds promise for developing innovative technologies for environmental sustainability, medical diagnostics, and bioproduction processes.

The Cattaneo–Christov heat and mass flux model is associated with Fourier’s [32] and Fick’s laws [33], incorporating non-local effects and accounting for finite thermal and mass propagation speeds. The Cattaneo theory [34] initially proposed adjustments to Fourier’s classical heat conduction model, incorporating the effects of thermal relaxation stresses arising from the finite speed of heat propagation. Building upon this foundation, Christov [35] expanded the model by substituting partial time differentiation with the Oldroyd upper convective derivative. This modification allowed for the consideration of non-local effects, thereby enhancing the accuracy of predictions in scenarios where traditional models fall short. In this model, the main equations include time derivatives of the respective gradients, introducing a relaxation time parameter that characterizes the delay in heat or mass propagation relative to the temperature or concentration gradient. Waqas et al. [36] analyzed comparatively the nanofluid flow by using the impacts of the Cattaneo–Christov model on fluid flow. This model is particularly relevant in situations where the classical Fourier and Fick laws fail to accurately describe thermal and mass transport, such as in systems with high thermal gradients or non-equilibrium conditions. By considering finite propagation speeds, the Cattaneo–Christov model captures transient and non-local effects, yielding more accurate predictions of temperature and concentration distributions in complex fluid flows [37]. This model has applications in several fields, like biological tissues and porous media, as well as mass transfer in chemical reactors and environmental transport phenomena [38]. Waqas et al. [39] applied the Cattaneo–Christov model to simulate computationally the production of entropy for fluid flow on a surface with thermal radiation. Ramesh et al. [40] discussed MHD effects on mass and thermal transportation for fluid flow on a stretched sheet using the Cattaneo–Christov model.

In continuation with the stated literature, this work evaluates blood-based hybrid nanofluid flow, including Au and Cu nanoparticles, across an expanding sheet using the impacts of microorganisms. Bioconvection patterns in the fluid flow across the stretched sheet can be induced by gyrotactic microorganisms. The structured movement of microorganisms as a result of their group relationships and reaction to environmental gradients is known as bioconvection. This phenomenon has the ability to alter mixing and fluid flow patterns further. Because of the way that their movement patterns change in response to fluid flow, gyrotactic bacteria can improve fluid mixing. Increased dispersion and mixing of materials in the fluid may result from this, which may have an impact on chemical reactions, ecological interactions, and the delivery of nutrients. The temperature distribution around the stretched sheet can change when gyrotactic microorganisms are present in heat-transfer settings, including thermal boundary layers. Their motion can have an impact on the rate of heat transmission by affecting the convective heat transfer coefficient. To illustrate the non-Newtonian performance of the blood-based hybrid nanofluid flow, a non-Newtonian model known as the Carreau–Yasuda model is used. The hybrid nanofluid flow problem is formulated under the influence of thermal radiation, magnetic field, Brownian motion, thermophoresis, and chemical reaction.

2 Formulation of problem

Assume the blood-based hybrid nanofluid flow containing Cu and Au nanoparticles on an extending sheet. A two-dimensional Cartesian coordinates system, with u and v in x - and y -directions, is chosen to investigate the hybrid nanofluid flow. The sheet stretches in the x -direction u w ( x ) = a x as its velocity, where a is constant. The temperature, concentration, and microorganism of the fluid, surface, and free stream are denoted by T , C , and N ; T w , C w , and N w ; T ∞ , C ∞ , and N ∞ (see Figure 1). Furthermore, it is supposed that the T f > T w , C f > C w , and N f > N w due to convection conditions at the sheet surface. A non-Newtonian model called Carreau–Yasuda is used to discuss the non-Newtonian behavior of the blood-based hybrid nanofluid flow. The non-Newtonian Carreau–Yasuda is defined as

(1) τ = A 1 ( ( 1 + Γ γ ̇ ) D ) n − 1 D ( ∇ 0 − ∇ ∞ ) + ∇ ∞ ,

where ∇ 0 is the viscosity at a zero shear rate and ∇ ∞ is the viscosity at an infinite share rate. Also, Γ and D are the Carreau–Yasuda liquid factors. A 1 denotes the Rivlin–Ericksen tensor while the extra stress tensor is τ . Furthermore, A 1 = grad V + ( grad V ) T and γ ̇ = 1 2 trace ( A 1 2 ) 1 / 2 .

Figure 1 Geometry for the flow scheme.
Figure 1

Geometry for the flow scheme.

For ∇ ∞ = 0 , equation (1) is reduced to:

(2) τ = ∇ 0 ( ( 1 + Γ γ ̇ ) D ) n − 1 D A 1 .

The flow behavior assumptions mentioned above generate the following equations [41,42]:

(3) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(4) u ∂ u ∂ x + v ∂ u ∂ y = ν hnf ( n − 1 ) D ( D + 1 ) Γ D ∂ u ∂ y D ∂ 2 u ∂ y 2 + σ hnf B 0 2 ρ hnf u ,

(5) u ∂ T ∂ x + v ∂ T ∂ y + λ T u 2 ∂ 2 T ∂ x 2 + u ∂ u ∂ x ∂ T ∂ x + v ∂ T ∂ x ∂ u ∂ y + v 2 ∂ 2 T ∂ y 2 + 2 u v ∂ 2 T ∂ x ∂ y + u ∂ T ∂ y ∂ v ∂ x + v ∂ v ∂ y ∂ T ∂ y = k hnf ( ρ C p ) hnf + 16 σ ⁎ T ∞ 3 3 k ⁎ ( ρ C p ) hnf ∂ 2 T ∂ y 2 + ( ρ C p ) np ( ρ C p ) hnf D B ∂ T ∂ y ∂ C ∂ y + ∂ T ∂ y 2 D T T ∞ ,

(6) u ∂ C ∂ x + v ∂ C ∂ y + λ C u 2 ∂ 2 C ∂ x 2 + v 2 ∂ 2 C ∂ y 2 + u ∂ u ∂ x ∂ C ∂ x + v ∂ u ∂ y ∂ C ∂ x + 2 u v ∂ 2 C ∂ x ∂ y + u ∂ C ∂ y ∂ v ∂ x + v ∂ C ∂ y ∂ v ∂ y = D B ∂ 2 C ∂ y 2 + D T T ∞ ∂ 2 T ∂ y 2 − k c ( C − C ∞ ) ,

(7) u ∂ N ∂ x + v ∂ N ∂ y − D m ∂ 2 N ∂ y 2 = − b W c ( C w − C ∞ ) ∂ ∂ y N ∂ C ∂ y .

With constraints at boundary,

(8) u = a x , v = 0 , − k hnf ∂ T ∂ y = h f ( T f − T ) , − D B ∂ C ∂ y = h C ( C f − C ) , − D m ∂ N ∂ y = h N ( N f − N ) at y = 0 , u → 0 , T → T ∞ , C → C ∞ , N → N ∞ as y → ∞ .

The thermophysical features of the hybrid nanofluid can be characterized as follows with their computational values portrayed in Table 1:

(9) k hnf k f = ƛ 1 k 1 + ƛ 2 k 2 ƛ 1 + ƛ 2 + 2 k f + 2 ( k 1 ƛ 1 + k 2 ƛ 2 ) − 2 ( ƛ 2 + ƛ 1 ) k f k 1 ƛ 1 + k 2 ƛ 2 ƛ 1 + ƛ 2 + 2 k f − ( k 1 ƛ 1 + k 2 ƛ 2 ) + ( ƛ 2 + ƛ 1 ) k f , ρ hnf = ( 1 − ( ƛ 1 + ƛ 2 ) ) ρ f + ρ 2 ƛ 2 + ƛ 1 ρ 1 , μ hnf = μ f ( 1 − ( ƛ 2 + ƛ 1 ) ) 2.5 , ( ρ C p ) hnf = ( 1 − ( ƛ 2 + ƛ 1 ) ) ( ρ C p ) f + ƛ 1 ( ρ C p ) 1 + ƛ 2 ( ρ C p ) 2 , σ hnf σ f = ƛ 1 σ 1 + ƛ 2 σ 2 ƛ 1 + ƛ 2 + 2 σ f + 2 ( ƛ 1 σ 1 + ƛ 2 σ 2 ) − 2 ( ƛ 1 + ƛ 2 ) σ f σ 1 ƛ 1 + σ 2 ƛ 2 ƛ 1 + ƛ 2 + 2 σ f − ( σ 1 ƛ 1 + σ 2 ƛ 2 ) + ( ƛ 2 + ƛ 1 ) σ f . .

Table 1

Thermophysical features of blood, Cu, and Au [43]

Base fluid and nanoparticles Properties
C p ( J/kg K ) ρ ( kg / m 3 ) k ( W/m K ) σ ( S/m )
Blood 3,594 1,063 0.492 0.667
Au 129 19,320 314 4.10 × 107
Cu 385 8,933 401 5.96 × 106

Variables for similarity are described as

(10) u = α x f ′ ( ξ ) , v = − a v f f ( ξ ) , ϕ ( ξ ) = C − C ∞ C f − C ∞ , ψ ( ξ ) = N − N ∞ N f − N ∞ , θ ( ξ ) = T − T ∞ T f − T ∞ , ξ = y a ν f .

Using equation (11) in equations (3)–(8), we have

(11) μ hnf ρ hnf f ‴ ( ξ ) + μ hnf ρ hnf ( n − 1 ) D ( 1 + D ) W e d ( f ″ ( ξ ) ) D f ‴ ( ξ ) + f ″ ( ξ ) f ( ξ ) − f ′ 2 ( ξ ) − M σ hnf ρ hnf f ′ ( ξ ) = 0 ,

(12) 1 Pr 1 ( ρ C p ) hnf / ( ρ C p ) f k hnf k f + Rd θ ″ ( ξ ) − δ T ( θ ″ ( ξ ) f 2 ( ξ ) − f ′ ( ξ ) f ( ξ ) θ ′ ( ξ ) ) + f ( ξ ) θ ′ ( ξ ) + 1 ( ρ C p ) hnf / ( ρ C p ) f { Nb θ ′ ( ξ ) ϕ ′ ( ξ ) + Nt θ ′ 2 ( ξ ) } = 0 ,

(13) ϕ ″ ( ξ ) − Sc δ C ( f 2 ( ξ ) ϕ ″ ( ξ ) − f ( ξ ) f ′ ( ξ ) ϕ ′ ( ξ ) ) + Sc f ( ξ ) ϕ ′ ( ξ ) + Nt Nb θ ″ ( ξ ) − Sc δ ϕ ( ξ ) = 0 ,

(14) ψ ″ ( ξ ) + Lb [ ψ ′ ( ξ ) f ( ξ ) − ψ ( ξ ) f ′ ( ξ ) ] − Pe { ( ψ ( ξ ) + Ω ) ϕ ″ ( ξ ) + ϕ ′ ( ξ ) ψ ′ ( ξ ) } = 0 ,

The transformed constraints are

(15) f ( ξ = 0 ) = 0 , f ′ ( ξ = 0 ) = 1 , f ′ ( ξ → ∞ ) = 0 , k hnf k f θ ′ ( ξ = 0 ) = Bi T { θ ( ξ = 0 ) − 1 } , θ ( ξ → ∞ ) = 0 , ϕ ′ ( ξ = 0 ) = Bi C { ϕ ( ξ = 0 ) − 1 } , ϕ ( ξ → ∞ ) = 0 , ψ ′ ( ξ = 0 ) = Bi N { ψ ( ξ = 0 ) − 1 } , ψ ( ξ → ∞ ) = 0 .

The emerging factors appearing in the above equations are

(16) We = a u w Γ ν f , M = σ f B 0 2 a ρ f , Rd = 4 σ ⁎ T ∞ 3 k f k ⁎ , Nb = ( ρ C p ) np ( ρ C p ) f D B ( C f − C ∞ ) ν f , Bi T = h f k f ν f a , Nt = ( ρ C p ) np ( ρ C p ) f D T ( T f − T ∞ ) ν f T ∞ , Ec = u w 2 ( ρ C p ) f ( T f − T ∞ ) , Sc = ν f D B , δ = k r a , Bi C = h C D B ν f a , Pr = ν f ( ρ C p ) f k f , δ T = λ T a , δ C = λ C a , Lb = ν f D m , Pe = b W c D m , Ω = N ∞ N f − N ∞ , Bi N = h N D m ν f a .

The main quantities of engineering interest are described as

(17) C fx = τ w ρ hnf u w 2 ,

(18) Nu x = x q w k f ( T f − T ∞ ) ,

(19) Sh x = x q m D B ( C f − C ∞ ) ,

(20) Dn x = x q n D m ( N f − N ∞ ) ,

where τ w , q w , q m , and q n are defined as

(21) τ w = μ hnf 1 + n − 1 D Γ D ∂ u ∂ y D ∂ u ∂ y y = 0 .

(22) q w = − k hnf k f + Rd ∂ T ∂ y y = 0 .

(23) q m = − D B ∂ C ∂ y y = 0 .

(24) q n = − D m ∂ N ∂ y y = 0 .

Using equations (21)–(24) in equations (17)–(20), we have

(25) C f = μ hnf μ f 1 + n − 1 2 We D f ″ D ( 0 ) f ″ ( 0 ) .

(26) N u = − k hnf k f + Rd θ ′ ( 0 ) .

(27) S u = − ϕ ′ ( 0 ) .

(28) n u = − ψ ′ ( 0 ) .

Here, Re x = u w x ν f is the local Reynolds number. Also, C f = Re x C fx , N u = Nu x Re x , S u = Sh x Re x , and n u = Dn x Re x .

3 Homotopic approach

In order to introduce the homotopic solution, we have defined the following initial guesses:

(29) f 0 ( ξ ) = 1 − exp ( − ξ ) , θ 0 ( ξ ) = Bi T A 5 + Bi T exp ( − ξ ) , ϕ 0 ( ξ ) = Bi C 1 + Bi C exp ( − ξ ) , ψ 0 ( ξ ) = Bi N 1 + Bi N exp ( − ξ ) .

The operators in linear form are given as

(30) L f ( ξ ) = f ‴ ( ξ ) − f ′ ( ξ ) , L θ ( ξ ) = θ ″ ( ξ ) − θ ( ξ ) , L ϕ ( ξ ) = ϕ ″ ( ξ ) − ϕ ( ξ ) , L ψ ( ξ ) = ψ ″ ( ξ ) − ψ ( ξ ) ,

with the following properties:

(31) L f ( χ 1 + χ 2 exp ( − ξ ) + χ 3 exp ( ξ ) ) = 0 , L θ ( χ 4 exp ( − ξ ) + χ 5 exp ( ξ ) ) = 0 , L ϕ ( χ 6 exp ( − ξ ) + χ 7 exp ( ξ ) ) = 0 , L ψ ( χ 8 exp ( − ξ ) + χ 9 exp ( ξ ) ) = 0 ,

where χ z ( z = 1 , … , 9 ) are constants of a general solution. The zeroth-order deformations are as follows:

(32) ( 1 − P ℜ ) L f [ f ( ξ ; P ℜ ) , f 0 ( ξ ) ] = P ℜ ℏ f N f [ f ( ξ ; P ℜ ) ] ,

(33) ( 1 − P ℜ ) L θ [ θ ( ξ ; P ℜ ) , θ 0 ( ξ ) ] = P ℜ ℏ θ N θ [ θ ( ξ ; P ℜ ) , f ( ξ ; P ℜ ) , ϕ ( ξ ; P ℜ ) ] ,

(34) ( 1 − P ℜ ) L ϕ [ ϕ ( ξ ; P ℜ ) , ϕ 0 ( ξ ) ] = P ℜ ℏ ϕ N ϕ [ ϕ ( ξ ; P ℜ ) , f ( ξ ; P ℜ ) , θ ( ξ ; P ℜ ) ] ,

(35) ( 1 − P ℜ ) L ψ [ ψ ( ξ ; P ℜ ) , ψ 0 ( ξ ) ] = P ℜ ℏ ψ N ψ [ ψ ( ξ ; P ℜ ) , f ( ξ ; P ℜ ) , ϕ ( ξ ; P ℜ ) ] ,

with boundary conditions

(36) f ( ξ = 0 ; P ℜ ) = 0 , f ′ ( ξ = 0 ; P ℜ ) = 1 , f ′ ( ξ → ∞ ; P ℜ ) = 0 .

(37) k hnf k f θ ′ ( ξ = 0 ; P ℜ ) = Bi T { θ ( ξ = 0 ; P ℜ ) − 1 } , θ ( ξ → ∞ ; P ℜ ) = 0 .

(38) ϕ ′ ( ξ = 0 ; P ℜ ) = Bi C { ϕ ( ξ = 0 ; P ℜ ) − 1 } , ϕ ( ξ → ∞ ; P ℜ ) = 0 .

(39) ψ ′ ( ξ = 0 ; P ℜ ) = Bi N { ψ ( ξ = 0 ; P ℜ ) − 1 } , ψ ( ξ → ∞ ; P ℜ ) = 0 .

The nonlinear operators are defined as

(40) N f [ f ( ξ ; P ℜ ) ] = μ hnf ρ hnf ∂ 3 f ( ξ ; P ℜ ) ∂ ξ 3 + μ hnf ρ hnf ( n − 1 ) D ( D + 1 ) W e d ∂ 2 f ( ξ ; P ℜ ) ∂ ξ 2 D ∂ 3 f ( ξ ; P ℜ ) ∂ ξ 3 + ∂ 2 f ( ξ ; P ℜ ) ∂ ξ 2 f ( ξ ; P ℜ ) − ∂ f ( ξ ; P ℜ ) ∂ ξ 2 − M σ hnf ρ hnf ∂ f ( ξ ; P ℜ ) ∂ ξ ,

(41) N θ θ ( ξ ; P ℜ ) , f ( ξ ; P ℜ ) , ϕ ( ξ ; P ℜ ) = 1 Pr 1 ( ρ C p ) hnf / ( ρ C p ) f k hnf k f + Rd ∂ 2 θ ( ξ ; P ℜ ) ∂ ξ 2 + f ( ξ ; P ℜ ) ∂ θ ( ξ ; P ℜ ) ∂ ξ − δ T ∂ 2 θ ( ξ ; P ℜ ) ∂ ξ 2 ( f ( ξ ; P ℜ ) ) 2 − f ( ξ ; P ℜ ) ∂ θ ( ξ ; P ℜ ) ∂ ξ ∂ f ( ξ ; P ℜ ) ∂ ξ + 1 ( ρ C p ) hnf / ( ρ C p ) f Nb ∂ ϕ ( ξ ; P ℜ ) ∂ ξ ∂ θ ( ξ ; P ℜ ) ∂ ξ + Nt ∂ θ ( ξ ; P ℜ ) ∂ ξ 2 ,

(42) N ϕ ϕ ( ξ ; P ℜ ) , θ ( ξ ; P ℜ ) , f ( ξ ; P ℜ ) = − Sc δ C ( f ( ξ ; P ℜ ) ) 2 ∂ 2 ϕ ( ξ ; P ℜ ) ∂ ξ 2 − f ( ξ ; P ℜ ) ∂ ϕ ( ξ ; P ℜ ) ∂ ξ ∂ f ( ξ ; P ℜ ) ∂ ξ + ∂ 2 ϕ ( ξ ; P ℜ ) ∂ ξ 2 + Sc ∂ ϕ ( ξ ; P ℜ ) ∂ ξ f ( ξ ; P ℜ ) + Nt Nb ∂ 2 θ ( ξ ; P ℜ ) ∂ ξ 2 − Sc δ ϕ ( ξ ; P ℜ ) ,

(43) N ψ ψ ( ξ ; P ℜ ) , ϕ ( ξ ; P ℜ ) , f ( ξ ; P ℜ ) = ∂ 2 ψ ( ξ ; P ℜ ) ∂ ξ 2 + Lb f ( ξ ; P ℜ ) ∂ ψ ( ξ ; P ℜ ) ∂ ξ − ψ ( ξ ; P ℜ ) ∂ f ( ξ ; P ℜ ) ∂ ξ − Pe ∂ 2 ϕ ( ξ ; P ℜ ) ∂ ξ 2 ( Ω + ψ ( ξ ; P ℜ ) ) + ∂ ϕ ( ξ ; P ℜ ) ∂ ξ ∂ ψ ( ξ ; P ℜ ) ∂ ξ ,

where P ℜ ∈ [ 0 1 ] and ℏ f , ℏ θ , ℏ ϕ , and ℏ ψ are the embedding factors.

The mth-order deformations are as follows:

(44) L f [ f m ( ξ ) − ϖ m f m − 1 ( ξ ) ] = ℏ f R m f ( ξ ) ,

(45) L θ [ θ m ( ξ ) − ϖ m θ m − 1 ( ξ ) ] = ℏ θ R m θ ( ξ ) ,

(46) L ϕ [ ϕ m ( ξ ) − ϖ m ϕ m − 1 ( ξ ) ] = ℏ ϕ R m ϕ ( ξ ) ,

(47) L ψ [ ψ m ( ξ ) − ϖ m ψ m − 1 ( ξ ) ] = ℏ ψ R m ψ ( ξ ) .

with constraints as

(48) f m ( ξ = 0 ) = f m ′ ( ξ = 0 ) = f ′ ( ξ = ∞ ) = 0 ,

(49) k hnf k f θ m ′ ( ξ = 0 ) − Bi T { θ m ( ξ = 0 ) − 1 } = θ m ( ξ = ∞ ) = 0 ,

(50) ϕ m ′ ( ξ = 0 ) − Bi C { ϕ m ( ξ = 0 ) − 1 } = ϕ ( ξ = ∞ ) = 0 ,

(51) ψ m ′ ( ξ = 0 ) − Bi N { ψ m ( ξ = 0 ) − 1 } = ψ m ( ξ = ∞ ) = 0 ,

(52) R m f ( ξ ) = μ hnf ρ hnf f m − 1 ′ ′ ′ ( ξ ) + μ hnf ρ hnf ( n − 1 ) D ( D + 1 ) W e d ( f m − 1 ′ ′ ( ξ ) ) D f m − 1 ′ ′ ′ ( ξ ) + ∑ k = 0 m − 1 f m − 1 − k ( ξ ) f m ′ ′ ( ξ ) − ∑ k = 0 m − 1 f m − 1 − k ′ ( ξ ) f m ′ ( ξ ) − M σ hnf ρ hnf f m − 1 ′ ( ξ ) ,

(53) R m θ ( ξ ) = 1 Pr 1 ( ρ C p ) hnf / ( ρ C p ) f k hnf k f + Rd θ m − 1 ′ ′ ( ξ ) + ∑ k = 0 m − 1 f m − 1 − k ( ξ ) θ m ′ ( ξ ) − δ T ∑ k = 0 m − 1 f m − 1 − k ( ξ ) ∑ l = 0 k f k − l ( ξ ) θ m ′ ′ ( ξ ) − ∑ k = 0 m − 1 f m − 1 − k ( ξ ) ∑ l = 0 k f k − l ′ ( ξ ) θ m ′ ( ξ ) + 1 ( ρ C p ) hnf / ( ρ C p ) f Nb ∑ k = 0 m − 1 ϕ m − 1 − k ′ ( ξ ) θ m ′ ( ξ ) + Nt ∑ k = 0 m − 1 θ m − 1 − k ′ ( ξ ) θ m ′ ( ξ ) ,

(54) R m ϕ ( ξ ) = ϕ m − 1 ′ ′ ( ξ ) + Sc ∑ k = 0 m − 1 f k − 1 − m ( ξ ) ϕ m ′ ( ξ ) + Nt Nb θ m − 1 ′ ′ ( ξ ) − Sc δ ϕ m − 1 ( ξ ) − Sc δ C ∑ k = 0 m − 1 f m − 1 − k ( ξ ) ∑ l = 0 k f k − l ( ξ ) ϕ m ′ ′ ( ξ ) − ∑ k = 0 m − 1 f m − 1 − k ( ξ ) ∑ l = 0 k f k − l ′ ( ξ ) ϕ m ′ ( ξ ) ,

(55) R m ψ ( ξ ) = ψ m − 1 ′ ′ ( ξ ) + Lb ∑ k = 0 m − 1 f k − 1 − m ( ξ ) ψ m ′ ( ξ ) − ∑ k = 0 m − 1 f k − 1 − m ′ ( ξ ) ψ m ( ξ ) − Pe ∑ k = 0 m − 1 ψ m ( ξ ) ϕ k − 1 − m ′ ′ ( ξ ) + Ω ϕ m − 1 ′ ′ ( ξ ) + ∑ k = 0 m − 1 ϕ k − 1 − m ′ ( ξ ) ψ m ′ ( ξ ) .

For P ℜ = 1 and P ℜ = 0 , we can write

(56) f ( ξ ; 0 ) = f 0 ( ξ ) , f ( ξ ; 1 ) = f ( ξ ) ,

(57) θ ( ξ ; 0 ) = θ 0 ( ξ ) , θ ( ξ ; 1 ) = θ ( ξ ) ,

(58) ϕ ( ξ ; 0 ) = ϕ 0 ( ξ ) , ϕ ( ξ ; 1 ) = ϕ ( ξ ) .

(59) ψ ( ξ ; 0 ) = ψ 0 ( ξ ) , ψ ( ξ ; 1 ) = ψ ( ξ ) .

When P ℜ varies from 0 to 1, then f ( ξ ; P ℜ ) , θ ( ξ ; P ℜ ) , ϕ ( ξ ; P ℜ ) , and ψ ( ξ ; P ℜ ) vary from initial to final solutions. Using Taylor series expansion, we have

(60) f ( ξ ; P ℜ ) = f 0 ( ξ ) + ∑ m = 1 ∞ f m ( ξ ) P ℜ m , f m ( ξ ) = 1 m ! ∂ m f ( ξ ; P ℜ ) ∂ P ℜ m P ℜ = 0 ,

(61) θ ( ξ ; P ℜ ) = θ 0 ( ξ ) + ∑ m = 1 ∞ θ m ( ξ ) P ℜ m , θ m ( ξ ) = 1 m ! ∂ m θ ( ξ ; P ℜ ) ∂ P ℜ m P ℜ = 0 ,

(62) ϕ ( ξ ; P ℜ ) = ϕ 0 ( ξ ) + ∑ m = 1 ∞ ϕ m ( ξ ) P ℜ m , ϕ m ( ξ ) = 1 m ! ∂ m ϕ ( ξ ; P ℜ ) ∂ P ℜ m P ℜ = 0 ,

(63) ψ ( ξ ; P ℜ ) = ψ 0 ( ξ ) + ∑ m = 1 ∞ ψ m ( ξ ) P ℜ m , ψ m ( ξ ) = 1 m ! ∂ m ψ ( ξ ; P ℜ ) ∂ P ℜ m P ℜ = 0 ,

At P ℜ = 1 , the above series (60)–(63) converge, i.e.

(64) f ( ξ ) = ∑ m = 1 ∞ f m ( ξ ) + f 0 ( ξ ) .

(65) θ ( ξ ) = ∑ m = 1 ∞ θ m ( ξ ) + θ 0 ( ξ ) .

(66) ϕ ( ξ ) = ∑ m = 1 ∞ ϕ m ( ξ ) + ϕ 0 ( ξ ) .

(67) ψ ( ξ ) = ψ 0 ( ξ ) + ∑ m = 1 ∞ ψ m ( ξ ) .

4 Convergence of HAM

Homotopy analysis method (HAM) ensures the convergence of a series of solutions. The factor ℏ plays an important role in regulating the convergence area of the solution in series. Figure 2(a)–(d) is used to determine the convergence of velocity f ′ ( ξ ) , temperature θ ( ξ ) , concentration ϕ ( ξ ) , and microorganism ψ ( ξ ) profiles. From these figures, we determined that the convergence area at the 15th-order approximation is − 1.2 ≤ ℏ f ≤ 0.5 for the velocity, − 1.5 ≤ ℏ θ ≤ 0.3 for the temperature panels, − 1.5 ≤ ℏ ϕ ≤ 0.28 for the concentration panels, and − 1.5 ≤ ℏ ψ ≤ 0.28 for the microorganism profiles.

Figure 2 (a)–(d) ℏ − \hslash - curves for f ″ ( 0 ) f^{\prime\prime} (0) , θ ′ ( 0 ) \theta ^{\prime} (0) , ϕ ′ ( 0 ) \phi ^{\prime} (0) , and ψ ′ ( 0 ) \psi ^{\prime} (0) .
Figure 2

(a)–(d) ℏ − curves for f ″ ( 0 ) , θ ′ ( 0 ) , ϕ ′ ( 0 ) , and ψ ′ ( 0 ) .

5 HAM and numerical comparison

This section presents the matching of HAM and numerical methods for velocity f ′ ( ξ ) , temperature θ ( ξ ) , concentration ϕ ( ξ ) , and microorganism ψ ( ξ ) profiles. The HAM technique is conducted at 15th-order approximation and then compared with numerical techniques. The results of both HAM and numerical methods are presented in Tables 2–5 and Figure 3(a)–(d). From these tables and figures, we have proven that both the HAM and numerical methods solve the modeled equations, which validates both the solution methodologies.

Table 2

HAM vs numerical solution for f ′ ( ξ )

ξ f ′ ( ξ )
HAM solution Numerical solution |HAM – numerical|
0.0 1.0 1.0 0.0
0.5 0.6136901962688692 0.6136822412282511 7.95504061812391 × 10−6
1.0 0.3748557659094872 0.3748480144375560 7.75147193116110 × 10−6
1.5 0.2283304529018467 0.2283246747987874 5.77810305935067 × 10−6
2.0 0.1388458725670490 0.1388419719122374 3.90065481162271 × 10−6
2.5 0.0843454101726432 0.0843428986042557 2.51156838748656 × 10−6
3.0 0.0512063178535998 0.0512047409097268 1.57694387303841 × 10−6
3.5 0.0310759484660214 0.0310749722826395 9.76183381892755 × 10−7
4.0 0.0188550441765591 0.0188544448373504 5.99339208692817 × 10−7
4.5 0.0114385641402669 0.0114381979540425 3.66186224471515 × 10−7
5.0 0.0069387234111651 0.0069385003262628 2.23084902310592 × 10−7
5.5 0.0042088735305942 0.0042087378616030 1.356689912556815 × 10−7
6.0 0.0025529304155957 0.0025528479953302 8.242026547713989 × 10−8
Table 3

HAM vs numerical solution for θ ( ξ )

ξ θ ( ξ )
HAM solution Numerical solution |HAM – numerical|
0.0 0.4731921487603305 0.4731920548545173 9.39058132365389 × 10−8
0.5 0.2806483026704129 0.2806482259117134 7.67586995076289 × 10−8
1.0 0.1682851385238433 0.1682850875489276 5.09749157329535 × 10−8
1.5 0.1014473447866921 0.1014473128829621 3.19037300328177 × 10−8
2.0 0.0613218438290716 0.0613218242585015 1.95705701505683 × 10−8
2.5 0.0371211279766323 0.0371211160573974 1.19192349393175 × 10−8
3.0 0.0224894457738898 0.0224894385335563 7.24033357921195 × 10−9
3.5 0.0136313222557260 0.0136313178617980 4.39392805473556 × 10−9
4.0 0.0082644740725819 0.0082644714069845 2.66559735742633 × 10−9
4.5 0.0050114389675497 0.0050114373506615 1.61688819191180 × 10−9
5.0 0.0030391457967907 0.0030391448160713 9.8071940889469 × 10−10
5.5 0.0018431717342258 0.0018431711393834 5.9484244786808 × 10−10
6.0 0.0011178801901255 0.0011178798293340 3.6079153380164 × 10−10
Table 4

HAM vs numerical solution for ϕ ( ξ )

ξ ϕ ( ξ )
HAM solution Numerical solution |HAM – numerical|
0.0 0.3150925925925926 0.3150870398134266 5.55277916597324 × 10−6
0.5 0.1846265939364708 0.1846212382393886 5.35569708214425 × 10−6
1.0 0.1096207934773012 0.1096168138244716 3.97965282955548 × 10−6
1.5 0.0656255225196950 0.0656228397233275 2.68279636748966 × 10−6
2.0 0.0394877332036752 0.0394860070403693 1.72616330589882 × 10−6
2.5 0.0238344953145856 0.0238334119364077 1.08337817751951 × 10−6
3.0 0.0144137318001615 0.0144130613046167 6.70495544740267 × 10−7
3.5 0.0087267051692205 0.0087262935659354 4.11603286650856 × 10−7
4.0 0.0052872545184834 0.0052870030558631 2.51462620245636 × 10−7
4.5 0.0032047637871987 0.0032046106005885 1.53186610248196 × 10−7
5.0 0.0019430084140240 0.0019429152563384 9.31576856184056 × 10−8
5.5 0.0011782076025241 0.0011781510092871 5.65932369677481 × 10−8
6.0 0.0007145136180279 0.0007144792592955 3.43587324906354 × 10−8
Table 5

HAM vs numerical solution for ψ ( ξ )

ξ ψ ( ξ )
HAM solution Numerical solution |HAM – numerical|
0.0 0.33410437710437707 0.3341088269287378 4.44982436008288 × 10−6
0.5 0.20291700716968594 0.2029197098537764 2.70268409044274 × 10−6
1.0 0.12317351610064704 0.1231751567052734 1.64060462643538 × 10−6
1.5 0.07474414488862194 0.0747451404537535 9.95565131611806 × 10−7
2.0 0.04534761872759574 0.0453482227465475 6.04018951827489 × 10−7
2.5 0.02750948136719789 0.0275098477884470 3.66421249158721 × 10−7
3.0 0.01668708987451511 0.0166873121441665 2.22269651437676 × 10−7
3.5 0.01012187278235457 0.0101220076045005 1.34822145964695 × 10−7
4.0 0.00613946178345131 0.0061395435604457 8.17769944729673 × 10−8
4.5 0.00372385842297197 0.0037239080244136 4.96014416415113 × 10−8
5.0 0.00225866615772337 0.0022586962429550 3.00852317082316 × 10−8
5.5 0.00136996198944089 0.0013699802372169 1.82477760854463 × 10−8
6.0 0.00083092825820074 0.0008309393260954 1.10678946837554 × 10−8
Figure 3 (a)–(d) HAM and numerical comparisons for f ′ ( ξ ) f^{\prime} (\xi ) , θ ( ξ ) \theta (\xi ) , ϕ ( ξ ) \phi (\xi ) , and ψ ( ξ ) \psi (\xi ) .
Figure 3

(a)–(d) HAM and numerical comparisons for f ′ ( ξ ) , θ ( ξ ) , ϕ ( ξ ) , and ψ ( ξ ) .

6 Discussion

Figure 4 shows the impact of ( M ) on f ′ ( ξ ) : increasing M decreases f ′ ( ξ ) . Essentially, the Lorentz forces are formed at higher M , which opposes the fluid velocity. This opposing force, due to the Lorentz force, enhances the friction force on the sheet’s surface, which results in decreasing the hybrid nanofluid flow velocity. Figure 5 shows the effect of Weissenberg number ( We ) on f ′ ( ξ ) . It is found the increasing We increases f ′ ( ξ ) . The Weissenberg number is actually the ratio of elastic to resistive forces. In fact, the fluid elasticity increases when We exceeds the relaxation time of the hybrid nanofluid particles, which causes the particles to slow down. This action leads to an improvement in the velocity outline. Figure 6 shows the impact of thermal Biot number Bi T on θ ( ξ ) . It is found that an increase in Bi T increases θ ( ξ ) . Convective heat transmission is more important than conductive heat transfer when Bi T increases. When the convective heat transfer coefficient increases, this frequently happens. Convective heat transmission overtakes conductive heat transfer in situations when Bi T is higher. As a result, the temperature gradient near the surface increases, raising the thermal panels of the fluid flowing across the expanding surface. Figure 7 shows the impact of the concentration Biot number Bi C on ϕ ( ξ ) . It is found that increasing Bi C enhances ϕ ( ξ ) . The higher Bi C increases the concentration gradient close to the surface, raising the concentration profile of the fluid flowing across the expanding surface. Figure 8 shows the impact of the microorganism Biot number Bi N on ψ ( ξ ) . It is found that increasing Bi N enhances ψ ( ξ ) . The higher Bi N increases the concentration gradient close to the surface, raising the microorganism profile of the fluid flowing across the expanding surface. Figure 9 depicts the effect of the temperature relaxation time factor ( δ T ) on θ ( ξ ) . From this figure, it is found that the larger δ T decreases θ ( ξ ) . The relationship between thermal distribution and thermal relaxation parameters is affected. The analysis also shows that the width of the thermal layer at the frontiers diminishes. Basically, the particles of materials require more time to convey heat to their nearby particles as δ T increase. In other words, we may state that a material exhibits non-conducting characteristics at larger values of δ T , which decreases θ ( ξ ) . Figure 10 shows the effect of the concentration relaxation time factor ( δ C ) on ϕ ( ξ ) . From this figure, it is found that higher δ C reduces ϕ ( ξ ) . The relationship between concentration distribution and concentration relaxation parameter is affected. The analysis also shows that the width of the concentration layer at frontiers diminishes. It is because the particles of the material require large time to transmit mass to their nearby particles as δ C increases. In other words, we may state that a material exhibits non-conducting characteristics at larger values of δ C , which decreases the concentration distribution. With progression in ( Nb ) there is a corresponding increase in the thermal distribution, as shown in Figure 11. Actually, the increase in ( Nb ) corresponds to a greater random movement of particles within a fluid, which in turn leads to an increase in the thermal distribution. Physically, this means that as ( Nb ) increases, the individual particles within the fluid experience more frequent collisions and interactions, resulting in greater dispersion of thermal energy throughout the system. Essentially, the particles become more energetic and diffuse more rapidly, contributing to an overall increase in θ ( ξ ) . This phenomenon has significant implications for various processes, such as heat transfer in colloidal suspensions, where Brownian motion plays a crucial role in enhancing thermal conductivity and promoting efficient energy transfer. An increase in ( Nb ) typically corresponds to a decrease in ϕ ( ξ ) , as shown in Figure 12. This counterintuitive phenomenon arises from the enhanced random motion of particles within a fluid, which promotes more efficient mixing and diffusion. As Brownian motion intensifies, particles experience greater collisions and erratic movement, leading to greater dispersion throughout the fluid. Consequently, regions of higher concentration become less pronounced as particles spread out more uniformly, while the areas of lower concentration experience influxes of particles due to increased diffusion. This results in a narrowing of concentration gradients and a more homogeneous distribution. This phenomenon has significant implications in various fields, including chemical engineering processes like mixing and dispersion, where understanding the role of Brownian motion is crucial for optimizing performance and product quality. As the thermophoresis factor ( Nt ) increases, there is a corresponding retardation in the thermal distribution due to the effects of thermophoresis on the particle movement within the fluid, as shown in Figure 13. Physically, thermophoresis refers to the motion of particles induced by temperature gradients, where particles tend to transfer from a zone of higher to lower temperature. When the thermophoresis factor increases, this migration becomes extra prominent, causing the accumulation of particles in zones of lower temperature. Subsequently, thermal distribution is decreased as the movement of particles tends to counteract the natural heat flow within the system, creating localized deviations from thermal equilibrium. This phenomenon has significant implications in various applications, including combustion processes and aerosol dynamics, where understanding and controlling thermophoresis effects are essential for optimizing thermal management and combustion efficiency. With an increase in the thermophoresis factor ( Nt ) , there is typically an increase in ϕ ( ξ ) , as shown in Figure 14 due to the effect of thermophoresis on particle migration within the fluid. When ( Nt ) increases, it indicates a stronger influence of temperature on the movement of particles. Consequently, this causes more concentration distribution. Essentially, as ( Nt ) increases, the particles exhibit an increased tendency to migrate toward the regions of differing temperatures, thereby altering the concentration distribution pattern within the fluid. This phenomenon can have significant implications in various fields, like particle transport in biological or industrial processes. Figure 15 shows the influence of radiation factor ( Rd ) on thermal profiles θ ( ξ ) . When the radiative factor ( Rd ) increases, it suggests a larger impact of radiative heat transfer mechanisms within a system. In physical terms, this suggests that more thermal energy is being exchanged within the flow of fluid via radiation; consequently, θ ( ξ ) increases due to the dispersal of heat throughout the system. Essentially, higher ( Rd ) leads to more efficient and widespread transfer of thermal energy, resulting in a more uniform thermal distribution across the system. This process is crucial in the processes like heat transfer in engineering systems, atmospheric dynamics, and even in the behavior of celestial bodies where radiation plays a significant role in energy exchange. The effects of variation in the chemical reactivity factor ( δ ) on ϕ ( ξ ) is shown in Figure 16, where ϕ ( ξ ) decreased with an increase in ( δ ) . Clearly, when ( δ ) increases, it implies that molecules are more prone to undergo chemical reactions, which often involves the consumption or transformation of reactants into products. This increased reactivity leads to a faster consumption or transformation of the reacting species, thereby causing a decline in ϕ ( ξ ) . Physically, this can be interpreted as a scenario where molecules are engaging in reactions more rapidly, depleting their population in the system as they convert into different chemical species. Consequently, the concentration distribution becomes less uniform as reactants are converted into products at a faster rate, altering the overall chemical composition of the system. The impacts of variation in Schmidt number ( Sc ) on ϕ ( ξ ) is shown in Figure 17, where ϕ ( ξ ) decreased with an increase in ( Sc ) . When ( Sc ) increases, it indicates a decrease in the rate of mass transfer relative to the rate of momentum transfer in a fluid system. This means that diffusion becomes less efficient compared to fluid advection, resulting in a slower spread of molecules through the fluid. Consequently, the concentration gradient becomes less pronounced over time, leading to a decline in ϕ ( ξ ) . Physically, this can be interpreted as a scenario where the movement of molecules through the fluid is hindered relative to the bulk flow of the fluid itself, causing slower mixing and a reduction in the overall uniformity of concentration within the system. The impacts of variation in bio-convection Lewis number ( Lb ) on ψ ( ξ ) is shown in Figure 18, where ϕ ( ξ ) decreases with an increase in ( Lb ) . Increase in ( Lb ) suggests that the diffusivity of nutrients as well as other substances essential for microbial growth relative to the diffusivity of momentum in the fluid decreases. This implies that the transport of nutrients becomes less efficient compared to the movement of the surrounding fluid. As a result, microorganisms experience decreased access to essential nutrients required for their growth and proliferation. Physically, this can be interpreted as a scenario where the nutrient gradients in the fluid become less pronounced, decreasing the ability of microorganisms to efficiently acquire nutrients from their surroundings. Consequently, there is a decline in the distribution and abundance of microorganisms as their growth is limited by nutrient availability in the environment. The impacts of variation in Peclet number ( Pe ) on ψ ( ξ ) is shown in Figure 19, where ϕ ( ξ ) decreases with an increase in ( Pe ) . The increase in ( Pe ) suggests a domination of convective transportation over diffusive transportation in the fluid flow on a surface. This implies that the fluid velocity becomes more significant compared to molecular diffusion in transporting nutrients and other substances. As a result, microorganisms experience less time to interact with the surface, leading to reduced attachment and colonization. Essentially, the higher Peclet number accelerates the fluid flow, diminishing the residence time of microorganisms near the surface, thereby decreasing their profiles and potentially impacting processes like biofilm formation or microbial activity. Figures 20 and 21 show the streamlines performance of a viscous hybrid nanofluid flow and Carreau increase Yasuda hybrid nanofluid flow. From these figures, a clear difference between the behavior of a viscous hybrid nanofluid flow and Carreau–Yasuda hybrid nanofluid flow is obtained. Table 6 shows the influence of M and We on C f . A higher M and We enhances C f , as shown in Table 6. The reason is that the greater values of M yields the Lorentz forces that always acts in a reverse direction of fluid motion. This opposition is due to the greater friction force at sheet’s surface. Also, the higher values of We enhances C f . Table 7 shows the influences of Rd , δ T , Nt , and Nb on N u . The higher values of Rd and Nt enhance N u while the higher values of δ T and Nb decreases N u .

Figure 4 Impact of M M on f ′ ( ξ ) f^{\prime} (\xi ) .
Figure 4

Impact of M on f ′ ( ξ ) .

Figure 5 Impact of We \text{We} on f ′ ( ξ ) f^{\prime} (\xi ) .
Figure 5

Impact of We on f ′ ( ξ ) .

Figure 6 Impact of Bi T {\text{Bi}}_{\text{T}} on θ ( ξ ) \theta (\xi ) .
Figure 6

Impact of Bi T on θ ( ξ ) .

Figure 7 Impact of Bi C {\text{Bi}}_{\text{C}} on ϕ ( ξ ) \phi (\xi ) .
Figure 7

Impact of Bi C on ϕ ( ξ ) .

Figure 8 Impact of Bi N {\text{Bi}}_{\text{N}} on ψ ( ξ ) \psi (\xi ) .
Figure 8

Impact of Bi N on ψ ( ξ ) .

Figure 9 Impact of δ T {\delta }_{\text{T}} on θ ( ξ ) \theta (\xi ) .
Figure 9

Impact of δ T on θ ( ξ ) .

Figure 10 Impact of δ C {\delta }_{\text{C}} on ϕ ( ξ ) \phi (\xi ) .
Figure 10

Impact of δ C on ϕ ( ξ ) .

Figure 11 Impact of Nb \text{Nb} on θ ( ξ ) \theta (\xi ) .
Figure 11

Impact of Nb on θ ( ξ ) .

Figure 12 Impact of Nb \text{Nb} on ϕ ( ξ ) \phi (\xi ) .
Figure 12

Impact of Nb on ϕ ( ξ ) .

Figure 13 Impact of Nt \text{Nt} on θ ( ξ ) \theta (\xi ) .
Figure 13

Impact of Nt on θ ( ξ ) .

Figure 14 Impact of Nt \text{Nt} on ϕ ( ξ ) \phi (\xi ) .
Figure 14

Impact of Nt on ϕ ( ξ ) .

Figure 15 Impact of Rd \text{Rd} on θ ( ξ ) \theta (\xi ) .
Figure 15

Impact of Rd on θ ( ξ ) .

Figure 16 Impact of δ \delta on ϕ ( ξ ) \phi (\xi ) .
Figure 16

Impact of δ on ϕ ( ξ ) .

Figure 17 Impact of Sc \text{Sc} on ϕ ( ξ ) \phi (\xi ) .
Figure 17

Impact of Sc on ϕ ( ξ ) .

Figure 18 Impact of Lb \text{Lb} on ψ ( ξ ) \psi (\xi ) .
Figure 18

Impact of Lb on ψ ( ξ ) .

Figure 19 Impact of Pe \text{Pe} on ψ ( ξ ) \psi (\xi ) .
Figure 19

Impact of Pe on ψ ( ξ ) .

Figure 20 Streamlines behavior of a viscous hybrid nanofluid flow.
Figure 20

Streamlines behavior of a viscous hybrid nanofluid flow.

Figure 21 Streamlines behavior of a Carreau–Yasuda hybrid nanofluid flow.
Figure 21

Streamlines behavior of a Carreau–Yasuda hybrid nanofluid flow.

Table 6

Impacts of M and We on C f

M We C f
0.1 1.446446
0.2 1.497544
0.3 1.529986
0.1 1.475363
0.2 1.486436
0.3 1.497535
Table 7

Impacts of Rd , δ T , Nt and Nb on N u

Rd δ T Nt Nb N u
0.1 1.964367
0.2 1.975374
0.3 1.986363
0.1 1.753253
0.2 1.732135
0.3 1.718437
0.1 1.753568
0.2 1.798543
0.3 1.833684
0.1 1.987543
0.2 1.959224
0.3 1.924684

7 Conclusions

This study examines a time-independent and incompressible blood-based hybrid nanofluid flow containing Cu and Au nanoparticles on elongating surfaces. A non-Newtonian model called Carreau–Yasuda was used with effects of thermal radiation, magnetic field, Brownian motion, thermophoresis, and chemical reactivity. The main equations were evaluated through HAM. The convergence analysis of HAM was investigated, and the HAM and numerical analyses were compared. Based on the above assumptions, the following concluding points are obtained:

  • It was found that increasing the magnetic factor decreases the velocity profile while the higher Weissenberg number increases the velocity profile.

  • The higher thermal, concentration, and microorganism Biot numbers increased the thermal, concentration, and microorganism profiles, respectively.

  • The greater thermal and concentration relaxation time factors decreased the thermal and concentration panels, respectively.

  • The greater Brownian motion factor increased the thermal panels while it decreased the concentration panels.

  • The greater thermophoresis factor increased both the thermal and concentration panels.

  • The higher Schmidt number and chemical reactivity factor decreased the concentration profiles.

  • The higher bioconvection Lewis and Peclet numbers decreased the microorganism profiles.

  • The increasing magnetic factor and Weissenberg number increased the skin friction at the sheet surface.

  • Both the HAM and numerical methods solved the modeled equations, which validated both the solution methodologies.

8 Future recommendations

In the future, this study will be associated with a three-dimensional blood-based hybrid nanofluid flow containing Cu and Au nanoparticles on elongating surfaces. Also, different metallic and non-metallic nanomaterials can be considered to investigate the same model by considering different physical conditions like first- and second-order velocity slip, thermal slip, mass flux, and zero-mass flux constraints.

Acknowledgments

The authors acknowledge the 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 (GrantA153).

  1. Funding information: 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 (GrantA153).

  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: All data generated or analyzed during this study are included in this published article.

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Received: 2023-12-01
Revised: 2024-03-24
Accepted: 2024-04-21
Published Online: 2024-05-23

© 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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  143. Mesoporous silica-grafted deep eutectic solvent-based mixed matrix membranes for wastewater treatment: Synthesis and emerging pollutant removal performance
  144. Electrochemically prepared ultrathin two-dimensional graphitic nanosheets as cathodes for advanced Zn-based energy storage devices
  145. Enhanced catalytic degradation of amoxicillin by phyto-mediated synthesised ZnO NPs and ZnO-rGO hybrid nanocomposite: Assessment of antioxidant activity, adsorption, and thermodynamic analysis
  146. Incorporating GO in PI matrix to advance nanocomposite coating: An enhancing strategy to prevent corrosion
  147. Synthesis, characterization, thermal stability, and application of microporous hyper cross-linked polyphosphazenes with naphthylamine group for CO2 uptake
  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
  155. Mechanism exploration of ion-implanted epoxy on surface trap distribution: An approach to augment the vacuum flashover voltages
  156. Nanoscale engineering of semiconductor photocatalysts boosting charge separation for solar-driven H2 production: Recent advances and future perspective
  157. Excellent catalytic performance over reduced graphene-boosted novel nanoparticles for oxidative desulfurization of fuel oil
  158. Special Issue on Advances in Nanotechnology for Agriculture
  159. Deciphering the synergistic potential of mycogenic zinc oxide nanoparticles and bio-slurry formulation on phenology and physiology of Vigna radiata
  160. Nanomaterials: Cross-disciplinary applications in ornamental plants
  161. Special Issue on Catechol Based Nano and Microstructures
  162. Polydopamine films: Versatile but interface-dependent coatings
  163. In vitro anticancer activity of melanin-like nanoparticles for multimodal therapy of glioblastoma
  164. Poly-3,4-dihydroxybenzylidenhydrazine, a different analogue of polydopamine
  165. Chirality and self-assembly of structures derived from optically active 1,2-diaminocyclohexane and catecholamines
  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
  179. Beyond conventional therapy: Synthesis of multifunctional nanoparticles for rheumatoid arthritis therapy
https://doi.org/10.1515/ntrev-2024-0031
Schlagwörter für diesen Artikel
non-Newtonian fluid; nanofluid; hybrid nanofluid; MHD; chemical reaction; motile microorganisms; convective conditions; Cattaneo-Christov heat and mass flux model
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Research Articles
  2. Tension buckling and postbuckling of nanocomposite laminated plates with in-plane negative Poisson’s ratio
  3. Polyvinylpyrrolidone-stabilised gold nanoparticle coatings inhibit blood protein adsorption
  4. Energy and mass transmission through hybrid nanofluid flow passing over a spinning sphere with magnetic effect and heat source/sink
  5. Surface treatment with nano-silica and magnesium potassium phosphate cement co-action for enhancing recycled aggregate concrete
  6. Numerical investigation of thermal radiation with entropy generation effects in hybrid nanofluid flow over a shrinking/stretching sheet
  7. Enhancing the performance of thermal energy storage by adding nano-particles with paraffin phase change materials
  8. Using nano-CaCO3 and ceramic tile waste to design low-carbon ultra high performance concrete
  9. Numerical analysis of thermophoretic particle deposition in a magneto-Marangoni convective dusty tangent hyperbolic nanofluid flow – Thermal and magnetic features
  10. Dual numerical solutions of Casson SA–hybrid nanofluid toward a stagnation point flow over stretching/shrinking cylinder
  11. Single flake homo p–n diode of MoTe2 enabled by oxygen plasma doping
  12. Electrostatic self-assembly effect of Fe3O4 nanoparticles on performance of carbon nanotubes in cement-based materials
  13. Multi-scale alignment to buried atom-scale devices using Kelvin probe force microscopy
  14. Antibacterial, mechanical, and dielectric properties of hydroxyapatite cordierite/zirconia porous nanocomposites for use in bone tissue engineering applications
  15. Time-dependent Darcy–Forchheimer flow of Casson hybrid nanofluid comprising the CNTs through a Riga plate with nonlinear thermal radiation and viscous dissipation
  16. Durability prediction of geopolymer mortar reinforced with nanoparticles and PVA fiber using particle swarm optimized BP neural network
  17. Utilization of zein nano-based system for promoting antibiofilm and anti-virulence activities of curcumin against Pseudomonas aeruginosa
  18. Antibacterial effect of novel dental resin composites containing rod-like zinc oxide
  19. An extended model to assess Jeffery–Hamel blood flow through arteries with iron-oxide (Fe2O3) nanoparticles and melting effects: Entropy optimization analysis
  20. Comparative study of copper nanoparticles over radially stretching sheet with water and silicone oil
  21. Cementitious composites modified by nanocarbon fillers with cooperation effect possessing excellent self-sensing properties
  22. Confinement size effect on dielectric properties, antimicrobial activity, and recycling of TiO2 quantum dots via photodegradation processes of Congo red dye and real industrial textile wastewater
  23. Biogenic silver nanoparticles of Moringa oleifera leaf extract: Characterization and photocatalytic application
  24. Novel integrated structure and function of Mg–Gd neutron shielding materials
  25. Impact of multiple slips on thermally radiative peristaltic transport of Sisko nanofluid with double diffusion convection, viscous dissipation, and induced magnetic field
  26. Magnetized water-based hybrid nanofluid flow over an exponentially stretching sheet with thermal convective and mass flux conditions: HAM solution
  27. A numerical investigation of the two-dimensional magnetohydrodynamic water-based hybrid nanofluid flow composed of Fe3O4 and Au nanoparticles over a heated surface
  28. Development and modeling of an ultra-robust TPU-MWCNT foam with high flexibility and compressibility
  29. Effects of nanofillers on the physical, mechanical, and tribological behavior of carbon/kenaf fiber–reinforced phenolic composites
  30. Polymer nanocomposite for protecting photovoltaic cells from solar ultraviolet in space
  31. Study on the mechanical properties and microstructure of recycled concrete reinforced with basalt fibers and nano-silica in early low-temperature environments
  32. Synergistic effect of carbon nanotubes and polyvinyl alcohol on the mechanical performance and microstructure of cement mortar
  33. CFD analysis of paraffin-based hybrid (Co–Au) and trihybrid (Co–Au–ZrO2) nanofluid flow through a porous medium
  34. Forced convective tangent hyperbolic nanofluid flow subject to heat source/sink and Lorentz force over a permeable wedge: Numerical exploration
  35. Physiochemical and electrical activities of nano copper oxides synthesised via hydrothermal method utilising natural reduction agents for solar cell application
  36. A homotopic analysis of the blood-based bioconvection Carreau–Yasuda hybrid nanofluid flow over a stretching sheet with convective conditions
  37. In situ synthesis of reduced graphene oxide/SnIn4S8 nanocomposites with enhanced photocatalytic performance for pollutant degradation
  38. A coarse-grained Poisson–Nernst–Planck model for polyelectrolyte-modified nanofluidic diodes
  39. 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
  40. The LyP-1 cyclic peptide modified mesoporous polydopamine nanospheres for targeted delivery of triptolide regulate the macrophage repolarization in atherosclerosis
  41. Synergistic effect of hydroxyapatite-magnetite nanocomposites in magnetic hyperthermia for bone cancer treatment
  42. The significance of quadratic thermal radiative scrutinization of a nanofluid flow across a microchannel with thermophoretic particle deposition effects
  43. Ferromagnetic effect on Casson nanofluid flow and transport phenomena across a bi-directional Riga sensor device: Darcy–Forchheimer model
  44. Performance of carbon nanomaterials incorporated with concrete exposed to high temperature
  45. Multicriteria-based optimization of roller compacted concrete pavement containing crumb rubber and nano-silica
  46. Revisiting hydrotalcite synthesis: Efficient combined mechanochemical/coprecipitation synthesis to design advanced tunable basic catalysts
  47. Exploration of irreversibility process and thermal energy of a tetra hybrid radiative binary nanofluid focusing on solar implementations
  48. Effect of graphene oxide on the properties of ternary limestone clay cement paste
  49. Improved mechanical properties of graphene-modified basalt fibre–epoxy composites
  50. Sodium titanate nanostructured modified by green synthesis of iron oxide for highly efficient photodegradation of dye contaminants
  51. Green synthesis of Vitis vinifera extract-appended magnesium oxide NPs for biomedical applications
  52. Differential study on the thermal–physical properties of metal and its oxide nanoparticle-formed nanofluids: Molecular dynamics simulation investigation of argon-based nanofluids
  53. Heat convection and irreversibility of magneto-micropolar hybrid nanofluids within a porous hexagonal-shaped enclosure having heated obstacle
  54. Numerical simulation and optimization of biological nanocomposite system for enhanced oil recovery
  55. Laser ablation and chemical vapor deposition to prepare a nanostructured PPy layer on the Ti surface
  56. Cilostazol niosomes-loaded transdermal gels: An in vitro and in vivo anti-aggregant and skin permeation activity investigations towards preparing an efficient nanoscale formulation
  57. Linear and nonlinear optical studies on successfully mixed vanadium oxide and zinc oxide nanoparticles synthesized by sol–gel technique
  58. Analytical investigation of convective phenomena with nonlinearity characteristics in nanostratified liquid film above an inclined extended sheet
  59. Optimization method for low-velocity impact identification in nanocomposite using genetic algorithm
  60. Analyzing the 3D-MHD flow of a sodium alginate-based nanofluid flow containing alumina nanoparticles over a bi-directional extending sheet using variable porous medium and slip conditions
  61. A comprehensive study of laser irradiated hydrothermally synthesized 2D layered heterostructure V2O5(1−x)MoS2(x) (X = 1–5%) nanocomposites for photocatalytic application
  62. Computational analysis of water-based silver, copper, and alumina hybrid nanoparticles over a stretchable sheet embedded in a porous medium with thermophoretic particle deposition effects
  63. A deep dive into AI integration and advanced nanobiosensor technologies for enhanced bacterial infection monitoring
  64. Effects of normal strain on pyramidal I and II 〈c + a〉 screw dislocation mobility and structure in single-crystal magnesium
  65. Computational study of cross-flow in entropy-optimized nanofluids
  66. Significance of nanoparticle aggregation for thermal transport over magnetized sensor surface
  67. A green and facile synthesis route of nanosize cupric oxide at room temperature
  68. Effect of annealing time on bending performance and microstructure of C19400 alloy strip
  69. Chitosan-based Mupirocin and Alkanna tinctoria extract nanoparticles for the management of burn wound: In vitro and in vivo characterization
  70. Electrospinning of MNZ/PLGA/SF nanofibers for periodontitis
  71. Photocatalytic degradation of methylene blue by Nd-doped titanium dioxide thin films
  72. Shell-core-structured electrospinning film with sequential anti-inflammatory and pro-neurogenic effects for peripheral nerve repairment
  73. Flow and heat transfer insights into a chemically reactive micropolar Williamson ternary hybrid nanofluid with cross-diffusion theory
  74. One-pot fabrication of open-spherical shapes based on the decoration of copper sulfide/poly-O-amino benzenethiol on copper oxide as a promising photocathode for hydrogen generation from the natural source of Red Sea water
  75. A penta-hybrid approach for modeling the nanofluid flow in a spatially dependent magnetic field
  76. Advancing sustainable agriculture: Metal-doped urea–hydroxyapatite hybrid nanofertilizer for agro-industry
  77. Utilizing Ziziphus spina-christi for eco-friendly synthesis of silver nanoparticles: Antimicrobial activity and promising application in wound healing
  78. Plant-mediated synthesis, characterization, and evaluation of a copper oxide/silicon dioxide nanocomposite by an antimicrobial study
  79. Effects of PVA fibers and nano-SiO2 on rheological properties of geopolymer mortar
  80. Investigating silver and alumina nanoparticles’ impact on fluid behavior over porous stretching surface
  81. Potential pharmaceutical applications and molecular docking study for green fabricated ZnO nanoparticles mediated Raphanus sativus: In vitro and in vivo study
  82. Effect of temperature and nanoparticle size on the interfacial layer thickness of TiO2–water nanofluids using molecular dynamics
  83. Characteristics of induced magnetic field on the time-dependent MHD nanofluid flow through parallel plates
  84. Flexural and vibration behaviours of novel covered CFRP composite joints with an MWCNT-modified adhesive
  85. Experimental research on mechanically and thermally activation of nano-kaolin to improve the properties of ultra-high-performance fiber-reinforced concrete
  86. Analysis of variable fluid properties for three-dimensional flow of ternary hybrid nanofluid on a stretching sheet with MHD effects
  87. Biodegradability of corn starch films containing nanocellulose fiber and thymol
  88. Toxicity assessment of copper oxide nanoparticles: In vivo study
  89. Some measures to enhance the energy output performances of triboelectric nanogenerators
  90. Reinforcement of graphene nanoplatelets on water uptake and thermomechanical behaviour of epoxy adhesive subjected to water ageing conditions
  91. Optimization of preparation parameters and testing verification of carbon nanotube suspensions used in concrete
  92. Max-phase Ti3SiC2 and diverse nanoparticle reinforcements for enhancement of the mechanical, dynamic, and microstructural properties of AA5083 aluminum alloy via FSP
  93. Advancing drug delivery: Neural network perspectives on nanoparticle-mediated treatments for cancerous tissues
  94. PEG-PLGA core–shell nanoparticles for the controlled delivery of picoplatin–hydroxypropyl β-cyclodextrin inclusion complex in triple-negative breast cancer: In vitro and in vivo study
  95. Conduction transportation from graphene to an insulative polymer medium: A novel approach for the conductivity of nanocomposites
  96. Review Articles
  97. Developments of terahertz metasurface biosensors: A literature review
  98. Overview of amorphous carbon memristor device, modeling, and applications for neuromorphic computing
  99. Advances in the synthesis of gold nanoclusters (AuNCs) of proteins extracted from nature
  100. A review of ternary polymer nanocomposites containing clay and calcium carbonate and their biomedical applications
  101. Recent advancements in polyoxometalate-functionalized fiber materials: A review
  102. Special contribution of atomic force microscopy in cell death research
  103. A comprehensive review of oral chitosan drug delivery systems: Applications for oral insulin delivery
  104. Cellular senescence and nanoparticle-based therapies: Current developments and perspectives
  105. Cyclodextrins-block copolymer drug delivery systems: From design and development to preclinical studies
  106. Micelle-based nanoparticles with stimuli-responsive properties for drug delivery
  107. Critical assessment of the thermal stability and degradation of chemically functionalized nanocellulose-based polymer nanocomposites
  108. Research progress in preparation technology of micro and nano titanium alloy powder
  109. Nanoformulations for lysozyme-based additives in animal feed: An alternative to fight antibiotic resistance spread
  110. Incorporation of organic photochromic molecules in mesoporous silica materials: Synthesis and applications
  111. A review on modeling of graphene and associated nanostructures reinforced concrete
  112. A review on strengthening mechanisms of carbon quantum dots-reinforced Cu-matrix nanocomposites
  113. Review on nanocellulose composites and CNFs assembled microfiber toward automotive applications
  114. Nanomaterial coating for layered lithium rich transition metal oxide cathode for lithium-ion battery
  115. Application of AgNPs in biomedicine: An overview and current trends
  116. Nanobiotechnology and microbial influence on cold adaptation in plants
  117. Hepatotoxicity of nanomaterials: From mechanism to therapeutic strategy
  118. Applications of micro-nanobubble and its influence on concrete properties: An in-depth review
  119. A comprehensive systematic literature review of ML in nanotechnology for sustainable development
  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
  122. Current advances of anticancer drugs based on solubilization technology
  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
  126. Microstructural, mechanical, and corrosion characteristics of Mg–Gd–x systems: A review of recent advancements
  127. Functionalized nanostructures and targeted delivery systems with a focus on plant-derived natural agents for COVID-19 therapy: A review and outlook
  128. Mapping evolution and trends of cell membrane-coated nanoparticles: A bibliometric analysis and scoping review
  129. Nanoparticles and their application in the diagnosis of hepatocellular carcinoma
  130. In situ growth of carbon nanotubes on fly ash substrates
  131. Structural performance of boards through nanoparticle reinforcement: An advance review
  132. Reinforcing mechanisms review of the graphene oxide on cement composites
  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
  137. Special Issue on Advanced Nanomaterials and Composites for Energy Conversion and Storage - Part II
  138. Highly safe lithium vanadium oxide anode for fast-charging dendrite-free lithium-ion batteries
  139. Recent progress in nanomaterials of battery energy storage: A patent landscape analysis, technology updates, and future prospects
  140. Special Issue on Advanced Nanomaterials for Carbon Capture, Environment and Utilization for Energy Sustainability - Part II
  141. Calcium-, magnesium-, and yttrium-doped lithium nickel phosphate nanomaterials as high-performance catalysts for electrochemical water oxidation reaction
  142. Low alkaline vegetation concrete with silica fume and nano-fly ash composites to improve the planting properties and soil ecology
  143. Mesoporous silica-grafted deep eutectic solvent-based mixed matrix membranes for wastewater treatment: Synthesis and emerging pollutant removal performance
  144. Electrochemically prepared ultrathin two-dimensional graphitic nanosheets as cathodes for advanced Zn-based energy storage devices
  145. Enhanced catalytic degradation of amoxicillin by phyto-mediated synthesised ZnO NPs and ZnO-rGO hybrid nanocomposite: Assessment of antioxidant activity, adsorption, and thermodynamic analysis
  146. Incorporating GO in PI matrix to advance nanocomposite coating: An enhancing strategy to prevent corrosion
  147. Synthesis, characterization, thermal stability, and application of microporous hyper cross-linked polyphosphazenes with naphthylamine group for CO2 uptake
  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
  155. Mechanism exploration of ion-implanted epoxy on surface trap distribution: An approach to augment the vacuum flashover voltages
  156. Nanoscale engineering of semiconductor photocatalysts boosting charge separation for solar-driven H2 production: Recent advances and future perspective
  157. Excellent catalytic performance over reduced graphene-boosted novel nanoparticles for oxidative desulfurization of fuel oil
  158. Special Issue on Advances in Nanotechnology for Agriculture
  159. Deciphering the synergistic potential of mycogenic zinc oxide nanoparticles and bio-slurry formulation on phenology and physiology of Vigna radiata
  160. Nanomaterials: Cross-disciplinary applications in ornamental plants
  161. Special Issue on Catechol Based Nano and Microstructures
  162. Polydopamine films: Versatile but interface-dependent coatings
  163. In vitro anticancer activity of melanin-like nanoparticles for multimodal therapy of glioblastoma
  164. Poly-3,4-dihydroxybenzylidenhydrazine, a different analogue of polydopamine
  165. Chirality and self-assembly of structures derived from optically active 1,2-diaminocyclohexane and catecholamines
  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
  179. Beyond conventional therapy: Synthesis of multifunctional nanoparticles for rheumatoid arthritis therapy
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