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Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface

  • Ebrahem A. Algehyne , Showkat Ahmad Lone , Anwar Saeed and Gabriella Bognár EMAIL logo
Published/Copyright: March 16, 2024
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Open Physics
From the journal Open Physics Volume 22 Issue 1

Abstract

This article presented micropolar hybrid nanofluid flow comprising copper and alumina nanoparticles over a flat sheet. The mixed convection phenomenon is studied under the effect of gravity. Some additional forces such as magnetic field, thermal radiation, Eckert number, heat source, and thermal slip condition are adopted in this analysis. The leading equations are transformed into dimensionless format by employing appropriate variables and then evaluated by homotopy analysis method (HAM). The obtained results are compared with published results and found a good agreement with those published results. Also, the results of HAM are compared with those of numerical method and found a good agreement as well. The fluctuations within the flow profiles are showcased utilizing figures and tables, followed by an in-depth discussion and analysis. The outcomes of this work show that the higher volume fractions of copper and alumina nanoparticles improved the hybrid nanofluid viscosity, which results in the augmenting variation in the velocity profiles. The higher volume fractions of copper and alumina nanoparticles improved the hybrid nanofluid thermal conductivity, which results in the augmenting variation in thermal distribution. The growing mixed convection factor amplifies the buoyancy force toward the stagnation point flow, which enlarges the velocity panel. The effects of hybrid nanoparticles (Cu-Al2O3/water) at the surface are smaller on friction force and larger in case of thermal flow rate when compared to the nanofluids (Cu/water and Al2O3/water).

Keywords: micropolar fluid; hybrid nanofluid; mixed convection; thermal slip condition; stretching surface; HAM solution

1 Introduction

Micropolar fluid flow is a specialized study within fluid dynamics that considers fluids with both translational and microrotational motions at a microscopic level. The idea of this fluid was introduced by Eringin [1]. Unlike conventional fluids, which only account for translational momentum, micropolar fluids exhibit a unique behavior, where each fluid particle possesses an intrinsic angular momentum or microrotation [2]. This introduces an extra degree of freedom that significantly influences the overall flow characteristics. The governing equations of micropolar fluid dynamics extend the Navier–Stokes equations to incorporate microrotation effects, resulting in a more comprehensive representation of fluid behavior [3]. This augmentation leads to intricate flow patterns and enhanced vorticity due to the interaction between translational movement and microrotational tendencies. Applications of micropolar fluid flow encompass various fields, including biomechanics, microfluidics, and material processing, where the inclusion of microrotation provides a more accurate understanding of fluid–particle interactions [4,5]. Kocić et al. [6] inspected the thermal transportation of micropolar flow of fluid through a horizontal conduit and have noted that expansion in magnetic factor has retarded the velocity transportation and has supported the temperature distribution of fluid. Abbas et al. [7] inspected the effects of thermal and slip flow for micropolar fluid flow on a nonlinear Riga sheet. Ahmad et al. [8] inspected the thermal transmission for bio-convective micropolar fluid flow with viscous dissipative and microrotational characteristics and noted that concentration has augmented for progression in permeability factor, while upsurge in microrotational factor has caused augmentation in Nusselt number. Khan et al. [9] inspected the micropolar bio-convective fluid flow on a thin needle with dissipative and chemical reactivity effects and have noted that fluid concentration has retarded with the progression in chemical reactivity factor.

Heat transfer in nanofluid flow involves the exchange of thermal energy between a base fluid and nanoparticles suspended within it, driven by temperature differences. The indication of nanoparticle suspension in pure fluid was discussed first by Choi [10]. The enriched thermal conductivity of nanoparticles leads to better heat dissipation and transmission rates, making nanofluids promising for applications such as thermal exchangers, efficient cooling, and thermal management in electronics [11]. The altered flow behavior and improved conduction pathways offered by nanofluids contribute to their potential to enhance overall heat transfer performance, addressing challenges in diverse fields requiring efficient heat dissipation and regulation [12]. Shahid et al. [13] examined the activated energy for bio-convective nanoparticle flow on a penetrable sheet and noted that flow panel has weakened and thermal transportation has expanded with progression in nanoparticle concentration. Khan et al. [14] debated on thermal fluid flow for nanofluid through a conduit using impacts of microorganism. Bhatti et al. [15] examined microbes’ impact on Williamson magnetohydrodynamics (MHD) nanofluid flow placed in a permeable medium. The enhanced thermal conductivity and altered flow behavior can lead to improved heat dissipation, making nanofluids attractive for applications such as electronics cooling, automotive radiators, and solar thermal systems [16]. One of the primary motivations for using nanofluids is the potential to significantly increase the thermal conductance of pure fluid. Anjum et al. [17] debated on MHD nanofluid flow on a plate with effect of microorganisms and examined that fluid thermal distribution has supported by nanoparticle concentration and magnetic effects. Upreti and Pandey [18] studied the tangent hyperbolic fluid flow past a stretching surface with Cattaneo–Christov heat flux model. Upreti et al. [19] examined the highly magnetized Casson nanofluid flow containing gold nanoparticle past a stretching surface. They examined different shapes of the gold nanoparticles and found that the cylindrical-shaped gold nanoparticle has greater influence on temperature profile compared to platelet and blade-shaped nanoparticles. Upreti et al. [20] investigated the stagnation point flow of Au-blood fluid past an extending sheet with suction/injection impacts. They examined different shapes of the Au nanoparticles and found that the blade-shaped Au nanoparticle has greater influence on temperature profile. Pandey and Upreti [21] investigated the two-dimensional nanofluid flow over a convectively heated stretching surface.

Hybrid nanofluid flow describes the behavior of a mixture that combines nanoscale particles and traditional fluids, creating a unique fluid medium with enhanced thermal properties. Nanofluids are engineered colloidal suspensions where nanoparticles are distributed in a pure fluid [22]. The introduction of nanoparticles can significantly modify the thermophysical and transportation characteristics of fluid. Sundar and Shaik [23] discussed the impacts of thermal transmission for diamond nanoparticle flow in a heat exchanger. Thermal flow in hybrid nanofluid flow is a complex phenomenon where the synergistic effects of combining several varieties of nanoparticles with a pure fluid lead to enhanced thermal conductivity and upgraded temperature transmission performance [24]. The incorporation of multiple nanoparticle types in the hybrid nanofluid further enhances this effect by utilizing their distinct thermal properties. This combination enables fine-tuning of the fluid’s thermophysical characteristics to optimize heat transfer efficiency for specific applications, such as electronics cooling, automotive thermal management, and renewable energy systems [25]. Mahmood et al. [26] analyzed the hybrid nanofluid flow on a curved stretched sheet using magnetic effects and have noted that when suction factor varied from 2.0 to 2.5, the rate of heat flow has enlarged from 34 to 39%. Atashafrooz et al. [27] inspected the collective radiative–convective effects on thermal flow transportation for hybrid nanofluid flow on a surface. Raizah et al. [28] discussed the hybrid nanofluid flow in a conduit with impacts of activated energy, chemical reactivity, and Soret/Dufour effects. Latha et al. [29] addressed the stagnation point flow of a ternary hybrid nanofluid. They found that the skin friction and local Nusselt number are greatly influenced by the embedded factor as compared to hybrid and nanofluids. Prekash at al. [30] investigated the ternary hybrid nanofluid flow over a porous wedge under the influences of transverse magnetic and electric fields. They found that both the electric and magnetic fields’ factors have declining impacts on the velocity profile of the ternary hybrid nanofluid flow. Further related studies can be seen in refs. [31–36].

The flow of fluid on stretching surface is a classical problem in fluid mechanics and has uses in various scientific areas. This problem involves the investigation of how a fluid flows on a sheet that is continuously stretching or contracting. Heat transference for flow of fluid on an extending surface involves the study of how heat is exchanged between a fluid and a surface that is continuously stretching or contracting [37]. The behavior of the thermal layer at boundary, influenced by stretching function of the surface, plays an essential role in thermal flow phenomenon [38]. The study of fluid flow on a stretching sheet is an active region of research, and researchers continue to explore different aspects of this problem, contributing to the area of dynamics of fluid and its applications. Bhatti et al. [39] computed the spectral relaxation for Maxwell fluid flow on a quadratic convection stretching surface and have observed that higher thermal relaxation factor and Prandtl number have retarded the thermal distribution. Alqatani et al. [40] inspected the mass and thermal transportation for MHD fluid flow on an extended sheet and found that the growth in suction parameter and the Darcy–Forchheimer effect significantly diminished the energy transfer rate of nanoliquids. Noor et al. [41] analyzed thermal flow for fluid flowing on a stretched sheet using the first- and second-order velocity slip constraints. Mahabaleshwar et al. [42] discussed the fluid axisymmetric flow on an elongated sheet and noted that skin friction has upsurge for a progression in magnetic factor. Sharma et al. [43] analyzed the fluid flow on an elongated sheet using chemical reactivity. Hussain and Sheremet [44] discussed the radiative and convective fluid flow on an extending surface using inclined magnetic effects.

In fluid dynamics, slip conditions for fluid flow on a surface play a significant role in thermal flow analysis. These conditions describe how the fluid molecules interact with the solid boundary and how their velocity is affected at the boundary. It is of worth mentioning that the choice of slip condition can have a substantial impression on the predicted performance of fluid flows, especially at micro- or nanoscales. In many practical applications, the no-slip condition is used because it simplifies the analysis and is appropriate for most macroscopic scenarios [45]. However, as the study of fluid dynamics at very small scales (nanofluidics) becomes more important, researchers are exploring the slip conditions to account for the unique behaviors that arise due to molecular interactions at the fluid–solid interface [46]. Patel et al. [47] tested the impacts of MHD on fluid flow on exponentially shrinking and enlarging sheet using slip condition and have noted that the skin friction coefficient has diminished across the contracting surface area, but conversely, the expanding surface has demonstrated an opposing impact as the velocity slip factor is augmented; meanwhile, the Nusselt number exhibited a contrary outcome. Zainodin et al. [48] discussed higher-order chemical reactivity slip constraint impacts on fluid flow on a Darcy medium and perceived that the existence of thermal and concentration slips caused a reduction in both mass and heat transmission rates, consequently resulting in a postponement of boundary-layer separation. Mahmood et al. [49] evaluated computationally the effects of MHD and slip constraints regarding nanofluid flow on shrinking and elongating sheet using heat sink/source. Yasin et al. [50] inspected experimentally the computational impacts of Hall current on fluid flow on a surface using slip constraints and Ohmic thermal effects. Ramzan et al. [51] compared a modeled based nanofluid flow with slip constraints and Darcy–Forchheimer effects and have revealed that the fluid temperature increased as the Eckert number and opposing buoyancy force increased, while it decreased with an elevation in the thermal jump parameter. Shahzadi et al. [52] discussed the impressions of slip conditions on ternary nanoparticles’ blood flow in an artery for drug dispersal system.

Based on the upstairs observed literature, we are confident that there is very less work done on the stagnation point flow of hybrid nanofluid past a flat sheet. It is important to mention that in the present analysis, the surface has no stretching velocity at all; however, there is a free-stream velocity above the surface. Therefore, the authors examined the micropolar hybrid nanofluid flow on a flat surface. Water is taken as pure fluid, whereas copper and alumina nanoparticles are used to form the hybrid nanofluid. The analysis is considered under the impact of gravitational force, which we called the mixed convection phenomenon. Furthermore, the magnetic field, radiation, and heat source impacts are taken into consideration. The complete article is designed in section-wise, i.e., problem is formulated in Section 2, with solution in Section 3 by homotopy analysis method (HAM). Section 4 shows the validation of the present results, Section 5 shows results/discussion, and Section 6 presents the conclusion.

2 Formulation of problem

Assume the 2D flow of a micropolar hybrid nanofluid comprising copper ( Cu ) and alumina ( Al 2 O 3 ) nanoparticles on a flat sheet. The flow is influenced by the magnetic field having strength B 0 , which is applied normal to the flow direction. The fluid flow is considered in the x-direction, whereas the y-direction is taken perpendicularly. The flat surface has no stretching velocity at all, whereas the free stream velocity is u e = c x ( c > 0 is constant). T , T w , and T ∞ denote the fluid, surface, and free stream temperatures, respectively. Furthermore, the thermal slip condition is adopted to study the rate of heat transmission. The graphical view of fluid flow is depicted in Figure 1. Following are the flow assumptions:

  • The stagnation point flow on a flat surface is adopted.

  • The mixed convection phenomenon is adopted under the effect of gravity.

  • Water is taken as pure fluid, whereas copper and alumina nanoparticles are used to form a hybrid nanofluid.

  • Thermal radiation, heat source, and viscous dissipative effects are considered in the temperature equation.

Figure 1 Flow configuration.
Figure 1

Flow configuration.

Keeping in mind the aforementioned assumptions, the leading equation are as follows [53,54]:

(1) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(2) u ∂ u ∂ x + v ∂ u ∂ y = u e d u e d x + μ hnf ρ hnf 1 + K 1 μ hnf ∂ 2 u ∂ y 2 + K 1 ρ hnf ∂ N ∂ y − σ hnf ρ hnf B 0 2 ( u − u e ) + g ( ρ β T ) hnf ρ hnf ( T − T ∞ ) ,

(3) u ∂ N ∂ x + v ∂ N ∂ y = γ hnf ρ hnf ∂ 2 N ∂ y 2 − K 1 ρ hnf j 2 N + ∂ u ∂ y ,

(4) u ∂ T ∂ x + v ∂ T ∂ y = k hnf ( ρ C p ) hnf + 1 ( ρ C p ) hnf 16 σ ⁎ T ∞ 3 3 k ⁎ ∂ 2 T ∂ y 2 + Q 0 ( ρ C p ) hnf ( T w − T ∞ ) + μ hnf ( ρ C p ) hnf ∂ u ∂ y 2 ,

with constraints at boundary:

(5) u = 0 , v = 0 , N = − n ∂ u ∂ y , T = T w + L 2 ∂ T ∂ y ,  at  y = 0 , u → u e , N → 0 , T → T ∞ ,  as  y → ∞ .

The flow components along the x and y -axes are u and v , N is the microrotation, T is the temperature, γ hnf ( = μ hnf + K 1 / 2 ) j is the gyration gradient, K 1 is the material factor, j is the micro-inertia density, and n is the constant that lies between [ 0 , 1 ] .

The thermophysical relations are given below with computation values in Table 1 [55,56,57,58,59]:

(6) μ hnf = μ f ( 1 − ϕ 1 ) 2.5 ( 1 − ϕ 2 ) 2.5 , ρ hnf = ρ s 2 ϕ 2 + ρ f ( 1 − ϕ 2 ) ( 1 − ϕ 1 ) + ϕ 1 ρ s 1 ρ f , ( ρ β T ) hnf = ( ρ β T ) s 2 ϕ 2 + ( ρ β T ) f ( 1 − ϕ 2 ) ( 1 − ϕ 1 ) + ϕ 1 ( ρ β T ) s 1 ( ρ β T ) f , ( ρ C p ) hnf = ( ρ C p ) s 2 ϕ 2 + ( ρ C p ) f ( 1 − ϕ 2 ) ( 1 − ϕ 1 ) + ϕ 1 ( ρ C p ) s 1 ( ρ C p ) f , σ hnf σ nf = σ s 2 + 2 σ nf − 2 ϕ 2 ( σ nf − σ s 2 ) σ s 2 + 2 σ nf + ϕ 2 ( σ nf − σ s 2 ) , σ nf = σ f σ s 1 + 2 σ f − 2 ϕ 1 ( σ f − σ s 1 ) σ s 1 + 2 σ f + ϕ 1 ( σ f − σ s 1 ) , k hnf k nf = k s 2 + 2 k nf − 2 ϕ 2 ( k nf − k s 2 ) k s 2 + 2 k nf + ϕ 2 ( k nf − k s 2 ) , k nf = k f k s 1 + 2 k f − 2 ϕ 1 ( k f − k s 1 ) k s 1 + 2 k f + ϕ 1 ( k f − k s 1 ) .

Table 1

Thermophysical features of nanoparticles and pure fluid [55,56,57,58,59]

Properties H 2 O Cu Al 2 O 3
ρ 997.1 8,933 3,970
C p 4,179 385 765
k 0.613 400 40
σ 0.05 5.96 × 107 1 × 10−10
β T 2.1 × 10−4 7.65 × 10−5 8.5 × 10−6

The following set of appropriate variables has used:

(7) u = c x f ′ ( ξ ) , v = − c ν f f ( ξ ) , ξ = c ν f y , N = c x c ν f g ( ξ ) , θ ( η ) = T − T w T w − T ∞ .

Using Eq. (7), the transformed equations are as follows:

(8) 1 + η 1 K η 1 f ‴ ( ξ ) + η 2 ( 1 + f ′ ( ξ ) f ( ξ ) − f ′ 2 ( ξ ) ) + K g ′ ( ξ ) − M η 3 ( f ′ ( ξ ) − 1 ) + λ η 4 θ ( ξ ) = 0 ,

(9) 2 + η 1 K 2 η 1 g ″ ( ξ ) + η 2 ( f ( ξ ) g ′ ( ξ ) − g ( ξ ) f ′ ( ξ ) ) − K ( 2 g ( ξ ) + f ″ ( ξ ) ) = 0 ,

(10) 1 η 6 ( η 5 + Rd ) θ ″ ( ξ ) + η 1 η 6 EcPr f ″ 2 ( ξ ) + 1 η 6 Pr Q θ ( ξ ) = 0 ,

with boundary conditions:

(11) f ′ ( ξ = 0 ) = 0 , f ( ξ = 0 ) = 0 , g ( ξ = 0 ) = − n f ″ ( ξ = 0 ) , θ ( ξ = 0 ) = 1 + ψ θ ′ ( ξ = 0 ) , f ′ ( ξ → ∞ ) → 1 , θ ( ξ → ∞ ) → 0 , H ( ξ → ∞ ) → 0 .

In the aforementioned equations, K shows the micropolar factor, λ shows the mixed convection factor, Q represents the heat source factor, Pr signifies the Prandtl number, M shows the magnetic factor, Rd signifies the thermal radiation factor, and Ec is the Eckert number. These factors are defined as:

(12) η 1 = μ f μ hnf , η 2 = ρ hnf ρ f , η 3 = σ hnf σ f , η 4 = ( ρ β T ) hnf ( ρ β T ) f , η 5 = k hnf k f , η 6 = ( ρ C p ) hnf ( ρ C p ) f , K = K 1 μ f , λ = Gr e Re x 2 , Gr e = g β T x 3 ( T w − T ∞ ) ν f 2 , Re x = x u e ν f , Q = Q 0 a ( ρ C p ) f , Pr = ν f ( ρ C p ) f k f , M = σ f B 0 2 a ρ f , Rd = 16 σ ⁎ T ∞ 3 3 k ⁎ k f , Ec = u e 2 ( C p ) f ( T w − T ∞ ) ,

For engineering interest, the skin friction and Nusselt number are demarcated as:

(13) C f x = 1 ρ hnf u 2 ( μ hnf + K ) ∂ u ∂ y + K N y = 0 , Nu x = − x k f ( T w − T ∞ ) k hnf ∂ T ∂ y + 4 σ ⁎ 3 k ⁎ ∂ T 4 ∂ y y = 0 .

Using Eq. (7), we have

(14) C f = C f x R e 0.5 = 1 η 2 1 + η 1 ( 1 − n ) K η 1 f ″ ( 0 ) , Nu = Nu x R e − 0.5 = − ( ( η 5 + Rd ) ) θ ′ ( 0 ) .

3 HAM solution

To solve the aforementioned nonlinear Eqs. (8)–(10) with boundary conditions (11), a semi-analytic approach HAM is utilized. For the proposed solution, we have used Mathematica 12.0 software. The initial guesses and linear operators are defined as:

(15) f 0 ( ξ ) = ξ − 1 + e − ξ , g 0 ( ξ ) = 0 θ 0 ( ξ ) = 1 1 + ψ e − ξ ,

(16) L f ( ξ ) = f ‴ − f ′ , L g ( ξ ) = g ″ − g , L θ ( ξ ) = θ ″ − θ ,

with properties:

(17) { L f ( d 1 + d 2 e − ξ + d 3 e ξ ) = 0 , L g ( d 4 e − ξ + d 5 e ξ ) = 0 , L θ ( d 6 e − ξ + d 7 e ξ ) = 0 } ,

where d 1−d 7 are the constants in general solution.

For the present model, the zeroth-order deformation is as follows:

(18) ( 1 − R ) L f [ f ( ξ ; R ) − f 0 ( ξ ) ] ( 1 − R ) = R χ f ℏ f [ f ( ξ ; R ) , g ( ξ , R ) , θ ( ξ , R ) ] ,

(19) ( 1 − R ) L g [ g ( ξ ; R ) − g 0 ( ξ ) ] ( 1 − R ) = R χ g ℏ g [ g ( ξ ; R ) , f ( ξ ; R ) ] ,

(20) ( 1 − R ) L θ [ θ ( ξ ; R ) − θ 0 ( ξ ) ] ( 1 − R ) = R χ θ ℏ θ [ θ ( ξ ; R ) , f ( ξ ; R ) ] ,

along with boundary conditions:

(21) f ( ξ ;R ) = 0 , h ( ξ ;R ) = − η f ″ ( 0 ) , f ′ ( ξ ;R ) = 0 , θ ( ξ ;R ) = 1 + ψ [ ( ξ ;R ) θ ′ ] } ξ = 0 , f ′ ( ξ ;R ) = 0 , h ( ξ ;R ) = 0 , θ ( ξ ;R ) = 0 } ξ = ∞ .

Here, R = 0 ≤ R ≤ 1 is the embedding parameter. Besides nonlinear operators χ f , χ H , and χ θ are defined as:

(22) χ f ( f ( ξ ; R ) , h ( ξ ; R ) , θ ( ξ ; R ) ) = 1 + η 1 K η 1 ∂ 3 f ( ξ ; R ) ∂ ξ 3 + η 2 1 + f ( ξ ; R ) ∂ f ( ξ ; R ) ∂ ξ − ∂ f ( ξ ; R ) ∂ ξ 2 + K ∂ h ( ξ ; R ) ∂ ξ − M η 3 ∂ f ( ξ ; R ) ∂ ξ − 1 + λ η 4 ∂ θ ( ξ ; R ) ∂ ξ ,

(23) χ H ( h ( ξ ; R ) , f ( ξ ; R ) ) = 2 + η 1 K 2 η 1 ∂ 2 h ( ξ ; R ) ∂ ξ 2 + η 2 f ( ξ ) ∂ h ( ξ ; R ) ∂ ξ − H ( ξ ) ∂ f ( ξ ; R ) ∂ ξ − K ∂ 2 f ( ξ ; R ) ∂ ξ 2 − 2 h ( ξ ; R ) ,

(24) χ θ ( θ ( ξ ;R ) , f ( ξ ;R ) ) = 1 η 6 ( η 6 + Rd ) ∂ 2 θ ( ξ ;R ) ∂ ξ 2 + η 1 η 6 Ec Pr ∂ 2 f ( ξ ;R ) ∂ ξ 2 2 + 1 η 6 Pr Q θ ( ξ ;R ) .

By choosing R = 1 and R = 0 , we obtain

(25) f ( ξ ; 1 ) = f ( ξ ) , g ( ξ ; 1 ) = g ( ξ ) , θ ( ξ ; 1 ) = θ ( ξ ) , f ( ξ ; 0 ) = f 0 ( ξ ) , g ( ξ ; 0 ) = g 0 ( ξ ) , θ ( ξ ; 0 ) = θ 0 ( ξ ) .

Expanding by Taylor series for w.r.t. R , we have

(26) f ( ξ ; R ) = f 0 ( ξ ) + ∑ r = 1 ∞ f r ( ξ ) R r , h ( ξ ; R ) = h 0 ( ξ ) + ∑ r = 1 ∞ h r ( ξ ) R r , θ ( ξ ; R ) = θ 0 ( ξ ) + ∑ r = 1 ∞ θ r ( ξ ) R r ,

where

(27) f r ( ξ ) = 1 r ! ∂ r f ( ξ ; R ) ∂ R r R = 0 , h r ( ξ ) = 1 r ! ∂ r h ( ξ ; R ) ∂ R r R = 0 , θ r ( ξ ) = 1 r ! ∂ r θ ( ξ ; R ) ∂ R r R = 0 .

The Y th -order deformation is as follows:

(28) L f [ f Y ( ξ ) − q Y f Y − 1 ( ξ ) ] = ℏ f ℜ Y f ( ξ ) ,

(29) L g [ g Y ( ξ ) − q Y g Y − 1 ( ξ ) ] = ℏ g ℜ Y g ( ξ ) ,

(30) L θ [ θ Y ( ξ ) − q Y θ Y − 1 ( ξ ) ] = ℏ θ ℜ Y θ ( ξ ) ,

where

(31) ℜ Y f ( ξ ) = 1 + η 1 K 2 η 1 f Y − 1 ′ ″ − η 3 M ∑ n = 0 Y − 1 f Y − 1 ′ − 1 + η 2 ∑ n = 0 Y − 1 f Y − 1 − n f n ′ − ∑ n = 0 Y − 1 f Y − 1 − n ′ f n ′ + 1 + K ∑ n = 0 Y − 1 g Y − 1 ′ + λ η 4 ∑ n = 0 Y − 1 θ ′ Y − 1 ,

(32) ℜ Y g ( ξ ) = 2 + η 1 K 2 η 1 g Y − 1 ″ + η 2 ∑ n = 0 Y − 1 f Y − 1 − n g n ′ − ∑ n = 0 Y − 1 g Y − 1 − n f n ′ − K ∑ n = 0 Y − 1 f Y − 1 ″ − 2 ∑ n = 0 Y − 1 g Y − 1 ,

(33) R Y θ ( ξ ) = 1 η 6 ( η 6 + Rd ) θ Y − 1 ″ + η 1 η 6 Ec Pr ∑ n = 0 Y − 1 f Y − 1 − n ″ f n ″ + 1 η 6 Pr Q ∑ n = 0 Y − 1 θ Y − 1 .

The boundary conditions are as follows:

(34) f Y ( 0 ) = 0 , f Y ′ ( 0 ) = 0 , g Y ( 0 ) = − η f Y ″ ( 0 ) , θ Y ( 0 ) = [ ψ θ Y ′ ( 0 ) + 1 ] , f Y ′ ( ∞ ) = 1 , g Y ( ∞ ) = 0 , θ Y ( ∞ ) = 0 , ,

where

(35) q Y = 0 ,  if  R ≤ 1 1 , if R > 1 .

4 Validation

To validate the present solution with those of the published results by Mahabaleshwar et al. [42], Table 2 is presented. This solution is compared with those established results and has found a very close solution in the present analysis. Thus, we confirm that the solution of the present model is valid.

Table 2

Solution of f ( ξ ) for varying ξ keeping other factors as zero

ξ Ashraf and Ashraf [53] Das [54] Present results
0.0 0.0 0.0 0.0
0.6 0.159171 0.159854 0.159568
1.2 0.555414 0.555403 0.555554
1.8 1.069900 1.067750 1.068654
2.4 1.635279 1.624300 1.624446
3.0 2.221620 2.212280 2.211435
3.6 2.816397 2.808153 2.808096
4.2 3.414474 3.410679 3.411246
4.8 4.013803 4.006367 4.009859
5.4 4.613592 4.606245 4.609869
6.0 5.213549 5.216228 5.214543

5 Discussion of results

This section presents the physical discussion about the obtained results. The obtained results are displayed in Figures 2–12 and Tables 3–7. The default values of the embedded factor are chosen as K = 0.5 , λ = 0.5 , Q = 0.2 , Pr = 6.2 , M = 1.2 , Rd = 0.3 , Ec = 0.1, ϕ 1 = 0.04 , and ϕ 2 = 0.04 . Figure 2(a) and (b) depicts the deviations in velocity profiles via nanoparticle volume fractions of the Cu ( ϕ 1 ) and Al 2 O 3 ( ϕ 2 ) nanoparticles, respectively. From both the figures, we see that the velocity profiles increase against ϕ 1 and ϕ 2 . The reason is that growth in volume fraction and the fluid viscosity improves. In this process, the density of fluid upsurge offers more resistivity to fluid flow. Thus, in the present case, both ϕ 1 and ϕ 2 enhance the water-based hybrid nanofluid flow viscosity, which, as a result, reduces the velocity panels. Figure 3(a) and (b) displays the variation in temperature profiles via nanoparticle volume fractions of the Cu ( ϕ 1 ) and Al 2 O 3 ( ϕ 2 ) nanoparticles, respectively. From both figures, it has revealed that temperature profiles intensify against ϕ 1 and ϕ 2 . This is because when more solid particles are suspended in pure fluid, the thermal conductivity of the fluid improves, increasing the heat transmission. The stiffening of the thermal layer at boundary is caused by this rise in temperature profiles. Comparing both the figures, we see that the larger impact of volume fraction of the solid nanoparticle on thermal panels is revealed for Cu–water nanofluid as of Al 2 O 3 –water. Actually, copper nanoparticle has better thermal conductance than the alumina nanoparticle. Thus, the highest impact is found in Figure 3(a). Figure 4(a) and (c) demonstrates the impression of micropolar parameter ( K ) on f ′ ( ξ ) , g ( ξ ) , and θ ( ξ ) , respectively. It can be noted that as the micropolar factor escalates, the velocity and temperature profiles experience a decrease. On the other hand, the microrotation velocity profile increases with the growth in micropolar factor. The reason is that the micropolar factor diminishes the dynamic viscosity of fluid, which consequently increases the microrotation velocity profile with the increasing micropolar factor. Figure 5(a) and (b) shows that higher magnetic factor escalates the flow panels. Conversely, the higher magnetic has dual nature for the microrotation profile. On the sheet’s surface, an elevated magnetic factor amplifies the counteracting force, leading to a decrease in the microrotation profile, while above the surface, the opposing force diminishes and the microrotation profile gets increase. Generally, the magnetic factor produces the Lorentz force that plays against the flow of liquid. This is only applicable for stretching sheet. In the present case, there is no stretching velocity of the sheet at all. Of course, there is an ambient velocity due to the presence of stagnation point. Figure 6(a) and (b) displays the outcome of mixed convection factor on velocity and microrotation panels. Growth in velocity distribution is noted for upsurge in λ as portrayed in Figure 6(a). Actually, increasing mixed convection factor amplifies the buoyancy force toward the stagnation point flow, which enlarges the width of momentum layer at boundary. Therefore, the higher λ escalates f ′ ( ξ ) . On the other hand, a dual impact of λ on microrotation profile is depicted. Near the surface of flat sheet, the buoyancy force increases the microrotation profile, while at some distance from the sheet surface, this effect behaves oppositely. Figure 7 shows the Ec effects on θ ( ξ ) with a growing behavior of θ ( ξ ) . It is shown that the width of thermal layer at boundary and temperature profiles improves as Ec increases. Physically, it is discovered that Ohmic heating effect results in a rise in temperature of fluid and, as a result, a reduction in thermal flow. Furthermore, it is discovered that viscous heating has an impact on temperature, with the occurrence of a stronger impact magnetic field. Figure 8 shows the impression of Rd on θ ( ξ ) . From this figure, it is perceived that the higher values of Rd increase θ ( ξ ) . The thermal radiation factor escalates the heat of the flow system that causes augmentation in thermal profiles. Therefore, higher Rd heightens the temperature profile. Similarly, the consequence of heat source factor Q on thermal profile is exposed in Figure 9. Also, an increasing impact is observed here. The reason is that the higher heat source increases the heat transfer rate, which consequently increases the thermal boundary-layer thickness and temperature profile as well. Thus, higher Q increases θ ( ξ ) . Figure 10(a)–(d) shows the streamlines and contour lines of the hybrid nanofluid for K = 0 and K = 1 , respectively. From Figure 11(a) and (b), we observed that the streamlines become closer to each other when K = 1 as compared to K = 0 , since the higher K increases the dynamic viscosity, which results in higher friction force at the sheet surface, and as a result, the streamlines become closer to each other. Also, when the dynamic viscosity increases with higher K , the boundary-layer thickness reduces as well. This effect is observed in Figure 10(c) and (d). Figure 11(a)–(d) shows the streamlines and contour lines of the hybrid nanofluid for M = 0 and M = 10 , respectively. From Figure 11(a) and (b), we observed that the streamlines become apart from each other when M = 10 as compared to M = 0 since the higher M increases the velocity profile due to reducing friction force at the sheet surface. Thus, streamlines become apart from each other. Also, when the skin friction reduces with higher M , the boundary-layer thickness increases as well. This effect is observed in Figure 11(c) and (d). Figure 12(a)–(c) and Tables 3–5 show the comparison of the HAM and numerical solutions for f ′ ( ξ ) , g ( ξ ) , and θ ( ξ ) . Form these figures and tables, we see that both the semi-analytical and numerical methods have close relation. Table 6 shows the impacts of ϕ 1 , ϕ 2 , ϕ 1 = ϕ 2 , M , and K on C f . From this table, it is perceived that nanoparticle volume fraction diminishes C f . The reason is explained in the above figures. Here, the important point is that the effect is smaller for the case of hybrid nanofluid flow (Cu-Al2O3/water) when compared to the nanofluids (Cu/water and Al2O3/water). Furthermore, upsurge in M diminishes the friction force at the surface, while the greater micropolar factor escalates it at the surface of the flat. Table 7 displays the effects of ϕ 1 , ϕ 2 , ϕ 1 = ϕ 2 , Rd, Ec, Q, and ψ on Nu. This table demonstrates that growth in nanoparticle volume fractions and heat transfer rate. Actually, upsurge in ϕ 1 and ϕ 2 increases the thermal conductivities of the nanofluids. The important point is here to mention that the heat transfer rate of the hybrid nanofluid is more than that of nanofluids. Furthermore, the Eckert number, thermal radiation, and heat source factors escalate the thermal flow rate, while the thermal slip reduces the heat transfer rate.

Figure 2 (a and b) Impacts of ϕ 1 {\phi }_{1} and ϕ 2 {\phi }_{2} on f ′ ( ξ ) f^{\prime} (\xi ) .
Figure 2

(a and b) Impacts of ϕ 1 and ϕ 2 on f ′ ( ξ ) .

Figure 3 (a and b) Impacts of ϕ 1 {\phi }_{1} and ϕ 2 {\phi }_{2} on θ ( ξ ) \theta (\xi ) .
Figure 3

(a and b) Impacts of ϕ 1 and ϕ 2 on θ ( ξ ) .

Figure 4 (a–c) Impact of K K on f ′ ( ξ ) f^{\prime} (\xi ) , g ( ξ ) g(\xi ) , and θ ( ξ ) \theta (\xi ) .
Figure 4

(a–c) Impact of K on f ′ ( ξ ) , g ( ξ ) , and θ ( ξ ) .

Figure 5 (a and b) Impact of M M on θ ( ξ ) \theta (\xi ) and g ( ξ ) g(\xi ) .
Figure 5

(a and b) Impact of M on θ ( ξ ) and g ( ξ ) .

Figure 6 (a and b) Impact of λ \lambda on f ′ ( ξ ) f^{\prime} (\xi ) and g ( ξ ) g(\xi ) .
Figure 6

(a and b) Impact of λ on f ′ ( ξ ) and g ( ξ ) .

Figure 7 Impact of Ec \text{Ec} on θ ( ξ ) \theta (\xi ) .
Figure 7

Impact of Ec on θ ( ξ ) .

Figure 8 Impact of Rd \text{Rd} on θ ( ξ ) \theta (\xi ) .
Figure 8

Impact of Rd on θ ( ξ ) .

Figure 9 Impact of Q Q on θ ( ξ ) \theta (\xi ) .
Figure 9

Impact of Q on θ ( ξ ) .

Table 3

Comparison of the HAM and numerical methods for f ′ ( ξ )

ξ f ′ ( ξ )
HAM Numerical
0.0 2.78 × 10−17 2.78 × 10−17
0.5 0.450228 0.452234
1.0 0.713433 0.715388
1.5 0.865786 0.867243
2.0 0.951572 0.952556
2.5 0.996976 0.997609
3.0 1.018325 1.018723
3.5 1.025983 1.026229
4.0 1.026412 1.026563
4.5 1.02348 1.023572
5.0 1.019374 1.019430
5.5 1.015233 1.015268
6.0 1.011575 1.011596
6.5 1.008573 1.008586
7.0 1.006224 1.006231
7.5 1.004445 1.004450
8.0 1.003133 1.003136
8.5 1.002184 1.002185
9.0 1.001507 1.001509
9.5 1.001032 1.001033
10.0 1.000701 1.000702
Table 4

Comparison of the HAM and numerical methods for g ( ξ )

ξ g ( ξ )
HAM Numerical
0.0 2.78 × 10−17 2.78 × 10−17
0.5 0.044816 0.044782
1.0 0.043734 0.043730
1.5 0.032915 0.032925
2.0 0.022511 0.022523
2.5 0.014693 0.014703
3.0 0.009343 0.009350
3.5 0.005848 0.005853
4.0 0.003623 0.003626
4.5 0.002230 0.002232
5.0 0.001366 0.001367
5.5 0.000834 0.000835
6.0 0.000508 0.000509
6.5 0.000309 0.000310
7.0 0.000188 0.000188
7.5 0.000114 0.000114
8.0 6.94 × 10−5 6.94 × 10−5
8.5 4.21 × 10−5 4.21 × 10−5
9.0 2.55 × 10−5 2.56 × 10−5
9.5 1.55 × 10−5 1.55 × 10−5
10.0 9.4 × 10−10 9.41 × 10−10
Table 5

Comparison of the HAM and numerical methods for θ ( ξ )

ξ θ ( ξ )
HAM Numerical
0.0 1.082509 1.085835
0.5 0.758072 0.760936
1.0 0.492794 0.494739
1.5 0.310022 0.311254
2.0 0.191899 0.192660
2.5 0.117761 0.118225
3.0 0.071917 0.072200
3.5 0.043798 0.043970
4.0 0.026630 0.026734
4.5 0.016176 0.016239
5.0 0.009820 0.009858
5.5 0.005959 0.005983
6.0 0.003616 0.003630
6.5 0.002193 0.002202
7.0 0.001331 0.001336
7.5 0.000807 0.000810
8.0 0.000490 0.000491
8.5 0.000297 0.000298
9.0 0.000180 0.000181
9.5 2.09 × 10−5 2.11 × 10−5
10.0 6.63 × 10−10 6.65 × 10−10
Table 6

Impacts of ϕ 1 , ϕ 2 , ϕ 1 = ϕ 2 , M , and K on C f

ϕ 1 ϕ 2 M K C f
0.04 0.0 0.5 0.5 0.723974
0.05 0.647360
0.06 0.579946
0.0 0.04 0.927634
0.05 0.877409
0.06 0.830391
0.04 0.04 0.577452
0.05 0.05 0.489587
0.06 0.06 0.415749
0.04 0.04 0.2 0.635134
0.3 0.615907
0.4 0.596679
0.5 0.2 0.410437
0.3 0.463797
0.4 0.519461
Table 7

Impacts of ϕ 1 , ϕ 2 , ϕ 1 = ϕ 2 , Rd , Ec, Q, and ψ on Nu

ϕ 1 ϕ 2 Rd Ec Q ψ Nu
0.01 0.0 0.3 0.1 0.3 0.2 1.66916
0.02 1.73014
0.03 1.79444
0.0 0.01 1.66663
0.02 1.72481
0.03 1.78603
0.01 0.01 1.82924
0.02 0.02 1.91029
0.03 0.03 2.00660
0.04 0.04 0.3 2.17084
0.4 2.35076
0.5 2.53313
0.3 0.2 2.29782
0.3 2.42483
0.4 2.55184
0.1 0.1 1.97354
0.2 2.07217
0.3 2.17084
0.3 0.3 1.81323
0.4 1.50284
0.5 1.23022
Figure 10 (a) Streamlines when K = 0 K=0 , (b) streamlines when K = 1 K=1 , (c) contour lines when K = 0 K=0 , and (d) contour lines when K = 1 K=1 .
Figure 10

(a) Streamlines when K = 0 , (b) streamlines when K = 1 , (c) contour lines when K = 0 , and (d) contour lines when K = 1 .

Figure 11 (a) Streamlines when M = 0 M=0 , (b) streamlines when M = 10 M=10 , (c) contour lines when M = 0 M=0 , and (d) contour lines when M = 10 M=10 .
Figure 11

(a) Streamlines when M = 0 , (b) streamlines when M = 10 , (c) contour lines when M = 0 , and (d) contour lines when M = 10 .

Figure 12 (a–c) Comparison of the HAM and numerical solutions for f ′ ( ξ ) f^{\prime} (\xi ) , g ( ξ ) g(\xi ) , and θ ( ξ ) \theta (\xi ) .
Figure 12

(a–c) Comparison of the HAM and numerical solutions for f ′ ( ξ ) , g ( ξ ) , and θ ( ξ ) .

6 Conclusion

In this article, we have studied the micropolar hybrid nanofluid containing copper ( Cu ) and alumina ( Al 2 O 3 ) nanoparticles past a flat surface. The mixed convection phenomenon is studied under the effect of gravity. Some additional forces such as magnetic field, thermal radiation, Eckert number, heat source, and thermal slip condition are adopted in the present analysis. The HAM is applied for the solution of the present analysis, which is validated with those of published results and a numerical method. On the completion of this analysis, the following concluding points are extracted:

  • The higher volume fractions of copper and alumina nanoparticles improved the hybrid nanofluid viscosity, which results in the augmenting variation in the velocity profiles.

  • The higher volume fractions of copper and alumina nanoparticles improved the hybrid nanofluid thermal conductivity, which results in the augmenting variation in the temperature profiles and heat transfer.

  • The micropolar factor has increased the dynamic viscosity of the hybrid nanofluid flow, which, as a result, increases the microrotation velocity profile. On the other hand, the micropolar factor has declining impacts on the velocity and temperature profiles.

  • The increasing mixed convection factor amplifies the buoyancy force toward the stagnation point flow, which enlarges the momentum boundary-layer thickness, and as a result, the velocity profile gets improved.

  • The higher Eckert number, thermal radiation, and heat source factor have increased the temperature profiles of the hybrid nanofluid flow.

  • The effects of the volume fractions of copper and alumina nanoparticles on friction force at the surface are smaller for the case of hybrid nanofluid flow (Cu-Al2O3/water) when compared to the nanofluids (Cu/water and Al2O3/water).

  • The effects of the volume fractions of copper and alumina nanoparticles on heat transfer rates are higher for the case of hybrid nanofluid flow (Cu-Al2O3/water) when compared to the nanofluids (Cu/water and Al2O3/water).

7 Future recommendations

In the future, the present model can be extended for a three-dimensional stagnation point flow of a micropolar fluid containing different types of nanoparticles such as CuO, Fe2O3, Fe3O4, and TiO2. Also, some different types of base fluids that exhibit Newtonian behavior can be considered in future work. The fluid flow can be examined by adopting the velocity slip, thermal convective, mass flux, and zero-mass flux conditions in the future.

  1. Funding information: This study was supported by Project No. 129257 implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the K 18 funding scheme.

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

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

  4. Data availability statement: The data that support the findings of this study are available from the corresponding author upon a reasonable request.

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Received: 2023-11-08
Revised: 2024-01-15
Accepted: 2024-01-29
Published Online: 2024-03-16

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

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

Articles in the same Issue

  1. Regular Articles
  2. Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
  3. Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
  4. High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
  5. Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
  6. Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
  7. Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
  8. The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
  9. Growth, characterization, and anti-bacterial activity of l-methionine supplemented with sulphamic acid single crystals
  10. A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium
  11. Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
  12. Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
  13. Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
  14. Numerical analysis of thermophoretic particle deposition on 3D Casson nanofluid: Artificial neural networks-based Levenberg–Marquardt algorithm
  15. Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator
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  18. Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
  19. Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
  20. Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
  21. Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
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https://doi.org/10.1515/phys-2023-0201
Keywords for this article
micropolar fluid; hybrid nanofluid; mixed convection; thermal slip condition; stretching surface; HAM solution
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
  3. Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
  4. High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
  5. Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
  6. Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
  7. Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
  8. The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
  9. Growth, characterization, and anti-bacterial activity of l-methionine supplemented with sulphamic acid single crystals
  10. A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium
  11. Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
  12. Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
  13. Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
  14. Numerical analysis of thermophoretic particle deposition on 3D Casson nanofluid: Artificial neural networks-based Levenberg–Marquardt algorithm
  15. Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator
  16. Bayesian estimation of equipment reliability with normal-type life distribution based on multiple batch tests
  17. Chaotic control problem of BEC system based on Hartree–Fock mean field theory
  18. Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
  19. Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
  20. Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
  21. Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
  22. Microscopic seepage simulation of gas and water in shale pores and slits based on VOF
  23. Model of conversion of flow from confined to unconfined aquifers with stochastic approach
  24. Study of fractional variable-order lymphatic filariasis infection model
  25. Soliton, quasi-soliton, and their interaction solutions of a nonlinear (2 + 1)-dimensional ZK–mZK–BBM equation for gravity waves
  26. Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering
  27. Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions
  28. Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model
  29. Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function
  30. Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints
  31. Super-resolution reconstruction method of the optical synthetic aperture image using generative adversarial network
  32. A two-stage framework for predicting the remaining useful life of bearings
  33. Influence of variable fluid properties on mixed convective Darcy–Forchheimer flow relation over a surface with Soret and Dufour spectacle
  34. Inclined surface mixed convection flow of viscous fluid with porous medium and Soret effects
  35. Exact solutions to vorticity of the fractional nonuniform Poiseuille flows
  36. In silico modified UV spectrophotometric approaches to resolve overlapped spectra for quality control of rosuvastatin and teneligliptin formulation
  37. Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems
  38. Substituent effect on the electronic and optical properties of newly designed pyrrole derivatives using density functional theory
  39. A comparative analysis of shielding effectiveness in glass and concrete containers
  40. Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation
  41. Analytical and numerical investigation for viscoelastic fluid with heat transfer analysis during rollover-web coating phenomena
  42. Influence of variable viscosity on existing sheet thickness in the calendering of non-isothermal viscoelastic materials
  43. Analysis of nonlinear fractional-order Fisher equation using two reliable techniques
  44. Comparison of plan quality and robustness using VMAT and IMRT for breast cancer
  45. Radiative nanofluid flow over a slender stretching Riga plate under the impact of exponential heat source/sink
  46. Numerical investigation of acoustic streaming vortices in cylindrical tube arrays
  47. Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion
  48. Fractional view analytical analysis of generalized regularized long wave equation
  49. Dynamic simulation of non-Newtonian boundary layer flow: An enhanced exponential time integrator approach with spatially and temporally variable heat sources
  50. Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source
  51. Estimation of monotone α-quantile of past lifetime function with application
  52. Numerical simulation for the slip impacts on the radiative nanofluid flow over a stretched surface with nonuniform heat generation and viscous dissipation
  53. Study of fractional telegraph equation via Shehu homotopy perturbation method
  54. An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
  55. Establishing breather and N-soliton solutions for conformable Klein–Gordon equation
  56. An electro-optic half subtractor from a silicon-based hybrid surface plasmon polariton waveguide
  57. CFD analysis of particle shape and Reynolds number on heat transfer characteristics of nanofluid in heated tube
  58. Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation
  59. A short report on a probability-based interpretation of quantum mechanics
  60. Study on cavitation and pulsation characteristics of a novel rotor-radial groove hydrodynamic cavitation reactor
  61. Optimizing heat transport in a permeable cavity with an isothermal solid block: Influence of nanoparticles volume fraction and wall velocity ratio
  62. Linear instability of the vertical throughflow in a porous layer saturated by a power-law fluid with variable gravity effect
  63. Thermal analysis of generalized Cattaneo–Christov theories in Burgers nanofluid in the presence of thermo-diffusion effects and variable thermal conductivity
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  67. Thermal analysis of extended surfaces using deep neural networks
  68. Steady-state thermodynamic process in multilayered heterogeneous cylinder
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  74. Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method
  75. Effectiveness of microwave ablation using two simultaneous antennas for liver malignancy treatment
  76. Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
  77. Reliability of two-dimensional steady magnetized Jeffery fluid over shrinking sheet with chemical effect
  78. Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
  79. Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field
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  85. A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
  86. Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles
  87. A novel tetra hybrid bio-nanofluid model with stenosed artery
  88. Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
  89. Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
  90. A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
  91. New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method
  92. Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
  93. Inflation and the principle of equivalence
  94. The use of a supercontinuum light source for the characterization of passive fiber optic components
  95. Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
  96. A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
  97. Photon balance in the fiber laser model
  98. Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics
  99. Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology
  100. Utilizing slip conditions on transport phenomena of heat energy with dust and tiny nanoparticles over a wedge
  101. Bismuthyl chloride/poly(m-toluidine) nanocomposite seeded on poly-1H pyrrole: Photocathode for green hydrogen generation
  102. Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
  103. On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
  104. Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
  105. Review Article
  106. Transformer-based intelligent fault diagnosis methods of mechanical equipment: A survey
  107. Special Issue on Predicting pattern alterations in nature - Part II
  108. A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method
  109. On the existence and numerical simulation of Cholera epidemic model
  110. Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions
  111. Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
  112. Prediction of COVID-19 spread with models in different patterns: A case study of Russia
  113. Study of chronic myeloid leukemia with T-cell under fractal-fractional order model
  114. Accumulation process in the environment for a generalized mass transport system
  115. Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
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