Flow and heat transfer insights into a chemically reactive micropolar Williamson ternary hybrid nanofluid with cross-diffusion theory

Muhammad Naveed Khan; Shafiq Ahmad; Zhentao Wang; Mohamed Hussien; Abdullah M. S. Alhuthali; Hassan Ali Ghazwani

doi:10.1515/ntrev-2024-0081

Artikel Open Access

Flow and heat transfer insights into a chemically reactive micropolar Williamson ternary hybrid nanofluid with cross-diffusion theory

Muhammad Naveed Khan , Shafiq Ahmad , Zhentao Wang , Mohamed Hussien , Abdullah M. S. Alhuthali und Hassan Ali Ghazwani

Veröffentlicht/Copyright: 17. September 2024

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Aus der Zeitschrift Nanotechnology Reviews Band 13 Heft 1

Abstract

The need for efficient nanotechnology has led to unexpected developments. Conserving continuous thermal propagation is essential in many industrial and thermal systems because it improves the efficiency of thermal engineering engines and machinery. Therefore, a promising platform to increase thermal power energy is the hybridization of magnetic nanoparticles in a heat-supporting, non-Newtonian fluid. In light of the above applications, a mathematical model is established to analyze the variable fluid features of the thermally radiative and chemically reactive flow of a micropolar Williamson ternary hybrid nanofluid with electromagnetohydrodynamic and electroosmosis forces on a porous stretching surface. Stratification boundary conditions and variable fluid properties were used to analyze the thermal and solutal behavior of the fluid flow. Furthermore, to measure the disorder of the flow system, entropy generation was considered by the impact of Joule heating and viscous dissipation. To develop the numerical scheme BVP4C in MATLAB, we first converted the mathematical flow model into two ordinary differential equations using a suitable transformation. The graphical and numerical results were determined against several parameters of a ternary hybrid nanofluid ( MWCNT , A l 2 O 3 , SiC ) and unary nanofluid ( A l 2 O 3 ) . The results indicate that the heat transfer rate is more prominent in the ternary hybrid nanofluid than in the unary nanofluid because the addition of nanofluids to the base fluid is used to improve the heat transport rate. It can be seen from the figures that a greater estimation of the magnetic and electric field parameters improves the fluid velocity because, for low values of M ≤ 1 , the aiding force is dominant compared to the retarding force, which results in an improvement in the velocity profile. Furthermore, the entropy generation rate increases for higher values of the Brinkman number and temperature ratio parameter because more heat is produced due to the greater values of Br .

Keywords: variable fluid properties; micropolar-Williamson ternary nanofluid model; entropy generation with Soret and Dufour effect

Nomenclature

y , x: axes of coordinates
( ρC p ) THNF: heat capacity
σ THNF: electrical conductivity
F ⁎: inertia coefficient
B o: magnetic field intensity
T ∞: ambient temperature
k 1: vortex viscosity
E x: electroosmotic forces
j: density of microinertia
k THNF: thermal conductivity
D T: thermal diffusivity
K T: thermal diffusion ratio
c s: concentration susceptibility
E a: modified Arrhenius function
k r: chemical reaction constant
E 0: electric field
m e: electroosmotic parameter
F r: Forchheimer number
K: material parameter
U HS: Helmholtz–Smoluchowski speed without dimensions
M: magnetic parameter
D u: Dufour number
S u: Soret number
W e: Weissenberg number
E 1: electric parameter
D a: porosity parameter
N _G: entropy generation
R d: radiation parameter
Br: Brinkman number
EOF: electroosmosis forces
u 1 , u 2: velocity components
K p: permeability of the porous medium
u 1 ∞ , u 1 w: free stream and wall velocities
T w: wall temperature
ρ THNF: tri-hybrid nanofluid density
α THNF: tri-hybrid nanofluid thermal diffusivity
υ THNF: kinematic viscosity of the tri-hybrid nanofluid
ρ e: entire energy density
ϕ THNF: hybrid nanoparticle volume fractions
μ THNF: dynamic viscosity
Pr: Prandtl number
σ ⁎: Stefan–Boltzmann constant
ε: variable thermal conductivity parameter
θ ω: temperature ratio parameter
γ 1: thermal stratification parameter
ε 2: variable thermal diffusivity parameter
γ 2: solutal stratification parameter
γ THNF: spin gradient viscosity
s 1: suction/injection parameter
k ⁎: coefficient of mean absorption
Al 2 O 3: aluminum oxide
MWCNT: multiwalled carbon nanotubes
SiC: silicon carbide
ϕ 1: volume fraction of the nanoparticle ( Al 2 O 3 )
ϕ 2: volume fraction of the nanoparticle (SiC)
ϕ 3: volume fraction of the nanoparticle (MWCNT)
S c: Schmidt number
S 0: characteristic entropy generation rate
E c: Eckert number
EMHD: electromagnetohydrodynamic

1 Introduction

In recent years, nanotechnology has emerged as a groundbreaking field, with applications spanning various industries, including engineering, medicine, electronics, and energy. One of the fascinating areas of research in nanotechnology is the development and study of nanofluids. Nanofluids are colloidal suspensions consisting of nanoscale solid particles dispersed in a base fluid, offering unique and enhanced properties compared with conventional fluids. The significant use of nanofluids in advanced energy-related technologies improves heat transfer and mechanical properties. There are many types of nanofluids, among which trihybrid nanofluids (THNFs) exhibit exciting features; they are formed by the composition of more than two nanoparticles in the regular fluid. The mixture of more nanoparticles in the base fluid opens a new area for researchers and engineers to design nanofluids with specific applications. Ramadhan et al. [1] discussed the stability analysis on the THNFs with a mixture of H₂O and ethylene glycol (EG). The thermal characteristics of a THNF flowing through a heated surface within a complex fluid flow were investigated by Nazir et al. [2]. The finite element method was used to obtain the numerical results of this study. Two spinning porous surfaces were considered by Rauf et al. [3] to investigate the heat and mass transfer of micropolar THNFs with magnetic impact. Sarangi et al. [4] built a mathematical model to examine the consequences of the Darcy–Forchheimer on the rotating motion of THNF and unary nanofluid (UNF) toward a stretched surface. Patil et al. [5] observed the chemically reactive and thermally radiative Prandtl hybrid nanofluid over a stretchable surface with double diffusion impact. They also analyzed magnetized flow with chemical reactions, thermal radiation, viscous dissipation, and ohmic heating effects on slip boundary conditions. Shamshuddin et al. [6] inspected the bio-convective flow of the radiative Casson hybrid nanofluid across an exponentially stretching sheet under the impact of ohmic heating. The analytical Chebyshev collocation method (CCM) is applied to the flow model solution. The magnetized flow of Maxwell and Williamson nanofluids in the presence of nanoparticles was numerically investigated by Kanimozhi et al. [7]; they employed multiple slip boundary conditions on the porous stretching surface under the impact of ohmic heating and chemical reaction terms. Mohana and Kumar [8] presented the hydromantic flow of Cu-water base nanofluid influenced by thermal radiation and heat source on a nonlinear stretching surface. Ahmad et al. [9] considered the spinning sphere with magnetic and heat source/sink effects to analyze the transport of heat and mass of hybrid nanoparticles numerically. Recently, many researchers have discussed the improvement in heat and mass transport owing to the use of hybrid nanofluids (see previous studies [10–14]).

Entropy generation is a central concept in thermodynamics and statistical mechanics and plays a crucial role in understanding the behavior of several physical systems. It provides venerated insights into the irreversibility and inefficiency of processes and is closely related to the second law of thermodynamics. In simple terms, entropy is a measure of the disorder or randomness of a system, and the disorderness of the system is increased owing to the stronger entropy. Furthermore, according to the second law of thermodynamics, the entropy of a system also increases with time. The irreversibility (a process that cannot occur with an external agent) in the physical process generates entropy. Entropy generation has many applications in the fields of chemistry, biology, engineering, and environmental science. The optimization of the efficiency of machine processes and minimization of energy losses have led many engineers and scientists to focus on the investigation of entropy generation. Bejan [15] proposed a thorough investigation of entropy generation. Khan et al. [16] analyzed the magnetized flow of Williamson hybrid nanoparticles (titanium oxide and cobalt ferrite) with the impact of surface-catalyzed reactions and entropy generation across the vertical needle. The heat and mass transport of the magnetized flow of a non-Newtonian nanofluid under the impact of entropy generation was studied by Kumar et al. [17] using a stretching surface. This study accounts for the joint impacts of Joule heating and viscous dissipation in the system. Khan et al. [18] explored the heat and mass transfer characteristics of a Williamson nanofluid flow, considering the upshots of Joule heating and chemical reactions with entropy generation. Muhammad Raza Shah Naqvi et al. [19] discussed the combined effect of radiative and magnetized flow on hybrid nanofluid with entropy generation across the stretching/shrinking surface numerically. Some research related to the thermal performance of the Williamson hybrid nanofluid flow in the presence of entropy generation can be found in previous studies [20,21].

The aim of this research is to analyze the chemically reactive and thermally radiative flow of micropolar Williamson ternary hybrid nanofluid across the stretching surface with the consideration of Darcy–Forchheimer porous medium, incorporating Soret and Dufour effects, Joule heating, viscous dissipation, and double stratification. From the novelty point of view, the collective effect of electromagnetohydrodynamic (EMHD) and electroosmosis forces (EOF) on the non-Newtonian nanofluid by the impacts of entropy generation and variable fluid properties was thoroughly analyzed in the existing study. Nondimensional similarity transformations are used to transform the flow model equations into a system of nonlinear coupled ordinary differential equations (ODEs). These coupled ODEs were solved numerically with the help of the BVP4C approach. The graphical discussion and tabulated data are formulated based on the variation in the values of the developing parameters.

2 Mathematical construction

This study focuses on a steady, incompressible, two-dimensional micropolar Williamson ternary hybrid nanofluid flow with suction/injection impact passing through a porous stretching surface. The investigation incorporated entropy generation and considered the influence of cross-diffusion and EOF in the system. Heat and mass transfer investigations are discussed in terms of double stratification, variable thermal conductivity and diffusivity, Joule heating, activation energy, and nonlinear thermal radiation effects. It is assumed that the fluid temperature and concentration are T and C , respectively, while at the wall they are T w and C w , with the fluid velocity represented by V ¯ = ( u ¯ 1 ( x , y ) , u ¯ 2 ( x , y ) ) in Figure 1.

Figure 1

The flow diagram of the problem.

Under the aforementioned premise, the following are the governing equations for the mass, momentum, temperature, concentration, and microorganisms [4,21,22]:

(1) ∂ u ¯ 1 ∂ x = − ∂ u ¯ 2 ∂ y ,

(2) u ¯ 1 ∂ u ¯ 1 ∂ x = − u ¯ 2 ∂ u ¯ 1 ∂ y + μ THNF + k 1 ρ F ∂ 2 u ¯ 1 ∂ y 2 + 2 ω ∂ u ¯ 1 ∂ y ∂ 2 u ¯ 1 ∂ y 2 + σ THNF ρ THNF ( E 0 B 0 − B 0 2 u ¯ 1 ) − F ⁎ u ¯ 1 2 + k 1 ρ THNF ∂ N ∂ y + ρ e E x ρ THNF − μ THNF u ¯ 1 ρ THNF K p ,

(3) ρ THNF j u ¯ 1 ∂ N ∂ x + u ¯ 2 ∂ N ∂ y = γ THNF ∂ 2 N ∂ y 2 − k 1 2 N + ∂ u ¯ 1 ∂ y ,

(4) u ¯ 1 ∂ T ∂ x + u ¯ 2 ∂ T ∂ y = σ THNF ( ρ F C p ) THNF ( B 0 u ¯ 1 − E 0 ) 2 + 1 ( ρ F C p ) THNF × ( μ THNF + k 1 ) 1 + ω 2 ∂ u ¯ 1 ∂ y ∂ u ¯ 1 ∂ y 2 × 1 ( ρ F C p ) THNF ∂ ∂ y k THNF ( T ) ∂ T ∂ y + D T K T C p c s ∂ 2 C ∂ y 2 + 16 σ ⁎ 3 k ⁎ ( ρ F C p ) THNF ∂ ∂ y T 3 ∂ T ∂ y ,

(5) u ¯ 1 ∂ C ∂ x + u ¯ 2 ∂ C ∂ y = ∂ ∂ y D ( C ) ∂ C ∂ y − k r 2 T T ∞ m exp − E a k B T ( C − C ∞ ) + D T K T T m ∂ 2 T ∂ y 2 .

The convenient boundary conditions are defined in the form [22,23],

(6) u ¯ 1 = u ¯ 1 w , u ¯ 2 = u ¯ 2 w , N = − n ∂ u ¯ 1 ∂ y , T = T w + a 1 x , C = C w + b 1 x , at y = 0 . u ¯ 1 → 0 , N → 0 , T → T ∞ + a 2 x , C → C ∞ + b 2 x , at y → ∞

In equations (2)–(6), E x denotes the electroosmotic forces, σ THNF is the electric conductivity of ternary hybrid nanofluid, ρ e is the entire energy density, ρ THNF is the magnified density of the ternary hybrid nanofluid, E 0 is the electric field, c s is the concentration susceptibility, D T is the thermal diffusivity, q r represents the radiative heat flux, K T is the thermal diffusion ratio, K r is the chemical reaction constant, k 1 denotes the vortex viscosity, γ THNF = μ THNF + k 1 2 j is the spin gradient viscosity, j = υ f a is the density of microinertia, and F ⁎ = C b x K p is the inertia coefficient.

2.1 Using flow along the boundary layer to calculate the electroosmotic forces

Gauss law is used to obtain

(7) ∇ ⋅ E = ρ e ε ⁎ ,

where ρ e is the total ionic energy density and ε ⁎ is the dielectric characteristics. The potential for electricity with the supposition of the electrical field is conservative and can be expressed as follows:

(8) E = − ∇ Γ ⁎ .

Combining equations (7) and (8), the Poisson formula for the potential electrical distributions is written as follows:

(9) ∇ ⋅ ∇ Γ ⁎ = ∇ 2 Γ ⁎ = − ρ e ε ⁎ .

The net density charge of Boltzmann follows its distribution and is as follows:

(10) ρ e = − Z v ⁎ e ( n − − n + ) ,

where cations ( n + ) and anions ( n − ) have the following definitions:

(11) n − = n 0 exp − e Z v ⁎ T av K B ⁎ Γ ⁎ , n + = n 0 exp + e Z v ⁎ T av K B ⁎ Γ ⁎ ,

where n 0 is the bulk concentration, K B ⁎ is the Boltzmann constant, Z v ⁎ is the charge balance, e is the electric charge, and T av is the average temperature.

The Debye–Huckel linearization concept yields − e Z v ⁎ T av K B ⁎ ≪ 1 . Equation (10) is reduced to

(12) ρ e = ε ⁎ λ e 2 Γ ⁎ .

Since the fluid motion takes place in the flow’s boundary layer, the expression for ∂ / ∂ x → 0 on the left side hand of equation (9) becomes

(13) d 2 Γ ⁎ d y 2 = Γ ⁎ λ e 2 ,

where λ e = ( e Z v ⁎ ) − 2 T av K B ⁎ ε ⁎ 2 n 0 .

Making use of the following dimensionless equations:

(14) Γ = Γ ⁎ ς , ζ = υ F a y .

Equation (13) therefore turns into

(15) d 2 Γ d ζ 2 = Γ m e 2 ,

where m e 2 = υ F c λ e 2 denotes the electroosmotic parameter. The analytical solution that depends on boundary conditions for equation (15) can be obtained as follows: Γ = 1 at ζ = 0 and Γ → ∞ at ζ → ∞ .

(16) Γ = exp ( − m e ζ ) .

2.2 Similarity transformation

The following list contains the pertinent similarity transformations [22]:

(17) ζ = a ν F 1 2 y , u ¯ 1 = a x f ′ ( ζ ) , u ¯ 2 = − a ν F f ( ζ ) , N = a x a ν F g ( ζ ) , θ ( ζ ) = T − T ∞ T w − T ∞ , h ( ζ ) = C − C ∞ C w − C ∞ , D ( C ) = D ∞ 1 + ε 2 C − C ∞ C w − C ∞ , k THNF ( T ) = k THNF 1 + ε T − T ∞ T w − T ∞ .

Equations (2)–(6) in the dimensionless form are transformed using the aforementioned approaches of Chandel and Sood [24]:

(18) μ THNF μ F + K ( 1 + W e f ″ ) f ‴ + ρ THNF ρ F ( f f ″ − f ′ 2 − F r f ′ 2 ) + U HS m e 2 exp ( − m e ζ ) − σ THNF σ F M ( f ′ − E 1 ) + K g ′ − μ THNF μ F D a f ′ = 0 ,

(19) μ THNF μ F + K 2 g ″ + ρ THNF ρ F ( f g ′ − g f ′ ) − K ( f ″ + 2 g ) = 0 ,

(20) k THNF k F ( 1 + ε θ ) + 4 3 R d ( 1 + ( θ ω − 1 ) θ ) 3 θ ″ + k THNF k F ε + 4 R d ( 1 + ( θ ω − 1 ) θ ) 2 ( θ ω − 1 ) θ ' 2 + Pr ( ρC p ) THNF ( ρC p ) F { D u h ″ + f θ ′ − ( γ 1 + θ ) f ′ } + Pr μ THNF μ F + K 1 + W e 2 f ″ f ″ 2 + σ THNF σ F M E c ( f ′ − E 1 ) 2 = 0 ,

(21) ( 1 + ε 2 h ) h ″ + ε 2 h ′ 2 − Sc ( γ 2 + h ) f ′ − f h ′ − S u θ ″ + K r ( 1 + δ θ ) m 1 exp − E 1 ⁎ 1 + δ θ h = 0 .

The concern boundary conditions are

(22) f ( ζ ) = s 1 , f ′ ( ζ ) = 1 , g ′ ( ζ ) = − n f ″ ( ζ ) , θ ( ζ ) = 1 − γ 1 , h ( ζ ) = 1 − γ 2 at ζ = 0 .

(23) f ′ ( ζ ) → 0 , g ( ζ ) → 0 , θ ( ζ ) → 0 , h ( ζ ) → 0 , at ζ → ∞ .

In the aforementioned equations, W e represent the Weissenberg number, K is the material parameter, U HS is the elector-osmatic velocity parameter, D a is the porosity parameter, F r is the Forchheimer number, D u is the Dufour number, s 1 is the suction/injection parameter, ( γ 1 , γ 2 ) is the thermal and solutal stratification parameter, θ ω is the temperature ratio parameter, m e 2 is the electroosmotic parameter, ε is the variable thermal conductivity parameter, and ε 2 is the variable thermal diffusivity parameter.

These parameters are defined in dimensionless forms as

(24) Pr = ν F α F , δ = ( T w − T ∞ ) T ∞ , K = k 1 υ F , γ 1 = a 2 a 1 , γ 2 = b 2 b 1 , M = σ F B 0 2 a ρ F , S c = ν F D B , U HS = − ε ⁎ ξ E x u 1 w μ F , R d = 16 σ ∗ T ∞ 3 3 k ⁎ κ , W e = 2 ω u 1 w a ν F , E 1 ⁎ = E a T ∞ k B , S u = D T K T ( T w − T ∞ ) ν F T m ( C w − C ∞ ) , D u = D m K T ( C w − C ∞ ) ν F c s C p ( T w − T ∞ ) , F r = C b K p , s 1 = u 2 w a ν F , m e 2 = ν F a λ e 2 , λ e = ( e Z v ⁎ ) − 2 T av K B ⁎ ε ⁎ 2 n 0 , θ w = T w T ∞ , K r = k r 2 a , D a = υ F a K p .

2.3 Physical quantities

The concept of physical quantities like the skin friction, the Nusselt number, and the Sherwood number is identified as [22]

(25) C f x = τ w ρ F u w 2 , Nu x = x q w k F ( T w − T ∞ ) , Sh x = x q m D ( C w − C ∞ ) .

The shear stress, heat flux, and mass flux are expressed as follows:

(26) τ w = ( μ THNF + k 1 ) ∂ u ¯ 1 ∂ y + ω 2 ∂ u ¯ 1 ∂ y 2 y = 0 + k 1 N ∣ y = 0 , q w = q r − k THNF ( T ) ∂ T ∂ y y = 0 , q m = − D ( C ) ∂ C ∂ y y = 0 .

The dimensionless forms of these physical quantities are expressed as follows:

(27) Re x C f x = μ THNF μ F + K f ″ ( 0 ) + W e 2 f ″ ( 0 ) 2 − n K f ″ ( 0 ) , Sh x = − Re x ( 1 + ε 2 h ( 0 ) ) h ′ ( 0 ) , Nu x = − Re x ( 1 + ε θ ( 0 ) ) k THNF k F + 4 3 R d ( 1 + ( θ ω − 1 ) θ ( 0 ) 3 ) θ ′ ( 0 ) .

Here, Re x = x u w ν F is the Reynolds number.

2.4 Thermophysical properties of the ternary hybrid nanofluid

The first nanoparticle in this study is Al 2 O 3 (spherical), the second nanoparticle is SiC (spherical), and the third nanoparticle is MWCNT (cylindrical). Blood was considered the working fluid. The thermophysical properties of ternary hybrid nanofluids, such as the density, viscosity, heat capacity, thermal conductivity, and electrical conductivity, are defined as follows [4,24] (Table 1).

Table 1

Models employed to describe the thermophysical description of the ternary hybridized nanofluid

ρ (density)	ρ THNF ρ F = ( 1 ‒ ϕ 1 ‒ ϕ 2 ‒ ϕ 3 ) + ϕ 1 ρ 1 ρ F + ϕ 2 ρ 2 ρ F + ϕ 3 ρ 3 ρ F
μ (viscosity)	μ THNF = ( ϕ 1 μ NF , 1 + ϕ 2 μ NF , 2 + ϕ 3 μ NF , 3 ) ϕ 1 + ϕ 2 + ϕ 3 , here μ NF , i = μ F ( 1 + B i ( ϕ 1 + ϕ 2 + ϕ 3 ) + C i ( ϕ 1 + ϕ 2 + ϕ 3 ) 2 )
ρC p (heat capacity)	( ρC p ) THNF ( ρC p ) F = ( 1 ‒ ϕ 1 ‒ ϕ 2 ‒ ϕ 3 ) + ϕ 1 ( ρC p ) 1 ( ρC p ) F + ϕ 2 ( ρC p ) 2 ( ρC p ) F + ϕ 3 ( ρC p ) 3 ( ρC p ) F
k (thermal conductivity)	k THNF = ( k NF , 1 ϕ 1 + k NF , 2 ϕ 2 + k NF , 3 ϕ 3 ) ( ϕ 1 + ϕ 2 + ϕ 3 ) , here k NF , i k F = k p . i + ( m i ‒ 1 ) k F + ( m i ‒ 1 ) ( ϕ 1 + ϕ 2 + ϕ 3 ) ( k p , i ‒ k F ) k p . i + ( m i ‒ 1 ) k F ‒ ( ϕ 1 + ϕ 2 + ϕ 3 ) ( k p , i ‒ k F ) .
σ (electric conductivity)	σ THNF σ HNF = ( 1 + 2 ϕ 1 ) σ 1 + ( 1 ‒ 2 ϕ 1 ) σ HNF ( 1 ‒ ϕ 1 ) σ 1 + ( 1 + ϕ 1 ) σ HNF , here σ HNF σ NF = ( 1 + 2 ϕ 2 ) σ 2 + ( 1 ‒ 2 ϕ 2 ) σ NF ( 1 ‒ ϕ 2 ) σ 2 + ( 1 + ϕ 2 ) σ NF , and σ NF σ F = ( 1 + 2 ϕ 3 ) σ 3 + ( 1 ‒ 2 ϕ 3 ) σ F ( 1 ‒ ϕ 3 ) σ 3 + ( 1 + ϕ 3 ) σ F

ϕ 1 , ϕ 2 , and ϕ 3 are the volume fractions of nanoparticles 1, 2, and 3, respectively.

Table 2 discusses the thermophysical properties of ordinary fluids and nanoparticles, whereas factors for establishing the shape and properties of nanoparticles are listed in Table 3.

Table 2

Thermophysical features of the nanoparticles [4,25,26]

Physical features	Regular fluid	Nanoparticles
Physical features	Blood	Al 2 O 3	MWCNT	SiC
C p ( J / kgK )	4180.0	0.773	3,000	1,340
ρ ( kg / m 3 )	1000.0	3,970	2,600	3,210
ϕ	—	0.03	0.01	0.02
k ( W / mK )	0.492	40	740	350

Table 3

Characteristics and shape of the nanoparticles [4,25]

Shape of nanoparticles	B	C	m
Platelets	37.10	612.60	5.70
Spherical	2.50	6.20	3.0
Cylindrical	13.50	904.40	4.90

3 Entropy generation

The local entropy generation of irreversible heat transfer, thermal radiation, viscous dissipation, and EMHD with electroosmotic forces is defined as follows [17,18,21]:

(28) S G = k F T ∞ 2 k THNF k F ( 1 + ε θ ) + 16 σ ⁎ T ∞ 3 3 k ⁎ k F ∂ T ∂ y 2 + μ THNF + k 1 T ∞ 1 + ω 2 ∂ u ¯ 1 ∂ y ∂ u ¯ 1 ∂ y 2 + μ THNF K p u ¯ 1 + σ THNF T ˆ ∞ { ( B 0 u ¯ 1 − E 0 ) 2 } + RD T ∞ ∂ T ∂ y ∂ C ∂ y + RD C ∞ ∂ C ∂ y ∂ C ∂ y .

The correlated relationship can structure dimensionless entropy generation:

(29) N G ( ζ ) = S G S 0 .

After employing the similarity transformation (10), the dimensionless form of entropy generation is developed as follows:

(30) N G ( ζ ) = k THNF k F ( 1 + ε θ ) + R d ( 1 + ( θ w − 1 ) θ ) 3 × Re θ ′ 2 + Re α 1 α 2 2 h ′ 2 + Re Φ c α 1 α 2 θ ′ h ′ , + Re Br α 1 σ THNF σ F M ( E − f ′ ) 2 + D a f ′ + μ THNF μ F + K ( 1 + W e f ′ ) f ″ 2 .

The parameters used in the above equation are defined as

(31) α 1 = Δ T T ∞ , α 2 = Δ C C ∞ , Br = Pr E c , Φ c = RD ( C w − C ∞ ) k T ∞ .

4 Numerical algorithm

This section provides a detailed explanation of the numerical solution technique used to solve the governing equations (18)–(21) and the related boundary conditions (22) and (23) using the bvp4c methodology. After careful consideration, the convergence criteria 10 − 6 and a step size of 0.001 were selected. To facilitate efficient computations using the BVP4C function, the given equations must be converted into a system of first-order ODEs (Figure 2). Several variables were added to the transformation in order to ensure that the BVP4C function operates correctly on this system of first-order ODEs.

Figure 2

Numerical scheme.

The following components make up the first-order ODE system:

(32) f = y ( 1 ) , f ′ = y ( 2 ) , f ″ = y ( 3 ) , f ‴ = y y 1 , g = y ( 4 ) , g ′ = y ( 5 ) , g ″ = y y 2 , θ = y ( 6 ) , θ ′ = y ( 7 ) , θ ″ = y y 3 , h = y ( 8 ) , h ′ = y ( 9 ) , h ″ = y y 4 ,

(33) y y 1 = 1 μ THNF μ F + K ( 1 + W e y ( 3 ) ) − ρ THNF ρ F ( y ( 1 ) y ( 3 ) − y ( 2 ) y ( 2 ) − F r y ( 2 ) y ( 2 ) ) − Ky ( 5 ) − U HS m e 2 exp ( − m e ζ ) + σ THNF σ F M ( y ( 2 ) − E 1 ) − μ THNF μ F D a y ( 2 ) ,

(34) y y 2 = 1 μ THNF μ F + K 2 ρ THNF ρ F ( y ( 4 ) y ( 2 ) − y ( 1 ) y ( 5 ) ) + K ( y ( 3 ) + 2 y ( 4 ) ) ,

(35) y y 3 = 1 k THNF k F ( 1 + ε y ( 6 ) ) + 4 3 R d ( 1 + ( θ ω − 1 ) y ( 6 ) ) 3 − S c D u S u Pr ( ρC p ) THNF ( 1 + ε 2 y ( 8 ) ) ( ρC p ) F × − k THNF k F ε + 4 R d ( 1 + ( θ ω − 1 ) y ( 6 ) ) 2 ( θ ω − 1 ) y ( 7 ) 2 − Pr μ THNF μ F + K 1 + W e 2 y ( 3 ) y ( 3 ) 2 + σ THNF σ F M E c ( y ( 2 ) − E 1 ) 2 − Pr ( ρC p ) THNF ( ρC p ) F D u ( 1 + ε 2 y ( 8 ) ) Sc ( γ 2 + y ( 8 ) ) y ( 2 ) − y ( 1 ) y ( 9 ) + K r ( 1 + δ y ( 6 ) ) m 1 exp − E 1 ⁎ 1 + δ y ( 6 ) y ( 8 ) − ε 2 y ( 9 ) 2 + y ( 1 ) y ( 7 ) − ( γ 1 + y ( 6 ) ) y ( 2 ) ,

(36) y y 4 = 1 ( 1 + ε 2 y ( 8 ) ) Sc ( γ 2 + y ( 8 ) ) y ( 2 ) − y ( 1 ) y ( 9 ) − S u y y 3 + K r ( 1 + δ y ( 6 ) ) m 1 exp − E 1 ⁎ 1 + δ y ( 6 ) y ( 8 ) − ε 2 y ( 9 ) 2 .

The boundary conditions become

(37) y 1 ( 0 ) = s 1 , y 2 ( 0 ) = 1 , y 5 ( 0 ) = − n y 3 ( 0 ) , y 6 ( 0 ) = 1 − γ 1 , y 8 ( 0 ) = 1 − γ 2 y 2 ( ∞ ) → 0 , y 4 ( ∞ ) → 0 , y 6 ( ∞ ) → 0 , y 8 ( ∞ ) → 0 .

It is clear from Table 4 that the same results of heat transfer rate ( − θ ' ( 0 ) ) are obtained by Eldabe et al. [22], Wang [27], and Nadeem and Hussain [28] in the absence of ternary nanofluids and microrotation motion for several values of Pr , as M = 0 = m e = U HS = γ 1 = θ w = S u = D u .

Table 4

Comparisons of the Nusselt number ‒ θ ' ( 0 ) for different values of Pr

Pr	Eldabe et al. [22]	Wang [27]	Nadeem and Hussain [28]	Present results
0.07	0.066	0.066	0.066	0.06610
0.20	0.169	0.169	0.169	0.16921
0.70	0.454	0.454	0.454	0.45420
2.00	0.911	0.911	0.911	0.91150

5 Graphical and numerical results

In this section, we investigate radiative and chemically reactive flows of a 2D Williamson micropolar ternary hybrid nanofluid flow with Soret and Dufour theory influenced by varied fluid properties and entropy generation on a stretching surface numerically and graphically. The BVP4C MATLAB scheme is used to obtain the numerical and graphical results. The graphical outcomes against several parameters on the linear and angular velocity, temperature, concentration distribution, and entropy generation due to UNF and THNF are shown. Furthermore, the dashed line used for UHF and solid lines used for THNF indicate that the magnitude of THNF is greater than that of the UHF. Figure 3(a)–(d) shows the graphical changes in the linear velocity along the W _e (Weissenberg number), K (material parameter), M (magnetic parameter), and E 1 (electric parameter). Figure 2(a) and (b) shows the trend of We and K on the linear velocity profile. The results indicate that stronger values of W _e reduce the fluid velocity while an opposite trend is observed for K for both THNF and UNF. The reason for an increase in the velocity is that due to enhancement in the Williamson parameter, an additional shear stress enters the fluid, which increases the fluid viscosity. Moreover, stronger K collisions in internally structured particles reduce the viscosity of the fluid and, therefore, increase the fluid velocity. The impact of the magnetic and electric parameters on the fluid velocity is shown in Figure 3(c) and (d). The plots indicate that the estimation of M and E 1 improves the fluid velocity for both THNF and UNF. Physically, the aiding force is stronger than the retarding force for M ≤ 1 ; therefore, the fluid velocity decreases. With a further increase in the magnetic parameter, the aiding and retarding forces reach an equilibrium condition, which is called the critical point, and beyond the critical point, the retarding force is dominant, and the fluid velocity decreases.

$Figure 3 (a)–(d) Changes in the linear velocity graphs for W e, K , M K,M , and E 1 {E}_{1} , respectively.$

Figure 3

(a)–(d) Changes in the linear velocity graphs for W _e, K , M , and E 1 , respectively.

Figure 4(a)–(d) shows the impact of porosity ( D a ), suction, and injection parameter ( s 1 ), and dimensionless Helmholtz–Smoluchowski velocity ( U HS ) on the velocity distribution. The velocity of the fluid and associated boundary layer thickness decrease due to an increase in D a (Figure 4(a)). The reason for the decrease in the velocity field is that the permeability of fluid increases due to stronger D a , which reduces the drag of fluid particles. Figure 4(b) and (c) shows the influence of the suction and injection parameters on the fluid velocity, respectively. It is clear that the velocity profile tends to decrease with larger values of s 1 . The fluid closest to the boundary increases for s 1 < 0 (injection), increasing the flow velocity and molecule-to-molecule collisions, which increase the internal kinetic energy. In contrast, when s 1 > 0 (suction), the fluid close to the boundary is drawn in, resulting in the creation of porosity in the vicinity of the boundary and a subsequent decrease in the velocity profile. Figure 4(d) shows the changes in the velocity of the dimensionless Helmholtz–Smoluchowski on the flow fluid. It is observed that the increasing values of U HS improves the fluid flow and the corresponding BL thickness for both THNF and UNF.

$Figure 4 (a)–(d) Changes in the linear velocity graphs for D a {D}_{a} , − s 1 {-s}_{1} , − s 1 -{s}_{1} , and U HS {U}_{{\rm{HS}}} , respectively.$

Figure 4

(a)–(d) Changes in the linear velocity graphs for D a , − s 1 , − s 1 , and U HS , respectively.

Figure 5(a)–(d) shows the changes in the various parameters of the microrotation profile. Figure 5(a) and (b) shows that due to the enhancement of the Weissenberg number and material parameter, the angular velocity of the fluid decreases for higher values of We and improves for greater K . This is because the increase in K generates more rotation in the fluid particles, which results in the increase of the microrotation sketch for both UNF and THNF. Figure 5(c) and (d) shows the impact of suction and injection parameters on the angular velocity profile for both THNF and UNF. These figures suggest that the micromotion component decreases with increasing s 1 > 0 (suction), but increases slightly with increasing s 1 < 0 (injection).

$Figure 5 (a)–(d) Changes in the angular velocity graphs for W e, K K , − s 1 {-s}_{1} , and − s 1 , -{s}_{1,} respectively.$

Figure 5

(a)–(d) Changes in the angular velocity graphs for W _e, K , − s 1 , and − s 1 , respectively.

Figure 6(a)–(e) shows the variation in the fluid temperature subject to the D _u (Dufour number), ∈ (variable conductivity parameter), E _c (Eckert number), R d (radiation parameter), and γ 1 (thermal stratification parameter) for both UNF and THNF. It is shown in Figure 6(a)–(c) that the temperature of the fluid increases due to the enhancement of D _u, ∈, and E _c. The reason for the increase in the temperature is that thermal energy converted into K.E. by the increase in ϵ and E _c, which increases the temperature of the fluid. The upshot of the radiation is observed in Figure 6(d). The increase in the heat transfer rate was significantly influenced by the thermal radiation parameter. This can be linked to the phenomenon where, with higher thermal radiation, the mean absorption coefficient decreases, leading to a diffusion flux driven by the temperature gradient. The effect of the thermal stratification parameter on temperature distribution is shown in Figure 6(e). It was found that the temperature of the boundary layers decreased with increasing thermal stratification values. Thermal stratification is the formation of two distinct layers of water at varying temperatures in a lake. Greater values of S resulted in greater temperature variations between the surfaces, which eventually lowered the temperature of the fluid.

$Figure 6 (a)–(e) Changes in temperature graphs for D u, ∈, E c, R d {R}_{{\rm{d}}} , and γ 1 , {\gamma }_{1,} respectively.$

Figure 6

(a)–(e) Changes in temperature graphs for D _u, ∈, E _c, R d , and γ 1 , respectively.

As shown in Figure 7(a)–(e), the variation in the concentration distribution is due to different values of the parameters like S _u (Soret number), ∈ ₂ (variable mass diffusivity), S _u (Schmidt number), K r (chemical reaction), and γ 2 (concentration stratification), respectively. From Figure 7(a), we conclude that an improvement occurs in the concentration profile due to an increase in S _u. Physically, S _u variation is related to the temperature gradient; therefore, higher values of S _u correspond to a stronger temperature gradient, which results in the improvement in the concentration graph. The concentration field is plotted versus the mass diffusivity parameter (∈ ₂) in Figure 7(b). This demonstrates that as ∈ ₂ increases, the mass diffusivity also increases, indicating an increase in the concentration. Figure 7(c) shows the impact of the Schmidt number S _c on the temperature profile. The figure clearly shows that the concentration field decreases with higher values of S _u. Physically, the momentum viscosity rate increases by the greater estimation of the Schmidt number; as a result, the nanoparticle concentration decreases. Figure 7(d) shows the impact of K r on the concentration profile. This demonstrates that stronger values of K r decrease the nanoparticle concentration and the related boundary layer thickness. The concentration field was compared to the solutal-stratified parameter, as shown in Figure 7(e). A higher γ 2 causes a reduction in the concentration. Physically, as γ 2 increases, the difference between the ambient and surface concentrations decreases. Thus, the concentration field becomes smaller for both the UNF and the THNF.

$Figure 7 (a)–(e) Changes in the concentration field graph for S u, ∈ 2, S c, K r , {K}_{{\rm{r}}},\hspace{0.25em} and γ 2 , {\gamma }_{2,} respectively.$

Figure 7

(a)–(e) Changes in the concentration field graph for S _u, ∈ ₂, S _c, K r , and γ 2 , respectively.

The effect of Br (Brinkman number) and α 1 (temperature difference parameter) on the entropy generation profile for both cases of UNF and THNF is shown in Figure 8(a) and (b). Figure 8(a) shows that higher values of Br produce more heat between the fluid layers and boundary region, which shows an improvement in the entropy generation. Moreover, Figure 8(b) exhibits that the degree of disorderness increases for larger values of α 1 , which is an indication of the increase in entropy generation. To observe the flow behavior, heat transfer rate, and mass transfer numerically, we obtained comparative numerical results for THNF and UNF, as shown in Tables 5–7. The results in Table 5 show that larger values of U HS and E 1 reduce the skin friction, but shear stress improves for various values of W _e, M, and K because the retardation effect increases the friction effect for both THNF and UNF. Table 6 represents the variation in the heat transfer rate for several values of ∈, E _c, D _u , and γ 1 . It is worth mentioning that larger estimation of ϵ , Ec , Du , and γ 1 diminishes the heat transfer rate for both THNF and UNF. The mass transfer rate variation against the E 1 * , S _u, S _c , and ∈ ₂ is listed in Table 7. The tabulated values describe that due to the growth of S _u and S _c , the mass transfer rate increases, while a reverse behavior of mass transfer is observed for the growth of E 1 * and ∈ ₂ for both THNF and UNF. Furthermore, from the numerical data, one can see that the results of the ternary hybrid nanofluid are more prominently associated with the UNF.

$Figure 8 (a) and (b) Changes in the entropy generation graph for Br {\rm{Br}} and α 1 {\alpha }_{1} .$

Figure 8

(a) and (b) Changes in the entropy generation graph for Br and α 1 .

Table 5

Skin friction ( Re x C f x ) values for the ternary hybrid nanofluid and UNF

W e	M	U HS	E 1	K	Ternary hybrid nanofluid ( ‒ Re x C f x )	UNF ( ‒ Re x C f x )
0.0	0.01	0.02	0.4	0.5	2.01258	1.48973
0.1					2.18684	1.46557
0.2					2.21952	1.43912
0.1	0.0				2.08193	1.46269
	0.1				2.13086	1.49226
	0.2				2.17919	1.52296
	0.01	0.0			2.08881	1.46751
		0.5			2.07898	1.45790
		1.0			2.06922	1.44847
		0.02	0.5		2.45645	1.36759
			0.6		2.45182	1.36544
			0.7		2.44723	1.35349
		0.1	0.5	0.0	2.02899	1.36997
				0.2	2.29067	1.78143
				0.4	2.55386	2.13359

Table 6

Nusselt numbers for several parameters

ε	E c	D _u	γ 1	Ternary hybrid nanofluid ( Nu x )	UNF ( Nu x )
0.0	0.5	0.2	0.01	1.957657	0.657157
1.0				1.784173	0.584273
2.0				1.450656	0.450456
0.1	0.0			1.116276	0.320140
	2.0			1.108936	0.210789
	4.0			1.097924	0.196762
	0.5	0.0		1.995376	0.895661
		1.0		1.833928	0.755410
		2.0		1.672258	0.619921
		0.2	0.0	1.519932	0.297709
			0.1	1.496053	0.217802
			0.2	1.436654	0.146418

Table 7

Sherwood numbers for several parameters

E 1 ⁎	S _u	ε 2	S c	Ternary hybrid nanofluid ( Sh x )	UNF ( Sh x )
0.0	0.1	0.1	0.1	1.323321	0.301579
0.5				1.317669	0.295599
1.0				1.309449	0.286909
0.5	0.0			1.316919	0.294986
	1.0			1.324421	0.301126
	2.0			1.331940	0.310374
	0.1	0.0		1.377145	0.295599
		0.5		1.304136	0.239074
		0.9		1.260425	0.206175
		0.1	0.0	1.162235	0.162235
			0.2	1.555366	0.427208
			0.5	1.963987	0.792321

6 Conclusion

In this investigation, we elaborate on the numerical study of the 2D radiative and thermally reactive flow of a micropolar Williamson THNF under the effect of EOF and EMHD on the stretching surface. The suction/injection effect along the stratification boundary was also analyzed in this study. Moreover, the impact of cross-diffusion with variable fluid properties was used to examine the thermal features of the problem. The main outcomes of this study are as follows.

The comparison results of the ternary hybrid nanofluid and UNF on the velocity field for various parameters are presented.
The dimensionless Helmholtz-Smoluchowski velocity produced an increment in the momentum boundary layer.
A larger estimation of the magnetic and electric field parameters increases the fluid velocity because the aiding force opposes the retarding force.
Greater collusion and rotation in internally structured particles occur owing to the enlargement of K , which increases both the linear and angular velocity of the fluid.
Higher values of D _u and S _u number correspond to stronger temperature gradients, which results in improved temperature and concentration, respectively.
Stronger heat is produced between the fluid and boundary region due to an increase of Br , thereby improving the entropy generation for both UNF and THNF.
The skin friction rate reduces for a larger estimation of U HS and E 1 , while it is larger for W e , M, and K for both UNF and THNF.
The heat and mass transport rates in the UNF were lower than those in the ternary hybrid nanofluid.

7 Future work

This study can be extended to examine hybrid nanofluids with two base fluids, investigate stagnation point flow over various geometries under unsteady conditions, and explore the behavior of other non-Newtonian fluids under different physical conditions. The findings of this study can be applied to high-flow potential flow wells, hydraulically fractured wells, reservoirs, and other situations involving fluid flows. This model is also applicable to gas turbines, aircraft technology, rotating machinery, medical devices, and other fields.

Acknowledgments

The author would like to acknowledge Deanship of Graduate Studies and Scientific Research, Taif University for funding this work. The authors acknowledge the Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education (No. ARES-2023-05), and Senior Talent Starting Foundation of Jiangsu University Grant (No. 21JDG014)

Funding information: This study was funded by Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education (No. ARES-2023-05) and Senior Talent Starting Foundation of Jiangsu University Grant (No. 21JDG014).
Author contributions: 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: The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2023-12-17

Revised: 2024-04-25

Accepted: 2024-07-21

Published Online: 2024-09-17

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

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https://doi.org/10.1515/ntrev-2024-0081

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variable fluid properties; micropolar-Williamson ternary nanofluid model; entropy generation with Soret and Dufour effect

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