Rheological and irreversibility analysis of ternary nanofluid flow over an inclined radiative MHD cylinder with porous media and couple stress

Nomenclature

g

Gravity acceleration (m/s²)

v w

Mass surface velocity (m/s)

L

Characteristic length

T

Ternary-hybrid nanofluid temperature (K)

T ∞

Ambient temperature (K)

β

Variable electrical conductivity parameter

ϕ 2

Nanoparticle volume fraction of MgO

Ec

Eckert number

S

Mass suction parameter

Cf x

Coefficient of skin friction

Pr

Prandtl number

α 1

Temperature difference parameter

m

Shape factor

γ

Curvature parameter

λ

Stretching cylinder parameter

A

Velocity slip parameter

ε

porosity parameter

μ

Dynamic viscosity of the fluid (kg/ms)

s 2

Magnesium oxide’s solid components

MHD

magnetohydrodynamic

Thnf

Ternary hybrid nanofluid

2D

Two-dimensional

c p

Specific heat at constant pressure (J kg⁻¹ K⁻¹)

ρ

Density (kg/m³)

β T

Thermal expansion (K⁻¹)

∞

At the free stream region

u , v

Components of velocity (m/s)

c 1 , c 2

Constants

T 0

Reference temperature

T w

Surface temperature of sphere (K)

B 0

Magnetic field

ϕ 1

Nanoparticle volume fraction of Ag

ϕ 3

Nanoparticle volume fraction of Fe₃O₄

δ

Mixed convection parameter

M

Magnetic field parameter

Nu x

Nusselt number

Br

Brinkmann number

u e

Free stream velocity

k

Thermal conductivity of the fluid (W/mK)

ω

Angle inclination parameter

Rd

Thermal radiation parameter

B

Thermal slip parameter

Cs

Couple stress parameter

s 1

Silver’s solid components

s 3

Iron oxide’s solid components

ODEs

Ordinary differential equations

PDEs

Partial differential equations

Ψ

Stream function

ν

Kinematic viscosity (m² s⁻¹)

σ

Electrical conductivity (ω/m)

w

At wall

ψ

Shape factor sphericity

1 Introduction

The geometric configuration of nanoparticles markedly affects the viscosity and thermal conductivity of nanofluids. Because they have more surface area and contact friction, non-spherical nanoparticles like blades, platelets, or bricks often make it harder for fluids to flow than spherical nanoparticles. This is called higher viscosity. These elongated or flat geometries improve thermal conductivity by establishing more extensive percolation networks and channels for heat transmission, hence enhancing thermal performance. Spherical nanoparticles decrease viscosity while slightly enhancing heat conductivity, providing a balanced trade-off. So, picking the right form factor is important for getting the best balance between viscosity and thermal conductivity [1]. Jiang et al. [2] described the dynamics of nanofluids influenced by thermo-capillary convection resulting from five distinct nanoparticle shapes: sphere, blade, brick, cylinder, and platelet. It was found that thermo-capillary convection was strongest in nanofluids with spherical nanoparticles and weakest in those with platelet-shaped nanoparticles. Sreenivasulu and Bhuvana Vijaya [3] investigated the influence of many features, including changing viscosity, chemical reactions, heat radiation, and the form factor of nanoparticles, on the flow of a hybrid nanofluid (H₂O + MWCNT + CuO) over a flat plate.

Mixed convection flow, which is used in various technical and industrial applications like heat exchangers, solar collectors, and electronic cooling systems, happens when forced and natural convection forces work together. When buoyancy effects from temperature gradients mix with externally imposed flow fields to produce complicated thermal and fluid dynamic behavior, this flow regime becomes important [4]. The Richardson number often determines whether forced or free convection is more important, which helps to understand how strong buoyancy is compared to inertial forces. Prediction of heat and mass transfer rates in systems where thermal stratification and flow recirculation may arise depends on accurate modelling of mixed convection. Ragulkumar et al. [5] used a finite difference scheme to study how free convection nanofluid moves around an upright cone while considering magnetic fields. This mathematical model investigates magnetohydrodynamic (MHD), viscous dissipation, radiation, chemical reactions, and suction/injection processes using the heat and mass flow pattern. Ragulkumar et al. [6] presented a computational methodology for analyzing the two-dimensional boundary layer heat and mass transfer of water-based nanofluids over a vertical cone permeated by porous media, incorporating heat generation/absorption, thermal radiation, chemical reactions, the Soret–Dufour number, viscous dissipation, and a magnetic field. Ramesh et al. [7] conducted modelling and calculations to investigate the mixed convective flow of ND-Co₃O₄/EG hybrid nanoliquid via a vertical porous cylinder. The flow analysis and formulation include slip effects and the influences of homogeneous and heterogeneous reactions.

A new class of improved working fluids called ternary hybrid nanofluids suspends three distinct types of nanoparticles in a base fluid. The unique thermal and rheological features of the nanoparticles make this combination outperform typical and hybrid nanofluids in heat transfer. Incorporating a third kind of nanoparticle significantly enhances the fluid’s stability, improves its thermal conductivity, and increases its heat retention capacity. Because of this, it is ideal for use in biological applications, cooling technologies, and energy systems [8]. A recent study by Ullah et al. [9] investigated the heat transfer properties of nanofluids, ternary hybrids, and hybrids containing nanoparticles of copper (Cu), silver (Ag), and alumina (Al₂O₃) between two elastic spinning discs maintained at a constant distance from one another. The study evaluated how thermal radiation, a heat source, Joule heating, and Arrhenius activation energy affected the new material’s flow stability and temperature equations. Incorporating the influences of melting, mixed convection, Thomson, and Troian slip, Madhukesh and Ramesh [10] compared two base liquids (kerosene/water) with ternary nanoparticle transport over a stretched vertical cylinder. Jawad et al. [11] look into the Cattaneo-Christov model of Electromagnetohydrodynamic tri-hybrid nanofluid (THNF) flow with gyrotactic microorganisms, Arrhenius activation energy, and natural convection in wedge and cone shapes. Using a medium with varying permeability, Lone et al. [12] examined three-dimensional flows of a ternary hybrid nanofluid on a stretchable sheet. They also considered heat source and thermal radiation. Although they hold significant promise, the complex interaction among particle concentration, morphology, and base fluid characteristics needs more research to fully realize their advantages. Khan et al. [13] want to understand heat transmission better, which might affect many physical processes. Researchers are studying how the tetra hybrid Tiwari and Das nanofluid models affect fluid dynamics. By studying nanoparticle-rich fluids’ complicated behaviors, researchers hope to find potential uses. This study looks at how a sodium alginate-based Casson tetra hybrid nanofluid moves through porous materials and between two flat surfaces, including the effects of couple stress. The work by Madhukesh et al. [14] investigated the thermal and mass transport properties of two different forms of ternary hybrid nanofluids, using water and kerosene as the base liquids inside a vertically orientated cylinder. Their study was driven by the need to improve thermal performance in energy, industrial, and biomedical applications where flow control and thermal management are essential. The study integrated the processes of melting phenomena, heat radiation, and chemical reaction.

It is important to use thermal radiation when studying heat transfer, especially when nanofluid and ternary hybrid nanofluid flows are happening at high temperatures. The radiative heat flow, dictated by the Rosseland approximation, substantially affects the energy equation by including non-linear phenomena that augment heat transport [15]. The synergistic interaction among numerous nanoparticles in ternary hybrid nanofluids enhances thermal conductivity, thereby intensifying radiative heat transfer effects. Ahmad et al. [16] investigated the oblique flow and heat transfer characteristics of a micropolar ternary hybrid nanofluid across a lubricated surface, including a convective boundary condition and thermal radiation. Obalalu et al. [17] use the Galerkin-weighted residual approach to solve the modelled equations and conduct a comprehensive analysis of the energy transition to improve heat transport mechanisms in renewable energy systems using ternary nanofluid flow. The flow problem also looks at the effect of non-Fourier heat flux, multiple slips, thermal radiation, and MHDs on surfaces that are very thin. The work by Wahid et al. [18] conducted a computational investigation of the unsteady flow of a water-based ternary hybrid nanofluid (alumina, copper, and titanium dioxide) across a permeable, biaxial, shrinking sheet while accounting for the effects of heat radiation. Jamrus et al. [19] studied axisymmetric flow of a nonlinearly extending or shrinking disc using Al₂O₃–Cu–TiO₂/H₂O ternary hybrid nanofluid. The study addresses convective boundary conditions and thermal radiation. Guedri et al. [20] looked into the convective flow of a two-dimensional ternary hybrid nanofluid over a nonlinear stretching sheet. Adding Brownian motion and thermophoresis to the energy and mass equations stabilizes the new composition’s flow and thermal characteristics. The energy equation includes heat absorption, production, and nonlinear thermal radiation. The study by Khan et al. [21] seeks to provide an examination of irreversibility in the unsteady flow of viscous fluid across a vertical flat plate characterized by a ramping wall temperature and arbitrary wall shear stress while considering the effects of thermal radiation. The Darcy–Forchheimer and time-dependent flow of a Casson hybrid nanofluid composed of single-walled and multi-walled carbon nanotubes across a Riga plate subjected to velocity slip was studied by Senthilvadivu et al. [22] in their study of water and glycerin. The nonlinear thermal radiation and viscous dissipation effects are included in the energy equation. The study by Khan et al. [23] seeks to investigate the impact of wall shear stress on dust-laden fluids exhibiting fluctuating flow. The analysis is conducted between two non-conducting parallel plates subjected to thermal radiation, a magnetic field, and mixed convection. The fluid flow is induced by heat transfer. Each dust particle is assumed to be uniformly dispersed and spherical within the base fluid. The perturbation solution is derived using the Poincare–Lighthill perturbation technique. Abbas et al. [24] looked at how heat radiation affects both uniform and mixed chemical substances, along with the movement of a non-Newtonian fluid over a curved surface that expands exponentially. The study by Madhu et al. [25] investigated the effects of activation energy, quadratic, nonlinear, and linear thermal radiation on the flow of hybrid nanofluids over a cylinder. The results show that, in comparison to linear and non-linear thermal radiation situations, quadratic thermal radiation exhibits the least uniform distribution of temperatures.

For modern energy systems, ternary hybrid nanofluids’ thermal and flow characteristics are dynamic due to their electrical conductivity fluctuation. Because of the temperature-dependent conductivity and synergistic effects of several nanoparticles, ternary hybrid nanofluids exhibit complex interactions not seen in ordinary fluids. Since this variance affects the flow’s electromagnetic properties, heat transfer rate, and thermal conductivity, accurate projections need extensive research [26,27]. Understanding these processes helps nanofluid-based systems perform better in cooling, energy harvesting, and electronic thermal management. Afridi et al. [28] performed a numerical study of entropy generation in the dissipative, unstable oscillatory flow of nanofluids, distinguished by variable electrical conductivity and magnetic heating influences. Mushahary and Ontela [29] examined how electrical conductivity variations affect the irreversibility of a couple-stress hybrid nanofluid flow in a vertical porous channel. This is quadratically mixed convective flow. Reddy et al. [30] concentrate on how temperature affects electrical current and top-layer nanofluid mobility. Convective heat and mass transport affect electrical conductivity, viscosity, thermo diffusion, thermal radiation, and absorption. Copper and alumina–water nanofluids fill a cylindrical annulus.

Examining entropy formation in ternary hybrid nanofluid flow is crucial for evaluating the thermodynamic efficiency of sophisticated heat transfer systems. Production of entropy, which has a significant impact on system performance, is the result of the interaction between nanoparticles, heat conduction, and viscous dissipation. Understanding these concepts reduces irreversibility’s and boosts thermal efficiency, optimizing energy systems. A study by Hayat et al. [31] looked at how an incompressible magnetized THNF moved over a curved, porous surface. Also included are the effects of entropy formation on Darcy–Forchheimer medium and natural convection heat flow. In their study, Gireesha and Anitha [32] investigated the effects of quadratic heat radiation on convective ternary hybrid nanoliquid flow in an upright microchannel. They examine the effects on thermal performance of the Darcy-Forchheimer rule, heat source coefficient, and no-slip condition. Thermodynamic and entropy studies of oblique channel flows of ternary couple stress hybrid nanoliquids in water and couple stress hybrid nanoliquids in water were the main areas of interest for Gireesha and Anitha [33]. Their study examined the couple’s stress ternary liquid by analyzing the effects of heat flow, porous medium, buoyant force, magnetic field, and convective state. Regarding the melting point in a ternary-hybrid nanoliquid flowed under the influence of a stretched surface, Hayat et al. [34] conducted an investigation. Radiation, heat production, and heat dissipation make up the energy equation. The pace at which entropy is generated is studied. Hayat et al. [35] investigated the flow of an incompressible, magnetized, ternary nanofluid in a Darcy–Forchheimer medium, including spontaneous convection over a curved surface. These result shows that when the levels of unsteadiness and variable porosity parameters increase, entropy production improves. In response to the growing need for better heat transfer in thermal systems, Liang et al. [36] came up with a new way to use a ternary nanofluid to improve heat transfer efficiency and reduce entropy formation. Additionally, a constant pressure gradient, a magnetic field, Joule heating, and thermal radiation all have an impact on the flow dynamics. A study by Khan et al. [37] built on the concept of entropy creation in a dusty tetra hybrid nanofluid that moves freely between two vertical, parallel plates that are not moving. Both buoyant force and free convection generate the flow, which in turn transfer’s heat. Ali et al. [38] investigated the convective flow of cross fluid containing carboxymethyl cellulose in water across a stretched sheet with convective heating. The examination investigates minimizing entropy formation. The primary objective of Anitha and Gireesha [39] work is to examine the flow characteristics and irreversibility analysis of water-based CuO–Fe₃O₄ hybrid nanofluid and CuO–Fe₃O₄–Ag ternary hybrid nanofluid inside an inclined microchannel. The impact of the heat source coefficient, Hall current, nonlinear radiative heat flux, and slip condition on the flow of ternary nanofluid in a channel has been examined, taking into account buoyant force and the Buongiorno model.

2 Problem formulation

This study examines the effects of buoyancy on the stagnation point flow of an Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluid across an inclined cylinder in a steady, incompressible MHD mixed convection situation, as shown in Figure 1. Along the axial and radial directions of the inclined cylinder, respectively, are the coordinates (x, r) formed. The problem formulation is based on the following assumptions:

• The thermophysical characteristics of the ternary hybrid nanofluid are regarded as uniform, with the exception of electrical conductivity.

• We assume that the surface temperature of the cylinder, denoted as T w = T ∞ + T 0 x L , where L is the characteristic length.

• The mass transfer velocity is articulated as v w = − S R r c 2 ν f L , where S > 0 implies suction.

• The surface velocity of the cylinder is expressed as u w = c 2 x L , with elongation occurring when c 2 > 0 and contraction when c 2 < 0 . The free stream velocity is defined as u e = c 1 x L , where c 1 > 0 .

• The study incorporates Joule heating, thermal radiation, velocity and thermal slip conditions, couple stress effects, porous media, and viscous dissipation.

• The influence of nanoparticle shape factors (spherical and non-spherical) for ternary hybrid nanoparticles is included.

• Entropy generation is also considered in the analysis.

Figure 1

Physical conformation of an inclined cylinder.

The governing equation in dimensional form appropriate for modelling the transport phenomenon is expressed as follows (see [40]):

The following form describes the boundary conditions (see [40]):

This research uses the variables u and v to represent the velocity components along the x -axis and r -axis, respectively. The variables g and ω stand for the gravitational acceleration and angle of inclination, respectively. B 0 is uniform magnetic field. Along with T representing the temperature of the ternary hybrid nanofluid, the parameters L 1 , K , and K 1 = K 0 x L stand for the permeability, velocity slip factor, and thermal slip factor of the porous medium, respectively.

To address the variance in electrical conductivity in the aforementioned equation, the below formula may be used (see [8]):

where β represent a positive constants.

Table 1 enumerates the thermophysical properties of the base fluid (H₂O) and the nanoparticles (Ag, MgO, and Fe₃O₄). Table 2 delineates the correlation parameters for the Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluid, including the form factor m = 3 / ψ , with ψ denoting nanoparticle sphericity. Table 3 presents the coefficients A 1 and A 2 used in the effective viscosity model for non-spherical particles, reflecting the influence of various geometries. It also includes dynamic viscosity correlations for both spherical and non-spherical nanoparticle geometries.

Table 3

Assessment of nanoparticle morphologies and associated coefficients for viscosity augmentation (see [8])

Nanoparticles shapes	Sphere	Bricks	Platelets	Cylinder	Blade
Shape structure
Sphericity ψ	1	0.81	0.52	0.62	0.36
Shape factor m	3	3.7	5.7	4.8	8.6
A 1		1.9	37.1	13.5	14.6
A 2		471.4	612.6	904.4	123.3

For similarity solutions, the application of a similarity transformation produces the following outcomes (refer to [40]):

Stream function is denoted by Ψ , and u = 1 r ∂ Ψ ∂ r and v = − 1 r ∂ Ψ ∂ x , which satisfies equation (1).

The application of the similarity variables delineated in equation (6) inherently fulfils equation (1) and converts equations (2)–(4) into the following set of ordinary differential equations (ODEs):

In the above equations, the Prandtl number is Pr = μ f C p k f , M = L σ f B 0 2 ρ f c 1 represent the magnetic parameter, γ = 1 R ν f L c 1 is the curvature parameter, and ω is the angle inclination. Ec = u e 2 ( T w − T ∞ ) ( C p ) f = c 1 2 x 2 ( C p ) f T 0 x 2 l 2 = c 1 2 l 2 ( C p ) f ( T 0 ) is the Eckert number. λ = c 1 c 2 , where λ > 0 designates a stretched cylinder, S is the mass suction, δ = Gr R e 2 = g β f T 0 x 2 l 2 c 1 2 x 2 / ν f = g β f ( T 0 ) c 1 2 l 2 / ν f represents the mixed convection parameter. A = L 1 c 2 ν f is the velocity slip parameter, Rd = 4 σ * T ∞ 3 3 k * k f is the thermal radiation parameter and B = K c 2 ν f is the thermal slip parameter, ε = ν f L c 1 K 0 is the porosity parameter and Cs = c 1 η 0 ν f is the couple stress parameter.

3 Engineering quantities of interest

The skin friction coefficient, which denotes flow resistance, and the local Nusselt number, which signifies the rate of heat transfer, are critical engineering metrics for assessing flow resistance and heat transfer efficiency at the surface. These quantities are represented numerically as follows:

Through similarity transformation, the aforementioned equations may be expressed as follows:

where local Reynold’s number is labeled as Re x = u e x υ f = c 1 x 2 υ f .

4 Numerical procedure for solution

Equations (7)–(9) were numerically resolved in MATLAB using the boundary value problem solver bvp4c. This solver uses the collocation formula from the three-stage Lobatto IIIa method to guarantee a continuous solution with fourth-order accuracy. An accurate initial solution estimate and an optimized boundary layer thickness for the problem parameters are the two most crucial inputs for the solver. The ‘bvp4c’ function in MATLAB is very beneficial for addressing boundary value issues because of its:

1. Adaptive mesh refinement automatically modifies the node distribution to meet a defined tolerance, enhancing accuracy in areas with steep slopes.

2. Proficiency in effectively managing stiff and non-linear systems, particularly prevalent in fluid mechanics and heat transfer issues represented by coupled ODEs.

We reformulated the governing equations using the system of first-order ODEs to facilitate the numerical solution. The computations were done with a tolerance error of 10⁻⁶. Here are the steps that outline the process:

Table 4 presents the numerical results derived from the current investigation and juxtaposes them with the findings provided by Zainal et al. [41] and Rafique et al. [42]. This comparison validates the precision and dependability of the present computational findings by contrasting them with known data from prior investigations.

5 Results and discussion

A sequence of numerical simulations has been performed to examine the physical behavior of flow, heat transfer, and entropy formation. The results are shown via charts and tables, emphasizing the impact of different embedded factors on the governing parameter values. The following are the ranges of the considered parameters: velocity slip ( A ) : 0.0 < A ≤ 1.0 , magnetic ( M ) : 0.0 < M ≤ 1.0 , mixed convection parameter ( δ ) : 0.0 ≤ δ ≤ 4.0 , mass suction ( S ) : 1.0 ≤ S ≤ 2.5 , thermal slip parameter ( B ) : 0.0 < B ≤ 1.0 , Eckert number parameter ( Ec ) : 0.0 < Ec ≤ 1.0 , porous media parameter ( ε ) : 0.0 < ε ≤ 2.0 , stretching parameter ( λ = 0.5 ) , couple stress parameter ( Cs ) : 0.5 < Cs ≤ 2.5 , variable electrical conductivity parameter ( β ) : 0.0 < β ≤ 2.0 , nanoparticle volume fraction of Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluid parameter ( ϕ 3 ) : 0.0 ≤ ϕ 3 ≤ 0.03 , ϕ 1 = ϕ 2 = 0.01 , angle of inclination ( ω ) : 0 ° < ω ≤ 90 ° and Pr = 6.2 .

Table 5 displays the numerical findings for skin friction C f Re x 1 / 2 of ternary hybrid nanofluids of various shapes (cylindrical, spherical, brick, platelet, cylinder) in relation to various technical factors.

5.12 Impression of thermal radiation parameter Rd on Nu x Re x − 1 / 2 of distinct shape factor of ternary hybrid nanofluid

The Nu x Re x − 1 / 2 gets bigger in ternary Ag + MgO + Fe₃O₄/H₂O nanofluid flow as the thermal radiation parameter (Rd) goes from 0.2 to 1.0. This is because the system’s radiative heat transfer becomes more efficient. The net heat transfer rate from the cylinder’s surface to the fluid around it rises as (Rd) grows because radiative energy exchange starts to have a bigger impact. The total efficiency of convective heat transfer is enhanced because the surface temperature gradient is amplified by this increased radiative heat flow. The effects of tilting and thermal radiation work together to make the redistribution of energy inside the boundary layer even better. This results in larger Nusselt numbers and a faster heat transfer rate.

Figure 2(a) and (b) validate the effect of varying mass suction parameter S on the skin friction C f Re x 1 / 2 and Nusselt number Nu x Re x − 1 / 2 of ternary hybrid nanofluid with different shape factors. Both skin friction C f Re x 1 / 2 and the Nusselt number Nu x Re x − 1 / 2 increase with higher values of the parameter S . In ternary Ag + MgO + Fe₃O₄/H₂O nanofluid flow, the C f Re x 1 / 2 and Nu x Re x − 1 / 2 both upsurge with higher values of the mass suction parameter (S = 1.0, 1.5, 2.0, 2.5). This is because suction influences the boundary layer dynamics, which in turn removes low-momentum fluid particles from the near-wall region, thinning the momentum and thermal boundary layers. This thinning enhances the velocity gradients at the surface, leading to higher magnitudes of skin friction. The decrease in thermal boundary layer thickness simultaneously elevates the temperature differential at the cylinder’s surface, hence enhancing heat transfer and increasing the Nusselt number. The cumulative impact of these events underscores.

Figure 2

Impact of S = 1.0, 1.5, 2.0, 2.5 on (a) C f Re x 1 / 2 and (b) Nu x Re x − 1 / 2 of ternary hybrid nanofluid.

Figure 3(a) and (b) show how skin friction and Nusselt number over inclined cylinder vary with Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluid volume fraction ϕ 3 of a different form. These values reveal that Nusselt number rises while skin friction decreases for form factor. The decrease in skin friction and rise in Nusselt numbers that happen at the same time are caused by higher nanoparticle volume fraction parameters (ϕ ₃ = 0.005, 0.01, 0.015, 0.02) in Ag + MgO + Fe₃O₄/H₂O nanofluid flow over an inclined cylinder. This is because nanoparticles have unique effects on flow and thermal properties. A higher level of ϕ 3 increases the effective viscosity of the nanofluid, which lowers the speed differences near the cylinder surface and decreases skin friction. A higher number of nanoparticles significantly improves thermal conductivity, leading to better heat transfer and a steeper temperature difference at the surface, which is indicated by a higher Nusselt number. This effect underscores the dual impact of nanoparticle volume fraction on momentum and heat transfer in the flow.

Figure 3

Influence of ϕ ₃ = 0.005, 0.01, 0.015, 0.02 on (a) C f Re x 1 / 2 and (b) Nu x Re x − 1 / 2 of distinct shape factor of ternary hybrid nanofluid.

Figure 4(a) exemplifies the consequence of the couple stress parameter (Cs) on the velocity profile f ′ ( η ) of a Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluid with a different form, whereas Figure 4(b) shows the effect of the porous media parameter ( ε ) on the same profile. The f ′ ( η ) profile of a Ag + MgO + Fe₃O₄/H₂O nanofluid moving across an inclined cylinder shows unique features that are affected by changing the coupling stress and the parameters of the porous medium. The f ′ ( η ) profile gets flatter as the couple stress parameter (Cs = 1.0, 1.3, 1.5, 1.7) goes up. This is because higher couple stress increases resistance in the fluid, which is caused by microstructural interactions inside the fluid that stop it from moving. In contrast, the f ′ ( η ) profile rises as the values of the porous media parameter (ε = 0.5, 1.0, 1.5, 2.0) climb. This is because the porous medium experiences less drag resistance, leading to increased permeability and smoother flow. This two-pronged effect highlights the complementary roles of pair stress and porous medium in controlling the fluid’s momentum transfer.

Figure 4

(a) Effect of Cs = 1.0, 1.3, 1.5, 1.7 on f ′ ( η ) , and (b) effect of ε = 0.5, 1.0, 1.5, 2.0 on f ′ ( η ) .

5.13 Effect of curvature parameter γ on velocity f ′ ( η ) and temperature θ ( η ) profiles of distinct shape ternary hybrid nanofluid

Figure 5(a) and (b) exhibit the impact of the parameter ( γ ) on the velocity f ′ ( η ) and temperature θ ( η ) profiles of Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluid flow across inclined cylinders of different forms. While Figure 5(b) shows the same patterns at all levels of curvature, Figure 5(a) shows the dual behavior. The f ′ ( η ) profile of Ag + MgO + Fe₃O₄/H₂O nanofluid flow over an inclined cylinder demonstrates dual behavior with increasing parameter (γ = 0.5, 1.0, 1.5, 2.0). As curvature rises, f ′ ( η ) decreases due to centripetal pressures and physical restrictions that impede fluid motion. Surface momentum transfer and f ′ ( η ) rise when fluid flow matches geometry at higher curvature levels. Concurrently, as the curvature increases, the θ ( η ) profile rises. This is because a higher curvature allows the thermal boundary layer to expand, which in turn improves the heat conduction from the surface to the fluid. Both effects show how fluid dynamics and thermal behavior interact intricately in curved geometries.

Figure 5

Impression of γ = 0.5, 1.0, 1.5, 2.0 on (a) f ′ ( η ) and (b) θ ( η ) .

5.14 Consequence of magnetic parameter M on f ′ ( η ) and Eckert number Ec on θ ( η ) profiles of distinct shape ternary hybrid nanofluid

Figure 6(a) demonstrates the influence of the magnetic parameter ( M ) on the velocity f ′ ( η ) profile, whereas Figure 6(b) shows the effect of the (Ec) on the temperature θ ( η ) profile of ternary Ag + MgO + Fe₃O₄/H₂O nanofluid flow with different shapes of nanoparticles. As the parameter values (M = 0.1, 0.3, 0.5, 0.7) increase, the f ′ ( η ) profile of the Ag + MgO + Fe₃O₄/H₂O nanofluid flow over an inclined cylinder becomes steeper. This is because the magnetic field produces a Lorentz force, which acts as a driving force, propelling the fluid in the direction of flow. In contrast, the θ ( η ) profile increases with higher Eckert number values (Ec = 0.1, 0.3, 0.5, 0.7), as the Eckert number represents the conversion of kinetic energy into thermal energy through viscous dissipation, which enhances the fluid’s thermal energy and raises its temperature. These findings highlight the combined effect of electromagnetic forces and viscous dissipation on the fluid and thermal dynamics inside the system.

Figure 6

(a) Impression of M = 0.1, 0.3, 0.5, 0.7 on f ′ ( η ) , and (b) impression of Ec = 0.1, 0.3, 0.5, 0.7 on θ ( η ) .

5.15 Effect of nanoparticle volume fraction ( ϕ 3 ) on velocity f ′ ( η ) and temperature θ ( η ) profiles of distinct shape factor

Figure 7(a) and (b) portray the stimulus of the nanoparticle volume fraction ( ϕ 3 ) on the velocity f ′ ( η ) and temperature θ ( η ) profiles of an Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluid traversing an inclined cylinder of various geometries. Rising parameters (ϕ ₃ = 0.005, 0.01, 0.015, 0.02) cause the f ′ ( η ) profile of Ag + MgO + Fe₃O₄/H₂O nanofluid flow over an inclined cylinder to decrease. This is because the ternary nanofluid’s increased density and viscosity increase flow resistance and decrease fluid velocity. Conversely, the temperature θ ( η ) profile increases with higher nanoparticle volume fractions, as the addition of nanoparticles enhances the fluid’s thermal conductivity, leading to improved heat transfer and a more uniform distribution of thermal energy within the system. The interplay between momentum and thermal transport demonstrates the dual role of nanoparticles in altering fluid dynamics and heat transfer characteristics.

Figure 7

Impression of ϕ ₃ = 0.005, 0.01, 0.015, 0.02 on (a) f ′ ( η ) and (b) θ ( η ) .

5.16 Effect of mass suction parameter ( S ) on velocity f ′ ( η ) and temperature θ ( η ) profiles of distinct shape ternary hybrid nanofluid

The impact of the mass suction parameter ( S ) on the velocity f ′ ( η ) and temperature θ ( η ) profiles of the Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluid is shown in Figure 8(a) and (b). In Ag + MgO + Fe₃O₄/H₂O nanofluid flow over an inclined cylinder, the f ′ ( η ) profile increases with higher mass suction parameter values (S = 1.0, 1.5, 2.0, 2.5). Eliminating slower fluid layers near the surface decreases boundary layer thickness and boosts momentum transfer. As suction rises, fluid evacuation lowers thermal boundary layer thickness, resulting in a drop in θ ( η ) profile. Because less heat can build up at the surface, the temperature gradient steepens, lowering fluid temperatures. This shows that suction significantly alters thermal boundary layers and momentum.

Figure 8

Impact of S = 1.0, 1.5, 2.0, 2.5 on (a) f ′ ( η ) and (b) θ ( η ) .

5.17 Effect of mixed convection parameter ( δ ) and radiation on temperature profiles of distinct shape ternary hybrid nanofluid

Both Figure 9(a) and (b) show how the mixed convection parameter ( δ ) and the thermal radiation parameter (Rd) affect the temperature profile of the Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluid, respectively. The temperature distribution of a ternary hybrid nanofluid traversing an inclined cylinder diminishes as the mixed convection parameter (δ = 1.0, 1.5, 2.0, 2.5) increases. Buoyancy-driven convective cooling facilitates the dissipation of heat from the surface. At values of Rd = 0.2, 0.5, 0.7, and 1.0, the temperature profile rises with increasing thermal radiation. Radiation increases the total thermal energy inside the boundary layer by energizing the fluid. The interplay between convection-driven cooling and radiation-induced heating highlights how these two processes affect the thermal dynamics of the system in different ways.

Figure 9

(a) Bearing of δ = 0.5, 1.0, 1.5, 2.0 on θ ( η ) , and (b) bearing of Rd = 0.2, 0.4, 0.6, 0.8 on θ ( η ) .

Figure 10 displays the effects of several parameters on the generation of entropy for ternary hybrid nanofluids of different shapes over an inclined cylinder. Figure 10(a) shows the effects of the Brinkmann number parameter (Br), Figure 10(b) shows the effects of the porous media parameter ( ε ) , Figure 10(c) shows the effects of the nanoparticle volume fraction parameter ( ϕ 3 ) , and Figure 10(d) shows the effects of the radiation parameter (Rd). The entropy generation profile of ternary hybrid nanofluid flow over an inclined cylinder escalates with elevated values of the (Br), ( ε ), ( ϕ 3 ), and parameter (Rd). An elevation in (Br) indicates augmented viscous dissipation effects, thereby directly enhancing thermal irreversibilities within the system. Likewise, an increase in ( ε ) amplifies flow resistance inside the porous medium, resulting in heightened frictional entropy formation. Increased nanoparticle volume fractions ( ϕ 3 ) enhance thermal conductivity, thereby elevating heat transfer rates and subsequently increasing entropy production. Furthermore, an augmentation in (Rd) enhances radiative heat flux, resulting in greater temperature gradients that further facilitate entropy generation. The cumulative effect highlights the preeminence of viscous, thermal, and radiative irreversibility’s within the flow system.

Figure 10

(a) Impact of Br = 0.2, 0.4, 0.6, 0.8 on N s , (b) impact of ε = 0.5, 1.0, 1.5, 2.0 on N s , (c) impact of ϕ ₃ = 0.005, 0.01, 0.015, 0.02 on N s , and (d) impact of Rd = 0.2, 0.4, 0.6, 0.8 on N s .

6 Conclusions

This work examined the behavior of mixed convection stagnation point flow of Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluid across an inclined cylinder. The influence of magnetic fields, porous media, coupling stress, velocity and thermal slip conditions, mass suction, viscosity dissipation, Joule heating, variable electrical conductivity, form factor, and thermal radiation. We numerically calculated and visually and tabulatively displayed the impacts of the dimensionless parameters on the entropy production, skin friction coefficient, and heat transfer rate at the surface. The following is a list of the summary findings:

• The influence of shape factor on the flow of Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluids across an inclined cylinder is examined, revealing that platelet shapes surpass other geometries for Nusselt number and velocity profiles.

• The velocity profile of Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluid flow with varying form factors diminishes as the coupling stress and nanoparticle volume fraction parameters grow, whereas the velocity profile enhances with rising mass suction, magnetic, and porous medium parameters. The curvature parameter exhibits dual behavior in the velocity profile.

• The temperature profile of Ag + MgO + Fe₃O₄/H₂O ternary hybrid nanofluid flow with varying shapes diminishes with elevated mixed convection and mass suction parameters, while it increases with higher values of radiation, Eckert number, curvature, and nanoparticle volume fraction parameters.

• An Ag + MgO + Fe₃O₄/H₂O nanofluid flow with different shaped components has an entropy generation profile that grows as the Brinkmann number, porous media, nanoparticle volume percentage, and thermal radiation parameters rise.

• The skin friction of Ag + MgO + Fe₃O₄/H₂O nanofluid flow with varying morphologies escalates with higher mass suction, curvature, porous medium, and magnetic parameters, while it diminishes with rising velocity slip parameter, coupling stress, and nanoparticle volume fraction parameters.

• An Ag + MgO + Fe₃O₄/H₂O nanofluid flow with different shapes has an increasing Nusselt number for large values of mass suction, inclination angle, nanoparticle volume fraction, curvature, and radiation parameter, and a decreasing value for large values of thermal slip, Eckert number, mixed convection, and variable electrical conductivity.

7 Future suggestions

Subsequent investigations may expand upon this work by

1. Examining the effects of time-dependent flow and transient heat transfer in analogous systems to investigate unsteady-state behavior.

2. Examining various geometries, including conical and elliptical surfaces, to evaluate the influence of form on heat transport and entropy creation.

3. Integrating hybrid heating methods, like variable surface heating or non-uniform magnetic fields, to improve thermal efficiency.

4. Investigating the interplay of chemical processes or phase change events in the ternary hybrid nanofluid to enhance industrial application.

5. Making use of optimization or machine learning methods to predict the right form parameters and fluid properties to improve heat transport and decrease entropy generation.