Stability analysis of unsteady ternary nanofluid flow past a stretching/shrinking wedge

Yun Ouyang; Md Faisal Md Basir; Kohilavani Naganthran; Ioan Pop

doi:10.1515/phys-2025-0148

Article Open Access

Stability analysis of unsteady ternary nanofluid flow past a stretching/shrinking wedge

Yun Ouyang , Md Faisal Md Basir , Kohilavani Naganthran and Ioan Pop

Published/Copyright: May 9, 2025

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From the journal Open Physics Volume 23 Issue 1

Abstract

Slit die extrusion depends highly on fluid temperature and flow properties, which play a crucial role in determining material quality. This research aims to enhance product quality in extrusion processes by deriving a mathematical model from the extrusion process, ensuring practical relevance to industrial applications. The study focuses on the stability analysis of unsteady ternary nanofluid flow past a stretching/shrinking wedge, incorporating viscous dissipation and Joule heating. A key novelty of this work lies in identifying the critical values for the existence of dual solutions and conducting a comprehensive stability analysis. The findings reveal that the first solution is stable, whereas the second is unstable. Critical values are determined using the boundary value problem solver using 4th-order collocation method function in Matlab, and the effects of key parameters – such as the wedge parameter, Eckert number, suction/injection parameter ( S ), and hybridity – are analyzed through graphical representations. Results show that for a shrinking wedge, the skin friction coefficient and Nusselt number increase with higher values of the unsteadiness parameter, nanoparticle volume fraction, and S . When λ > − 3.23 (shrinking wedge), the ternary nanofluid demonstrates superior thermal transfer compared to binary and mono nanofluids. This study provides a critical foundation for validating advanced models and optimizing heat transfer performance in industrial processes, paving the way for enhanced applications in extrusion and thermal management systems.

Keywords: ternary nanofluid; unsteady flow; wedge flow; viscous dissipation; ohmic heating; stability analysis

Nomenclature

HNF: binary hybrid nanofluid
NF: mono hybrid nanofluid
THNF: ternary hybrid nanofluid
ODEs: ordinary differential equations
PDEs: partial differential equations
BVPs: boundary value problems
: Roman letters
A: unsteadiness parameter
a: positive constant
c: non-negative constant
C f: skin friction coefficient
C p: specific heat capacity (J kg⁻¹ K⁻¹)
k: thermal conductivity (W mK⁻¹)
l: characteristic length (m)
m: wedge angle parameter
Nu x: Nusselt number
Pr: Prandtl number
Re x: Reynolds number
S: suction/injection parameter
T: temperature (K)
T ∞: temperature far away from the sheet (K)
u , v: velocity component in the x and y directions (m s − 1 )
U w: stretching surface velocity (m s − 1 )
v w: mass flux velocity (m s − 1 )
u e: ambient velocity (m s − 1 )
ρ C p: heat capacitance ( J m − 3 K − 1 )
: Greek symbols
β: dimensionless wedge gradient parameter
η: dimensionless similarity variable
λ: stretching/shrinking parameter
μ: dynamic viscosity (N s m − 2 )
ν: kinematic viscosity ( m 2 s − 1 )
ϕ 1: cupric nanoparticle volume fraction
ϕ 2: magnesia nanoparticle volume fraction
ϕ 3: titania nanoparticle volume fraction
ρ: density (kg m − 3 )
θ: dimensionless temperature
: Subscripts
f: fluid
hnf: hybrid nanofluid
nf: nanofluid
thnf: ternary hybrid nanofluid
1: cupric
2: magnesia
3: titania
w: wall

1 Introduction

Slit die extrusion requires precise control of the fluid's thermal and flow properties to achieve optimal product characteristics and enhance processing efficiency. In this context, the addition of ternary hybrid nanofluids, which comprise three types of nanoparticles dispersed in a base fluid, plays a crucial role. Nanoparticles improve the fluid's thermal conductivity and viscosity, enabling better heat transfer and flow behavior during the extrusion process. For instance, Said et al. [1] studied ternary nanofluids composed of rGO- Fe 3 O 4 - Ti O 2 hybrid nanocomposites in ethylene glycol to determine the optimal nanofluid composition for maximizing heat transfer. Their findings, supported by machine learning analysis, revealed that higher nanoparticle concentrations significantly enhance heat transfer performance by modifying viscosity and density under varying conditions. Slit die extrusion also benefits from identifying ideal nanoparticle combinations and stability to improve processing outcomes. Mousavi et al. [2] investigated C -type nanofluids with a mass ratio of 6:3:1 (CuO:MgO: Ti O 2 ) and 2:1:1 volume fraction. Their study showed that CuO-MgO- Ti O 2 /water-based nanofluids exhibit excellent thermal conductivity, making them suitable for enhancing heat transfer in extrusion processes. Such findings underscore the potential of carefully designed nanofluids for achieving superior thermal and flow performance during slit die extrusion.

Slit die extrusion processes are deeply connected to foundational fluid dynamics concepts, such as boundary layer theory. Introduced by Prandtl [3] in 1904, this theory provides the framework for understanding the distinction between steady and unsteady flows. Unsteady flows, characterized by temporal variations in velocity and pressure, have applications across multiple industries. Pop et al. [4] reviewed the significance of unsteady boundary layer flows in fields like aerospace, turbine engineering, and nanofluid applications. Their work highlights how nanofluids enhance thermal conductivity in permeable materials and microparticle fluids, transforming governing equations into ordinary differential equations (ODEs) to analyze mass transfer, micropolar effects, and flow dynamics. Ouyang et al. [5] investigated an Al 2 O 3 -Cu/water nanofluid mixture to improve water cleanliness by enhancing thermal transfer and facilitating material breakdown. Their study, focusing on flow dynamics in stagnation regions within permeable media, demonstrates the potential of unsteady hybrid nanofluid flows in optimizing water systems and addressing environmental challenges.

In 1931, Falkneb and Skan [6] expanded on Prandtl's boundary layer theory by developing wedge flow theory, describing fluid flow in directions not parallel to the primary flow. This theory has been extensively applied in various fields. Notably, Seddeek et al. [7] investigated steady Falkner–Skan (F-S) flow and thermal transfer over a wedge with variable viscosity and heat conductivity, significantly advancing the understanding of such flows. The F-S boundary layer flow equation, which addresses boundary layer flow with a streamwise pressure gradient, has been widely studied. For example, Fang and Zhang [8] and Weidman et al. [9] independently conducted numerical analyses of the F-S equation cases. Abbasbandy and Hayat [10] also explored different aspects of these solutions, contributing valuable insights into fluid flow behavior. Zhang et al. [11] examined the flow and thermal characteristics of a radiating homogeneous hybrid nanofluid over a stationary wedge surface, highlighting the effects of heat buoyancy forces, Lorentz forces, and Darcy forces on the cooling performance of the hybrid nanofluid. Riley and Weidman [12] analyzed pressure-driven flow over a stretching boundary, revealing multiple solution possibilities and the existence of similar solutions. Ishak et al. [13] studied the boundary layer flow of a conductive fluid perpendicular to a changing magnetic field along a moving wedge, providing insights into complex flow conditions.

Joule heating, a mechanism for converting electrical energy into thermal energy, involves heat generation through a decay layer. Research by Shagaiya Daniel examined radiation and ohmic heating effects on nanofluids. Teh and Ashgar [14] studied 3D mixed nanofluid flow across a rotationally deformable surface under ohmic heating, finding that increasing magnetic parameters increased the velocity but decreased the temperature. Yan et al. [15] analyzed Joule heating effects on mixed nanoparticle fluid flow over an exponential surface, noting that Re x 1 ∕ 2 ; C f increased with magnetic and suction parameters while decreasing with velocity slip factor. They also found that temperature in the dual solution increased with Eckert number, but boundary layer separation remained unaffected. This was consistent with Khashi’ie et al. [16], who observed that the Eckert number did not influence the separation point. According to Kishore et al. [17], thermal dissipation in natural convective flow, characterized by the Eckert number, is crucial in both natural and industrial processes. High-viscosity fluid flow along elongated structures involves significant viscous and ohmic dissipations, essential for heat transfer. Zeeshan et al. [18] investigated Ohmic heating and viscous dissipation in magnetic nanofluids over a sinusoidal wavy surface. Zainal et al. [19] explored viscous dissipation and hybrid nanofluid flow over a surface undergoing exponential stretching or shrinking, finding that viscous dissipation increased the thermal state, enlarging the temperature boundary layer. Ouyang et al. [20] researched time-dependent MHD ternary nanofluid flow over a moving wedge, developing a comprehensive model that included viscous dissipation and Joule heating. Their findings highlighted enhanced heat transfer and improved surface finish, applicable to industrial processes like slit die extrusion. This study extends the understanding of MHD effects in nanofluid dynamics and provides a theoretical basis for optimizing manufacturing techniques involving complex fluid behaviors.

Stability analysis is used to determine the response of the solution to perturbations. Stable solutions can be used in actual production. Stability analysis can help inform our understanding of the physical properties of different solutions. Merkin [21] applied stability analysis framework to his earlier work [22] on boundary layer and mixed repulsion flow against an upright plane in a fully permeated porous environment. These insights set a precedent, leading many researchers to adopt Merkin's stability analysis and results in their examination of dual solutions' stability. Subsequent studies, such as those by Bachok et al. [23], Harris et al. [24], Roşca and Pop [25,26], Najib et al. [27], and Yasin et al. [28], have followed Merkin’s [21] analytical approach to determine the stability of solutions in their respective investigations. This framework has since been extended to encompass the stability analysis of two solutions in nanofluid flow (including NF, HNF, and THNF), marking a significant evolution in the study of fluid dynamics within boundary layers.

Slit die extrusion requires precise control of fluid thermal and flow properties, making it essential to incorporate nanoparticles for enhancing thermal conductivity and viscosity. The current study explores the stability analysis of unsteady ternary nanofluid flow over a stretching/shrinking wedge, incorporating viscous dissipation and Joule heating – an area that has been insufficiently addressed in prior research. A key novelty of this study lies in identifying the critical values for the existence of dual solutions and conducting a comprehensive stability analysis. By understanding these critical values, the study provides insights into predicting temperature distributions and maintaining thermal stability, which are vital for ensuring product quality in extrusion processes. Optimizing these factors enables manufacturers to achieve greater uniformity, reduce defects, and enhance efficiency in sectors such as polymers, food processing, and composite materials. This research not only addresses a significant gap in the literature but also lays a robust foundation for validating advanced models and optimizing heat transfer processes in industrial applications. This research aims to improve thermal efficiency in slit die extrusion and addresses the following key questions:

Which solution is stable?
How do viscous dissipation and Joule heating impact heat transfer?
How do factors like fluid type, unsteadiness parameter ( A ), nanoparticle volume fraction ( ϕ ), suction/injection parameter ( S ), and wedge angle ( β ) affect fluid dynamics and heat transfer?

2 Mathematical formulation

Consider dual solutions of the unsteady boundary layer nanofluid flow over a moving wedge with viscous dissipation and Joule heating, as shown in Figure 1, where the flow being at y ≥ 0 . The velocity of the stretching/shrinking wedge is U w ( x , t ) = u w ( x , t ) λ , where u w ( x , t ) = U 0 x m ∕ ( 1 − c t ) , and that of the far-field inviscous flow is U e ( x , t ) = U 0 x m ∕ ( 1 − c t ) , where U 0 , U ∞ , and m are constants, 0 ≤ m ≤ 1 , λ = U ∞ ∕ U 0 , λ < 0 stands for the shrinking wedge, λ = 0 for a steady wedge, λ > 0 for stretching, respectively. The variable mass flux velocity is v = v w ( x , t ) with v w ( x , t ) > 0 for suction and v w ( x , t ) < 0 for injection. The variable surface temperature at the surface of the wedge is T w ( x , t ) = T ∞ + T 0 ( x ∕ l ) 2 m ∕ ( 1 − c t ) 2 , where T 0 is the temperature characteristic of the surface of the wedge, l is the length characteristic of the wedge, and T ∞ represents the free stream temperature. The fluid is CuO-MgO- Ti O 2 /water. In the context of this study, it is also assumed that the thermal equilibrium is maintained between the water and the dispersed nanoparticles. Moreover, it is presumed that the nanoparticles exhibit uniform characteristics in terms of both size and shape, and that they are in a state of thermal equilibrium.

Figure 1

The physical model and coordinate system.

Using the above statements and assumptions, the control equations can be set forth as [29]

(1) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(2) ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y = ∂ u e ∂ t + u e ∂ u e ∂ x + μ thnf ρ thnf ∂ 2 u ∂ y 2 ,

(3) ∂ T ∂ t + u ∂ T ∂ x + v ∂ T ∂ y = k thnf ( ρ C p ) thnf ∂ 2 T ∂ y 2 + μ thnf ( ρ C p ) thnf ∂ u ∂ y 2 ,

subject to

(4) At y = 0 t < 0 : u = 0 , v = 0 , T = T ∞ t ≥ 0 : u = U w ( x , t ) = u w ( x ) 1 − c t λ = U 0 x m 1 − c t λ , v = v w ( t ) , T w ( x , t ) = T ∞ + T 0 ( x ∕ l ) 2 m ( 1 − c t ) 2 when y → ∞ , u = u e ( x , t ) → U 0 x m 1 − c t , T → T ∞ ,

where ( u , v ) are the velocity components of the hybrid nanofluid along ( x , y ) -axis, T is the temperatures of the hybrid nanofluid. The pressure gradient parameter m , which represents the exponent, is determined by the wedge angle parameter. In this case, we consider the total apex angle of the wedge as β π and denote it as m = β 2 − β or β = 2 m m + 1 . White and Majdalani [30] defined the wedge parameter as a parameter that is related to the pressure gradient. Non-negative values of the wedge parameter indicate that when β = 0 ( m = 0 ), it illustrates the boundary layer flow past a parallel flat plate. On the other hand, when β = 1 ( m = 1 ), it correlates with a vertical plate.

Further, the ternary nanofluid’s physical characteristics μ thnf , ρ thnf , k thnf , ( ρ C p ) thnf , and σ thnf are described by [4]

(5) μ thnf μ f = 1 ( 1 − ϕ 1 ) 2.5 ( 1 − ϕ 2 ) 2.5 ( 1 − ϕ 3 ) 2.5 ρ thnf ρ f = ( 1 − ϕ 3 ) ( 1 − ϕ 2 ) ( 1 − ϕ 1 ) + ϕ 1 ρ 1 ρ f + ϕ 2 ρ 2 ρ f + ϕ 3 ρ 3 ρ f ( ρ C p ) thnf ( ρ C p ) f = ( 1 − ϕ 3 ) ( 1 − ϕ 2 ) ( 1 − ϕ 1 ) + ϕ 1 ( ρ C p ) 1 ( ρ C p ) f + ϕ 2 ( ρ C p ) 2 ( ρ C p ) f + ϕ 3 ( ρ C p ) 3 ( ρ C p ) f k thnf k hnf = k 3 + 2 k hnf − 2 ϕ 3 ( k hnf − k 3 ) k 3 + 2 k hnf + ϕ 3 ( k hnf − k 3 ) ,

where k hnf k nf = k 2 + 2 k nf − 2 ϕ 2 ( k nf − k 2 ) k 2 + 2 k nf + ϕ 2 ( k nf − k 2 ) , k nf k f = k 1 + 2 k f − 2 ϕ 1 ( k f − k 1 ) k 1 + 2 k f + ϕ 1 ( k f − k 1 ) ,

where ϕ represents the volume fraction of the nanofluid. When ϕ equals 0, it corresponds to a regular fluid, often referred to as a classical viscous fluids. The dynamic viscosity is denoted as μ , densities as ρ , thermal diffusivities as k , heat capacities as ρ C p , and electrical conductivities as σ . Additionally, C p represents the thermal capacity. In Table 1, the physical characteristics of the water and nanoparticles are provided.

Table 1

Thermophysical properties (Adnan and Ashraf [31])

Physical characteristics	H 2 O	CuO	MgO	Ti O 2
ρ ( kg m − 3 )	997.1	6,320	3,560	4,250
C p ( J kg − 1 K )	4,179	531.8	955	686.2
k ( W m − 1 K )	0.613	76.5	45	8.953

To solve Eqs (1)–(3), it is appropriate to present the ensuing similarity transformations

(6) u = U w ( x ) 1 − c t f ′ ( η ) , v = − ( m + 1 ) ν f U w ( x ) 2 x ( 1 − c t ) f + m − 1 m + 1 η f ′ T ( x , t ) = T ∞ + T 0 ( x ∕ l ) 2 m θ ( η ) ∕ ( 1 − c t ) 2 , η = y ( m + 1 ) U w ( x ) 2 ν f x ( 1 − c t ) ,

and

(7) v w ( x , t ) = − ( m + 1 ) ν f U w ( x ) 2 x ( 1 − c t ) S .

In this passage, the symbol prime represents differentiation with respect to variable η . The symbol S represents the constant mass flux velocity, with a positive value of S indicating suction of the fluid, and a negative value indicating injection of the fluid.

By substituting Eq. (6) in Eqs (2)–(4), we can derive the subsequent equations for similarity.

(8) μ thnf ∕ μ f ρ thnf ∕ ρ f f ″ ′ + f f ″ + β ( 1 − f ′ 2 ) − A f ′ + η 2 f ″ − 1 = 0

(9) 1 Pr k thnf ∕ k f ( ρ C p ) thnf ∕ ( ρ C p ) f θ ″ + f θ ′ − 2 β f ′ θ − A 2 θ + η θ ′ 2 + Ec ( ρ C p ) thnf ∕ ( ρ C p ) f μ thnf μ f ( f ″ ) 2 = 0

along with

(10) f ( 0 ) = S , f ′ ( 0 ) = λ , θ ( 0 ) = 1 f ′ ( η ) → 1 , θ ( η ) → 0 as η → ∞ ,

where Pr is the Prandtl number, A is the unsteadiness parameter, β is the wedge parameter, and Ec is the Eckert number, described as

(11) Pr = ( μ C p ) f k f , A = 2 x c ( m + 1 ) U 0 x m , β = 2 m m + 1 , Ec = u w ( x , t ) 2 ( C p ) f ( T w − T ∞ ) = U 0 2 x 2 m ∕ ( 1 − c t ) 2 T 0 ( x ∕ l ) 2 m ( C p ) f ∕ ( 1 − c t ) 2 = U 0 2 l 2 m T 0 ( C p ) f ,

where C f and Nu x are characterized as

(12) C f = μ thnf ρ f u e 2 ( x , t ) ∂ u ∂ y y = 0 , Nu x = − x k thnf k f [ T w ( x ) − T ∞ ] ∂ T ∂ y y = 0 .

Using Eqs (6) and (12), we obtain

(13) Re x 1 ∕ 2 C f = μ thnf μ f m + 1 2 f ″ ( 0 ) , Re x − 1 ∕ 2 Nu x = − k thnf k f m + 1 2 θ ′ ( 0 ) ,

where Re x = u w ( x ) x ∕ ν f is the local Reynolds number.

3 Stability analysis

The stability of the non-unique solutions provided by Eqs (8)–(10) is evaluated in this section. τ = c A t ( 1 − c t ) is introduced.

(14) u ( x , y , t , τ ) = U 0 x 1 − c t ∂ f ∂ η ( η , τ ) , v = v w ( t , τ ) = − a ν f 1 − c t f ( η , τ ) , θ ( η , τ ) = T − T ∞ T w − T ∞ , η = y a ν f ( 1 − c t ) , τ = c A t ( 1 − c t ) .

Thus,

(15) μ thnf ∕ μ f ρ thnf ∕ ρ f f η η η + f f η η + β ( 1 − f η 2 ) − A f η + η 2 f η η − 1 − ( 1 + A τ ) f η τ − τ 2 ( m − 1 ) m + 1 f η f η τ = 0 ,

(16) 1 Pr k thnf ∕ k f ( ρ C p ) thnf ∕ ( ρ C p ) f θ η η + Ec ( ρ C p ) thnf ∕ ( ρ C p ) f μ thnf μ f ( f η η ) 2 + f θ η − 2 β f η θ − A 2 θ + η θ η 2 − θ τ − τ 2 ( m − 1 ) m + 1 f η θ τ = 0

together with

(17) f ( 0 , τ ) = S , ∂ f ∂ η ( 0 , τ ) = λ , θ ( 0 , τ ) = 1 , lim η → ∞ ∂ f ∂ η ( η , τ ) = 0 , lim η → ∞ θ ( η , τ ) = 0 .

The robustness of the similarity solutions f ( η ) = f 0 ( η ) and θ ( η ) = θ 0 ( η ) is examined by employing the following perturbation equation:

(18) f ( η , τ ) = f 0 ( η ) + e − ε τ F ( η , τ ) , θ ( η , τ ) = T − T ∞ T f − T ∞ = θ 0 ( η ) + e − ε τ G ( η , τ ) ,

where ε is an unknown eigenvalue, F ( η , τ ) and G ( η , τ ) are relatively small than f 0 ( η ) and θ 0 ( η ) . By substituting Eq. (18) in Eqs (15)–(17), the following equations are obtained:

(19) μ thnf ∕ μ f ρ thnf ∕ ρ f F η η η + f 0 F η η + f ″ 0 F − 2 β f ′ 0 F η − A F η + η 2 F η η + ( 1 + A τ ) ( ε F η − F η τ ) − τ 2 ( m − 1 ) m + 1 ( f ′ 0 F η τ − ε f ′ 0 F η ) = 0

(20) 1 Pr k thnf ∕ k f ( ρ C p ) thnf ∕ ( ρ C p ) f G η η + 2 Ec ( ρ C p ) thnf ∕ ( ρ C p ) f × μ thnf μ f f ″ 0 F η η + f 0 G η + F θ ′ 0 − A 2 G + η G η 2 − 2 β ( f ′ 0 G + F η θ 0 ) + ( 1 + A τ ) ( ε G − G τ ) − τ 2 ( m − 1 ) m + 1 ( f ′ 0 G τ − ε f ′ 0 G ) = 0

together with

(21) F ( 0 , τ ) = 0 , F η ( 0 , τ ) = 0 , G ( 0 , τ ) = 0 , lim η → ∞ F η ( η , τ ) → 0 , lim η → ∞ G ( η , τ ) → 0 .

Let τ = 0 .

(22) μ thnf ∕ μ f ρ thnf ∕ ρ f F 0 ‴ + f 0 F ″ 0 + f ″ 0 F 0 − 2 β f ′ 0 F ′ 0 − A F ′ 0 + η 2 F ″ 0 + ε F ′ 0 = 0

(23) 1 Pr k thnf ∕ k f ( ρ C p ) thnf ∕ ( ρ C p ) f G ″ 0 + 2 Ec ( ρ C p ) thnf ∕ ( ρ C p ) f μ thnf μ f f ″ 0 F ″ 0 + f 0 G ′ 0 + F 0 θ ′ 0 − 2 β ( f ′ 0 G 0 + F ′ 0 θ 0 ) − A 2 G 0 + η G ′ 0 2 + ε G 0 = 0

with

(24) F 0 ( 0 ) = 0 , F 0 ′ ( 0 ) = 0 , G 0 ( 0 ) = 0 , F 0 ′ ( η ) = 0 , G 0 ( η ) = 0 , as η → ∞ .

Following Harris et al. [24], F ′ 0 ( ∞ ) → 0 is eased into F ″ 0 ( 0 ) = 1 .

4 Results and discussion

The boundary value problem solver using 4th-order collocation (BVP4C) method function in MATLAB is developed to solve non-linear boundary value problems. This function is established on the three-stage Lobatto IIIa formula. The core idea behind this method is to transform BVPs into a set of ODEs, as shown in Eqs (8)–(10). Solutions to these problems are illustrated in Figures 2, 3, 4, 5, 6, 7. The substantial agreement highlighted in Table 2 bolsters the confidence in the accuracy of our numerical outcomes, thereby reinforcing the credibility of the results.

$Figure 2 Physical properties for NF, HNF, THNF fluid flow: (a) Re x 1 ∕ 2 C f {{\rm{Re}}}_{x}^{1/2}{C}_{{\rm{f}}} and (b) Re x − 1 ∕ 2 Nu x {{\rm{Re}}}_{x}^{-1/2}{{\rm{Nu}}}_{x} .$

Figure 2

Physical properties for NF, HNF, THNF fluid flow: (a) Re x 1 ∕ 2 C f and (b) Re x − 1 ∕ 2 Nu x .

$Figure 3 Physical properties for different ϕ \phi values: (a) Re x 1 ∕ 2 C f {{\rm{Re}}}_{x}^{1/2}{C}_{{\rm{f}}} , (b) f ′ ( η ) {f}^{^{\prime} }(\eta ) , (c) Re x − 1 ∕ 2 Nu x {{\rm{Re}}}_{x}^{-1/2}{{\rm{Nu}}}_{x} , and (d) θ ( η ) \theta (\eta ) .$

Figure 3

Physical properties for different ϕ values: (a) Re x 1 ∕ 2 C f , (b) f ′ ( η ) , (c) Re x − 1 ∕ 2 Nu x , and (d) θ ( η ) .

$Figure 4 Physical properties for different S S values: (a) Re x 1 ∕ 2 C f {{\rm{Re}}}_{x}^{1/2}{C}_{{\rm{f}}} , (b) f ′ ( η ) {f}^{^{\prime} }(\eta ) , (c) Re x − 1 ∕ 2 Nu x {{\rm{Re}}}_{x}^{-1/2}{{\rm{Nu}}}_{x} , and (d) θ ( η ) \theta (\eta ) .$

Figure 4

Physical properties for different S values: (a) Re x 1 ∕ 2 C f , (b) f ′ ( η ) , (c) Re x − 1 ∕ 2 Nu x , and (d) θ ( η ) .

$Figure 5 Physical properties for different m m values: (a) Re x 1 ∕ 2 C f {{\rm{Re}}}_{x}^{1/2}{C}_{{\rm{f}}} , (b) f ′ ( η ) {f}^{^{\prime} }(\eta ) , (c) Re x − 1 ∕ 2 Nu x {{\rm{Re}}}_{x}^{-1/2}{{\rm{Nu}}}_{x} , and (d) θ ( η ) \theta (\eta ) .$

Figure 5

Physical properties for different m values: (a) Re x 1 ∕ 2 C f , (b) f ′ ( η ) , (c) Re x − 1 ∕ 2 Nu x , and (d) θ ( η ) .

$Figure 6 Physical properties for different A A values: (a) Re x 1 ∕ 2 C f {{\rm{Re}}}_{x}^{1/2}{C}_{{\rm{f}}} and (b) Re x − 1 ∕ 2 Nu x {{\rm{Re}}}_{x}^{-1/2}{{\rm{Nu}}}_{x} .$

Figure 6

Physical properties for different A values: (a) Re x 1 ∕ 2 C f and (b) Re x − 1 ∕ 2 Nu x .

$Figure 7 Physical properties for different Ec values: (a) Re x − 1 ∕ 2 Nu x {{\rm{Re}}}_{x}^{-1/2}{{\rm{Nu}}}_{x} and (b) θ ( η ) \theta (\eta ) .$

Figure 7

Physical properties for different Ec values: (a) Re x − 1 ∕ 2 Nu x and (b) θ ( η ) .

Table 2

Values of f ″ ( 0 ) for the shrinking wedge (Pr = 0.72; ϕ 1 = ϕ 2 = ϕ 3 = 0 ; S = 0 ; Ec = 0 ; A = 0 ; m = 1 )

λ	Wang [32]		Naganthran et al. [33]		Present result
	Upper branch	Lower branch	Upper branch	Lower branch	Upper branch	Lower branch
− 0.25	1.40224	—	1.4022407	—	1.4022408	—
− 0.5	1.49567	—	1.4956697	—	1.4956698	—
− 1.0	1.32882	0	1.3288168	0	1.3288169	0
− 1.1	—	—	1.1866805	0.0492290	1.1866803	0.0492289
− 1.15	1.08223	0.116702	1.0822311	0.1167020	1.0822312	0.1167021
− 1.2	—	—	0.9324733	0.2336496	0.9324735	0.2336497
− 1.2465	0.5543	—	0.5842759	0.5542976	0.5842819	0.5542962
− 1.24657	—	—	0.5745397	0.5640169	0.5745587	0.5640157

Figures 2–7 demonstrate the effects of fluid flow on the skin friction coefficient and the Nusselt number. The results indicate the presence of two solutions when λ c i ≤ λ . However, for λ ≤ λ c i ≤ 0 ( i = 1 , 2, …, 5), Eqs (8)–(10) lack any solutions. Consequently, Eqs (1)–(4) must be solved using numerical techniques. The critical points λ c i < 0 ( i = 1 , 2, 3, 4) indicate the conditions under which real solutions to Eqs (8)–(10) can be found. Stability analysis has shown that the first solution is physically realizable, whereas the second solution is not applicable in practice.

Figure 2 illustrates the enhancement of the Nusselt number but a decrease in the skin friction coefficient as the fluid transitions from NF to HNF to THNF. According to Myers et al. [34], the sum of the volume fractions ϕ 1 , ϕ 2 , and ϕ 3 should be less than 0.05. For this analysis, we assume ϕ 1 + ϕ 2 + ϕ 3 = 0.04 . The three nanofluid flows considered are: NF with ϕ 1 = 0.04 , ϕ 2 = ϕ 3 = 0 ; HNF with ϕ 1 = 0.03 , ϕ 2 = 0.01 , ϕ 3 = 0 ; and THNF with ϕ 1 = 0.02 , ϕ 2 = ϕ 3 = 0.01 as per Mousavi et al. [2].

The expression of the skin friction coefficient, representing a specific fluid dynamic characteristic, shows a decreasing trend from NF to HNF to THNF. This reduction likely reflects changes in the fluid's viscosity, thermal conductivity, or both, which impact the friction factor at the fluid's surface. In Figure 2(b), the Nusselt number increases as the fluid transitions from NF to HNF to THNF as λ > − 3.23 . This aligns with the concept that optimizing nanoparticle concentration can increase the convection mechanism, because surface temperature decreases with rising hybridity. The critical values for NF, HNF, and THNF are − 3.451212 , − 3.41996 , and − 3.396836 , respectively. This progression in critical values indicates accelerated boundary layer separation, suggesting changes in flow characteristics that could differently influence performance in various applications using these fluids.

Figure 3 illustrates the effect of nanoparticle volume fraction ( ϕ ) on various parameters, including the skin friction coefficient, velocity, the Nusselt number, and temperature profile. As ϕ increases, both the skin friction coefficient and velocity increase, while T decreases. The observed increase in the skin friction coefficient and velocity with rising ϕ can be attributed to the enhanced thermal conductivity and viscosity of the nanofluid, which improve momentum transfer and result in a more robust flow profile. The decrease in temperature ( T ) is due to the improved heat transfer capabilities of the nanofluid, which facilitates more efficient cooling. For the Nusselt number, the behavior varies depending on the value of λ . When λ < − 3.26 , the increased nanoparticle volume fraction enhances the convective heat transfer, leading to a rise in the Nusselt number ( Nu x ). Conversely, for λ > − 3.26 , the adverse pressure gradient may overpower the beneficial effects of enhanced thermal conductivity, resulting in a reduction in Nu x .

Additionally, the Nusselt number shows an increase for λ < − 3.26 and a decrease for λ > − 3.26 . The increase in ϕ causes the flow separation point to delay, shifting from − 3.36958 to − 3.384193 , and finally to − 3.396836 . The delay in flow separation with increasing ϕ can be explained by the improved stability of the boundary layer ascribable to the increasing viscosity and thermal conductivity of the nanofluid. These properties contribute to a more stable flow regime, allowing the fluid to adhere to the surface for a longer distance before separating. This delay in flow separation leads to improved aerodynamic performance and reduced drag, which is beneficial in various engineering applications.

Figure 4(a) illustrates the behavior of the skin friction coefficient in the upper branch solution as the wedge experiences shrinkage and S increases. Rising S effect enhances the velocity gradient, resulting in drop in the momentum boundary layer thickness (refer to Figure 4(b)). Conversely, in the lower branch solution, the skin friction coefficient decreases with increasing S . This suggests that the left-moving wedge cannot accommodate further fluid penetration, resulting in a drop in the velocity gradient and a thicker momentum boundary layer, as shown in Figure 4(b).

Both figures indicate that higher S values reduce the boundary layer thickness, as suction diminishes drag force and prevents boundary layer separation. As to temperature, the first solution shows a declining trend with S , while the second solution exhibits an escalating trend (refer to Figure 4(d)). This behavior is consistent with the temperature variations depicted in Figure 4(c). The reduction in temperature decreases the thermal diffusivity of the tri-hybrid nanofluid, gradually improving convective heat transfer. As S increases, boundary layer separation is delayed, with the critical value shifting from − 3.396836 to − 3.5282 and eventually to − 3.66521 .

Figure 5 illustrates the impact of m on the skin friction coefficient, the Nusselt number, dimensionless velocity, and temperature. As m increases, β rises, indicating that the angle of the wedge is enlarging. For λ > − 3.34 , the skin friction coefficient increases with the increase in m . In Figure 5(b), velocity is an increasing function of m , meaning that as the wedge’s angle rises, velocity also increases. This might be due to the increased interactions between the fluid and the surface, which impede fluid motion and create resistance to flow.

When λ c < λ < − 3.34 , the rapid stretching of the wedge causes the fluid velocity to drop as m rises, leading to a decline in the skin friction coefficient. The temperature increases with m in Figure 5(d), likely due to the opposition created by m , additional energy is converted into heat. As the fluid faces greater resistance, a portion of its kinetic energy is transformed into thermal energy, leading to a rise in temperature.

The Nusselt number decreases with the increase in m , as demonstrated in Figure 5(c). The Nusselt number, which relates convective to conductive heat transfer, decreases, indicating that convective heat exchange is decreasing compared to conductive thermal transfer. This may be attributed to the lowered fluid velocity and heightened resistance to flow, which hinder convective thermal transfer.

In Figure 6(a), it is observed that as A increases, the skin friction coefficient increases when the wedge is shrinking, whereas it decreases with a stretching wedge. This behavior can be attributed to the boundary layer characteristics influenced by the wedge's motion. When the wedge is shrinking, an increase in A enhances the adverse pressure gradient, thickening the boundary layer and increasing the skin friction coefficient, as shown by the rise in the skin friction coefficient. Conversely, for a stretching wedge, the favorable pressure gradient thins the boundary layer, reducing the skin friction and thus, C f decreases. These effects are pronounced due to the interaction between the wedge's motion and the local flow dynamics.

Additionally, in Figure 6(b), the increase in the Nusselt number with rising A can be attributed to enhanced thermal boundary layer effects. As the Reynolds number, R e x , decreases (due to the factor of Re x − 1 ∕ 2 ), the impact of viscous forces becomes more significant, leading to a thicker thermal boundary layer. This thicker layer facilitates greater thermal exchange between the surface and the fluid, thus increasing Nu x , which measures the convective thermal transfer on the surface. These results further show that increasing the intensity of unsteadiness enhances thermal transfer efficiency.

In Figure 7, the influence of Ec on thermal exchange distribution is presented. Ec considers the combined effects of Ohmic heating and the magnetic field. As shown in Figure 7(b), an increased Ec results in a higher temperature, indicating more effective convective thermal transfer and enhanced heat diffusivity, which together promote greater heat dissipation. However, despite these improvements, a higher Eckert number ultimately leads to a decrease in the convective heat exchange rate. This occurs because the increased internal temperature of the fluid reduces the temperature gradient between the fluid and its environment, making heat exchange less efficient. When Ec increases, the critical value and the skin friction coefficient remain unchanged.

From Table 3, the first solution is stable because ε > 0 , while the other one is unstable.

Table 3

ε for λ , m , and HNF/THNF when Pr = 6.2, m = 0.5

A	Ec	HNF/THNF	λ	1st solution ε	2nd solution ε
0.6	1	THNF	− 3.2689	0.018436	− 0.018354
			− 3.268	0.07144	− 0.070232
			− 3.26	0.22129	− 0.210064
0.8	1.5	THNF	− 3.3968	0.013348	− 0.0133016
			− 3.396	0.064373	− 0.0633063
			− 3.39	0.186729	− 0.1780104
0.8	1	NF	− 3.4512	0.0078496	− 0.0078338
			− 3.451	0.0325714	− 0.0323031
			− 3.45	0.0782518	− 0.0767201

5 Conclusion

This research focuses on developing a numerical solution for unsteady wedge flow, considering viscous dissipation and Joule heating in a ternary CuO – MgO – Ti O 2 / H 2 O nanofluid. Numerical computations are carried out using the BVP4C function in Matlab to analyze the effects of key parameters, including the wedge parameter, nanoparticle volume fraction, suction/injection parameter, and Eckert number. The findings reveal the existence of nonunique solutions over a broad range of control parameters for the ternary nanofluid. This analysis has direct applications in slit die extrusion, where managing fluid properties such as thermal conductivity and viscosity is essential for achieving uniform heat transfer and enhanced material quality. The key discoveries of this research can be listed:

The first solution is stable, whereas the second solution is unstable.
The Nusselt number drops as the Eckert number increases.
The convective heat transfer enhances from mono to binary to ternary nanofluid when λ > − 3.23 .
The boundary layer separation is delayed with rising unsteadiness parameter, nanoparticle volume fraction, and suction/injection parameter while decelerating with increasing type numbers of nanoparticles and wedge angle.
When the wedge surface is shrinking, the skin friction coefficient and the Nusselt number increase with the rising unsteadiness parameter, nanoparticle volume fraction, and suction/injection parameter.
When the wedge surface is shrinking, the skin friction coefficient increases with the rising wedge angle parameter, while the Nusselt number drops.

This research provides valuable insights into the flow behavior of ternary nanofluids, offering practical guidance for optimizing heat transfer and improving efficiency in extrusion processes across industries such as polymer manufacturing, food processing, and composite materials. The findings suggest that increasing parameters such as the unsteadiness parameter, nanoparticle volume fraction, and suction/injection parameter can help meet specific industrial requirements, enabling better control over thermal and flow properties.

However, a limitation of this study is its focus on a specific set of parameters, leaving room for further exploration. Future research could expand the scope by incorporating additional parameters and introducing new conditions, such as velocity slip (Ouyang et al. [35]), heat generation/absorption (Ouyang et al. [36]), Casson fluid (Naganthran and Nazar [37]), gyro tactic bio-convection flow (Basir et al. [38]), cone (Hanif et al. [39]), and Brownian motion (Saleem et al. [40]). These additions could further refine the understanding of nanofluid behavior and enhance its applicability in industrial processes.

Acknowledgments

The authors would like to acknowledge the financial support from Universiti Teknologi Malaysia for the funding under UTM fundamental Research (UTMFR) [grant number Q.J130000.3854.21H91], Malaysian Ministry of Higher Education [grant number FRGS/1/2023/STG06/UM/02/14], and Key Laboratory of AI and Information Processing (Hechi University), Education Department of Guangxi Zhuang Autonomous Region, China.

Funding information: The authors would like to acknowledge the financial support from Universiti Teknologi Malaysia for the funding under UTM fundamental Research (UTMFR) [grant number Q.J130000.3854.21H91], Malaysian Ministry of Higher Education [grant number FRGS/1/2023/STG06/UM/02/14], and Key Laboratory of AI and Information Processing (Hechi University), Education Department of Guangxi Zhuang Autonomous Region, China.
Author contributions: Yun Ouyang – resources, methodology, software, and writing; Md Faisal Md Basir – funding acquisition, writing, supervision, and project administration; Kohilavani Naganthran – writing, supervision, visualization, and validation; Ioan Pop – writing, methodology, formal analysis, and validation. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2024-10-27

Revised: 2024-12-16

Accepted: 2025-01-05

Published Online: 2025-05-09

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

Articles in the same Issue

https://doi.org/10.1515/phys-2025-0148

Keywords for this article

ternary nanofluid; unsteady flow; wedge flow; viscous dissipation; ohmic heating; stability analysis

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