Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source

Xiangning Zhou; Muhammad Amer Qureshi; Nargis Khan; Wasim Jamshed; Siti Suzilliana Putri Mohamed Isa; Nanthini Balakrishnan; Syed M. Hussain

doi:10.1515/phys-2024-0082

Article Open Access

Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source

Xiangning Zhou , Muhammad Amer Qureshi , Nargis Khan , Wasim Jamshed , Siti Suzilliana Putri Mohamed Isa , Nanthini Balakrishnan and Syed M. Hussain

Published/Copyright: September 9, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

In this work, the Marangoni convective flow of magnetohydrodynamic tangent hyperbolic ( F e 3 O 4 − Cu / ethylene glycol) hybrid nanofluids over a plate dipped in a permeable material with heat absorption/generation, heat radiation, elastic deformation and viscous dissipation is discussed. The impact of activation energy is also examined. Hybrid nanofluids are regarded as advanced nanofluids due to the thermal characteristics and emerging advantages that support the desire to augment the rate of heat transmission. The generalized Cattaneo–Christov theory, which takes into account the significance of relaxation times, is modified for the phenomena of mass and heat transfer. The fundamental governing partial differential equations are converted to ordinary differential equations (ODEs) by adopting similarity variables. The Runge–Kutta–Fehlberg-45 technique is utilized to solve nonlinear ODEs. Regarding the non-dimensional embedded parameters, a graphic investigation of the thermal field, concentration distribution, and velocity profile is performed. The results show that the increasing Marangoni ratio parameter enhances velocity and concentration distributions while decreases the temperature distribution. The velocity profile is decreased and the efficiency of heat transfer is improved as the porosity parameter is increased. Nusselt number is diminished with the rising values of the porosity variable.

Keywords: elastic deformation; tangent hyperbolic hybrid nanofluids; Cattaneo-Christov model; activation energy; Marangoni convection

Nomenclature

B 0: uniform magnetic field
C fx: skin friction
D B: mass diffusivity coefficient
Ec: Eckert number
E a: activation energy
K: porosity parameter
K = 6 π μ r: coefficient of drag Stokes
K *: chemical reaction co-efficient
K *: porous medium permeability
k +: mean absorption coefficient
k f: thermal conductivity of the fluid
k r 2: reaction rate
L: reference length
M: magnetic parameter
m: fitted rate constant
Ma: Marangoni ratio parameter
Mn: Marangoni number
n: power law index parameter
Nu x: Nusselt number
Q 0: temperature dependent source co-efficient
Q 1: temperature dependent source co-efficient
Q e: exponential dependent heat source parameter
Q t: temperature dependent heat source parameter
Pr: Prandtl number
q m: mass flux
q r: radiative heat flux
q w: heat flux
Rc: chemical reaction rate parameter
Rd: radiation parameter
Sc: Schmidt number
Sh x: Sherwood number
T 0: constants
We: Weissenberg number
( u , v ): velocity fields of fluid
( x , y ): Cartesian coordinates

Greek symbols

Φ 1: volume fraction of F e 3 O 4
Φ 2: volume fraction of Cu
μ f: dynamic viscosity
T ∞: fluid ambient temperature
γ 1: thermal relaxation parameter
γ 2: concentration relaxation parameter
γ C: surface tension coefficient for concentration
γ T: surface tension coefficient for temperature
ν f: kinematic viscosity
ρ f: fluid density
σ *: Stefan–Boltzmann constant
σ 0: surface tension
σ f: electrical conductivity
Γ: time constant
δ: elastic deformation parameter
σ: surface tension
ψ ( x , y ): stream functions

1 Introduction

The surface tension at the two fluids’ interface enables the Marangoni phenomenon to manifest in large quantities. This phenomenon was initially described in 1855. The surface tension gradient causes fluid particles to move from an area with lower surface tension to that with higher surface tension. Surface tension gradients are connected to temperature or concentration gradients. The Marangoni effect is used in a diversity of practical applications, including for convection cells, stabilization of soap films, crystal development, convection in Bénard cells, and melting of metals by an electron beam [1,2,3,4,5,6,7,8,9,10,11,12] . Of all the models of fluid with four constants, it is possible to employ the pseudoplastic fluid model to describe the shear thinning process in fluids. This phenomenon can be used in fluids that are tangent and hyperbolic. For example, soaked blood, ketchup, paint, nail polish, and whipped cream, to mention a few. Implementing the transfer of heat and the control of vertical asymmetric ducts, Nadeem and Akram [13] discussed the magnetohydrodynamic (MHD) peristaltic flow of tangent hyperbolic fluids. In the study conducted by Akbar et al. [14], it was identified that the same fluid flow can be modelled incorporating an extended surface. The physicochemical mechanisms of the tangent hyperbolic flow of nanoparticles in the MHD steady state conditions have been studied by Mahdy [15] while considering the effect of cylinder wall temperature variation. A study on the effect of partial slip on tangent hyperbolic fluid flow through an inclined cylinder has been conducted by Kumar et al. [16].

Choi [17] developed the idea of nanofluids, which are thermal conducting fluids consisting of 1–100 nm diameter nanoparticles dispersed in base fluids. Recent research has focused on mass and heat transfer in nanofluids because of their wide range of uses, including energy storage, cooling of electronic engines, solar collectors, heat exchangers, petroleum industries, photovoltaic cell thermal management, lubricants, machining technology, and drug delivery [18,19,20,21,22,23,24,25,26,27,28,29,30,31]. Figure 1 highlights the applications of nanoparticles.

Figure 1

Applications of nanoparticles.

Hybrid nanofluids are defined as mixtures of two dissimilar types of nanoparticles that are scattered in a base fluid. The base liquid’s thermal conductivity is different from that of hybrid nanoparticles, which can be formed from two different metals or nonmetallic materials. There are numerous uses for this kind of fluid that are connected to the process of heating, and such a fluid functions effectively where a high range of temperature is required for cooling. Several fields involve the potential uses of hybrid nanofluids in the fields of refrigeration, industry, solar energy, machining and manufacturing, space, defense, etc. [32,33,34,35]. Figure 2 depicts applications of hybrid nanofluids.

Figure 2

Applications of hybrid nanofluids.

The term “elastic deformation” describes a temporary change in a material’s shape that returns to its original state when the force is removed. Few scientists have discussed about how elastic deformation affects the MHD flow of nanofluids. A stretched surface with a heat source has been used to study the impact of elastic deformation on the MHD nanofluid flow by Ahmad et al. [36]. The outcome of elastic deformation on the flow of Walter’s-B fluid past an impermeable sheet was considered by Nandeppanavar et al. [37]. Khan et al. [38] examined the influence of elastic deformation on the flow of a viscoelastic fluid across a stretched sheet. Hakeem et al. [39] described how elastic deformation affected the flow of Walter’s-B fluid across a sheet with a heat source. Kalaivanan et al. [40] numerically analyzed the outcome of elastic deformation on second grade nanofluid flow across a stretched surface.

It is well recognized that when there is a temperature differential between two things or between several sections of one object, the natural phenomenon of heat exchange takes place. Fourier’s heat conduction equation has been successfully applied during the past few decades to explain the apparatus of heat transport. However, Cattaneo [41] proposed in 1948 that one might incorporate thermal relaxation time into Fourier’s model to produce an efficient heat transfer rate because different substances have various thermal relaxation times. The Cattaneo–Christov model is a time derivative model proposed by Christov [42]. Later, few scholars [43,44,45] used the Cattaneo–Christov model to describe the mechanisms of mass and heat transport.

To the authors’ knowledge, none of the aforementioned articles have yet addressed tangent hyperbolic ( F e 3 O 4 − Cu / EG) hybridized flow over a sheet with Marangoni convection. Marangoni convection finds application in a multitude of industrial, biological, and everyday circumstances, such as microfluidics, coating flow technologies, foams, and film drainage in emulsions. Therefore, this study’s primary goal is to examine the Marangoni driven boundary layer flow of tangent hyperbolic hybrid nanofluids with activation energy, elastic deformation, and binary chemical reaction. The following inquiries are answered by the current study using numerical and statistical methods:

What effects of the Marangoni ratio parameter (Ma) and Marangoni number (Mn) are seen on the solutal, velocity, and thermal profiles?
How do the rates of mass and heat transfer change as a result of the hybrid nanofluid?
How do thermal and solutal relaxation parameters affect thermal and solutal profiles?
How do the volume fraction parameters of nanoparticles affect the velocity as well as temperature distribution?
How does the thermal profile respond to the elastic deformation parameter?
In the presence of nanofluids, what effect does the power law index, porosity, and Weissenberg number ( We ) parameters have on the velocity profile?
How will the thermal profile be influenced by the an exponentially dependent parameter and heat source temperature?
How does the activation energy parameter affect the concentration profile?

2 Mathematical formulation

We investigate the behavior of a surface tension-driven boundary layer flow of tangent hyperbolic hybrid nanofluids with activation energy. The characteristics of mass and heat transmission in the tangent hyperbolic hybrid nanofluid flow over plate are examined by the Cattaneo–Christov mass and heat flux model. In the proposed model, we examined the two-dimensional incompressible hybrid nanofluid ( F e 3 O 4 − Cu / ethylene glycol) flowing over a surface with a magnetic field, as portrayed in Figure 3(c). In the thermal transfer examination, the functions of elastic deformation, thermal radiation, viscous dissipation and heat source are also presumed. Assume a rectangular coordinate system at y ≥ 0 . The properties of iron oxide ( Fe 3 O 4 ) and copper ( Cu ) nanoparticles with ethylene glycol as their base fluid are presented in Table 1. Table 2 shows theoretical representations of the nanofluid and hybrid nanofluid properties. Figure 3(a, b) display the implementations of iron oxide and Cu-nanomolecules. Surface tension varies throughout space and is affected by differences in temperature and concentration. The surface tension σ = σ 0 [ 1 − γ T ( T − T ∞ ) − γ C ( C − C ∞ ) ] is assumed to depend on linear variation with solutal and thermal profiles.

Figure 3

(a) and (b) Application of iron oxide and copper nanoparticles, respectively. (c) Flow description.

Table 1

Nanoparticles’ and base fluid’s thermophysical characteristics [48,49]

Constituents	c p ( J / kg K )	k ( W / m K )	σ ( Ω m ) ‒ 1	β ( 1 / K )	ρ ( kg / m 3 )
Fe 3 O 4	670	6	25 , 000	1.3	5,200
Cu	385	401	5.96 × 10 7	1.67	8,933
Ethylene glycol	2,415	0.252	5.5 × 10 ‒ 6	5.7	1,114

Table 2

Thermo-physical characteristics of a hybrid nanofluid [50]

Properties	Hybrid nanofluid
Dynamic viscosity, μ hnf	A 1 = μ hnf μ f = 1 ( 1 ‒ Φ 1 ) 2.5 ( 1 ‒ Φ 2 ) 2.5
Density, ρ hnf	A 2 = ρ hnf ρ f = Φ 2 ρ s 2 ρ f + ( 1 ‒ Φ 2 ) Φ 1 ρ s 1 ρ f + ( 1 ‒ Φ 1 )
Electrical conductivity, σ hnf	A 3 = σ hnf σ f = ( 1 + 3 σ 2 Φ 2 + σ 1 Φ 1 σ f ‒ ( Φ 1 + Φ 2 ) σ 2 Φ 2 + σ 1 Φ 1 σ f + 2 ‒ σ 2 Φ 2 + σ 1 Φ 1 σ f ‒ Φ 1 Φ 2 )
Thermal conductivity, k hnf	A 4 = k hnf k nf × k nf k f = ( k s 2 + 2 k nf ) ‒ 2 Φ 2 ( k nf ‒ k s 2 ) ( k s 2 + 2 k f ) + Φ 2 ( k nf ‒ k s 2 ) × ( 2 k f + k s 1 ) ‒ 2 Φ 1 ( k f ‒ k s 1 ) ( 2 k f + k s 1 ) + Φ 1 ( k f ‒ k s 1 )
Heat capacitance, ( ρ c p ) hnf	A 5 = ( ρ c p ) hnf ( ρ c p ) f = ( 1 ‒ Φ 2 ) Φ 1 ( ρ c p ) s 1 ( ρ c p ) f + ( 1 ‒ Φ 1 ) + Φ 2 ( ρ c p ) s 2 ( ρ c p ) f

γ T = − 1 σ 0 ∂ σ ∂ T T , γ C = − 1 σ 0 ∂ σ ∂ C C .

According to these assumptions, the modeled equations are organized as follows [46,47]:

(1) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(2) ρ hnf u ∂ u ∂ x + v ∂ u ∂ y = μ hnf ( 1 − n ) + 2 n Γ ∂ u ∂ y ∂ 2 u ∂ y 2 − μ hnf K * u − σ hnf B 0 2 u ,

(3) ( ρ C p ) hnf u ∂ T ∂ x + v ∂ T ∂ y = k hnf ∂ 2 T ∂ y 2 + Q 0 ( T − T ∞ ) + Q 1 T 0 X 2 exp − n y L − ∂ q r ∂ y + μ hnf ( 1 − n ) + n Γ 2 ∂ u ∂ y ∂ u ∂ y 2 − λ T u 2 ∂ 2 T ∂ x 2 + v 2 ∂ 2 T ∂ y 2 + 2 uv ∂ 2 T ∂ x ∂ y + u ∂ u ∂ x + v ∂ u ∂ y ∂ T ∂ x + u ∂ v ∂ x + v ∂ v ∂ y ∂ T ∂ y − δ k 0 ∂ u ∂ y ∂ ∂ y u ∂ u ∂ x + v ∂ u ∂ y ,

(4) u ∂ C ∂ x + v ∂ C ∂ y = D B ∂ 2 C ∂ y 2 − k r 2 ( C − C ∞ ) T T ∞ m exp − E a k * T − λ C u 2 ∂ 2 C ∂ x 2 + v 2 ∂ 2 C ∂ y 2 + 2 uv ∂ 2 C ∂ x ∂ y + u ∂ u ∂ x + v ∂ u ∂ y ∂ C ∂ x + u ∂ v ∂ x + v ∂ v ∂ y ∂ C ∂ y .

The relative boundary constraints are listed as follows [44]:

(5) μ hnf ∂ u ∂ y = ∂ σ ∂ x = − σ 0 γ T ∂ T ∂ x + γ C ∂ C ∂ x , v = 0 , T = T ∞ + T 0 X 2 , at y = 0 ,

C = C ∞ + C 0 X 2 , at y = 0 ,

u → 0 , T → T ∞ , C → C ∞ , at y → ∞ .

The proposed coordinates are taken in the following form:

(6) ψ = v f Xf ( η ) , η = y L , X = x L , T = T ∞ + T 0 X 2 θ ( η ) ,

u = ∂ ψ ∂ y , v = − ∂ ψ ∂ x , C = C ∞ + C 0 X 2 ϕ ( η ) .

These are mathematical equations in reduced non-dimensional forms.

(7) A 1 f ‴ ( η ) ( ( 1 − n ) + n We f ″ ( η ) ) + A 2 ( f ″ ( η ) f ( η ) − [ f ′ ( η ) ] 2 ) − A 3 M f ′ ( η ) − A 1 K f ′ ( η ) = 0 ,

(8) A 4 θ ″ ( η ) + Rd θ ″ ( η ) + Q t θ ( η ) + Q t exp ( − n η ) + A 5 Pr f ( η ) θ ′ ( η ) − 2 f ′ ( η ) θ ( η ) + PrEc A 1 ( ( 1 − n ) + n We f ″ ( η ) ) ( f ″ ( η ) ) 2 − Pr γ 1 4 f ′ 2 ( η ) θ ( η ) − f ( η ) f ″ ( η ) θ ( η ) − 3 f ( η ) f ′ ( η ) θ ′ ( η ) + f 2 ( η ) θ ″ ( η ) − δ K 1 Ec ( f ′ ( η ) ( f ″ ( η ) ) 2 − f ( η ) f ″ ( η ) f ‴ ( η ) ) = 0 ,

(9) ϕ ″ ( η ) + Sc ( f ( η ) ϕ ′ ( η ) − 2 f ′ ( η ) ϕ ( η ) ) − Sc γ 2 4 f ′ 2 ( η ) ϕ ( η ) − f ( η ) f ″ ( η ) ϕ ( η ) − 3 f ( η ) f ′ ( η ) ϕ ′ ( η ) + f 2 ( η ) ϕ ″ ( η ) − ScRc ( 1 + δ θ ( η ) ) m exp − E 1 + δ θ ( η ) ϕ ( η ) = 0 .

The boundary conditions that have been transformed are as follows:

(10) f ″ ( 0 ) = − 2 Mn ( 1 + Ma ) A 1 , f ( 0 ) = 0 , θ ( 0 ) = 1 , ϕ ( 0 ) = 1 ,

f ′ ( ∞ ) → 0 , θ ( ∞ ) → 0 , ϕ ( ∞ ) → 0 ,

where We = Γ X υ f L 2 is the Weissenberg number, K = L 2 K * is the porosity parameter, Pr = v f ( ρ C p ) f k f is the Prandtl number, M = σ 1 B 0 2 L 2 μ f is the magnetic parameter, S c = v f D f is the Schmidt number, E = E a k * T ∞ is the non-dimensional activation energy, Ec = ( v f ) 2 L 2 C p is the Eckert number, Q t = Q 0 L 2 ( ρ C p ) f v f is the temperature dependent heat source parameter, Rc = k r 2 L 2 v f is the chemical reaction rate parameter, Q e = Q 1 L 2 ( ρ C p ) f v f is the exponential dependent heat source parameter, γ 1 = v f λ T L 2 is the thermal relaxation parameter, δ * = T 0 X 2 T ∞ is the temperature difference, Ma = C 0 γ C T 0 γ T is the Marangoni ratio parameter, Mn = σ 0 T 0 γ T XL X μ f ν f is the Marangoni number, Rd = 16 σ * * T ∞ 3 3 k * k f is the radiation parameter, γ 2 = v f λ C L 2 is the solutal relaxation parameter, and K 1 = K 0 v f L 2 is the elastic deformation parameter.

3 Physical quantities of engineering interest

The rate of mass and heat transmission at the surface is often measured as follows:

(11) C fx = τ w ρ , Nu x = x q w k f ( T w − T ∞ ) , Sh x = x q m D B ( C w − C ∞ ) ,

(12) τ w = μ hnf ( 1 − n ) ∂ u ∂ y + n Γ 2 ∂ u ∂ y 2 ,

(13) q w = − k hnf + 16 T ∞ 3 σ * 3 k * ∂ T ∂ y y = 0 ,

(14) q m = − D m ∂ C ∂ y y = 0 .

Here, q w describes the heat flux, τ w denotes the skin friction, and q m shows the mass flux.

(15) C fx = A 1 ( ( 1 − n ) f ″ ( 0 ) + 1 2 n We ( f ″ ( 0 ) ) 2 ) ,

(16) Nu x ( Re x ) − 0.5 = − ( A 4 + Rd ) θ ′ ( 0 ) ,

(17) Sh x ( Re x ) − 0.5 = − ϕ ′ ( 0 ) .

4 Numerical method: Runge–Kutta–Fehlberg-45th (RKF-45) method

There does not seem to be a precise solution for the current model, which is highly non-linear. The RKF-45 approach’s high-level language and interactive environment are used to solve these equations numerically. Using the replacements, the ordinary differential equations (ODEs) (9)–(11) become the first-order ODEs (Eq. (19)).

f = y 1 , f ′ = y 2 , f ″ = y 3 , f ‴ = y 3 ′ , θ = y 4 , θ ′ = y 5 , θ ″ = y 5 ′ , ϕ = y 6 , ϕ ′ = y 7 , ϕ ″ = y 7 ′

in the following:

(18) y 1 ′ = y 2 , y 2 ′ = y 3 , y 3 ′ = 1 ( ( 1 − n ) + n We y 3 ) A 1 ( A 2 ( y 2 2 − y 1 y 3 ) + M A 3 y 2 + A 1 K y 2 ,

(19) y 4 ′ = y 5 , y 5 ′ = ( A 4 + Rd − Pr γ 1 ( y 1 ) 2 ) − 1 ( A 5 ( Pr ⁡ ( 2 y 2 y 4 − y 1 y 5 ) − Pr Ec A 1 ( ( 1 − n ) + n We y 3 ) ( y 3 ) 2 + Pr γ 1 ( 4 ( y 2 ) 2 y 4 − y 1 y 3 y 4 − 3 y 1 y 2 y 5 ) − Pr Q t y 4 − Pr Q e exp ( − n η ) + δ K 1 Ec ( y 2 ( y 3 ) 2 − y 1 y 3 y 3 ′ ) ,

(20) y 6 ′ = y 7 , y 8 ′ = 1 − Sc γ 2 ( y 1 ) 2 ) − 1 Sc [ 2 y 2 y 7 − y 1 y 8 ] + ScRc ( 1 + δ * y 4 ) m exp − E ( 1 + δ * y 4 ) y 6 + Sc γ 2 ( 4 ( y 2 ) 2 y 6 − y 1 y 3 y 6 − 3 y 1 y 2 y 7 ) .

Boundary conditions:

(21) y 1 ( 0 ) = 0 , y 2 ( 0 ) = m 1 , y 3 ( 0 ) = − 2 Mn ( 1 + Ma ) A 1 , y 4 ( 0 ) = 1 , y 5 ( 0 ) = m 2 ,

(22) y 6 ( 0 ) = 1 , y 7 ( 0 ) = m 3 .

The simulation procedure is portrayed in Figure 4.

Figure 4

Flow chart.

5 Results and discussion

The shooting method is effective in transforming BVPs into IVPs. The main emphasis is on evaluating dimensionless quantities, including f ′ ( η ) , θ ( η ) , and ϕ ( η ) profiles of thermal, velocity, and concentration, respectively, for numerous values of parameters, e.g., M a , We , n , K , M , Rd , γ 1 , γ 2 , δ , Mn , Rc , E , Φ 1 , and Φ 2 . The influences of n and We on f ′ ( η ) are shown in Figure 5(a) and (b). The fluid velocity is additionally slowed down as we increase We , as noted in Figure 5(a). In fact, the We is directly related to the relaxation time; higher values of We increase the relaxation time and provide more resistance to fluid motion, which affects the velocity of hybrid nanofluids and nanofluids. The fluids that are described by the power law index can be characterized by pseudoplastic fluids ( n < 1 ) or dilatant fluids ( n > 1 ) . According to the values of n for the shear thinning phenomena, Figure 5(b) shows a reduction in the velocity profile of hybrid nanofluids and nanofluids. This is because a rising power law index corresponds to a rise in viscosity, which causes a decrease in fluid velocity. The outcome of K on f ′ ( η ) is shown in Figure 6(a). The figure clearly shows that the velocity profile of hybrid nanofluids and nanofluids falls as K is increased. This is due to the fact that a porous medium’s widening pores provide resistive forces that work against flow and lower velocity profiles. The outcome of an M on f ′ ( η ) is shown in Figure 6(b). Statistics clearly show that the velocity profile depreciates as M augments. This is brought about by the Lorentz force, which opposes the flow and arises as M is increased.

$Figure 5 (a) and (b) Effect of We {\rm{We}} and n n on f ′ ( η ) {f}^{^{\prime} }\left(\eta ) .$

Figure 5

(a) and (b) Effect of We and n on f ′ ( η ) .

$Figure 6 (a) and (b) Effect of K K and M M on f ′ ( η ) {f}^{^{\prime} }\left(\eta ) .$

Figure 6

(a) and (b) Effect of K and M on f ′ ( η ) .

The MHD force enhances the thermal boundary layer and reduces the momentum boundary layer. The outcome of nanoparticle volume fraction of Fe 3 O 4 and Cu nanoparticles Φ 1 and Φ 2 on f ′ ( η ) and θ ( η ) is shown in Figures 7(a) and (b), 8(a) and (b); as we increase Φ 1 and Φ 2 , this causes an improvement in the velocity profile and declines the thermal profile of hybrid nanofluids and nanofluids. The responses of θ ( η ) against an increase in Rd are shown in Figure 9(a). The thermal radiation phenomena serve as energy sources for the fluid system; it is anticipated that θ ( η ) increases when the thermal radiation parameter ( Rd ) is increased. The influence of δ on θ ( η ) is shown in Figure 9(b). It is noticed that the θ ( η ) profile reduces with higher values of δ , the elastic deformation parameter. The influence of Q t and Q e on θ ( η ) is shown in Figure 10(a) and (b). It is clear that when heat generation is increased, the temperature profiles improve. When Q t and Q e are positive, the heat source factor shows how much heat is generated and is dispersed across the environment. It is demonstrated that energy generated by the boundary thermal layer causes the fluid’s temperature to rise for increasing values of Q t and Q e > 0.

$Figure 7 (a) and (b) Effect of Φ 1 {\Phi }_{1} and Φ 2 {\Phi }_{2} on f ′ ( η ) {f}^{^{\prime} }\left(\eta ) .$

Figure 7

(a) and (b) Effect of Φ 1 and Φ 2 on f ′ ( η ) .

$Figure 8 (a) and (b) Effect of Φ 1 {\Phi }_{1} and Φ 2 {\Phi }_{2} on θ ( η ) {\rm{\theta }}\left(\eta ) .$

Figure 8

(a) and (b) Effect of Φ 1 and Φ 2 on θ ( η ) .

$Figure 9 (a) and (b) Effect of Rd {\rm{Rd}} and δ \delta on θ ( η ) \theta \left(\eta ) .$

Figure 9

(a) and (b) Effect of Rd and δ on θ ( η ) .

$Figure 10 (a) and (b) Effect of Q t {Q}_{{\rm{t}}} and Q e {Q}_{{\rm{e}}} on θ ( η ) \theta \left(\eta ) .$

Figure 10

(a) and (b) Effect of Q t and Q e on θ ( η ) .

The impacts of γ 1 and γ 2 on θ ( η ) and ϕ ( η ) are described in Figure 11(a) and (b). They adequately show that θ ( η ) and ϕ ( η ) profiles as well as the associated concentration and thermal boundary layer thicknesses tend to decline with increasing relaxation times. Additionally, it is clear from these figures that, in contrast to the Cattaneo–Christov model, where mass and heat instantly diffuse throughout the medium, the thermal and solutal boundary layer thicknesses are larger for the traditional Fick’s law and Fourier’s law (i.e., γ 1 = γ 2 = 0 ). Figure 12(a) and (b) depicts the outcome of E and Rc on the concentration profile. Here, an increase in E and Rc enhances ϕ ( η ) . Each system that receives activation energy experiences a drop in acceleration and heat, which leads to a low reaction rate constant, according to the Arrhenius equation. The chemical reaction causes a rise in particle concentration. As the activation energy E rises, the Arrhenius mechanism declines. The generative chemical reaction is ultimately initiated by this, increasing the nanoparticles’ concentration profile. The outcomes of Mn number and Ma on f ′ ( η ) , θ ( η ) , and ϕ ( η ) profiles of the tangent hyperbolic hybrid nanofluid are depicted in Figures 13(a) and (b), 14(a) and (b). The graph illustrates how the increase in the value of Mn and Ma augments the velocity f ′ ( η ) and concentration ϕ ( η ) profiles of the hybrid nanofluid and nanofluid. These phenomena relate to the variation of the surface of a given material. For liquid streams, the thermosolutal Marangoni effect works as a pouring force; therefore, a higher Marangoni effect will always cause a more knotted velocity profile. This graph proves that as the value of Ma increases, the temperature profile is lowered by a considerable amount. The thermosolutal Mn is directly correlated with the surface tension. The bulk attraction of the liquid to the particles in the surface layer produces surface tension, a physical characteristic of liquid surfaces. As an outcome, the thermal profile decreases as the surface tension rises, enhancing the attraction between molecules on the surface. As a result, the temperature gradient is reduced (Tables 3–5).

$Figure 11 (a) and (b) Effect of γ 1 {\gamma }_{1} and γ 2 {\gamma }_{2} on θ ( η ) \theta \left(\eta ) .$

Figure 11

(a) and (b) Effect of γ 1 and γ 2 on θ ( η ) .

$Figure 12 (a) and (b) Effect of E E and Rc {\rm{Rc}} on ϕ ( η ) \phi \left(\eta ) .$

Figure 12

(a) and (b) Effect of E and Rc on ϕ ( η ) .

$Figure 13 (a) and (b) Effect of Mn {\rm{Mn}} and Ma {\rm{Ma}} on θ ( η ) \theta \left(\eta ) .$

Figure 13

(a) and (b) Effect of Mn and Ma on θ ( η ) .

$Figure 14 (a) and (b) Effect of Ma {Ma} on f ′ ( η ) {f}^{^{\prime} }\left(\eta ) and ϕ ( η ) \phi \left(\eta ) .$

Figure 14

(a) and (b) Effect of Ma on f ′ ( η ) and ϕ ( η ) .

Table 3

Outcome of numerous parameters on skin friction

							C fx = A 1 ( ( 1 ‒ n ) f ″ ( 0 ) + 1 2 n We ( f ″ ( 0 ) ) 2 )
							Hybrid nanofluid	Nanofluid
Ma	We	n	K	M	Φ 1	Φ 2	F e 3 O 4 + Cu ‒ C 2 H 6 O 2	F e 3 O 4 ‒ C 2 H 6 O 2
0.2	0.3	0.4	0.1	0.6	0.05	0.04	2.450192	2.126052
0.3							2.450297	2.126100
0.4							2.450327	2.126287
	0.3						2.450199	2.126052
0.4	0.4	0.3	0.1	0.6	0.05	0.04	2.637727	2.313587
	0.5						2.825247	2.501110
		0.2					2.971968	2.555241
0.1	2.0	0.3	0.1	0.6	0.05	0.04	2.241128	2.009590
		0.4					2.010678	1.964382
			0.2				2.450199	2.126052
0.2	2.0	0.3	0.3	0.6	0.05	0.04	2.637709	2.313556
			0.4				2.825220	2.501062
				0.2			2.450199	2.126052
0.2	2.0	0.3	0.1	0.3	0.05	0.04	2.637665	2.313555
				0.4			2.825145	2.501060
					0.02		2.450199	2.126052
0.2	2.0	0.3	0.1	0.6	0.04	0.04	3.756961	3.245493
					0.06		7.515934	6.465740
						0.01	2.450199	… … … … .
						0.03	2.710004	… … … … .
0.2	2.0	0.3	0.1	0.6	0.05	0.05	3.496819	… … … … .

Bold values are showing the variation of the current model.

Table 4

Change in the Nu x

									N u x = ‒ ( Re ) 1 / 2 [ A 4 + Rd ] θ ′ ( 0 )
									Hybrid nanofluid	Nanofluid
Ma	Q t	Q e	Ec	γ 1	Rd	Φ 1	Φ 2	δ	F e 3 O 4 + Cu ‒ C 2 H 6 O 2	F e 3 O 4 ‒ C 2 H 6 O 2
0.2	0.3	0.4	0.1	0.6	0.7	0.03	0.04	0.4	5.372312	5.082330
0.3									5.434445	5.082545
0.4									5.472839	5.083362
	0.3								5.372312	5.082330
0.4	0.4	0.3	0.1	0.6	0.7	0.03	0.04	0.4	5.472013	5.182040
	0.5								6.772014	6.280130
		0.2							5.846152	5.082330
0.1	2.0	0.3	0.1	0.6	0.7	0.03	0.04	0.4	5.848545	5.085030
		0.4							5.849315	5.088430
			0.2						5.033365	5.082142
0.2	2.0	0.3	0.3	0.6	0.7	0.03	0.04	0.4	5.372184	5.082330
			0.4						5.392286	5.082342
				0.2					5.033783	4.925243
0.2	2.0	0.3	0.1	0.3	0.7	0.03	0.04	0.4	5.034058	4.925441
				0.4					5.034110	4.925807
					0.2				5.372184	5.082142
0.2	2.0	0.3	0.1	0.6	0.3	0.03	0.04	0.4	6.741922	6.452015
					0.4				8.111659	7.821888
						0.02			5.372312	5.082330
0.2	2.0	0.3	0.1	0.6	0.7	0.04	0.04	0.4	6.244517	5.814387
						0.06			7.872433	7.181170
							0.01		5.372312	… … … … .
0.2	2.0	0.3	0.1	0.6	0.7	0.03	0.02	0.4	5.587332	… … … … .
							0.03		11.813890	… … … … .
								0.2	4.221601	4.144061
0.2	2.0	0.3	0.1	0.6	0.7	0.03	0.04	0.4	5.439877	5.315836
								0.6	6.658171	6.487623

Bold values are showing the variation of the current model.

Table 5

Influence of several parameters on Sh x

							S h x = ‒ ( Re ) 1 / 2 ϕ ′ ( 0 )
							Hybrid nanofluid	Nanofluid
Ma	Rc	Sc	γ 2	E	Φ 1	Φ 2	F e 3 O 4 + Cu ‒ C 2 H 6 0 2	F e 3 O 4 ‒ C 2 H 6 0 2
0.1	0.3	0.4	0.1	0.6	0.05	0.04	1.370334	1.370771
0.3							1.370602	1.371268
0.5							1.371030	1.371865
	0.3						1.370349	1.370629
0.2	0.6	0.3	0.1	0.6	0.05	0.04	1.370249	1.370529
	0.9						1.370144	1.370492
		0.2					1.371226	1.370870
0.2	2.0	0.3	0.1	0.6	0.05	0.04	1.271200	1.350848
		0.4					1.160249	1.260827
			0.1				1.370249	1.370629
0.2	2.0	0.3	0.3	0.6	0.05	0.04	1.375249	1.470820
			0.4				2.740302	1.570629
				0.2			2.398473	2.298177
0.2	2.0	0.3	0.1	0.4	0.05	0.04	1.370634	1.282461
				0.6			1.370438	1.200207
					0.02		1.370888	1.370771
0.2	2.0	0.3	0.1	0.6	0.04	0.04	2.740855	1.370644
					0.06		2.740989	1.371022
						0.01	1.370334	… … … … .
						0.03	1.370205	… … … … .
0.2	2.0	0.3	0.1	0.6	0.05	0.05	1.369677	… … … … .

Bold values are showing the variation of the current model.

6 Final remarks

In this investigation, we examined the flow of a tangent hyperbolic hybrid nanofluid across a surface with a permeable material and elastic deformation. The explanations of the concentration, thermal and velocity fields are shown for a variety of flow parameters. Following are the key findings of the current investigation:

The velocity and concentration profiles are improved, and the heat transfer is declined, when the Ma increases.
The thermal and concentration gradients are declined when the concentration and thermal relaxation parameter are augmented.
The thermal profile is declined and the velocity gradient is increased as the nanoparticle concentration enhances.
The heat transmission is improved by increasing the values of heat source and heat Rd .
The thermal gradient declines as the elastic deformation parameter increases.
A rise in the activation energy tends to elevate the concentration.
The Nu x , skin friction factor, and Sh x are increased when the Ma is increased.
An augmentation in the volume fraction of hybrid nanoparticles improves the friction factor, Nu x , and Sh x of tangent hyperbolic hybrid nanofluids and nanofluids.

Acknowledgments

This research was funded by a grant from Universiti Putra Malaysia (Project code: GP-IPS/2023/9782800).

Funding information: This research was funded by a grant from Universiti Putra Malaysia (Project code: GP-IPS/2023/9782800).
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: 2024-05-09

Revised: 2024-07-26

Accepted: 2024-08-09

Published Online: 2024-09-09

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

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0082

Keywords for this article

elastic deformation; tangent hyperbolic hybrid nanofluids; Cattaneo-Christov model; activation energy; Marangoni convection

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