Thermal transport in single, binary and tri-hybrid nanofluid over porous medium for heat and aeronautical applications

Abstract

This study examines the flow and heat transfer behavior of a tri-hybrid nanofluid over an exponentially stretching/shrinking surface, incorporating the influences of thermal radiation, viscous dissipation, internal heat generation, and magnetic fields. The working fluid consists of a suspension of Al₂O₃, Cu, and TiO₂ nanoparticles in water. Governing partial differential equations are reduced to an ordinary differential system through similarity transformations, subject to prescribed surface temperature (PST) and prescribed heat flux (PHF) conditions. The transformed equations are solved numerically using MATLAB’s bvp4c solver. Parametric effects on velocity, thermal characteristics, and Nusselt number are analysed and illustrated via tables and graphs for both PST and PHF cases. Comparative analysis highlights the efficiency of the tri-hybrid nanofluid Al₂O₃-Cu-TiO₂-H₂O over single and binary nanofluids. Results show enhanced heat transfer with increasing magnetic parameter, supporting applications in thermal and aeronautical engineering, including heat exchangers, solar absorbers, and lightweight cooling systems.

Introduction

The boundary level induced by a continually stretching/shrinking surface contributes an important part in industrial manufacturing procedures. From a technological perspective also, this system is utilized in several fabricating processes such as glass fibre line-drawing, plastic extrusion and continuous venting etc. Thermal performance can significantly increase with nanofluids and thermal efficacy is an essential part in micro-engineering concepts. Nanofluids, which are engineered gives colloid suspensions of nanomaterials with fundamental fluid, have expressive immersion in recent decades because of their enhanced thermal properties compared to conventional fluids. Their unique ability to improve heat transfer performance makes them highly valuable in several industrial and engineering applications. The content of nanofluid is first introduced by Choi [1]. Jamuna et al. [2] explored the entropy generation along permeable cavity containing nanofluid and gyrotactic microbes. Bodduna et al. [3] investigated the bio-convention of nanofluid with influence of inclination angle. Various metallic and non-metallic combinations with basic fluids such as H₂0, and C₂H₄(Ethylene) are known as Hybrid nanofluids resulting more ability in transfer of heat elevation with contrast to nanofluids. Mallesh et al. [4] studied about heat and radiation impacts for time dependent convective binary nanofluid past an up straight oscillating plane. Padma et al. [5] explained the unsteady natural convective heat circulation of hybrid nanofluid over an erect plane. Padma et al. [6] scrutinized the stability analysis with viscous dissipation on binary nanofluid on an exponential stretching/shrinking surface. A numerical analysis of heat transfer over Bi-viscous Bingham nanofluid in permeable surface is analyzed by Bisht and Maheshwari [7].

In recent years, researchers explored an incredible discovery on mixing three distinct nanoparticles into the fundamental fluid, leading a unique nanofluid classified as a trihybrid nanofluid. Because of its extreme thermo physical features, investigators have exhibited a significant amount of interest. The applications of tri hybrid nanoliquid flow results more efficacy in improvement of rate of heating/cooling. With effect of MHD, these fluids have several applications in solar energy transmission, micro-electronics, warming procedure in buildings, biomedicine etc. Abdul et al. [8] elaborated heat transfer and MHD flow of a tri-hybrid nanofluid in a rotating system. Jakeer et al. [9] worked the numerical analysis of EMHD system of nanofluids (Cu-CNT-Ti/H₂O) with radiation impact. A computational simulation of MHD stagnation flow along heat source/sink is given by Zafar et al. [10]. Impact of thermal radiation with biot number of tri hybrid nanoparticles in blood past a curved elaborating plane is examined by Nagaraja et al. [11]. A convectively heated curvature to amplify heat transmission in between fluid particles by Shinwari et al. [12]. Reddy et al. [13] numerically described uniform heat flux effect over a stretching cylinder on MHD dusty hybrid nanofluid flow.

In fluid mechanics and heat transfer, a Prescribed surface temperature (PST) and Prescribed heat flux (PHF) cases refers to situations where the temperature at the boundary surface of a system such as a wall or a surface of a solid object is specified, remains constant or follows a predefined variation with time or position and heat transfer rate per unit area across a surface is specified. With these situations many researchers presented their simulations as follows: Salah et al. [14] discussed the impact of PST and PHF cases on stir flow of trihybrid nanofluid over a rotating cone. An examination of three-dimensional (3D) radiative viscoelastic nanofluid flow over expandable permeable plane is given by Boujelbene et al. [15]. Nadeem et al. [16] elaborated thermal analysis of PST and PHF situations past a riga curved stretching plane.

Porous medium hangs an important role in rate of cooling. Although the physical characteristics of cooling medium also influence rate of cooling, real-world conditions necessitate for physical attributes with variable qualities. Suneetha et al. [17] numerically investigated the pulsatile blood-based hybrid nanofluid flow heat radiation effects on a permeable surface. The fluid flow behaviour of Prandtl number with gyrotactic microorganisms past an erect cylinder is explored by Shaik et al. [18]. Sekhar et al. [19] elaborated the binary chemical reactions on hydromagnetic stagnation point flow using ANN implementation with impact of thermophoresis. The heat and mass transfer of electrically conducting flow of Jeffrey fluid past a spiral disk is examined using Kellar-Box method by Sarfraz and Khursheed [20]. Sarfraz et al. [21] explained hall and joule heating effects for spherical-shaped particles using Kellar-Box method. A mathematical model observing the behaviour of a HNF flow between two infinite parallel plates with flow enhancement effect is scrutinized by Muhammad and Sarfraz [22].Umar et al. [23] scrutinized the unsteady rotating nanofluid with nanoparticle aggregation, slip conditions, and variable viscosity of fluid behaviour in advanced thermal and microfluidic systems. Rafique et al. [24] examined the entropy generation in the hydromagnetic radiative stagnation flow of tri hybrid nanofluid. The ternary hybrid nanofluid flow over a rotating sphere with thermal conductivity and variable viscosity, and several nano particles shapes like cylinder, brick and blade is investigated by Zafar et al. [25]. Magnetized second-grade fluid flow along a permeable surface with thermal non-equilibrium integrated model is explained by Ramzan et al. [26]. Muhammad et al. [27] explored the heat transfer analysis with slip boundary conditions considering the prescribed exponential surface (PEST) and prescribed exponential heat flux (PEHF) cases. The flow behaviour of aqueous-based nano liquid containing MWCNTs considering the well-established Yamada-Ota thermal conductivity model to evaluate the nanoparticle radius and length impacts is presented by Ramzan et al. [28]. Aziz [29] evaluated the characteristics of heat transmission on a mixed convection flow along an exponential stretchable sheet. The complex dynamics of pulsatile hydromagnetic flow of micropolar ternary hybrid nanofluid between vertical permeable walls is explored by Sajjan et al. [30].

Aforementioned literature is illustrational to research the heat transfer improvement with the influence of several parameters of a group of three nanomaterials , considering water as a base fluid with novelty of PST and PHF constraints. The interaction of porous effects with tri hybrid nanofluids under PST and PHF constraints fills the gap in the existing literature by examining the behaviour of fluid flow and thermal transfer of advanced nanofluids with in porous structures. The partial differential equations are converted into ordinary differential equations with self-similarity conversions and the outcomes are discussed graphically, which are obtained using a MATLAB software implementing bvp4c. The outcomes of this work follow the below mentioned bullets to address the research questions along numerical approach by tri-hybrid nanofluid in both scenarios,

• How does the magnetic parameter effects velocity and temperature consequences past an exponentially elaborating/shrinking surface of compared to single and binary nanofluid as well as its effectiveness in PST than PHF scenario?

• What the effects of space and time dependable parameters of heat generation/compression varies the momentum and thermal distributions of in the cases?

• What are the broader inferences of the current simulation for tri-hybrid nanofluid behaviour and how it contributes scientific knowledge in these areas?

Understanding heat transfer in tri-hybrid nanofluids contributes to the development of new materials with improved thermal conductivity properties. Insights into enhanced heat transfer can optimize the design and performance of micro-channel-based systems, nanofluid based solar water heaters and energy storages etc.

Gap analysis table

Author	Nano fluid type	Porous medium	PST/PHF conditions	Aeronautical applications
Rahman et al. [42]	Nanofluid	✔	X	X
Padma et al. [6]	Hybrid	✔	X	X
Zafar et al. [10]	Ternary	✔	X	X
Present study	Tri Hybrid	✔	✔	✔

Practical applications

Practical application is the process of using theoretical knowledge in real-world situations to solve problems and achieve outcomes. The stagnation-point flow of a tri-hybrid nanofluid over a non-linear elongated/compressed plane with slip conditions has a lot of real-world applications. The different foregrounds and attributes of tri-hybrid nanofluids makes a wide range of uses, such as enhancement of heat transfer efficacy in thermal engineering, material creation with advanced technology in aeronautical and sensor technology, cooling of semi-conductors, power systems in renewable energy systems etc.

Problem description

A cartesian co-ordinate system with x-axis normal to y-axis as presented in Fig. 1, for a steady, incompressible and laminar boundary layer fluid flow of tri-hybrid nanofluid over an exponentially porous stretching/shrinking sheet with an account of varied thermal conductivity and heat generation/absorption. The velocities in direction of x and y are u and v respectively. An applied magnetic field, with magnetic strength - is employed perpendicular to the surface . Tables 1 and 2 explains the values numerically and fluids heat transfer of three nanomaterials with as a fundamental fluid. Figure 1 depict a simplified schematic depiction of the work. The novelty of the current work lies in incorporating multiple interacting physical mechanisms, with porous media impacts, stretching/shrinking dynamics, and variable magnetic field effects, while simultaneously examining different nano particle combinations. This configuration enables a deeper understanding of how multi-component nanoparticle suspensions enhance thermal transport compared with conventional nanofluids. It is determined that sheets and surface velocity is represented as and respectively. A system of equations with above assumptions portrays the following continuity, momentum and energy equations of fluid flow with boundary conditions (Wahid et al. [35]).

Fig. 1.

Geometrical Interpretation

Table 1.

Numerical values of base water (H₂O) and tri-hybrid nanoparticles (Sajjan et al. [30])

Properties	Al2O3 (ϕ1)	Cu (ϕ2)	TiO2 (ϕ3)	H2O
ρ (Kg/m3)	3970	8933	4250	997.1
CP (J/Kg K)	765	385	686.2	4179.0
K (W/m k)	0.40	400	8.538	0.613
σ (Ω/m)	3.5 × 106	59.6 × 106	2.352 × 106	0.05
Pr	-	-	-	6.2

Table 2.

Thermal properties of ternary hybrid nanofluid (Padma et al. [6]; Zafar et al. [10])

Properties	Nanofluid	Hybrid nanofluid	Ternary hybrid nanofluid
Density
Heat capacitance
Dynamic viscosity
Electrical conductivity
Thermal conductivity
Variable thermal conductivity

Continuity equation:

1

Momentum equation:

2

Energy equation:

3

Boundary conditions:

4

where T is fluid temperature, C_P is specific heat at constant pressure.

For solving energy equation, we use two types of thermal boundary conditions:

• (i)
  PST Case: The boundary conditions in PST case are of the form

  

  5
• (ii)
  PHF Case: The boundary conditions in PHF case are of the form

  

  6

  where are constants based on thermal properties of the fluid.

Consider the thermal conductivity K varies linearly (Abel and Mahesha [36] Chiam [37], Chiam [38]) with temperature and it is of the form:

7

where ε = small parameter known as variable thermal conductivity parameter which is negative for solids and liquids and positive for gases (Nag [39]).

= thermal conductivity of the fluid.

θ = dimensionless temperature in prescribed surface temperature, PST – Scenario.

g = dimensionless temperature in heat flux, PHF – Case.

The non-uniform heat source/sink (Abel et al. [36]) is modelled as:

8

where and are the coefficient of space and temperature dependent heat source/sink respectively.

is the wall temperature.

is the temperature outside the dynamic region.

, > 0 shows internal heat generation.

, < 0 corresponds to internal heat absorption.

Similarity transformations (Padma et al. [6]):

9

Reduced equations:

10

PST case:

11

PHF case:

12

Boundary conditions:

13

Physical quantities

Physical quantities like the coefficient of skin-friction, heat flux and drag force have significant applications. For instance, the skin-friction coefficient helps in understanding the resistance faced by a moving object through a liquid, drag force is important for designing automobiles and sports tools, and heat flux plays vital role for creating efficient cooling systems in electronic devices. Nusselt number (Nu) and Skin-friction coefficient (C_f) are defined as (Aziz et al. [40]; Hamid et al. [41]; Usman et al. [42])and

14

Equation (14) is transformed as below using Eq. (9) (Abu-Hamdeh et al. [43]; Pal and Mandal [44])

15

in PST and PHF cases respectively,

and where denotes Reynolds number.

Numerical procedure and code validation

Solving nonlinear differential equations that arise in boundary layer flow problems can be accomplished using several approaches. Numerical, analytical, and semi-analytical techniques are commonly employed methods. The bvp4c solver in MATLAB is specifically designed for solving boundary value problems (BVPs) associated with ordinary differential equations (ODEs). In the present simulation, Eqs. (9–15) are transformed into a system of ODEs and solved using the numerical solver bvp4c. A computational code is developed to obtain the velocity, temperature, Nusselt number, and skin-friction coefficients, as well as to compare the present results with those available in existing studies.

In the numerical implementation, the absolute tolerance and relative tolerance are specified as 10⁻⁸ each to control the accuracy of the numerical solution, and a step size of 0.01 is employed to ensure numerical stability. The following transformations convert the governing equations into a first-order system that can be solved for both PST and PHF cases along with the corresponding boundary conditions. Figure 2 represents the working flow chart of the present simulation.

16

Fig. 2.

Working flow chart

Tables and graphs

Tables 1–2 provides the properties of water and nanoparticles to obtain the behavior of tri-hybrid nanofluids . Table 3 expose that the current numerical simulation is good in agreement when compared with the existing literatures of Zafar et al. [10] and Rahman et al. [36]. In present computation the tri-hybrid nanofluid is considered instead of simple fluid that subjected to Rahman et al. [45] and also PST & PHF cases as a novel computation are used against Zafar et al. [10].

Table 3.

Values for varying and for Cu–H₂O nanofluid on considering with Pr = 6.2

			Present	Zafar et al. [10]	Rahman et al. [45]
0	-0.5	0	2.118666077	2.1183	2.1182
		0.1	3.238919997	3.23824	3.2382
		0.2	4.50817173	4.50714	4.5071
	0	0	1.687218163	1.68722	1.6872
		0.1	2.579342119	2.57935	2.5793
		0.2	3.590121789	3.59014	3.5901
	0.5	0	0.960416075	0.97042	0.9845
		0.1	1.468240261	1.4973	1.5051
		0.2	2.043606901	2.09463	2.0950
0.1	-0.5	0	2.170795116	2.10784	2.1078
		0.1	3.296595826	3.31865	3.3186
		0.2	4.619290980	4.61913	4.6191
	0	0	1.717381110	1.71655	1.7165
		0.1	2.705925927	2.62415	2.6241
		0.2	3.741926531	3.65242	3.6524
	0.5	0	0.990538956	0.98832	0.9971
		0.1	1.534897137	1.51782	1.5243
		0.2	2.133346096	2.11852	2.1216
0.2	-0.5	0	2.222772863	2.22224	2.2222
		0.1	3.474241813	3.39713	3.3971
		0.2	4.790112667	4.72844	4.7284
0.2	0	0	1.746087223	1.74532	1.7453
		0.1	2.738508574	2.66815	2.6681
		0.2	3.780857296	3.71373	3.7137
	0.5	0	1.007014933	1.00786	1.0094
		0.1	1.528505574	1.53822	1.5432
		0.2	2.121567026	2.14694	2.1478

Results and discussion

This section depicts the significance of the results computed and the current numerical analysis of velocity, temperature, Nusselt and skin-friction properties and analyses the nature of tri-hybrid nanofluid with Pr = 6.2 as fixed in the entire study. Equations (9)–(15) are resolved using MATLAB software with bvp4c solver in PST and PHF constraints. A detailed view is given in preceding section. The pertinent parameters (Eckert number); (Variable thermal conductivity); (Shrinking case); (Stretching case); (Magnetic);(Thermal radiation);(Porosity); (Suction) are considered based on key references and promises of outcomes for PST and PHF cases. Figures 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16 provide a thorough view of the present simulation on illustrating the velocity, temperature, Nusselt number and skin-friction coefficient in different patterns.

Fig. 3.

Velocity profile varying a) M = 0, 1, 3, 5; b) P = 0.01, 0.02, 0.03, 0.04; c) S = 0, 1, 3, 5 when Pr = 6.2 in PST case

Fig. 4.

(a), (b) Velocity &Temperature profile varying shrinking parameter λ = -0.5, -1, -1.5, -1.63 in PST condition and (c), (d) Velocity & Temperature profile varying shrinking parameter λ = -0.1, -0.2, -0.3, -0.4 in PHF condition

Fig. 5.

(a), (b) Velocity and Temperature profile varied for stretching parameter λ = 0.1,0.2,.0.3,0.4 in PHF condition and (c), (d) Velocity & Temperature profile varying stretching parameter λ = 0.4, 0.6, 0.8,1 in PST condition

Fig. 6.

Temperature profile for A = 0.01, 0.02, 0.03, 0.04 in (a) PST and (b) PHF cases

Fig. 7.

Variation in B = 1, 3, 5, 7 describing the temperature profile in (a) PST and (b) PHF case

Fig. 8.

Temperature profile with varied Ec = 0, 0.3, 0.5, 0.9 in (a) PST case and (b) PHF case

Fig. 9.

Temperature profile varying ε = 0, 0.1, 0.2, 0.3 in (a) PST case and ε = 0,0.1,0.2,0.3,0.4 in (b) PHF case

Fig. 10.

Temperature profile varying for M = 1, 3, 5, 7 in (a) PST and (b) PHF cases

Fig. 11.

Temperature profile varying for Nr = 0, 0.2, 0.4, 0.6 in (a) PST case and Nr = 0.2, 0.4, 0.6, 0.8 in (b) PHF case

Fig. 12.

(a), (b) Temperature profile varying the porosity parameter P = 0.001, 0.002, 0.003, 0.004 and (c), (d) varying the suction parameter S = 0.1, 0.3, 0.5, 0.7 in PST and PHF conditions

Fig. 13.

Variation in with effect of (a) (b) (c) (d) (e) and (f) against in PST Case

Fig. 14.

Variation in with effect of (a) (b) against in PST case (c) (d) against in PHF Case

Fig. 15.

Bar graphs of (a) and (b) varying with in PST Case

Fig. 16.

Bar graphs of (a) and (b) varying with in PHF case

It is viewed in Fig. 3 that an elevation in velocity profile as M, S and P elevates for nanofluid, hybrid nanofluid and tri-hybrid nanofluid with other parameters as fixed values as displayed in graph , while the variation in shrinking parameter(λ) displays the decrease of momentum distribution for nanofluid, hybrid nanofluid and group of three hybrid nanofluid in case of PST. This is because the flow controlling equations for MHD flow entail a term Lorentz force, which arises due to the interaction between magnetic field and fluid motion. The presence of magnetic field stabilizes the boundary layer and reduce the flow, which results to amplified velocity. An elevation in porosity typically means a deceleration in the solid fraction of medium, which leads to low resistance and hence, allowing the fluid to flow more freely. This reduction drag on the liquid, thereby boosts up the velocity stencil. Suction stabilizes the boundary layer on removing low momentum fluid, elevating the velocity profile. The reason behind the enhanced performance of the is that the synergistic interaction of varied nanoparticles optimises the fluids thermal and physical attributes resulting in a more substantial reaction to changes in magnetic (M), porosity (P) and suction parameters(S). As the momentum equation is similar in PHF case same as PST case, not explored graphically subjecting as PHF case.

Figure 4(a), (c) displays the velocity profile with the effect of shrinking parameter λ < 0 as a decreasing function, whereas Fig. 4(b), (d) explores the temperature profile as an elevating function in PST & PHF conditions. Influence of shrinking alters the dynamics of boundary layer resulting to a thicker boundary layer and reduced velocity near the surface. Due to the frictional forces and enhanced viscosity of a more significant reduction in velocity come across within this boundary layer compared to single and binary nanofluid. The constant temperature boundary allows for consistent heat distribution throughout the fluid, resulting the transfer of heat more effective in PST case and interaction among various types of nanoparticles improves the mechanism of heat transmission leading to a superior distribution of heat and an enhancement in thermal profile.

While, from Fig. 5(a), (c) and (b), (d) reverse trend appears for velocity and temperature profiles on influence of stretching parameter λ > 0 in PST and PHF conditions. Since, in the existence of multiple nanoparticles, the viscosity effects lowers, improving the flow ability allowing for higher velocity consequences and enhanced properties make it more responsive to this change. The combination of enhanced heat transfer attributes and higher velocities allows the group of three hybrid nanofluid to remove the heat more efficacy, leading in a lower thermal distribution. Due to the variant in boundary conditions in PHF case from PST condition, the effectiveness varies.

Temperature consequence of tri-hybrid nanofluid flow varying for A > 0 in PST & PHF cases is explored graphically in Fig. 6. Here Fig. 6a represents PST case, whereas Fig. 6b represents PHF case. In both the cases it is seen that the thermal distribution amplifies compared to the nanofluid and hybrid nanofluid. This is due the synergistic effects of multiple nanoparticles, effective interaction of heat source/sink, stability and distribution of nanoparticle and an enhanced convection. The thermal conductivity improvement and the presence of several nanomaterials enhance the convective heat transfer within the fluid which leads to more effective spreading of heat, contributing to a higher temperature consequence when subjected to space-dependent heat source/sink. In case of simple and hybrid nanofluid, the convective heat transfer is less amplified compared to the tri-hybrid nanofluid, which results lower in thermal profile. PHF outcomes the less effective.

than in PST conditions since the constant heat flux in PHF allows for a rapid thermal dissipation, limiting the rise in temperature. In contrast, PST conditions create a continuous thermal gradient that better utilizes the intensified heat transfer attributed of tri-hybrid nanofluid, resulting a higher temperature.

The influence of time dependent heat source/sink parameter is portrayed as an increasing function of when compared to nanofluid and hybrid nanofluid in Fig. 7. exhibits a higher thermal profile with rising due to amplified thermal conductivity, effective nanoparticle interaction and dynamic thermal response. It is more effective in PHF conditions as the constant heat flux allows for a more significant manifestation of its superior heat transfer properties, whereas PST conditions limit the thermal profile due to its constant temperature constraint.

In Fig. 8, the temperature profile of with the rise in enhances due to enhanced thermal conductivity, improved energy conversion from viscous dissipation and interaction of complex nanoparticles, which leads in a higher thermal consequence compared to single and binary nanofluids. The fluid flow of is less effective in PHF case compared to PST conditions because the constant heat flux in PHF results to rapid heat dissipation, confining the rise in temperature. In opposite, PST constraints create a stable temperature gradient that better uses the amplified properties of heat transmission of three mixed nanomaterials plus water, resulting in higher thermal profile.

It is viewed in Fig. 9 that as ε varies in increasing order, the temperature profile in cases of PST and PHF amplifies in a group of tri nanoparticles flow with water as host fluid. This is due to the enhanced heat transfer & synergistic impacts of mixture of 3 nanomaterial leads to improvement in thermal conductivity resulting a higher temperature distribution throughout the fluid. Also, with the sustained temperature gradient, the amplified heat transmission properties are better utilized in maintaining the thermal gradient, resulting to a more consequent thermal profile rise in PST case than PHF conditions.

The temperature profile with impact of increase in magnetic parameter of fluid flow elevates on PHF boundary conditions whereas reverse trend appears on PST condition is clearly visible in Fig. 10. The mixture of exhibits an amplification in thermal conductivity than pure water and also it lacks the more impacts brought by metal oxide nanoparticles. In fluid flow, copper has additional thermal conductivity than and combination including has excellent stability in suspension and high thermal conductivity leading to enhancement in magnetic responsiveness. This is because from the thermodynamic outlook, upwind in gives in piling up the Lorentz force which catalyzes opposing circulation of fluid, leads in progress of stress, resulting temperature to builds up. In PST case, the rise in magnetic parameter results in more heat generation within the fluid, and since the surface temperature is constant, there is rise in temperature throughout the fluid, whereas the opposite trend is seen in case of PHF conditions due to the constant heat flux, which forces the system to disperse this heat differently. This behavior is due to the distinct methods of heat management under varied thermal boundary conditions (PST vs PHF) with the impression of magnetic field on heat source and fluid flow.

Figure 11 elaborates outlook of temperature distribution as a boosting function of fluid motion varying the parameter in cases of boundary conditions. Since, radiation contributes to the overall energy balance by transferring energy in the form of electromagnetic waves leading an upwind in fluid temperature, as additional energy is absorbed by the particles in fluid. Comparatively, the motion is higher than and flow with more thermal conductivity of for consuming and releasing thermal radiation. In PHF case, the growth in thermal stencil is less pronounced since the stable heat flux accompanies to a redistribution of energy, lessening the enhanced radiation impact on the temperature when compared to PST conditions. This is due to the energy management under these two different constraints and fluid’s ability to absorb.

Figure 12(a), (b) provides the behavior of temperature in constraints of fluid flow of as a decelerating function in PST case, whereas an accelerating function in PHF case as the porosity parameter . This difference appears due to the heat transfer mechanism under the boundary constrains considered in the current study and the enrich porosity, impacts the efficacy of convective transfer of heat within the fluid. The outcome thermal consequence of motion under the conditions of is portrayed in Fig. 12(c), (d) as a diminishing function with a comparative analysis of and flow under the influence of suction parameter The containing nanoparticles and its mixture makes susceptible to the impacts of suction, leading in more significant deceleration in the thermal profile compared to single and binary nanofluids. The temperature profile decreases more significantly in PST case because of constant surface temperature, combined with increased suction, efficiently removes heat, leading to a sharper reduction in temperature compared to PHF conditions.

The comportment of against under influences of a Eckert number , group of three nanoparticles on comparison of , and , Variable thermal parameter , Magnetic parameter , Radiation parameter and Porosity parameter is presented in Fig. 13 (a), (b), (c), (d), (e), (f) with pertinent boundary conditions. It is clearly seen, Nusselt number amplifies with variation of aforementioned pertinent parameters. It is due to the presence of multiple types of nanoparticles elevates the surface area of heat transfer and with more efficacy heat exchange enhances the Nusselt number. Thermal radiation is more significant with an amplification in the radiation parameter, which contributes to higher energy exchange between the fluid and its surroundings. In a tri-hybrid nanofluid, nanoparticles presence upsurges the absorption and emission of thermal radiation, further boosts up the Nusselt number. The combination of these factors in tri-hybrid nanofluids result to a more efficient process of heat transfer, leading in a higher Nusselt number compared to single and binary fluids.

Figure 14 with influence of (a) , , and (b) reveals the variation of skin-friction against of triple nanofluids in case of PST. It is seen that skin-friction coefficient increases as and respectively. The nanoparticles alter the thermal attributes of the fluid like Cu – nanoparticle have higher thermal conductivity, resulting elevation in heat transfer. However, with boost up in concentration of these nanoparticles, momentum transfer within the fluid also rises, resulting in stronger shear forces near the surface. As an increase in magnetism, the Lorentz force exerted on the fluid by strengthen of magnetism. This force opposes the motion of fluid, leading it to decelerate near the surface, which in turn up strengthen the friction between the fluid and plane (or boundary). As an outcome, the coefficient of skin-friction enhances. Whereas Fig. 14(c) and (d) describes the variation in of as a deceleration trend. This is because the combination of , and with create more stable suspension with upgrade thermal conductivity but lower momentum diffusion reducing skin-friction. In a PHF case, heat transfer rate is constant across the boundary layer, leads to better stable thermal gradients. Lorentz force resists the fluid flow, reducing velocity gradient near the plane, resulting to lower skin-friction. The impact is more pronounced in case of PHF since the boundary condition stabilizes the thermal field, lowering the influence of viscous dissipation compared to PST case. The interaction between thermal and velocity fields maintains a higher skin-friction coefficient relative to PHF case, while in PST case, the temperature at the wall maintains stronger thermal gradients which outcomes to rise in skin-friction as .

It is exposed in Fig. 15(a) and (b) in bar graph representation about the Nusselt number and skin-friction varying in PST case that are elevating for three group hybrid nanofluid , considering the values when compared to single and binary nanofluid. On the other hand, Fig. 16(a) and (b) describes the bar graph representation about and in case of PHF.

With aforementioned data here, the tri-hybrid nanofluid illustrates better improvement in heat transfer and skin-friction in comparison to the mono and hybrid nanofluid under constraints in cases of PST and PHF. This indicates that amplifying the system’s performance by strengthening the nanoparticles concentration in the fundamental fluid yields to improved results.

Conclusion

The current simulation aim is to elaborate the heat transfer analysis of tri-hybrid nanofluid flow on comparing with single and binary nanofluids flow in PST and PHF cases over an exponential stretching/shrinking surface. The key explorations of current numerical simulation are:

• As , the velocity consequence of enhances than the and in PST case because of the synergistic interaction of different nanoparticles improves the physical and thermal properties of fluid motion which leads a good reaction to changes in magnetic, porosity and suction parameters.

• The impact of shrinking parameter explores the velocity as a deceleration function and temperature accelerates, whereas a reverse trend is noticed when elaborating sheet parameter for velocity(accelerating) and temperature(diminishes) respectively in PST and PHF cases of group of three hybrid nanofluids compared to group of Cu and Al₂O₃ with water-H₂O as a host fluid.

• Thermal profile of with the influence of A > 0 boosts up in PST case than PHF, since earlier condition establish a continuous thermal gradient that more effectively leverages the amplified transfer of heat properties of , leading to a higher temperature.

• With the effect of B > 0, the temperature distribution elevates in PHG case than PST because its conditions as the stable heat flux permits significant phenomenon of its higher properties of heat transfer.

• The variation in Ec elevates the improved consequence in temperature more in fluid flow than the and in PST case, since the tri-hybrid nanofluid flow is effectively low in PHF confining elevation in thermal pattern.

• For ε > 0 and Nr > 0 the temperature profile seen to be amplified with its superior properties in PHF case of fluid flow.

• When the porosity and magnetic parameter enhance there is decrease in temperature of three group hybrid nanofluid flow in PST case whereas opposite phenomenon is noticed in PHF case because of its thermal attributes.

• With the impact of suction parameter S > 0, temperature profile diminishes in cases of fluid flow.

• Nusselt number upsurges on λ contrasting in all above discussed parameters of mixture of Al₂O₃, Cu and TiO₂ with H₂O in PST case as this combination of nanoparticles leads to a better significant process of heat transmission resulting amplifying Nusselt number compared to mono and binary hybrid nanofluid.

• The variation in skin-friction coefficient of against three nanoparticles and magnetic parameter presents an enhancement in PST case.

• With the influence of M > 0, C_f upsurges in PST case whereas it drops down in case of PHF as the boundary constraints stabilizes the thermal surface, lessening the influence of viscous dissipation compared to PST conditions.

The future scope of this article is to enhance the understanding of several properties of nano particles over varied geometrical convection flow in combination of more nanoparticles on using latest computing techniques, based on ANN model.

List of symbols

PST

Prescribed surface temperature

PHF

Prescribed heat flux

A and B

Coefficient of space and temperature dependent heat source/sink parameters

A₁

Velocity slip factor

B₀

Magnetic field strength (Telsa(T))

C_p

Specific heat (J Kg⁻¹ K⁻¹)

(ρC_p)_nf

Heat capacitance of nanofluid (J Kg⁻¹ K⁻¹)

(ρC_p)_hnf

HNF heat capacitance (J Kg⁻¹ K⁻¹)

(ρC_p)_trhnf

THNF heat capacitance (J Kg⁻¹ K⁻¹)

Nu_x

Non-dimensional heat transfer coefficient

C_f

Coefficient of skin-friction

L

Sheet length (m)

h_f

Coefficient of convective heat transfer (W/(m² K))

Ec

Eckert number

M

Magnetic parameter

Nr

Thermal radiation parameter

Pr

Prandtl number (μC_pk⁻¹)

q_r

Thermal radiative heat flux (W m⁻²)

q_w

Wall heat flux (W m⁻²)

Re_x

Reynolds number

S

Suction/Injection parameter

K

Thermal conductivity (Wm⁻¹ k⁻¹)

P

Porosity

T

Temperature (K)

T_w

Temperature at stretching/shrinking surface (K)

T_∞

Free Stream Temperature (K)

x, y

2D-space cartesian coordinates

u, v

Fluid velocity elements in x, y-direction

Greek letters

ϕ₁, ϕ₂

Nanoparticle volume fraction of Cu, Al₂O₃ respectively

η

Similarity variable

K

Thermal conductivity (Wm⁻¹ K)

μ

Dynamic viscosity (kg m⁻¹ S⁻¹)

ν

Kinematic viscosity (m² S⁻¹)

ρ

Density (Kg m⁻³)

λ

Stretching/Shrinking parameter

σ

Electrical Conductivity (Ω/m)

σ^*

Stefan-Boltzmann Constant (W m⁻² K⁻⁴)

θ

Dimensionless temperature in PST case

ψ

Stream function

ε

Variable thermal conductivity parameter

g

Dimensionless temperature in PHF case

Subscripts

f

Host fluid

nf

Nanofluid

hnf

Hybrid nanofluid

t_rhnf

Tri-hybrid nanofluid

Author contributions

Problem formulation, commit to paper and drafting by S.V. Padma, Programming and approach by Dr. M. Sanjalee, Assessment, revising and guidance by Dr. S.R.R. Reddy and shaik Jakeer has thoroughly reviewed the entire problem and assisted in its modelling.

Funding

This study receives no funding.

Data availability

No data was used for the research described in the article.

Declarations

Ethics approval and consent to participate

Approval not applicable.

Consent for publication

All the authors provide consent for publication.

Competing interests

The authors declare no competing interests.