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Investigation of solar radiation effects on the energy performance of the (Al2O3–CuO–Cu)/H2O ternary nanofluidic system through a convectively heated cylinder

  • Adnan EMAIL logo , Waseem Abbas , Refka Ghodhbani EMAIL logo , Kaouther Ghachem , Tadesse Walelign EMAIL logo , Yasir Khan , Mehdi Akermi and Rym Hassani
Published/Copyright: July 31, 2025
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Open Physics
From the journal Open Physics Volume 23 Issue 1

Abstract

Thermal transport in ternary nanofluid is a topic of interest in different engineering systems. These fluids have higher thermal conductivity than traditional nanofluids. Hence, the present study aims to develop a new ternary nanofluid model for a cylindrical working domain. For this, thermophysical properties of ternary nanoliquids and appropriate transformations are used. The problem is then investigated through a numerical approach and the comparative results are obtained. The ternary nanofluid shows an optimum decrease in the velocity due to the involvement of three types of nanoparticles. Suction of the fluid with strength α = 0.1 , 0.9 , 1.7 , 2.5 and Reynolds effects Re = 1.0 , 1.5 , 2.0 , 2.5 significantly control the motion and dominant behaviour is examined for a simple nanofluid. The thermal capability of the nanofluids is enhanced against the concentration factor ϕ 1 = 0.01 , 0.0.3 , 0.05 , 0.07 while suction phenomena resist the temperature. Inclusion of radiations ( Rd = 0.1 , 0.5 , 0.9 , 1.3 ) and convective transport ( B i = 0.01 , 0.02 , 0.03 , 0.04 ) contribute dominantly for thermal applications in nanofluids. The shear drag magnitude changes from 107.4995 to 162.287% (TNF), 113.427 to 170.666% (HNF), and 120.886 to 180.704% (SNF) for varying ϕ 1 from 1.0 to 7.0%. Further, the efficiency of TNF, HNF, and SNF showed a prominent increase from 42.0126 to 68.8055% (TNF), 40.6019 to 66.6076% (HNF), and 39.8879 to 65.5324% (SNF), for stronger Biot effects from 0.5 to 2.0. Hence, the study’s outcomes would help to address the heat transfer issues from multiple aspects.

Keywords: ternary nanofluid; heat transfer; cylinder; thermal radiation; convective surface

Nomenclature

w , u

components of the velocity (m/s)

p

pressure (Pa)

T , T ∞

fluid temperature and ambient temperature (K)

U w

surface velocity (m/s)

k tnf

thermal conductivity (W/m K)

ρ tnf

density (kg/m3)

μ tnf

dynamic viscosity (N s/m2)

c

arbitrary constant

η

dimensionless variable

F′

velocity dimensionless form

β

temperature dimensionless form

Re

Reynold number

Pr

Prandtl number

Rd

radiation number

B i

Biot number

Abbreviations

ODEs

ordinary differential equations

PDEs

partial differential equations

MHD

magnetohydrodynamic

TBL

thermal boundary layer

1 Introduction

Improved characteristics of nanomaterials make the primary solvents more realistic for engineering and thermal applications. The researchers made efforts towards the development of these engineered fluids. Mishra [1] scrutinized the importance of Fe 3 O 4 ,  CoFe 2 O 4 nanoparticles’ (NPs) impacts on the properties of ethylene glycol (EG). The study was conducted for a permeable surface under two types of slips, termed as Troian and Thompson. The investigation shows that the heat transfer rate decreases absolutely due to the changing parameters. Waqas et al. [2] scrutinized that with the greater estimation of the Biot number, the temperature of the fluid is increased. Song et al. [3] analyzed an incompressible, time-dependent Williamson nanoliquid flow through a stretching cylinder and observed the reduction in the velocity against the unsteady parameter. Nadeem et al. [4] explored the viscoelastic nanofluid [5,6] flow through stretching absorbent media with suction and injection cases. Some recent studies for nanoliquids under varying physical constraints are discussed by various researchers (see refs. [7,8,9]).

The nanofluid’s heat transport mechanism is of greater interest in engineering applications. These fluids are obtained by the dispersion of tiny particles in the traditional fluids. The heat transport rate in a nanoliquid flow under heat dissipative phenomena and heat transfer through a porous cylinder was scrutinized by Waqas et al. [2] and concluded that the heat performance enhances the B i effects increased. Song et al. [3] analyzed an incompressible, time-dependent Williamson liquid through a stretching cylinder with the influence of mixed convection and heat transfer effects and observed the reduction in the velocity against unsteady parameter. Nadeem et al. [4] explored the viscoelastic nanofluid [5,6] flow through stretching absorbent media for suction and injection cases. Some recent studies for nanoliquids under varying physical constraints are discussed by various researchers (see refs. [10–12]).

Javaid et al. [13] studied the dynamics of second-grade fluid past through cylinder placed vertically. They examined the flow nature and heat mechanism comprehensively via numerical approach. An unsteady convective flow over a circular cylinder with heating effects is explored by Shah et al. [14] by using Fourier’s law explained the behavior of velocity and temperature profile against Grashof and Prandtl number. Gholinia et al. [15] analyzed an incompressible, viscous mixed convection nanofluid [16,17], flow into a circular vertical cylinder with the influence of electrical conductivity. EG used as a base fluid while NPs included silver and copper in this study. Emam [18] presented an investigation on a steady, boundary layer, incompressible fluid flow over a semi-infinite cylinder which in immersed in the porous medium and move vertically.

Rehman et al. [19] performed an analysis of non-Newtonian BL (boundary layer) micropolar liquid with heat transfer into a vertical cylinder which is exponentially stretched and the results of the parameters under consideration are presented through graphs. Ganesh et al. [20] examined steady, incompressible, and laminar axi-symmetric boundary layer heat transport over a vertical cylinder influenced by Lorentz forces. They reported that TBL thickness increased with increase in magnetic field parameter and decreased with Prandtl number. Zokri et al. [21] discussed the heat transfer due to Grashof effects for Jeffrey nanofluid with heat dissipation effects over a horizontal cylinder.

Javed et al. [22] investigated the mixed convective flow of viscous nanofluid [23,24] with the effects of constant heat flux over an elliptical cylinder and concluded that Skin friction and Nusselt number increased with growing value mixed convection parameter. Similarly, a free convection boundary layer nanofluid flow with constant temperature over an elliptical cylinder with the impact of Brownian motion was securitized in a study by Cheng [25]. An MHD mixed convective flow of a ternary nanofluid consisting on NPs, including Cu, Al2O3, and TiO2 due to inner rotating cylinder was examined by Lahonian et al. [26]. Recently, Noreen et al. [27] recognized the CCM to express an MHD ternary nanofluid flow model by considering heating source and magnetic field effects through double discs.

Recently, researcher’s community inspired by the characteristics of advanced nanofluids and accelerated towards the analysis in this new direction. Mohanty et al. [28] discussed irreversibility characteristics of ternary nanofluids through an elastic cylinder and pointed there industrial uses. They included resistive heating and solar radiations of nonlinear nature to upsurges the novelty. The study revealed that ternary nanofluid prepared by MWCNTs, SWCNTs, and GO are efficient than the previously discussed nanofluids. Sarangi et al. [29] explored the performance of Bodewadt flow a fluid containing ternary NPs. The study accommodated the effects of stretching, dissipation, and radiations for the flow situation over a disk. Reduced surface drag due to ternary NPs, and minimized entropy of the system for smaller thermal differences are the main findings. Analysis of combined convection, permeability, and activation energy has been conducted by Mohanty et al. [30]. It is noticed that Sherwood number declines due to augmenting chemical and activation energy effects.

Pandey and Kumar [31] explored the collective impact of thermal radiation and convection flow of nanofluid consisting on Cu NPs and considered viscous dissipation past a porous stretching cylinder and discussed the Nusselt number against the various physical constraints. The viscous dissipation and heat transfer nanofluid flow with non-thermal radiations over a cylinder with radiations is examined in [32]. Mehmood et al. [33] investigated the transport of NPs [34,35] aluminum oxide in EG past a convectively heated cylinder. Heat performance rate of ternary nanofluid over a permeable cylinder is comprehensibly investigated in [36] with the effects of permeable medium, inclined geometry and heat source/sink on temperature distribution.

Fractional order approaches [37] and ANN schemes [38] are newly introduced mathematical tools for the analysis of nanofluid problems. Researchers put their efforts for the investigate nanofluids dynamics under varying controls and geometries for remarkable thermal transport using these schemes. Khan et al. [39], examined nanofluid inspired by the influences of an MHD in darcy media. The nonlinear problem discussed through fractional order derivative approach and comprehensively elaborated the outputs. The 56.51% increase in the temperature is acquired for Grashof effects and suggested that tackled issue is good to cope the problems of combustion engines, and maintaining temperature of electronic devices. Another relevant study by introducing the γ nanoliquid has been reported in [40]. The γAl 2 O 3 NPs utilized to augment the characteristics of H 2 O and C 2 H 6 O 2 along with adopted physical controls. They examined that γAl 2 O 3 ( C 2 H 6 O 2 ) is better to γAl 2 O 3 ( H 2 O ) from applications perspectives in power plants, turbines and heat exchanging devices.

This novelty and objectives of this research falls in three steps. Firstly, prolong the traditional problem for ternary nanofluids which has promising features than hybrid or conventional nanofluids. This will spotlight the importance of composition of three types of NPs for heat featuring applications. Secondly, to investigate the impacts on radiations and suction on the comparative efficiency of ternary, hybrid and simple fluids performance. Thirdly, to introduce the convective condition and how it alter the behaviour of nanofluids through a cylinder and its impacts on the rate of heat transfer. The comparative study of three sort of nanofluids would help to predict the type of nanofluids for superior thermal applications and control of shear drag and rate of thermal transport. This will add valuable insights in the engineering and heat transfer applications.

2 Model development

Suppose an incompressible and steady nature flow of ternary nanofluid through a cylinder. The fluid components are Cu, CuO, and Al2O3, while water is taken as the primary solvent. Further, a be the radius along the extending axial direction, r makes a right angle with the z- axis , and the z -axis is parallel to the cylinder. The suction effects on the cylinder. The whole flow situation is configured in Figure 1, while the supporting governing laws for the continuity, momentum, and energy are given in Eqs. (1)–(6) along with fitting flow conditions [2]

(1) ∂ ∂ r ( ru ) + ∂ ∂ z ( rw ) = 0 ,

(2) ρ tnf u ∂ u ∂ r + w ∂ u ∂ z = − u r 2 + ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r μ tnf ,

(3) ρ tnf u ∂ w ∂ r + w ∂ w ∂ z = ∂ 2 w ∂ r 2 + 1 r ∂ w ∂ r μ tnf ,

(4) ( ρ c p ) tnf w ∂ T ∂ z + u ∂ T ∂ r = k tnf ∂ 2 T ∂ r 2 + 1 r ∂ T ∂ r + ∂ ∂ r 16 σ * T ∞ 3 3 k * ∂ T ∂ r .

Figure 1 Incompressible ternary nanofluid flow through a cylinder.
Figure 1

Incompressible ternary nanofluid flow through a cylinder.

The above equations are associated with the boundary conditions as follows:

(5) r → ∞ : T → T ∞ , w → 0 ,

(6) r = a : − k tnf ∂ T ∂ r = h f ( T f − T ) , w = w w , u = U w .

Here, U w = − a α c and w w = 2 z c .

The following ternary nanoliquid [41,42] models are preferred to check the performance of the ternary nanoliquid. The ternary NPs (Al2O3–CuO–Cu) are taken, which are uniformly dissolved in the host liquid. The detailed expressions for improved properties of ternary nanoliquids are described in Table 1. It is noteworthy that these models easily reduced for the previous classes (hybrid and nanofluids) by setting the concentration factor. Figure 2a–c shows the thermo-physical properties of various NPs and base fluid involved in this study. Further, Figure 3a–d present the impacts of NPs changing concentration on the effective properties.

Table 1

The enhanced characteristics of ternary nanoliquids

Enhanced characteristics Supporting model
Density ρ tnf = ζ ˜ 3 ζ ˜ 2 ζ ˜ 1 + φ 1 ρ s 1 ρ f + φ 2 ρ s 2 ρ f φ 3 ρ s 3 ρ f
ζ ˜ 1 = 1 − φ 1 , ζ ˜ 2 = 1 − φ 2 , ζ ˜ 3 = 1 − φ 3
Dynamic viscosity μ tnf = 1 ζ ˜ 1 2.5 ζ ˜ 2 2.5 ζ ˜ 3 2.5
ζ ˜ 1 = 1 − φ 1 , ζ ˜ 2 = 1 − φ 2 , ζ ˜ 3 = 1 − φ 3
Heat capacity ( ρ C p ) tnf = ( ρ C p ) f ζ ˜ 3 ζ ˜ 2 ζ ˜ 1 + φ 1 ( ρ C p ) s 1 ( ρ C p ) f + φ 2 ( ρ C p ) s 2 ( ρ C p ) f + φ 3 ( ρ C p ) s 3 ( ρ C p ) f  
Thermal conductivity k ˜ tnf k ˜ hnf = k ˜ s 3 + 2 k ˜ hnf − 2 φ 3 ( k ˜ hnf − k ˜ s 3 ) k ˜ s 3 + 2 k ˜ hnf + φ 3 ( k ˜ hnf − k ˜ s 3 )
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 )
Figure 2 Thermal–physical properties of the components in (a)–(c).
Figure 2

Thermal–physical properties of the components in (a)–(c).

Figure 3 Comparison of nanofluids characteristics against the increasing weight concentration in (a)–(d).
Figure 3

Comparison of nanofluids characteristics against the increasing weight concentration in (a)–(d).

To convert the basic model into final form, we introduced the following velocity and temperature variables, which support the current analysis

(7) u = − ac [ η − 0.5 F ( η ) ] , η = r a 2 , β ( η ) = T − T ∞ T f − T ∞ , w = 2 zc F ′ ( η ) .

In the next stage, the desired model is achieved by implementing the appropriate differentiation and ternary nanoliquid formulas in the primary governing laws:

(8) η F ‴ + F ″ − ρ tnf ρ f μ tnf μ f − 1 Re ( F ′ 2 − F F ″ ) = 0 ,

(9) 2 ( η β ″ + β ′ ) + ( ρ C p ) tnf ( ρ c p ) f k tnf k f − 1 Pr Re F β ′ + Rd ( 2 η β ″ + β ′ ) = 0 .

Moreover, the conditions imposed on the velocity and temperature of the ternary nanoliquid over a cylinder are reduced in the subsequent equations

(10) F ( 1 ) = α , F ′ ( 1 ) = 1 , F ′ ( ∞ ) → 0 ,

(11) k ∼ tnf k ∼ f β ′ ( 1 ) = − B i ( 1 − β ( 1 ) ) , β ( ∞ ) = 0 ,

where Pr = μ f ( C p ) f k f is the Prandtl number, Re = a 2 c Ѵ f is the Reynolds number, Rd = 16 σ ⁎ T ∞ 3 3 k ⁎ k f is the radiation parameter, and B i = a 2 h f 2 r k f is the Biot number and the suction parameter, respectively. Moreover, φ 1 , φ 2 , and φ 3 represent the NPs concentration.

The study of shear drag and heat transport rate at the working surface are important factors in many engineering systems. Therefore, these expressions for ternary nanofluids flow through a cylinder are described in Eqs. (12) and (13), respectively

(12) C f = τ w 1 2 ρ tnf u w 2 and Nu = r Q w k f ( T w − T ∞ ) ,

where

(13) τ w = μ tnf ∂ u ∂ r , and Q w = − k tnf + 16 σ * T ∞ 3 3 k * ∂ T ∂ r at r = a .

Now, the final expression for shear drag and heat transfer rate is

(14) C f = μ tnf μ f ρ tnf ρ f F ″ ( 1 ) and Nu = − 2 k tnf k f + Rd β ′ ( 1 ) .

3 Mathematical analysis

As the scientific problems are mostly modeled with the help of appropriate partial differential equations [43] or non-linear ordinary differential equations (ODEs) along with specific boundary conditions. The ternary nanofluid model, developed in this study, is necessary to solve for the description of a flow phenomenon against various physical constraints. The governing equations develop in the model are first reduced into system of ODEs.

Step I: Introduce the velocity and temperature transformative functions against the model order as given in Eqs. (8) and (9)

(15) Θ ̆ l 1 = F , Θ ̆ l 2 = F ′ , Θ ̆ l 3 = F ″ , Θ ̆ l 3 ′ = F ‴ ,

(16) Θ ̆ l 4 = β , Θ ̆ l 5 = β ′ , Θ ̆ l 5 ′ = β ″ .

Step II: Rearrangement of the problem is carried out in this step for the insertion of the functions from Eqs. (15) and (16)

(17) η F ‴ = − F ″ + ρ tnf ρ f μ tnf μ f − 1 Re ( F ′ 2 − F F ″ )

(18) F ‴ = 1 η − F ″ + ρ tnf ρ f μ tnf μ f − 1 Re ( F ′ 2 − F F ″ ) ,

(19) 2 ( η β ″ + β ′ ) = − ( ρ C p ) tnf ( ρ c p ) f k tnf k f − 1 Pr Re F β ′ − Rd ( 2 η β ″ + β ′ ) ,

(20) 2 ( η β ″ ) = − ( ρ C p ) tnf ( ρ c p ) f k tnf k f − 1 Pr Re F β ′ − Rd ( 2 η β ″ + β ′ ) − 2 β ′ ,

(21) ( β ″ ) = 1 2 η − ( ρ C p ) tnf ( ρ c p ) f k tnf k f − 1 Pr Re F β ′ − Rd ( 2 η β ″ + β ′ ) − 2 β ′ .

Step III: Reduce Eqs. (18) and (21) as follows using Eqs. (15) and (16), respectively

(22) Θ ̆ l 3 ′ = 1 η − Θ ̆ l 3 + ρ tnf ρ f μ tnf μ f − 1 Re ( ( Θ ̆ l 2 ) 2 − Θ ̆ l 1 Θ ̆ l 3 ) ,

(23) Θ ̆ l 5 ′ = 1 2 η − ( ρ C p ) tnf ( ρ c p ) f k tnf k f − 1 Pr Re Θ ̆ l 1 Θ ̆ l 5 − Rd ( 2 η Θ ̆ l 5 ′ + Θ ̆ l 5 ) − 2 Θ ̆ l 5 .

Step IV: In this step, the transformed problem in Eqs. (22) and (23) is analyzed using MATHEMATICA 13.0 and the output results are obtained. The convergence of the scheme is subject to the satisfaction of the conditions and asymptotic behavior.

4 Results and discussion

This section contains a comprehensive discussion of the results under varying levels of the physical parameters. These are shown in Figures 4–6 for the temperature.

Figure 4 The velocity performance for the mentioned constraints in (a)–(f).
Figure 4

The velocity performance for the mentioned constraints in (a)–(f).

Figure 5 The temperature performance against the mentioned constraints in (a)–(d).
Figure 5

The temperature performance against the mentioned constraints in (a)–(d).

Figure 6 The temperature efficiency for B i , Rd {B}_{i},\hspace{.25em}\text{Rd} , and Re \text{Re} in (a)–(f).
Figure 6

The temperature efficiency for B i , Rd , and Re in (a)–(f).

Figure 4 shows the velocity trends for ϕ 1 , α , and Reynolds number. The investigation reveals that the velocity fluctuation is very slow for increasing values of ϕ 1 . Physically, the density of primary fluid enhances when the NPs amount increases from 0.01 to 0.07 . Due to increase density, the mass per unit volume increases, which directly affects the movement of the ternary nanofluid. Further, the mass suction results drop in the velocity as shown in Figure 4b for all types of fluids under consideration. Physically, the mass suction through the surface attracts the fluid molecules outside the fluidic system, which reduces the momentum; as a consequence, the movement declines. The velocity approaches zero, which fulfills the flow condition far from the working domain. In nanofluid, the reduction in the velocity is slower than in hybrid and ternary fluids because this is less dense than the other two types, which allow the free movement. Moreover, the larger R e values is observed to control the motion in the current scenario.

The changes in the physical parameters alter the efficiency of the flow of fluid through a cylinder. Hence, Figures 5 and 6 are organized to analyze the temperature fluctuations in ternary, nano, and hybrid nanofluids against the mentioned ranges of the parameters. The temperature in all three types of nanofluids was augmented for ϕ 1 = 1.0 % to ϕ 1 = 7 % , while the mass suction effects depreciate it. The prominent change in the temperature due to ϕ 1 is associated with the thermal conductivity difference of these fluids. Physically, the simple nanofluid has the lowest thermal conductivity due to the role of only one type of nanoparticle. Due to this, the heat transfer ability of simple nanofluid is weaker than hybrid and ternary nanofluids. These results along with their 3D scenarios are shown in Figure 5a–d, respectively.

The two heat transfer ways, solar radiation and convective conditions, are of utmost interest in the study of nanofluids. Figure 6a–d illustrates the performance of nanofluids when B i changes from 0.01 to 0.04 and radiation number Rd from 0.1 to 1.3. It is scrutinized that both these two factors are crucial for heat transfer applications. Physically, imposed radiation effects augment the internal energy of the system, due to which the performance of the whole system enhances. Further, the convective condition shows dominant changes in the temperature near the surface of the cylinder. Physically, near the surface, conduction takes place, which heats up the particles near the surface. These particles gain energy due to the heated surface and move towards the free stream, and the other particles take their place. Due to this continuous process, the temperature significantly enhances near the surface. Further, Figure 6e and f reveal that the increasing ranges of Re are good to the temperature depreciation.

Tables 2 and 3 present the numeric computation of the skin friction and heat gradient. The fluctuation of shear drag against numerous values of φ 1 , Re , and α for ternary, hybrid, and simple nanofluid flow is computed in Table 2. For φ 1 , Re , and suction parameter α , the skin friction is greater for the ternary nanofluid than for the other two types. Physically, the ternary fluid has dominant denser effects due to which the shear drag is higher than hybrid and simple nanofluids. Hence, the skin friction coefficient gradually decreases from ternary to simple fluids. Moreover, the value of C f slightly increases for all three nanofluids with increasing value of φ 1 .

Table 2

Shear drag computation for multiple model quantities

Parametric ranges Comparative shear drag for nanofluids
φ 1 Re α Ternary nanoliquid Hybrid nanoliquid Nanoliquid
0.01 − 0.57500 − 0.61443 − 0.67097
0.03 − 0.57427 − 0.61420 − 0.67139
0.05 − 0.57334 − 0.61371 − 0.67148
0.07 − 0.57223 − 0.61299 − 0.67128
0.02 1.0 − 1.07499 − 1.13427 − 1.20886
1.5 − 1.27963 − 1.34795 − 1.43190
2.0 − 1.45964 − 1.53605 − 1.62856
2.5 − 1.62287 − 1.70666 − 1.80704
0.1 1.0 − 0.61689 − 0.65765 − 0.71493
1.5 − 0.64127 − 0.68262 − 0.74005
2.0 − 0.66630 − 0.70822 − 0.76577
2.5 − 0.69198 − 0.73447 − 0.79209
Table 3

Comparative computation for heat transfer rate against the model parameters

Model quantities Computation for heat transfer rate
φ 1 R e α R d B i Ternary Hybrid Nano
0.01 0.140929 0.136004 0.133445
0.03 0.140699 0.135765 0.133198
0.05 0.140495 0.135552 0.132977
0.07 0.140312 0.135362 0.132779
0.02 1.0 0.149639 0.144708 0.142235
1.5 0.151668 0.146644 0.144112
2.0 0.153016 0.147912 0.145324
2.5 0.153987 0.148816 0.146183
0.1 1.0 0.143328 0.138461 0.135972
1.5 0.144559 0.13971 0.137245
2.0 0.145684 0.140842 0.138391
2.5 0.146711 0.141869 0.139425
1.5 0.5 0.141668 0.136891 0.134458
1.0 0.139295 0.13456 0.132139
1.5 0.137697 0.132983 0.130563
2.0 0.136552 0.13185 0.129428
1.0 0.5 0.420126 0.406019 0.398879
1.0 0.567431 0.548888 0.53967
1.5 0.642526 0.621823 0.611632
2.0 0.688055 0.666076 0.655324

Table 3 reveals that the thermal radiation parameter is increasing the Nu values. When the value of the radiation parameter Rd is increased , the TC of the nanoliquid is also enhanced. As, the particle's strength in ternary nanofluid is larger than hybrid nanofluid, which has a greater concentration value as compared to the hybrid and common nanofluids. On the basis of weight concentration, the temperature of the ternary nanofluid rises more rapidly than hybrid nanofluid. That is why, with the increase in radiation parameter, the value of Nu is larger.

The involved constraints in the study greatly affect the flow and heat lines patterns. Thus, Figure 7a–h are furnished for different values of the physical controls and found that the stream contours are thick along the vertical axis. Further, the heatline contours are examined less at the inner part while they become strengthen in the rest of the portion.

Figure 7 Streamlines and isotherms changes against the model parameters in (a)–(h).
Figure 7

Streamlines and isotherms changes against the model parameters in (a)–(h).

The code and study are validated with the results of Waqas et al. [2] under restricted parametric values. The result plot for α = 1.0 and Re = 2.5 in Figure 8 and found that both the solutions are well aligned which provides the reliability of the study.

Figure 8 Code and study validation.
Figure 8

Code and study validation.

5 Conclusions

Investigation of convectively heated ternary nanofluid past a cylinder is performed. The permeability, radiation, and Reynolds number effects are also included in the problem. It is examined that

  • The stronger permeability of the surface ( α = 0.1 , 0.9 , 1.7 , 2.5 ) resists the flow significantly, and a rapid decline is scrutinized for the ternary case while nano and hybrid fluids show slow depreciation in the velocity.

  • The Al2O3 concentration in the range of 1.0 – 7.0 % provided a considerable increase in the temperature, and dominant trends are observed for the ternary nanofluid.

  • The addition of solar radiations ( Rd = 0.1 , 0.5 , 0.9 , 1.3 ) and convective condition ( B i = 0.01 , 0.02 , 0.03 , 0.04 ) are examined as good physical agents to augment the problem thermal efficiency.

  • The skin friction enhances absolutely from − 1.07499 to − 1.62287 when Re varies from 1.0 to 2.5 and − 0.616896 to − 0.691989 for α = 1.0 , 1.5 , 2.0 , 2.5 .

  • The heat gradient enhances from 0.149639 to 0.153987 and 0.420126 to 0.688055 for Re = 1.0 , 1.5 , 2.0 , 2.5 and B i = 0.5 , 1.0 , 1.5 , 2.0 , respectively.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA, for funding this research work through the project number “NBU-FFR-2025-2461-11.” Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R41), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

  1. Funding information: This research work was funded by the Deanship of Scientific Research at Northern Border University, Arar, KSA, through the project number “NBU-FFR-2025-2461-11.” Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R41), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

  2. Author contributions: Adnan and Waseem Abbas: conceptualization, software, methodology, writing original draft, writing review, and editing. Refka Ghodhbani, Kaouther Ghachem, and Tadesse Walelign: formulation, formal analysis, investigation, software, validation, writing review, and language editing. Yasir Khan, Mehdi Akermi, and Rym Hassani: visualization, formulation, Investigation, writing review, and editing. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

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

  4. Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2024-06-18
Revised: 2025-02-18
Accepted: 2025-06-12
Published Online: 2025-07-31

© 2025 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/phys-2025-0177
Keywords for this article
ternary nanofluid; heat transfer; cylinder; thermal radiation; convective surface
Creative Commons
BY 4.0

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