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Publicly Available Published by De Gruyter December 4, 2021

Theoretical Analysis of Activation Energy Effect on Prandtl–Eyring Nanoliquid Flow Subject to Melting Condition

  • Ikram Ullah , Rashid Ali , Hamid Nawab , Abdussatar , Iftikhar Uddin , Taseer Muhammad , Ilyas Khan EMAIL logo and Kottakkaran Sooppy Nisar

Abstract

This study models the convective flow of Prandtl–Eyring nanomaterials driven by a stretched surface. The model incorporates the significant aspects of activation energy, Joule heating and chemical reaction. The thermal impulses of particles with melting condition is addressed. The system of equations is an ordinary differential equation (ODE) system and is tackled numerically by utilizing the Lobatto IIIA computational solver. The physical importance of flow controlling variables to the temperature, velocity and concentration is analyzed using graphical illustrations. The skin friction coefficient and Nusselt number are examined. The results of several scenarios, mesh-point utilization, the number of ODEs and boundary conditions evaluation are provided via tables.

1 Introduction

Numerous applications of nanofluids have recently caught the attention of physicists, researchers and engineers from several areas of science and technology. The rapidly increasing interest in the research area of nanofluids is because of their unique chemical, mechanical and thermal properties. Nanoliquids are used for enhancement of thermal behavior of the traditional base liquids like oil, water and ethylene glycol mixtures. Besides the variety of applications in biomedicine and industry, these traditional and widely used fluids possess low and non-efficient thermal conductivity properties. Due to such characteristics the thermal processes become slow. A mixture of nanoparticles and conventional base liquids is recognized as nanoliquid/nanofluid. Generally, several types of nanoparticles can be used for making a nanofluid. These nanoparticles are metals (Au, Cu and Ag), metallic oxides (Al2O3, CuO, etc.) and carbon nanotubes. Recently, nanoparticles with a solar thermal system have turned into a new zone of investigation. On the microscale level, the assembly and storage of energy via geothermal and solar means has also increasingly made use of nanofluids. Based on the high number of applications, research studies have explored the various physical aspects of nanofluids. The concept of metallic nanomaterials suspended in a base fluid was initially introduced by Choi [1]. Buongiorno [2] then explored the thermal features of nanomaterials. Significant articles about nanomaterials include Refs. [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].

Activation energy is frequently taken into account in the study of several physical phenomena like oil storage and engineering. There are a few theoretical works available related to the role of activation energy in fluid dynamics. Numerous applications of Arrhenius activation energy along with chemical reaction have made the field of fluid dynamics attractive for researchers. At an elementary level, activation energy just starts a chemical reaction. This is because various chemical reactions require some amount of energy just to start. There are other applications of activation energy such as invention of compounds, atomic reactions and recovery of thermal lubricants. Activation energy is defined as the minimum quantity of energy that is essential for stimulating the particles or molecules in which physical transport takes place. Investigating Carreau nanoliquid, Hsiano [22] researched the activation energy and concluded there is higher proficiency in the development of a thermal energy extrusion structure. Furthermore, they examined the economic capability of the model. It was Bestman [23] who studied activation energy with vertical radiation in a pipe set. He conducted the analysis using a perturbation technique. In a recent work, Hayat et al. [24] analyzed the effects of activation energy on Carreau liquid taking cross-diffusion into account. Activation energy features on magnetized nanomaterials were discussed by Mustafa et al. [25]. More recent works related to activation energy are provided in Refs. [26], [27], [28], [29]. The aforementioned works motivated us to study the characteristics of steady nanofluid flow with revised conditions and activation energy.

In the current study, the Buongiorno model is considered to investigate the magnetohydrodynamic (MHD) mixed convection flow of nanomaterials subjected to melting condition. A non-Newtonian type fluid known as Prandtl–Eyring [30], [31], [32], which mimics more precisely the physical aspects of visco-inelastic liquids, is considered. This model possesses a linear relationship between the sine hyperbolic function and the function of the deformation rate and shear stress. It should be noted that the mathematical model of convective transport in nanofluids developed by Buongiorno [2] has some shortcomings because he considers that Brownian diffusion, thermophoresis and diffusiophoresis are separate and independent processes. It must be noted that thermophoresis is the result of Brownian movement when a temperature gradient is applied and that when the Brownian movement of nanoparticles is correctly simulated, the thermophoresis coefficients may be derived [33]. If there is a temperature gradient in the flow domain of a nanofluid, small particles disperse slower in colder areas and faster in hotter regions. Therefore, in the presence of a temperature gradient, particles move on average against the gradient. This averaged motion of particles is known as thermophoresis. The mathematical modeling of the flow design capitulates the nonlinear system of partial differential equations. The systems of ordinary differential equations (ODEs) are obtained through the application of variables. The achieved system is then solved numerically by means of Lobatto IIIA “bvp4c.” The behavior of more interesting variables is shown graphically. Moreover, the surface drag force and temperature gradient are also estimated numerically and elaborated in detail. The last section encloses the conclusion of the current analysis.

Figure 1 
Flow physical sketch.
Figure 1

Flow physical sketch.

2 Mathematical development

Here we considered 2D non-Newtonian incompressible Prandtl–Eyring nanoliquid flow by a stretched sheet. The liquid is limited in the area y > 0 and the sheet is located at y = 0, which is linearly stretched with velocity u w ( x ) = a x (where a is the dimensional constant). A magnetic field of strength B 0 is implemented in the orthogonal direction (see Fig. 1). The melting condition is taken into account at the sheet. The impacts of mixed convection, Joule heating, chemical reaction and activation energy are included. Furthermore, it is presumed that there is no relative movement between base liquid and nanoparticles. According to aforementioned assumptions, the governing expressions are [30], [31], [32], [33], [34]

(1) u x + υ y = 0 ,
(2) ρ n f u u x + υ u y = A 2 u y 2 σ B 0 2 u + A 2 c 1 3 u y 2 2 u y 2 + g β T ( T T m ) + g β C ( C C m ) ,
(3) u T x + υ T y = 1 ρ c p k + 16 3 δ 1 T 3 K 2 T y 2 + τ D B C y T y + σ B 0 2 u 2 ( ρ c p ) f + τ D T T T y 2 + k r 2 β 1 T T m m Exp E a κ T ( C C m ) ,
(4) u C x + υ C y = D B 2 T y 2 + D T T 2 T y 2 k r 2 T T m m exp E a κ T ( C C m ) ,
with

(5) u = u w ( x ) , υ = 0 , T = T m , C = C m , k T y = ρ λ 2 + c s ( T m T 0 ) υ for y = 0 , u 0 , C C , T T for y ,

where β T and β C are the coefficients of thermal and solutal expansions, g is the acceleration due to gravity, u and υ denote the velocity components in x- and y-directions, respectively, P is the pressure, T is the fluidic temperature, C is the nanoparticles’ volume fraction, A and c 1 are fluidic variables, ρ n f is the nanoliquid density, σ is the fluidic electrical conductivity, k is the thermal conductivity, τ is the ratio of heat capacities, D B and D T denote the coefficient of Brownian movement and thermophoretic diffusion, T stands for ambient temperatures, C is the ambient concentration of nanoparticles, k stands for thermal conductivity, k r is the rate of reaction, E a is the activation energy, κ is the Boltzmann constant and m designates the fitted rate constant.

We define the stream function as follows:

(6) u = y ψ , υ = x ψ .

In order to transform the above governing systems of the flow problem to a dimensionless form, the following variables are defined:

(7) ψ = a v x f ( η ) , ϕ = C C m C C m , θ = T T m T T m , η = y a ν .

Utilizing eq. (7) in eqs. (1)–(5) we obtain

(8) α f α β f ( f ) 2 ( f ) 2 + f f Mf + λ 1 ( θ + N 1 ϕ ) = 0 ,
(9) 1 + 4 3 Rd θ + Pr f θ + N b θ ϕ + N t ( θ ) 2 + Pr EcM ( f ) 2 + Pr λ γ 1 ϕ ( 1 + δ θ ) m Exp E 1 + δ θ = 0 ,
(10) ϕ + Sc ( f ϕ ) + N t N b θ γ 1 Sc ϕ ( 1 + δ θ ) m Exp E 1 + δ θ = 0 ,
(11) f ( η ) = 1 , θ ( η ) = 0 , ϕ ( η ) = 1 , Me θ ( η ) + Pr f ( η ) = 0 for η = 0 , f ( η ) 0 , θ ( η ) 1 , ϕ ( η ) 1 for η .

In the above system α, β are the material parameters, M is the Hartmann number, λ 1 is the mixed convection variable, N 1 is the buoyancy ratio variable, Pr is the Prandtl number, N b is the Brownian motion variable, N t is the thermophoresis variable, Ec is the Eckert number, Sc is the Schmidt number, γ 1 is the reaction rate, δ is the temperature difference variable, E is the dimensionless activation energy, Me is the melting variable, Rd is the radiation parameter and λ is the endothermic/exothermic reaction parameter. These variables can be expressed as follows:

(12) α = A c μ , β = a 3 x 2 2 ν c 2 , M = σ ρ a B 0 2 , λ 1 = G r x Re x 2 , G r x = g B T ( T T m ) x 3 γ 2 , Re x = u w x ν , N 1 = β c β T C C m T m T , Pr = ν α , N b = τ ( C C m ) D B α , N t = τ ( T T m ) D T α T m , Ec = u w 2 c p ( T m T ) , Sc = ν D B , γ 1 = k r 2 a , δ = T T m T m , E = E a κ T , Rd = 4 δ 1 T 3 κ K , Me = c p ( T T m ) λ 2 + c s ( T T m ) , λ = β 1 ( C C m ) c p ( T T m ) .

The important quantities are

(13) Cf x = τ w ρ n f u w 2 , Nu x = x q w k ( T T m ) .

Here

(14) τ w = A c u y y = 0 A 6 c 3 u y y = 0 3 , q w = k T y y = 0 + 16 3 δ 1 T 3 K T y y = 0 .

Putting eq. (14) into eq. (13) and then applying the transformations defined in eq. (7), we get

(15) 1 2 C f x Re x 1 2 = α f ( 0 ) α β 3 [ f ( 0 ) ] 3 , Nu x Re x 1 2 = θ ( 0 ) 1 + 4 3 Rd .

Here Re x depicts the local Reynolds number.

3 Solution approach

The nonlinear flow problem is executed as a built-in scheme known as bvp4c in MATLAB. The Lobatto IIIA formula is utilized in the present technique. To utilize this scheme, it is important to convert the nonlinear and higher-order system into first-order ODEs by setting new variables. Consider

(16) f = u 1 , f = u 2 , f = u 3 , f = u 3 , θ = u 4 , θ = u 5 , θ = u 5 , ϕ = u 6 , ϕ = u 7 and ϕ = u 7 .

This technique attains exceptional accuracy and is stable unconditionally. This scheme is used for highly nonlinear coupled problems.

Table 1

Variation of parameters.

Scenario Parameter Case 1 Case 2 Case 3 Case 4
1 β 0.2 0.5 0.8 1.1
2 M 0.1 0.3 0.5 0.7
3 λ 1 0.1 0.3 0.5 0.7
4 N b 0.1 0.3 0.5 0.7
5 Ec 0.4 0.8 1.2 1.6
6 γ 1 0.8 1.0 1.2 1.4
7 δ 0.2 0.8 1.4 2.2
8 E 0.2 1.2 2.2. 3.2
9 λ 0.1 0.4 0.7 1.0
10 Me 0.3 0.4 0.5 0.6
11 Rd 0.2 0.4 0.6 0.8

Table 2

Fixed values of the parameters appearing in the problem.

Scenario Parameter Default value
1 α 1.5
2 β 0.2
3 M 0.1
4 λ 1 0.1
5 N 1 0.1
6 Pr 1.0
7 N b 0.1
8 N t 0.1
9 Ec 1
10 Sc 1
11 γ 1 0.7
12 δ 0.4
13 m 0.4
14 E 1
15 λ 0.1
16 Me 0.4
17 Rd 0.2

4 Discussion of outcomes

The resulting system of ODEs, eqs. (8)–(10) with boundary conditions eq. (11), is solved numerically by utilizing features of MATLAB routine “bvp4c”. The effects of involved variables (α, β, M, λ 1 , N 1 , Pr, N b , N t , Ec , Sc , γ 1 , δ, m , E, Rd , Me and λ) on temperature, velocity and concentration are studied. Table 1 describes the variation of the 11 variables, presenting the corresponding 11 scenarios, in detail, whereas Table 2 elaborates the default values of the parameters. Numerical solutions for these scenarios of the nanofluid flow problem are derived using the input mesh points (see Table 3), while the maximum residual error is set as stopping criterion of the solver in Table 4. The numerical values of skin friction and Nusselt numbers are provided in Tables 5 and 6, respectively, for all 11 scenarios. Figure 2(a)–(d) is displayed to scrutinize the effects of Me , λ 1 , M and β on f ( η ). Figure 2(a) presents the effect of Me on f ( η ). It is obvious from this figure that velocity f ( η ) enhances for higher estimations of Me . Results of this figure are important because the melting parameter can enhance and slow down the fluid velocity. In many practical systems, a lower or higher velocity of the material is very significant. Figure 2(b) depicts the behavior of velocity f ( η ) versus λ 1 . Physically increasing the mixed convection parameter, the buoyancy force due to gravity enhances, which consequently increases velocity. The results in these figures are important for the enhancement of fluid velocity with less applied stress. Figure 3(c) is included to show the effects for M. The larger the Hartmann number, the stronger the Lorentz force, which has propensity to resist the transport rate of fluid. As a result, f ( η ) is diminished. MHD flows use a Lorentzian drag force which can be utilized to manage a variety of flow regimes. Moreover, M = 0 manifests that there is hydrodynamic flow. The effect of β on f ( η ) is depicted in Fig. 2(d). It is seen that larger values of β correspond to a lower velocity field.

Table 3

Analysis of mesh points.

Scenario Variable C-1 C-2 C-3 C-4
1 β 757 785 796 805
2 M 757 724 634 470
3 λ 1 757 800 825 854
4 N b 757 759 745 750
5 Ec 777 761 755 753
6 γ 1 759 764 768 771
7 δ 751 776 749 773
8 E 758 757 753 748
9 λ 757 765 772 782
10 Me 798 757 733 619
11 Rd 757 739 736 740

Table 4

Error analysis.

Scenario Variable C-1 C-2 C-3 C-4
1 β 5.451E−12 3.052E−11 4.391E−10 1.858E−09
2 M 5.451E−12 3.836E−11 3.089E−10 5.991E−10
3 λ 1 5.451E−12 1.928E−11 1.618E−12 1.576E−12
4 N b 5.451E−12 1.610E−12 1.558E−12 1.520E−12
5 Ec 5.596E−12 5.477E−12 5.454E−12 5.512E−12
6 γ 1 5.437E−12 5.450E−12 5.438E−12 5.373E−12
7 δ 5.667E−12 5.295E−12 3.998E−12 1.949E−12
8 E 5.389E−12 5.439E−12 5.515E−12 5.804E−12
9 λ 5.451E−12 5.560E−12 5.644E−12 5.669E−12
10 Me 2.436E−11 5.451E−12 5.517E−12 8.787E−12
11 Rd 5.451E−12 2.053E−11 2.032E−11 1.842E−11

Table 5

Analysis of skin friction coefficients.

Scenario Variable C-1 C-2 C-3 C-4
1 β 1.1189 1.1788 1.2422 1.2932
2 M 1.1189 1.2168 1.3083 1.3942
3 λ 1 1.1189 1.0134 0.9104 0.8095
4 N b 1.1189 1.1126 1.1075 1.1026
5 Ec 1.1231 1.1203 1.1175 1.1148
6 γ 1 1.1189 1.1188 1.1188 1.1188
7 δ 1.1190 1.1188 1.1188 1.1188
8 E 1.1188 1.1190 1.1193 1.1195
9 λ 1.1189 1.1099 1.1022 1.0958
10 Me 1.1383 1.1189 1.1014 1.0855
11 Rd 1.1189 1.1253 1.1308 1.1354

Table 6

Analysis of Nusselt number.

Scenario Variable C-1 C-2 C-3 C-4
1 β 0.5624 0.5593 0.5567 0.5552
2 M 0.5624 0.6182 0.6672 0.7106
3 λ 1 0.5624 0.5913 0.6177 0.6422
4 N b 0.5624 0.5901 0.6148 0.6392
5 Ec 0.5401 0.5549 0.5698 0.5847
6 γ 1 0.5629 0.5637 0.5643 0.5646
7 δ 0.5618 0.5633 0.5643 0.5649
8 E 0.5644 0.5618 0.5589 0.5570
9 λ 0.5624 0.5992 0.6301 0.6558
10 Me 0.5882 0.5624 0.5392 0.5183
11 Rd 0.5624 0.5332 0.5087 0.4878

Figure 2 
Effects of various physical quantities on the velocity profile.
Figure 2

Effects of various physical quantities on the velocity profile.

Figure 3 
Effects of various physical quantities on the temperature profile.
Figure 3

Effects of various physical quantities on the temperature profile.

The effects of physical variables ( Me , Ec , λ, M and Rd ) on temperature θ ( η ) are presented in Fig. 3(a)–(e). The effect of Me on θ ( η ) is presented in Fig. 3(a). It is found that θ ( η ) is lower for higher estimations of Me . The effect of Ec on nanoparticle temperature is presented in Fig. 3(b). This graph reveals that θ ( η ) is an increasing function of the Eckert number. The main reason behind this fact is that nanoliquid viscosity gains energy from nanofluid movement and this energy is converted into internal energy, which boosts up the temperature. This behavior is useful for enhancing the temperature. An increment in θ ( η ) is noticed for higher values of λ (see Fig. 3(c)). Since thermophoretic force has the potential to push the nanoparticles from the hot surface towards the cold regime, the temperature becomes higher. This shows that the performance of nanoliquid in plentiful noteworthy applications can be improved in different scientific and industrial fields like extraction of geothermal energy and hot rolling. Variation in θ ( η ) due to M is presented in Fig. 3(d). Here higher values of M increase θ ( η ). The central cause of this feature is magnetic force, which has the potency to generate the drag force know as Lorentz force. This force acts in the reverse direction of flow, which resists the fluid movement. Thus, the nanophase and liquid temperature significantly rises, which fully agrees with the physical situation. This result signifies that the Hartmann number controls the flow rate of heat transfer and the nanoparticle effect. Furthermore, this result can be used to reduce the entropy production in a system. In Fig. 3(e) it is shown that θ ( η ) is lower for higher values of Rd .

The effects of important variables ( γ 1 , δ, E and N b ) on ϕ ( η ) are disclosed in Fig. 4(a)–(d). The impact of γ 1 on ϕ ( η ) is portrayed in Fig. 4(a). Clearly, an increasing trend is seen for advanced estimations of γ 1 . A similar trend of ϕ ( η ) against δ is displayed in Fig. 4(b). Figure 4(c) interprets the effect of E on ϕ ( η ). Here the concentration diminishes for higher estimations of E. Contributions of N b to ϕ ( η ) are depicted in Fig. 4(d). Clearly, concentration ϕ ( η ) decays for higher values of N b . Physically, due to the concentration gradient more particles are shifted in the opposite direction to form a more homogeneous solution. Thus, a lower concentration estimation is noticed for higher values of N b . This decaying behavior of the Brownian motion parameter will be helpful in enhancing oil recovery and upstream oilfield applications.

The variation of 11 parameters out of 17 involved parameters represents 11 scenarios and there are 4 cases for each scenario, which show variation of each physical variable. In Table 1, the variation of 11 parameters is displayed, while Table 2 presents the fixed values of all 17 parameters. For numerical solution of problem, the mesh points taken during the analysis for each scenario and cases are provided in Table 3. Outcomes in Table 3 are quite important to conduct the mash point analysis in numerical solutions. Error analysis of the “bvp4c” method is conducted in terms of maximum residual, which is presented in Table 4. The residual is calculated for each case and each scenario during the analysis of the fluid problem. Error analysis of any method is significant to check whether the obtained solution is convergent and gives a better solution of the problem. To estimate the performance of different nanoliquid/fluidic as well as thermal devices, the heat transfer rate and drag force at the wall or surface are significant. Therefore, Tables 5 and 6 show the effects of various variables on skin friction and Nusselt number. In Table 5 the effect of sundry physical variables on skin fraction is discussed. Table 6 discloses the effects of interesting physical variables on the Nusselt number. It has been observed that the Nusselt number is an increasing function of M, λ, N b , Ec , γ 1 and δ, while opposite features are noticed for β, E, Me and Rd .

Figure 4 
Effects of various physical quantities on the temperature profile.
Figure 4

Effects of various physical quantities on the temperature profile.

5 Conclusions

Key features of the current study are mentioned below:

  1. A decaying behavior is seen of velocity f ( η ) for higher values of β and M while the opposite trend is observed for increasing Me and λ 1 .

  2. The nanoparticle concentration ϕ(ƞ) is an increasing function of λ and Ec , while it is a decreasing function of Me .

  3. Higher values of Me and λ 1 cause an increment in θ ( η ), while an opposite trend is observed for larger values of β and M.

  4. The Nusselt number increased for higher values of M, N b , Ec , λ and δ, while it decreases for higher values of E, Me and β.

Nomenclature

u w

Velocity of the sheet

B 0

Magnetic field strength

( u , υ )

Velocity components

a

A constant

A , c 1

Fluidic parameters

P

Fluid pressure

T

Fluid temperature

C

Nanoparticles volume fraction

ρ n f

Nanofluid density

σ

Electrical conductivity

τ

Ratio parameter of heat capacities of nanoparticles and fluid

D B

Coefficient of Brownian movement

D T

Coefficient of thermophoresis diffusion

T w

Temperature at wall

T , C

Ambient temperature and concentration

κ

Boltzmann’s constant

k

Thermal conductivity

k r

Reaction rate

K

Mean absorption coefficient

m

Fitted rate constant

E a

Activation energy

β T , β C

Thermal and solutal expansions

C f x

Local skin friction

N 1

Buoyancy ratio

( T 0 , T m )

Surface and melting temperatures

g

Gravitational acceleration

ψ

Stream function

y , x

Partial derivatives

η

Transformed variable

( f , θ , ϕ )

Dimensionless velocity, temperature and concentration

Re x

Local Reynolds number

Nu x

Local Nusselt number

ν n f

Kinematic viscosity of nanofluid

α , β

Material parameters of Prandtl–Eyring fluid

λ

Endothermic/exothermic parameter

λ 1

Mixed convection parameter

λ 2

Latent heat

N b

Brownian motion

β 1

Endothermic/exothermic reaction

Pr

Prandtl number

N t

Thermophoresis parameter

Me

Melting parameter

Ec

Eckert number

Sc

Schmidt number

γ 1

Reaction rate

δ

Temperature difference parameter

δ 1

Stefan–Boltzmann constant

Rd

Radiation parameter

c s

Solid surface specific heat

C m

Melting concentration

E

Dimensionless activation energy

M

Hartmann number

Funding source: King Khalid University

Award Identifier / Grant number: GRP/64/42

Funding statement: The authors express their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia for funding this work through general research group program under grant number GRP/64/42.

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Received: 2020-09-08
Revised: 2021-02-10
Accepted: 2021-03-29
Published Online: 2021-12-04
Published in Print: 2022-01-31

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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