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Magnetized water-based hybrid nanofluid flow over an exponentially stretching sheet with thermal convective and mass flux conditions: HAM solution

  • Showkat Ahmad Lone , Zehba Raizah , Anwar Saeed and Gabriella Bognár EMAIL logo
Published/Copyright: April 3, 2024
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Nanotechnology Reviews
From the journal Nanotechnology Reviews Volume 13 Issue 1

Abstract

The boundary-layer flow on a shrinking/contracting sheet has abundant industrial applications, which include continuous glass casting, metal or polymer extrusions, and wire drawing. In this regard, the present analysis focuses the hybrid nanofluid flow on an exponentially extending sheet. The water-based hybrid nanofluid flow contains CoFe2O4 and TiO2 nanoparticles. Heat transfer rate analysis involves the utilization of the Cattaneo–Christov heat flux model. Moreover, the Brownian motion and thermophoresis effects are used in this novel work. The mathematical model is presented in the form of system of partial differential equations, which is then transformed into system of ordinary differential equations (ODEs) using the similarity variables. The system of ODEs is evaluated by homotopy analysis method. The variation in the flow profiles has been investigated using figures and tables. The conclusions demonstrate that the effect of magnetic parameter is 52% better for hybrid nanofluid flow than for the pure water. Conversely, the increasing magnetic parameter diminishes the thermal transfer rates for water, TiO2–H2O, CoFe2O4–H2O, and TiO2–CoFe2O4/H2O. The increasing thermophoresis parameter upsurges the thermal flow rate of nanofluids and hybrid nanofluid, while the increasing Brownian motion parameter lessens the thermal transfer rates of nanofluids and hybrid nanofluid. The increasing effect of thermophoresis parameter is 39% better for hybrid nanofluid than for the base fluid. However, the declining impression of Brownian motion factor is 48% greater for hybrid nanofluid related to pure water.

Keywords: hybrid nanofluid flow; nanoparticles; three-dimensional; exponentially stretching sheet; thermal and mass flux conditions; Cattaneo–Christov heat flux model

Nomenclature

A

temperature exponent parameter

B

concentration exponent parameter

B 0

magnetic field strength ( kg s − 1 A − 1 )

Bi T

thermal Biot number

C

concentration ( mol m -3 )

C p

specific heat ( J kg − 1 K − 1 )

D B

coefficient of Brownian diffusion ( m 2 s − 1 )

D T

coefficient of thermophoresis diffusion ( m 2 s − 1 )

Ec

Eckert number

f 0 , g 0 , θ 0 , ϕ 0

initial guesses (−)

h f

coefficient of heat transfer ( W m − 2 K − 1 )

k

thermal conductivity ( W m − 1 K − 1 )

L

reference length (m)

L f , L g , L θ , L ϕ

linear operator (−)

M

magnetic parameter

m i ( i = 1 , 2 , 3 , … , 10 )

constants in general solution (−)

Nb , Nt

Brownian motion and thermophoresis factors

Sc , Pr

Schmidt and Prandtl numbers

w 1 , w 2 , w 3

velocity components ( m s − 1 )

x , y , z

coordinate axes (m)

Greek letters

ρ

density ( kg m − 3 )

μ

dynamic viscosity ( kg m − 1 s − 1 )

σ

electrical conductivity ( S m − 1 )

φ 1 , φ 2

nanoparticles’ volume fractions

Λ

thermal relaxation time parameter

α

ratio of rate parameter

Subscripts

∞

free stream

w

surface

f

fluid

nf

nanofluid

hnf

hybrid nanofluid

1 , 2

first and second nanoparticles

1 Introduction

As the technology continues advancement, heat transmission has been the most significant procedure. Transportation, thermal power generation, chemical processes, manufacturing, as well as several other applications and sectors requiring heat generation, demand effective thermal performance to obtain the best results. Various analyses have been attempted in recent years to expand the thermal transfer capabilities of the transmitting medium, which is referred to as thermal transmission of fluid. Maxwell [1] was the first one to present the concept of dissolving solid particles that have high thermal conductivity into a base fluid, which was later carried on by Hamilton and Crosser [2] to increase the thermal conductance of fluid. Despite this, numerous defects and restrictions remained, like clotting in the flow field channel. In response to this limitation, a rapid investigation was conducted, resulting in the development of nanofluids. Choi and Eastman [3] manufactured this innovative thermal transference fluid, and it is thought that owing to its unique characteristic, it will be able to avoid coagulation of the flow passage. With the continuous progress of technology, a novel type of thermal transfer fluid called hybrid nanofluid has emerged as advancement over traditional nanofluids. This innovative fluid is formulated by dispersing two types of particles (nanosized) with excellent thermal conductivity. Sarkar et al. [4] highlighted the dominance of hybrid nanofluid regarding their thermophysical properties, heat transmission and potential applications, and obstacles. According to the review, the appropriate hybridization of hybrid nanofluids might make them highly promising for heat transfer augmentation. Nabil et al. [5] offered a review article on the hybrid nanofluid flow. Sajid and Ali [6] presented a comprehensive review based on artificial neural networking, experimental and numerical studies of hybrid nanofluids. Devi and Devi [7] investigated the magnetized Cu-Al2O3/water hybrid nanofluid flow on elongating surface. Their results signified that the thermal transference rate of the Cu-Al2O3/water is better than that of the Cu/water. Zainal et al. [8] discussed hybrid nanoparticles’ flow containing Cu and Al2O3 nanoparticles. For the general synchronism of the electric and magnetic fields, they found that the magnetic field and suction slow down fluid movement. Furthermore, increases in the radiative heat parameter enhances the Nusselt number. Roy and Pop [9] inspected hybrid nanofluid flow over an extending sheet and showed that for the escalating values of magnetic, suction, Cu nanoparticles’ volume fraction, and second-grade parameters, the dual solutions’ existence region expands. Furthermore, stability analysis is conducted to identify the problem that has actually stable and unstable solutions. Sreedevi and Reddy [10] discussed Williamson hybrid nanoparticles’ fluid flow on a gyrating cylinder using the impacts of microorganisms and thermal as well as mass flux model suggested by Cattaneo-Christov. Babu et al. [11] investigated computationally the optimization of entropy for MHD nanofluid flow on a nonlinear elongating surface convective and slip constraints at the boundaries and have noted that nonlinear elongating features of the surface are supporting the thermal flow more effectively. Reddy and Sreedevi [12] examined the impacts of thermal as well as mass flux model suggested by Cattaneo–Christov on Maxwell hybrid nanofluid flow on a shrinking/elongating surface. Reddy and Sreedevi [13] explored thermal transportation for carbon nanotubes (CNTs) nanofluid flow in conduit using improved Fourier thermal flux. Harish Babu et al. [14] inspected influences and the impacts of inclined magnetic field for CNTs on exponentially elongating sheet using slip effects. Harish Babu et al. [15] discussed the collective influences of thermal flow and magnetic field on nanofluid flow over elastic surface. Further research on similar concept is based on previous studies [16–21].

The boundary-layer flow past a shrinking/elongating sheet has numerous applications toward manufacturing processes in the field of engineering, which include continuous glass casting, metal or polymer extrusions, and wire drawing. Crane [22] was the pioneer to investigate flow past a linearly extending sheet. Various flows over a stretched sheet and flows over a contracting surface have just recently been a focus of concern. Bhattacharyya [23] carried out the investigation that indicates the criteria for the presence of steady boundary-layer flow as a result of the exponential shrinking sheet. Their result showed that when the mass suction factor surpasses a specific critical value, steady flow is feasible. Mat Yuzut [24] investigated the heat transmission of MHD fluid flow on exponentially stretched surface. Their consequences showed that the transmission rate of heat was observed to be declined when the penetrability parameters were increased. Waini et al. [25] addressed the hybrid nanofluid flow on an extending/shrinking sheet. It was perceived that the heat transmission rate is augmented with the augmenting volume fraction of the nanoparticles. Further research on hybrid nanofluid flow on an extending/shrinking surfaces can be observed in the study by Ishak [26]. Abbas et al. [27] discussed computationally nonlinearly radiative MHD nanofluid flow on a vertical sheet and have noted that skin friction has augmented, while Nusselt number has weakened with expansion in concentration of nanoparticles. Abbas et al. [28] examined theoretically nanofluid flow on exponentially elongating surface with impacts of velocity at free stream. Abbas and Shatanawi [29] examined thermal and mass transportations for Casson nanofluid flow on a variable elongating Riga surface and have noted that velocity characteristics weakened with upsurge in micro-polar factor.

Heat and mass transfer is a major topic of research for its applications toward industries and engineering, such as food processing, biomedical sciences, business, and tissue conduction. Akbar et al. [30] discussed mixed convective and time-independent flow of a viscous fluid past an extending sheet with magnetic field impact. Their results showed that the augmenting species and thermal Grashof numbers have direct impact on thermal transference rate, while the thermophoresis and Schmidt number have reducing impact of heat transfer rate. Khan et al. [31] examined the squeezing flow over a sensor sheet. The results disclosed that growth in squeezing factor has improved the velocity panel of the non-Newtonian fluid. Sandeep et al. [32] examined the heat transmission of magnetohydrodynamic (MHD)-radiated flow of a dusty nanofluid containing Cu and CuO nanoparticles on an exponentially extending surface and have noted that thermal flow rate has augmented with escalation in nanoparticles’ volume fractions, while the wall fraction has reduced with augmenting nanoparticles’ volume fractions. Haq et al. [33] discussed the MHD flow of a thermally convective nanofluid past an exponentially expending surface with suction/injection effects. Heat and mass transference of the hybrid nanofluids flow toward different physical phenomena can be analyzed in previous studies [34,35,36,37,38,39,40,41].

The analysis of hybrid nanofluid flow past an exponentially extending sheet has many applications in various fields, especially in engineering, fluid dynamics, and heat transfer. Some of the these applications include heat exchangers, cooling systems, renewable energy systems, material processing, biomedical applications, aerospace engineering, oil and gas industries, and energy conversion systems. Keeping these applications in mind, the current analysis focuses on the flow of hybrid nanofluids over an exponentially extended surface with thermal convection and mass flux constraints. The aim of this analysis is to investigate the flow of hybrid nanofluids containing TiO 2 and CoFe 2 O 4 nanoparticles and H 2 O as a base fluid under thermal convection and mass flux conditions. Furthermore, the Cattaneo–Christov heat flow model is considered to analyze the heat and mass transfer flow. Furthermore, a strong magnetic field is considered to analyze the magnetized flow of the hybrid nanofluid over an extending sheet. Problem is formulated both physically and mathematically in Section 2. The homotopic solution of the present model is presented in Section 3, and the convergence of the applied technique is presented in Section 4. The physical discussion about the obtained results is presented in Section 5 and concluded in Section 6.

2 Formulation of the problem

Take a three-dimensional viscous flow of TiO 2 − CoFe 2 O 4 /H 2 O hybrid nanofluid on an exponentially elongating surface. The surface stretches in x - and y -directions with velocities W 1 w ( x + y ) = W 10 exp x + y L and W 2 w ( x + y ) = W 20 exp x + y L , respectively, where W 10 and W 20 are the stretching constants. A magnetic field B ( 0 , 0 , B 0 ) is practiced in normal direction to flow of fluid, where B 0 is the magnetic field strength. Figure 1 shows the physical representation of the flow problem. The temperature and concentration of fluid are represented by T and C . T w , T ∞ , C w , and C ∞ are the surface and free stream temperatures and concentrations, respectively. The stretching surface is also exposed to thermal convection and mass flux conditions. In order to examine the heat transfer rate, the Cattaneo–Christov heat flux model is used. The consequences of thermophoresis and Brownian motion are also assumed. Using the abovementioned suppositions, the principal equations are given in previous studies [42,43,44]:

(1) ∂ w 1 ∂ x + ∂ w 2 ∂ y + ∂ w 3 ∂ z = 0 ,

(2) ρ hnf w 1 ∂ w 1 ∂ x + w 2 ∂ w 1 ∂ y + w 3 ∂ w 1 ∂ z = μ hnf ∂ w 1 2 ∂ z 2 − σ hnf B 0 2 w 1 ,

(3) ρ hnf w 1 ∂ w 2 ∂ x + w 2 ∂ w 2 ∂ y + w 3 ∂ w 2 ∂ z = μ hnf ∂ w 2 2 ∂ z 2 − σ hnf B 0 2 w 2 ,

(4) w 1 ∂ T ∂ x + w 2 ∂ T ∂ y + w 3 ∂ T ∂ z + λ w 1 2 ∂ 2 T ∂ x 2 + w 2 2 ∂ 2 T ∂ y 2 + w 3 2 ∂ 2 T ∂ z 2 + 2 w 1 w 2 ∂ 2 T ∂ x ∂ y + 2 w 2 w 3 ∂ 2 T ∂ y ∂ z + 2 w 1 w 3 ∂ 2 T ∂ x ∂ z + ∂ T ∂ x w 1 ∂ w 1 ∂ x + w 2 ∂ w 1 ∂ y + w 3 ∂ w 1 ∂ z + ∂ T ∂ y w 1 ∂ w 2 ∂ x + w 2 ∂ w 2 ∂ y + w 3 ∂ w 2 ∂ z + ∂ T ∂ z w 1 ∂ w 3 ∂ x + w 2 ∂ w 3 ∂ y + w 3 ∂ w 3 ∂ z = k hnf ( ρ C p ) hnf ∂ 2 T ∂ z 2 + σ hnf ( ρ C p ) hnf ( w 1 2 + w 2 2 ) + ( ρ C p ) np ( ρ C p ) hnf D B ∂ T ∂ z ∂ C ∂ z + D T T ∞ ∂ T ∂ z 2 ,

(5) w 1 ∂ C ∂ x + w 2 ∂ C ∂ y + w 3 ∂ C ∂ z = D T T ∞ ∂ 2 T ∂ z 2 + D B ∂ 2 C ∂ z 2 ,

with boundary conditions [42,43,44]:

(6) w 1 = W 1 w ( x + y ) = W 10 exp x + y L , w 2 = W 2 w ( x + y ) = W 20 exp x + y L , w 3 = 0 , k hnf ∂ T ∂ z = − h f ( T f − T ) , D T T ∞ ∂ T ∂ z + D B ∂ C ∂ z = 0 , at z = 0 , w 1 → 0 , w 2 → 0 , T → T ∞ , C → C ∞ , } as z → ∞ .

Figure 1 Geometrical view of flow problem.
Figure 1

Geometrical view of flow problem.

The thermophysical characteristics of the nanofluid and hybrid nanofluid, respectively, are described as follows with their numerical values depicted in Table 1:

(7) μ hnf μ f = 1 ( 1 − φ 1 , 2 ) 2.5 , ρ hnf ρ f = ( 1 − φ 1 , 2 ) + ρ 1 , 2 φ 1 , 2 ρ f , ( ρ C p ) hnf ( ρ C p ) f = ( 1 − φ 1 , 2 ) + ( ρ C p ) 1 , 2 ( ρ C p ) 1 , 2 ( ρ C p ) f , σ hnf σ f = 1 + 3 σ 1 , 2 σ f − 1 φ 1 , 2 σ 1 , 2 σ f + 2 − σ 1 , 2 σ f − 1 φ 1 , 2 , k hnf k f = 1 + 3 k 1 , 2 k f − 1 φ 1 , 2 k 1 , 2 k f + 2 − k 1 , 2 k f − 1 φ 1 , 2 .

(8) μ hnf μ f = 1 ( 1 − φ 1 − φ 2 ) 2.5 , ρ hnf ρ f = ( 1 − φ 2 ) ( 1 − φ 1 ) + ρ 1 φ 1 ρ f + ρ 2 φ 2 ρ f ( ρ C p ) hnf ( ρ C p ) f = ( 1 − φ 2 ) ( 1 − φ 1 ) + ( ρ C p ) 1 φ 1 ( ρ C p ) f + ( ρ C p ) 2 φ 2 ( ρ C p ) f , σ hnf σ f = 1 + 3 ( φ 1 + φ 2 ) [ φ 1 σ 1 + φ 2 σ 2 − σ f ( φ 1 + φ 2 ) ] φ 1 σ 1 + φ 2 σ 2 + 2 σ f ( φ 1 + φ 2 ) − σ f ( φ 1 + φ 2 ) φ 1 σ 1 + φ 2 σ 2 − σ f ( φ 1 + φ 2 ) , k hnf k f = 1 + 3 ( φ 1 + φ 2 ) [ φ 1 k 1 + φ 2 k 2 − k f ( φ 1 + φ 2 ) ] φ 1 k 1 + φ 2 k 2 + 2 k f ( φ 1 + φ 2 ) − k f ( φ 1 + φ 2 ) φ 1 k 1 + φ 2 k 2 − k f ( φ 1 + φ 2 ) .

Table 1

Thermophysical features of the nanoparticles and pure fluid [45,46,47,48]

Base fluid/nanoparticles C p ρ k σ
H 2 O 4,179 997.1 0.6071 5.5 × 10−6
TiO 2 686.2 4,250 8.9538 2.38 × 106
CoFe 2 O 4 700 4,907 3.7 5.51 × 109

The following similarity transformations are according to flow assumptions:

(9) η = W 10 2 ν f L exp x + y 2 L z , w 1 = W 10 exp x + y L f ′ ( η ) , w 2 = W 20 exp x + y L g ′ ( η ) , w 3 = ν f W 10 2 L exp x + y 2 L { f ( η ) + g ( η ) + η ( f ′ ( η ) + g ′ ( η ) ) } , T = T ∞ + T 0 exp A ( x + y ) 2 L θ ( η ) , C = C ∞ + C 0 exp B ( x + y ) 2 L ϕ ( η ) .

By employing equation (9) in the leading equations (1)–(4), we have

(10) 1 ( 1 − φ 1 − φ 2 ) 2.5 f ‴ + ( 1 − φ 2 ) ( 1 − φ 1 ) + ρ 1 φ 2 ρ f + ρ 2 φ 2 ρ f f ″ { f + g } − 2 f ′ { f ′ + g ′ } − 1 + 3 ( φ 1 + φ 2 ) [ φ 1 σ 2 + φ 1 σ 2 − σ f ( φ 1 + φ 2 ) ] φ 1 σ 1 + φ 2 σ 2 + 2 σ f ( φ 1 + φ 2 ) − σ f ( φ 1 + φ 2 ) φ 1 σ 1 + φ 2 σ 2 − σ f ( φ 1 + φ 2 ) M 2 f ′ = 0 ,

(11) 1 ( 1 − φ 1 − φ 2 ) 2.5 g ‴ + ( 1 − φ 2 ) ( 1 − φ 1 ) + ρ 1 φ 2 ρ f + ρ 2 φ 2 ρ f g ″ { f + g } − 2 g ′ { f ′ + g ′ } − 1 + 3 ( φ 1 + φ 2 ) [ φ 1 σ 2 + φ 1 σ 2 − σ f ( φ 1 + φ 2 ) ] φ 1 σ 1 + φ 2 σ 2 + 2 σ f ( φ 1 + φ 2 ) − σ f ( φ 1 + φ 2 ) φ 1 σ 1 + φ 2 σ 2 − σ f ( φ 1 + φ 2 ) M 2 g ′ = 0 ,

(12) 1 + 3 ( φ 1 + φ 1 ) [ φ 1 k 1 + φ 2 k 2 − k f ( φ 1 + φ 2 ) ] φ 1 k 1 + φ 2 k 2 + 2 k f ( φ 1 + φ 2 ) − k f ( φ 1 + φ 2 ) φ 1 k 1 + φ 2 k 2 − k f ( φ 1 + φ 2 ) ( 1 − φ 2 ) ( 1 − φ 1 ) + ( ρ C p ) 1 φ 1 ( ρ C p ) f + ( ρ C p ) 2 φ 2 ( ρ C p ) f 1 Pr θ ″ + Nb θ ′ ϕ ′ + Nt Nb θ ″ − A θ { f ′ + g ′ } + A θ ′ { f + g } + Λ 2 { ξ ( f ′ + g ′ ) + ( 1 + 2 A ) ( f + g ) } ( f ′ + g ′ ) θ ′ − A ( A + 2 ) ( f ′ + g ′ ) 2 − ( f ″ + g ″ ) ( f + g ) θ − ( f + g ) 2 θ ″ − 1 + 3 ( φ 1 + φ 2 ) [ φ 1 σ 2 + φ 1 σ 2 − σ f ( φ 1 + φ 2 ) ] φ 1 σ 1 + φ 2 σ 2 + 2 σ f φ 1 + φ 2 − σ f φ 1 + φ 2 φ 1 σ 1 + φ 2 σ 2 − σ f ( φ 1 + φ 2 ) ( 1 − φ 2 ) ( 1 − φ 1 ) + ( ρ C p ) 1 φ 1 ( ρ C p ) f + ( ρ C p ) 2 φ 2 ( ρ C p ) f M 2 Ec { f ′ 2 + g ′ 2 } = 0 ,

(13) ϕ ″ − Sc B ( f ′ + g ′ ) ϕ + Sc ( f + g ) ϕ ′ + Nt Nb θ ″ = 0 ,

(14) f ( 0 ) = 0 , f ′ ( 0 ) = 1 , f ′ ( ∞ ) = 0 , g ( 0 ) = 0 , g ′ ( 0 ) = α , g ′ ( ∞ ) = 0 , k hnf k f θ ′ ( 0 ) = − Bi T ( θ ( 0 ) − 1 ) , θ ( ∞ ) = 0 , Nb ϕ ′ ( 0 ) + Nt θ ′ ( 0 ) = 0 , ϕ ( ∞ ) = 0 .

The dimensionless factors are defined as

(15) M 2 = 2 σ f L B 0 2 W 1 w ρ f , Pr = μ f ( ρ C p ) f k f , Ec = W 1 w 2 C p ( T w − T ∞ ) , Nb = ( ρ C p ) np ( ρ C p ) f D B ( C w − C ∞ ) ν f , Nt = ( ρ C p ) np ( ρ C p ) f D T ( T w − T ∞ ) ν f T ∞ , Sc = ν f D B , α = W 20 W 10 , Λ = λ W 1 w L , Bi T = γ = h f k f 2 ν f L W 1 w .

The skin friction coefficients along primary and secondary directions are illustrated as:

(16) C f x = τ w x 1 2 ρ f w 1 2 , where τ w x = μ hnf ∂ w 1 ∂ z z = 0 , C f y = τ w y 1 2 ρ f w 2 2 , where τ w y = μ hnf ∂ w 2 ∂ z z = 0 .

The dimensionless forms are written as

(17) Re x 2 exp − 3 ( x + y ) 2 L C f x = 1 ( 1 − φ 1 − φ 2 ) 2.5 f ″ ( 0 ) , Re x 2 exp − 3 ( x + y ) 2 L C f y = 1 ( 1 − φ 1 − φ 2 ) 2.5 g ″ ( 0 ) ,

Or

(18) C x = 1 ( 1 − φ 1 − φ 2 ) 2.5 f ″ ( 0 ) , C y = 1 ( 1 − φ 1 − φ 2 ) 2.5 g ″ ( 0 ) ,

where C x = Re x 2 exp − 3 ( x + y ) 2 L C f x and C y = Re x 2 exp − 3 ( x + y ) 2 L C f y . Also, Re x = w 1 w L ν f and Re y = w 2 w L ν f are the local Reynolds numbers.

The local Nusselt number is described as

(19) Nu x = x q w k f ( T w − T ∞ ) , where q w = − k hnf ∂ T ∂ z z = 0 .

In dimensionless form, we can write

(20) 2 L x exp − ( x + y ) 2 L Nu x Re x = − 1 + 3 ( φ 1 + φ 1 ) [ φ 1 k 1 + φ 2 k 2 − k f ( φ 1 + φ 2 ) ] φ 1 k 1 + φ 2 k 2 + 2 k f ( φ 1 + φ 2 ) − k f ( φ 1 + φ 2 ) φ 1 k 1 + φ 2 k 2 − k f ( φ 1 + φ 2 ) θ ′ ( 0 ) .

Or

(21) Nu = − 1 + 3 ( φ 1 + φ 2 ) [ φ 1 k 1 + φ 2 k 2 − k f ( φ 1 + φ 2 ) ] φ 1 k 1 + φ 2 k 2 + 2 k f ( φ 1 + φ 2 ) − k f ( φ 1 + φ 2 ) φ 1 k 1 + φ 2 k 2 − k f ( φ 1 + φ 2 ) θ ′ ( 0 ) .

where Nu = 2 L x exp − ( x + y ) 2 L Nu x Re x .

The Sherwood number is defined as

(22) Sh x = x q m D B ( C w − C ∞ ) , where q m = − D B ∂ C ∂ z z = 0 .

In dimensionless form, we can write

(23) 2 L x exp − ( x + y ) 2 L Sh x Re x = − ϕ ′ ( 0 ) .

Or

(24) Sh = − ϕ ′ ( 0 ) .

where Sh = 2 L x exp − ( x + y ) 2 L Sh x Re x .

3 Homotopy analysis method (HAM) solution

The HAM is the method used in this work to determine the solutions. The initial guesses and linear operators are defined as follows:

(25) f 0 ( ξ ) = 1 − e − ξ , g 0 ( ξ ) = α ( 1 − e − ξ ) , θ 0 ( ξ ) = k hnf k f Bi T 1 + Bi T e − ξ , ϕ 0 ( ξ ) = − Nt Nb Bi T 1 + Bi T e − ξ .

(26) L f ( f ) = f ‴ − f ′ , L g ( g ) = g ‴ − g ′ , L θ ( θ ) = θ ″ − θ , L ϕ ( ϕ ) = ϕ ″ − ϕ ,

with properties:

(27) L f ( ℓ 1 + ℓ 2 e − ξ + ℓ 3 e ξ ) = 0 , L g ( ℓ 4 + ℓ 5 e − ξ + ℓ 6 e ξ ) = 0 , L θ ( ℓ 7 e − ξ + ℓ 8 e ξ ) = 0 , L ϕ ( ℓ 9 e − ξ + ℓ 10 e ξ ) = 0 .

Here ℓ i ( i = 1 , 2 , 3 , … , 10 ) are the constants in general solution.

4 HAM convergence

In order to investigate the convergence of series solutions, HAM is applied. The liner auxiliary factors ℏ f , ℏ g , ℏ θ , and ℏ ϕ play a main role in regulating and adjusting the convergence areas of the modeled equations. Thus, the ℏ -curves for velocities, thermal, and concentration distributions are plotted in Figure 2. It can be understood from Figure 2 that these flow profiles converge in their respective regions.

Figure 2 ℏ \hslash -curves for f ″ ( 0 ) f^{\prime\prime} (0) , g ″ ( 0 ) g^{\prime\prime} (0) , θ ′ ( 0 ) \theta ^{\prime} (0) , and ϕ ′ ( 0 ) \phi ^{\prime} (0) .
Figure 2

ℏ -curves for f ″ ( 0 ) , g ″ ( 0 ) , θ ′ ( 0 ) , and ϕ ′ ( 0 ) .

5 Results and discussion

This segment deals the variations in velocities along x - and y -axes ( f ′ ( ξ ) , g ′ ( ξ ) ), temperature ( θ ( ξ ) ), and concentration ( ϕ ( ξ ) ) profiles via different physical parameters, which are displayed in Tables 3–5 and Figures 3–13. The ranges of the embedded factor are chosen as M = 0.4, Pr = 6.2, Ec = 0.2, Nb = 0.1, Nt = 0.1, Sc = 0.3, α = 1.0, Λ = 0.5, A = 0.2, B = 0.2, φ 1 = 0.04, φ 2 = 0.04, and Bi T = 0.5. Table 2 shows the assessment of the current consequences with previously issued reports and observed a strong promise with those published reports. Table 3 portrays the impact of M on skin friction along x-direction for water, TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O . It is found that the increase in M boosts the skin friction. Physically, growth in M augments Lorentz force that opposes velocity and hence the skin friction coefficients augment. Additionally, the effect of magnetic parameter is lower on pure water as compared to nanofluids and hybrid nanofluid. This is for mixing of solid nanoparticles in water, which increases the electrical conductance of base fluid, and hence, dominant impacts are found in nanofluids and hybrid nanofluid. Also, this effect is greater for TiO 2 − CoFe 2 O 4 / H 2 O than for CoFe 2 O 4 − H 2 O and is greater for CoFe 2 O 4 − H 2 O than for TiO 2 − H 2 O . Furthermore, the effect of magnetic parameter ( M = 1.5 ) is 52% greater for hybrid nanofluid flow than for the pure water. A similar impression of M is found on skin friction along y-direction for water, TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O , as portrayed in Table 4. Table 5 portrays the influence of various factors on Nusselt number for water, TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O . Growth in M reduces the heat transfer rates of water, TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O . Physically, the upsurge in M hikes the shear stress and causes a growing behavior in skin frictions and Nusselt number. Furthermore, Nusselt number is greater for TiO 2 − CoFe 2 O 4 / H 2 O , TiO 2 − H 2 O and CoFe 2 O 4 − H 2 O compared to pure water. Additionally, the greatest increasing behavior is found for TiO 2 − CoFe 2 O 4 / H 2 O . The increasing thermophoresis parameter increases the heat transfer rates of water, TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O , while the increasing Brownian motion parameter reduces the heat transfer rates of pure water, TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O . The increasing effect of thermophoresis parameter ( Nt = 0.4 ) is 39% higher for hybrid nanofluid than for pure fluid. Conversely, the declining impact of Nb = 0.4 is 48% higher for hybrid nanofluid. Figures 3 and 4 show the impression of M on velocity profiles ( f ′ ( ξ ) , g ′ ( ξ ) ). Growth in M lessens both f ′ ( ξ ) and g ′ ( ξ ) . The increasing M generates the Lorentz force that opposes the TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O profiles. This opposing force reduces the velocities profiles, as shown in Figures 3 and 4. Furthermore, the impact of M is greater for hybrid nanofluid flow in contrast of nanofluids. This effect is for greater electrical conductance of the hybrid nanofluid. Figures 5 and 6 demonstrate the impression of α on f ′ ( ξ ) and g ′ ( ξ ) . Growth in α diminishes the velocity profiles of the TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O along the x -direction, while reduces along the y -direction. The ratio parameter is defined as α = W 20 / W 10 , which clearly shows that there is a direct relation between the stretching constant W 20 and α , while an inverse relation between the stretching constant W 10 and α . Thus, the increasing α has direct relation to W 20 and reverse relation to W 10 . Therefore, the velocity profiles of the TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O along the x -direction reduce and the velocity profiles of the TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O along the y -direction augment. Figures 7 and 8 depict the impacts of Nt on θ ( ξ ) and ϕ ( ξ ) of the TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O . The increasing Nt means the strengthening of thermophoretic force. On surface of the stretching sheet, a higher concentration is witnessed, whereas the concentration reduces as the nanofluids and hybrid nanofluid moves away from the surface. Additionally, the higher impression is observed for both TiO 2 − H 2 O and CoFe 2 O 4 − H 2 O as compared to TiO 2 − CoFe 2 O 4 / H 2 O . On the other hand, the impact of Nt is slightly greater for TiO 2 − CoFe 2 O 4 / H 2 O as compared to TiO 2 − H 2 O and CoFe 2 O 4 − H 2 O . Figures 9 and 10 depict the effects of Nb on θ ( ξ ) and ϕ ( ξ ) of the TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O . Upsurge in Nb boosts θ ( ξ ) , while it diminishes ϕ ( ξ ) . The influence of Nb is better for TiO 2 − CoFe 2 O 4 / H 2 O as compared to TiO 2 − H 2 O and CoFe 2 O 4 − H 2 O . Figure 11 illustrates the effects of Λ on θ ( ξ ) of the TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O . The increasing Λ reduces the thermal profiles of TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O . The increasing Λ indicates that the nanofluids and hybrid nanofluid take extra time to transfer heat into surrounding particles, which consequently decline the thermal profiles. Additionally, the reducing impact is greater on TiO 2 − CoFe 2 O 4 / H 2 O as compared to C o F e 2 O 4 − H 2 O and TiO 2 − H 2 O . Figure 12 shows the impression of Bi T on thermal distribution of T i O 2 − H 2 O , C o F e 2 O 4 − H 2 O , and T i O 2 − C o F e 2 O 4 / H 2 O . The increasing Bi T escalates the thermal profiles. The increasing values of Bi T understand the augmentation in convective heating at the surface. It should be noted that the isothermal case is achieved for Bi T → ∞ and the isoflux wall condition is achieved for Bi T = 0 . It is obvious that the increasing Bi T has greater impact on the surface as revealed in Figure 12. Figure 13 illustrates the impression of Sc on concentration profiles of the TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O . Growth in Sc diminishes the concentration profiles. Schmidt number is defined as Sc = ν f D B . From here, we see that there is an inverse relation among Schmidt number and Brownian diffusion. So a hike in Schmidt number diminishes the Brownian diffusion of the fluid flow that ultimately diminishes the concentration distribution. Furthermore, the decreasing impact of Sc is dominant for TiO 2 − CoFe 2 O 4 / H 2 O .

Figure 3 Impact of M M on f ′ ( ξ ) f^{\prime} (\xi ) .
Figure 3

Impact of M on f ′ ( ξ ) .

Figure 4 Impact of M M on g ′ ( ξ ) g^{\prime} (\xi ) .
Figure 4

Impact of M on g ′ ( ξ ) .

Figure 5 Impact of α \alpha on f ′ ( ξ ) f^{\prime} (\xi ) .
Figure 5

Impact of α on f ′ ( ξ ) .

Figure 6 Impact of α \alpha on g ′ ( ξ ) g^{\prime} (\xi ) .
Figure 6

Impact of α on g ′ ( ξ ) .

Figure 7 Impact of Nt \text{Nt} on θ ( ξ ) \theta (\xi ) .
Figure 7

Impact of Nt on θ ( ξ ) .

Figure 8 Impact of Nt \text{Nt} on ϕ ( ξ ) \phi (\xi ) .
Figure 8

Impact of Nt on ϕ ( ξ ) .

Figure 9 Impact of Nb \text{Nb} on θ ( ξ ) \theta (\xi ) .
Figure 9

Impact of Nb on θ ( ξ ) .

Figure 10 Impact of Nb \text{Nb} on ϕ ( ξ ) \phi (\xi ) .
Figure 10

Impact of Nb on ϕ ( ξ ) .

Figure 11 Impact of Λ \Lambda on θ ( ξ ) \theta (\xi ) .
Figure 11

Impact of Λ on θ ( ξ ) .

Figure 12 Impact of Bi T {\text{Bi}}_{\text{T}} on θ ( ξ ) \theta (\xi ) .
Figure 12

Impact of Bi T on θ ( ξ ) .

Figure 13 Impact of Sc \text{Sc} on ϕ ( ξ ) \phi (\xi ) .
Figure 13

Impact of Sc on ϕ ( ξ ) .

Table 2

Comparison of the present results of θ ′ ( 0 ) with formerly reported outcomes, when φ 1 = φ 2 = 0.0

Pr A Liu et al. [49] Magyari and Keller et al. [50] Ramzan et al. [51] Present results
1 −1.5 0.37741256 0.377413 0.37741301 0.37714
0 −0.54964375 −0.549643 −0.54964339 −0.54964
1 −0.95478270 −0.954782 −0.95478277 −0.95478
3 −1.56029540 −1.560294 −1.56029499 −1.56029
5 −1.5 1.35324050 1.353240 1.35324055 1.35324
0 −1.52123900 −1.521243 −1.52123893 −1.52123
1 −2.50013157 −2.500135 −2.500135210 −2.50014
3 −3.88655510 −3.886555 −3.88655512 −3.88656
10 −1.5 −2.20002816 2.200000 2.20000798 2.20008
0 −2.25742372 −2.2574249 −2.25742910 −2.25743
1 −3.66037218 −3.660379 −3.66037911 −3.66038
3 −5.62819631 −5.635369 −5.635316812 −5.63532
Table 3

Impact of M on C x

ϕ 1 = ϕ 2 M TiO 2 ‒ H 2 O CoFe 2 O 4 ‒ H 2 O TiO 2 ‒ CoFe 2 O 4 / H 2 O
0.0 0.5 1.52250 1.52250 1.52250
0.0 1.0 1.70583 1.70583 1.70583
0.0 1.5 1.88917 1.88917 1.88917
0.05 0.5 1.83757 1.86401 2.23569
0.05 1.0 2.07889 2.10533 2.55160
0.05 1.5 2.32022 2.34665 2.86751
Table 4

Impact of M on C y

ϕ 1 = ϕ 2 M TiO 2 ‒ H 2 O CoFe 2 O 4 ‒ H 2 O TiO 2 ‒ CoFe 2 O 4 / H 2 O
0.0 0.5 0.15225 0.15225 0.15225
0.0 1.0 0.17058 0.17058 0.17058
0.0 1.5 0.18891 0.18891 0.18891
0.05 0.5 0.18375 0.18640 0.22356
0.05 1.0 0.20788 0.21053 0.25516
0.05 1.5 0.23202 0.23466 0.28675
Table 5

Impacts of M , Nt and Nb on Nu

ϕ 1 = ϕ 2 M Nt Nb TiO 2 ‒ H 2 O CoFe 2 O 4 ‒ H 2 O TiO 2 ‒ CoFe 2 O 4 / H 2 O
0.0 0.5 0.3 0.3 0.05032 0.05032 0.05032
0.0 1.0 0.05400 0.05400 0.05400
0.0 1.5 0.06022 0.06022 0.06022
0.0 0.4 0.08066 0.08066 0.08066
0.0 0.6 0.08302 0.08302 0.08302
0.0 0.8 0.08424 0.08424 0.08424
0.0 0.4 0.07532 0.07532 0.07532
0.0 0.6 0.06926 0.06926 0.06926
0.0 0.8 0.06320 0.06320 0.06320
0.05 0.5 0.3 0.3 0.05916 0.05758 0.11552
0.05 1.0 0.06056 0.05921 0.11593
0.05 1.5 0.06139 0.06019 0.11686
0.05 0.4 0.08027 0.04673 0.11244
0.05 0.6 0.08294 0.04933 0.11304
0.05 0.8 0.08433 0.05067 0.11335
0.05 0.4 0.07424 0.04087 0.11111
0.05 0.6 0.06740 0.03422 0.10960
0.05 0.8 0.06056 0.02757 0.10809

6 Conclusion

The present analysis focuses the hybrid nanofluid flow on an exponentially extending surface. The nanoparticles of CoFe2O4 and TiO2 are mixed in water to hybridize the fluid. Thermal flow rat involves the utilization of the Cattaneo–Christov heat flux model. The famous Buongiorno model has also used in this study. The mathematical model is evaluated by HAM. The concluding remarks are presented at the end of this analysis.

  1. It is found that upsurge in magnetic factor augments the skin friction coefficients along both directions. Furthermore, the effect of magnetic parameter ( M = 1.5 ) is 52% greater for hybrid nanofluid flow than for the pure water. However, the increasing magnetic parameter reduces the heat transfer rates of water, TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O . Additionally, the greatest reducing behavior is found for TiO 2 − CoFe 2 O 4 / H 2 O .

  2. The increasing thermophoresis parameter increases the heat transfer rates of water, TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O , while the increasing Brownian motion factor diminishes the heat transfer rates of pure water, TiO 2 − H 2 O , CoFe 2 O 4 − H 2 O , and TiO 2 − CoFe 2 O 4 / H 2 O . The increasing effect of thermophoresis parameter (Nt = 0.4) is 39% higher for hybrid nanofluid than for the pure fluid. On the contrary, the declining impact of Brownian motion factor ( Nb = 0.4 ) is 48% higher in case of hybrid nanofluid.

  3. The increasing magnetic parameter diminishes the velocity panels along both directions. Also, due to the greater electrical conductivity of fluid, the highest impact of magnetic parameter is found for hybrid nanofluid flow.

  4. The increasing ratio of rate factor reduces the velocity distribution of the nanofluid and hybrid nanofluid in the x-direction and augments along the y -direction.

  5. Increasing the thermophoresis, thermal Biot number, and Brownian motion factors raises the thermal panels of fluids, while increasing the thermal relaxation time lowers them.

  6. The concentration distributions of fluid are reduced when the Brownian motion factor and Schmidt number increase.

  1. Funding information: This study was supported by Project No. 129257 implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the K 18 funding scheme. The authors extends their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (RGP.2/505/44).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

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

  4. Data availability statement: The data that support the findings of this study are available from the corresponding author upon a reasonable request.

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Received: 2023-09-24
Revised: 2024-01-13
Accepted: 2024-02-22
Published Online: 2024-04-03

© 2024 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/ntrev-2023-0220
Keywords for this article
hybrid nanofluid flow; nanoparticles; three-dimensional; exponentially stretching sheet; thermal and mass flux conditions; Cattaneo–Christov heat flux model
Creative Commons
BY 4.0

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