[picture] | Fluid Dynamics & Materials Processing | [picture]

DOI: 10.32604/fdmp.2021.012789

ARTICLE

Buoyancy driven Flow of a Second-Grade Nanofluid flow Taking into Account the Arrhenius Activation Energy and Elastic Deformation: Models and Numerical Results

R. Kalaivanan¹, N. Vishnu Ganesh² and Qasem M. Al-Mdallal³,*

1 Department of Mathematics, Vivekananda College, Madurai, 625234, India
2 PG and Research Department of Mathematics, Ramakrishna Mission Vivekananda College, Chennai, 600004, India
3 Department of Mathematical Sciences, United Arab Emirates University, Al Ain, Abu Dhabi, United Arab Emirates
Corresponding Author: Qasem M. Al-Mdallal. Email: q.almdallal@uaeu.ac.ae
Received: 12 July 2020; Accepted: 11 January 2021

Abstract: The buoyancy driven flow of a second-grade nanofluid in the presence of a binary chemical reaction is analyzed in the context of a model based on the balance equations for mass, species concentration, momentum and energy. The elastic properties of the considered fluid are taken into account. The two-dimensional slip flow of such non-Newtonian fluid over a porous flat material which is stretched vertically upwards is considered. The role played by the activation energy is accounted for through an exponent form modified Arrhenius function added to the Buongiorno model for the nanofluid concentration. The effects of thermal radiation are also examined. A similarity transformations is used to turn the problem based on partial differential equations into a system of ordinary differential equations. The resulting system is solved using a fourth order RK and shooting methods. The velocity profile, temperature profile, concentration profile, local skin friction, local Nusselt number and local Sherwood number are reported for several circumstances. The influence of the chemical reaction on the properties of the concentration and momentum boundary layers is critically discussed.

Keywords: Arrhenius activation energy; buoyancy effects; chemical reaction; elastic deformation; nanofluid; nonlinear thermal radiation

1  Introduction

Nanofluids are artificially synthesized novel fluids, which are colloidal mixtures of 1–100-nm-sized inorganic solid particles and conventional base liquids. These fluids were initially popularized by Choi et al. [1]. They have various important applications in many engineering and medical fields, which can be attributed to their tri-biological, chemical, thermal, biological, mechanical, and magnetic properties. The choice of base fluids (water, molten salts, oil, glycols, etc.) and nanoparticles (carbon nanotubes, metals, metallic oxides, etc.) is mainly dependent on the industrial requirements. Recently, numerous studies have focused on nanofluid heat transfer in different geometries [2–10]. Researchers have introduced various theoretical models to thoroughly examine the properties of nanofluids. It is vital to study the non-Newtonian fluid behavior because of its numerous biological and industrial applications, including in food, mineral suspensions, cosmetics, pharmaceutical, and personal care products. The investigations on the boundary-layer flow characteristics of incompressible non-Newtonian fluids over a stretchable material are paramount in various engineering processes such as polymer processing, drawing of plastic films and wires, metallic plate cooling, glass fiber processing, food processing, and crystal growth. Outstanding performances can be achieved in the aforementioned processes by replacing ordinary fluids with nanofluids. Buongiorno proposed a non-Newtonian flow model of a nanofluid with important slip mechanisms [11]. Nield et al. [12] investigated the natural convective nanofluid flow with active control of the nanoparticle fraction and the temperature at the boundary (the nanoparticle volume fraction can be actively controlled without considering any mass flux boundary condition and other nanofluid parameters at the boundary). They considered the Darcian porous medium model in their study. Kuznetsov et al. [13,14] remodeled the above problem and the convective transport of nanofluids over a vertical plate with the passive control of nanoparticles (the nanoparticle volume fraction can be controlled passively at the wall by including Brownian motion and thermophoresis at the boundary). They reported that the use of the passive control of nanoparticles with respect to the boundary condition is physically realistic. This model is applicable to second-grade nanofluids with two types of nanoparticle controls: active and passive controls. Recently, various studies have been conducted on non-Newtonian second-grade nanofluids [15–20].

Arrhenius first observed the concept of activation energy in heat and mass transport problems in 1889. The energy obtained from molecules or atoms was employed to initiate a reaction in the chemical processes. The study of a chemical reaction along with the activation energy is useful in food processing, chemical engineering of emulsions of various suspensions, oil reservoirs, etc. The amount of activation energy required for a chemical reaction is mainly dependent on the application of the mass transport problem. There may be no need to activate a reaction with additional energy in some situations. The buoyancy forces play a crucial role in controlling the boundary layer in many practical situations involving convection problems, e.g., the boundary layer flows in the atmosphere, petroleum extraction, and solar collectors. The influences of the Arrhenius activation energy have been considered in the absence of the buoyancy effect in recent nanofluid studies [21–28].

Maleque [29] reported the impacts of the Arrhenius activation energy on the flat plate fluid flow problem and viscous dissipation, heat generation, and buoyancy. The local similar flow of a magnetonanofluid with activation energy and buoyancy effects was solved by Mustafa et al. [30]. They reported that the activation energy is directly proportional to the Brownian motion and inversely proportional to the occurrence of thermophoresis. Mustafa et al. [31] inspected the buoyant heat transport in Maxwell’s non-Newtonian fluid flow model and obtained an “S”-shaped temperature profile function for a large temperature difference in the presence of activation energy. Ramzan et al. [32] explored micropolar nanofluid flow when considering the buoyancy and activation energy impacts. They observed that the concentration profile diminished with increasing solute stratification. Khan et al. [33] documented the buoyancy and activation energy impacts on entropy generation in the magnetohydrodynamic (MHD) Casson nanofluid model with heat generation/absorption and thermal radiation. Their work mainly focused on entropy generation and the effects of Bejan number. A computational assessment of the Jeffery nanofluid flow with the Arrhenius activation energy and buoyancy effects was performed by Ahmad et al. [34]. Dhalmini et al. [35] studied the impacts of the activation energy and buoyancy on the time-dependent flow of nanofluids over a heated surface with an infinite boundary length and concluded that the Biot number increases with the increasing concentration of the chemical species. Irfan et al. [36] analyzed the three-dimensional (3D) flow of the Carreau non-Newtonian nanofluid with the Arrhenius activation energy and buoyancy effects and reported that the activation energy was considerably influential. The same problem along with the thermal radiation impacts was analyzed in another study [37]. The Darcy–Forchheimer flow over a curved stretching surface with buoyancy, entropy generation, and activation energy was discussed by Muhammad et al. [38]. Alghamdi [39] focused on the MHD nanofluid flow using a rotating disk in three dimensions with assumptions of activation energy and buoyancy. An entropy generation analysis of the Casson nanofluid, which passed over a stretchable surface with activation energy, buoyancy, and viscous dissipation effects, was conducted by Khan et al. [40]. Recently, Ijaz et al. [41] examined the MHD stagnation point flow of the Walter-B nanofluid flow with activation energy under convective boundary conditions.

This study mainly focuses on the integrated impacts of the Arrhenius activation energy, elastic deformation, chemical reaction, and buoyancy on the MHD second-grade non-Newtonian nanofluid flow. This is an important problem that has not yet been investigated. Further, a computational model is developed by considering the second-grade nanofluid two-phase model, magnetic field, nonlinear thermal radiation, Arrhenius activation energy, binary-order chemical reaction, buoyancy effects, and active and passive control of nanoparticles. The notable results obtained via a numerical experiment are discussed in this study.

2  Problem Formulation

2.1 Flow Formulation

Let us consider a time-independent two-dimensional (2D) slip flow of a non-Newtonian nanofluid over a porous flat material that is stretched vertically upward. The nanofluid flow is considered to be laminar and incompressible. Heat transport analysis is conducted with a nonlinear form of thermal radiation, elastic deformation, and internal heat generation/absorption. The Arrhenius activation energy is assumed to initiate the chemical reaction. The binary-order chemical reaction is assumed to preserve the surface temperature. The flow is subjected to the combined effects of the magnetic field of strength B0 in the transverse direction, suction, and velocity slip. The induced magnetic field is disregarded for a low magnetic Reynolds number. The stretching surface velocity is assumed as U=(V1)w=dx , where d>0 is a constant. The stretching sheet surface is preserved at a uniform temperature Tw and concentration Cw when assuming Tw – T∞ > 0 and Cw − C∞ > 0, where C∞ and T∞ are the ambient concentration and temperature, respectively (Fig. 1). Active as well as passive controls of the nanoparticles are imposed on the boundary conditions. Further, a thermal equilibrium is maintained between the fluid phase and nanoparticles, and the external forces and pressure gradient are ignored.

Figure 1: Physical layout of the model considered in this study

The governing equations of the present problem can be modeled using the following partial differential equations, as presented in previous studies [17–19].

∂V1∂x+∂V2∂y=0, (1)

V1∂V1∂x+V2∂V1∂y+σB02ρV1−ϑ∂2V1∂y2−βC(C−C∞)g−βT(T−T∞)g+k0∂V1∂y∂∂x(∂V1∂y)−k0{V1∂∂x(∂2V1∂y2)+V2∂∂y(∂2V1∂y2)+∂V1∂x(∂2V1∂y2)}=0, (2)

V1∂T∂x+V2∂T∂y−kρcp∂2T∂y2−δk0cp{∂V1∂y∂∂y(V1∂V1∂x+V2∂V1∂y)}−τ{DB∂T∂y∂C∂y+DTT∞(∂T∂y)2}+1ρcp∂qr∂y−Q0ρcp(T−T∞)=0, (3)

V1∂C∂x+V2∂C∂y−DB∂2C∂y2−DTT∞∂2T∂y2+Γ2(C−C∞)(TT∞)m*e−Eak*T=0, (4)

where V1 and V2 are the velocities along the x and y axes, respectively. k0 represents the elastic parameter, k represents the thermal conductivity, cp represents the specific heat capacity, ϑ represents the kinematic viscosity, δ represents the coefficient of elastic deformation, ρ represents the density, τ represents the ratio of the effective heat capacity of the nanoparticle material and the heat capacity of the base fluid, g represents the acceleration due to gravity, βT represents the coefficient of thermal expansion, DT represents the thermophoresis diffusion coefficient βC represents the coefficient of thermal expansion with concentration, DB represents the Brownian diffusion coefficient, Q0 represents the heat generation/absorption coefficient, Γ represents the chemical reaction rate, and (TT∞)m*e−Eak*T represents the modified Arrhenius function, where k* = 8.61 × (1 × 10−5), eV/K is the Boltzmann constant, m* = −1 < m* < 1, and Ea is the activation energy.

The radiation term qr can be expressed in a nonlinear form, similar to that in a previous study [4].

qr=−(4σSB*3k0*)∂∂y(T4)=−(16σSB*3k0*)T3∂∂y(T), (5)

where σSB* and k0* are the Stefan–Boltzmann constant and the coefficient of mean absorption, respectively.

2.2 Boundary Condition

The boundary conditions are given below.

At the stretching material wall ( y=0 ),

V1=(V1)w(x)+σ*τw,V2=(V2)w.

Here, σ* corresponds to Navier’s constant slip length and τw is the shear stress at the surface of the sheet, which is given as τw=μ∂V1∂x+ρk0{2∂V1∂x∂V1∂y+u∂2V1∂x∂y} .

T−Tw=0, and

C−Cw=0 (for active control)

−DB∂C∂y=DTT∞∂T∂y (for passive control)

The following condition can be observed far away from the stretching material wall, where the flow attains free stream velocity ( y→∞) .

V1→0,C−C∞→0,T−T∞→0. (6)

2.3 Similarity Transformations and Dimensionless Forms

To perform a similarity analysis, similarity transformations based on dimensionless nanofluid temperature and nanofluid concentration are introduced.

V1=(dx)Gη(η),V2=(−dϑ)1/2G(η),andη=(dϑ)1/2y. (7)

Twθ(η)−T∞θ(η)=T−T∞

Cwϕ(η)−C∞ϕ(η)=C−C∞(dimensionlessconcentrationfortheactivecontrolofnanoparticles),and

C∞ϕ(η)=C−C∞,(dimensionlessconcentrationforthepassivecontrolofnanoparticles). (8)

When considering the nonlinear thermal radiation, the nanofluid temperature is assumed to be T=(−(θT∞−θT∞θw)+T∞) , where T∞θw=Tw is the temperature ratio parameter with T∞<Tw .

By invoking the above assumptions into Eqs. (1)–(4) and Eq. (6), the similarity transformations in Eq. (7) are proved to satisfy the continuity in Eq. (1). Further, Eqs. (2)–(4) are transformed into the following dimensionless forms based on the physical parameters that govern the flow.

Gηηη+GGηη−Gη2+k1(2GηGηηη−GGηηηη−Gηη2)−MGη+λ1θ+λ2φ=0, (9)

(1+Rd(1+θ(θw−1))3)θηη+3Rd(θw−1)(1+θ(θw−1))2θη2+PrGθη−2PrGηθ+δk1EcPrGηη(GηGηη−GGηηη)+NbPrθηϕη+NtPrθη2+λ3Prθ=0, (10)

ϕηη−2LeGηϕ+LeGφη−γϕLe(1+θ(θw−1))m*e−E1+θ(θw−1)+NtNbθηη=0. (11)

Consequently, the BCs in Eq. (6) adopt the following dimensionless form.

G(0)−s=0,Gη(0)=1+σ(1+3k1Gη(0))Gηη(0),Gη(∞)=0,andGηη(∞)=0. (12)

θ(0)−1=0andθ(∞)=0. (13)

ϕ(0)−1=0,(masstransferboundaryconditionforactivecontrol)

Nbϕη(0)=−Ntθη(0)(masstransferboundaryconditionforpassivecontrol)andφ(∞)=0 (14)

where the parameters are as follows:

M=σρ−1B02d−1 → magnetic parameter

k1=k0ϑ1d → viscoelastic parameter

σ=σ*dϑ → slip parameter

λ1=(V1)w−1gβT(Tw−T∞) → thermal buoyancy parameter

λ2=gβC(V1)w−1(Cw−C∞) → concentration buoyancy parameter

λ3=ρ−1Q0cp−1, → uniform heat source/sink parameter

3Rd=16k1σ*T∞3k0*−1 → radiation parameter

Nb=(Cw−C∞)τDBϑ−1 → Brownian motion parameter

E=k*−1EaT∞−1 → nondimensional activation energy

γ=d−1Γ2 → chemical reaction parameter

Ec=cp−1(V1)w2(Tw−T∞)−1 → Eckert number

Nt=τϑ−1DT(Tw−T∞)T∞−1 → thermophoresis parameter

Pr=μk−1cp → Prandtl number

Le=DB−1ϑ → Lewis number

The following essential expressions are derived for the interest of engineering and industrial processes.

CfRex=2(1+3k1Gη(0))Gηη(0),(localskinfriction) (15)

NuxRex=−(1+Rdθw3)θη(0),(localNusseltnumber) (16)

ShxRex=−ϕη(0),(Sherwoodnumberfortheactivecontrol) (17)

ShxRex=NtNbθη(0).(Sherwoodnumberforthepassivecontrol) (18)

3  Numerical Solution

To solve the nonlinear ordinary differential equations (Eqs. (9)–(11)) along with the BCs in Eqs. (12)–(14) using an iterative power-series method with a shooting strategy [42,43], the higher-order BVP equations must be converted into first-order IVP equations; additionally, an appropriate finite value of η→∞ , i.e., η∞ , must be selected. Therefore, the following first-order systems are established using Eqs. (9)–(14).

(P1)η=P2,

(P2)η=P3,

(P3)η=P4,

(P4)η=−1P1(P32−2P2P4+1k1(P22−P1P3−P4+MP2−λ1P5−λ2P7)),

(P5)η=P6,

(P6)η=11+Rd(1+P5(θw−1))3(Pr(2P2P5−P1P6−δk1EcP3(P2P3−P1P4)−NbP6P8−NtP62−λ3P5)−3Rd(θw−1)(1+P5(θw−1))2P62),

(P7)η=P8,

(P8)η=Le(2P2P7−P1P8+γP7(1+P5(θw−1))m*e−E1+P5(θw−1))−(NtNb)(P6)η. (19)

withP1(0)−s=0,P2(0)=1+σ(1+3k1P2(0))P3(0),

P5(0)−1=0,and

P7(0)−1=0(foractivecontrol)and

NbP8(0)=−NtP6(0)(forpassivecontrol) (20)

To solve Eq. (19) by considering Eq. (20) as an IVP equation, the values for P3(0) , i.e., Gηη(0) , P6(0) , i.e., θη(0) , and P8(0) , i.e., ϕη(0) are required. The primary assumptions for Gηη(0) , θη(0) , and ϕη(0) are made, and the IPS method is employed to obtain numerical results. Subsequently, the computed values of Gη(0) , θη(0) , and ϕη(0) at η∞ when Gη(η∞)=0 , Gηη(η∞)=0 , θη(η∞)=0 , and ϕη(η∞)=0 are compared, and the values of Gηη(0) , θη(0) , and ϕη(0) are adjusted using the shooting strategy to obtain an improved estimate of the solution. This procedure is repeated until an accuracy level of 10−8 is achieved.

4  Discussion

The performance of the dimensionless parameters is analyzed by solving Eq. (19) based on Eq. (20) using the above numerical method. The following values are fixed throughout the investigation: σ=1 , M=0.5, k1=0.5, s = 1, Ec = 0.5, Rd = 0.5, Pr = 1.2, θw = 1.5, δ = 5, Nb = 0.5, γ = 1, Le = 2, m* = 0.5, λ1 = 0.4, λ2 = 0.4, λ3 = 0.5, Nt = 0.5, and E = 1. The performance of the pertinent parameter is observed based on plots by varying each parameter. A remarkable agreement is observed in this study when comparing the −Gηη(0) values with the results of Cortell [44] (see Tab. 1).

Table 1: Validation of −Gηη(0) based on the study of Cortell [44]

This study mainly explores the impact of the activation energy and chemical reaction on Gn(η) , θ(η) , and ϕ(η) . The values of CfRex (local skin friction) and NuxRex (local Nusselt number) are calculated as a function of the viscoelastic and radiation parameters, respectively. The value of ShxRex (local Sherwood number) is reported for active as well as passive control of the nanoparticles.

The activation energy, chemical reaction, radiation, and elastic deformation parameters have an impact on Gη(η) , θ(η) , and ϕ(η) because of the presence of the thermal and concentration buoyancy parameters in the momentum balance equation, i.e., Eq. (2). The impacts of the activation energy and chemical reaction on the Gη(η) , θ(η) , and ϕ(η) profiles are presented in Fig. 2. E and γ show similar trends in the velocity and concentration profiles and an opposite trend in the temperature profile. The influences of E and γ are more significant in the concentration profile than in the other profiles. An increasing trend is observed in the velocity and concentration profiles and the corresponding boundary layers with increasing E. However, a decreasing trend is observed in the temperature profile as E increases. The rate of mass transport increases based on the chemical reaction parameter, considerably decreasing the concentration profile. A slight increase can be observed in the thermal boundary layer thickness and a decline is noted in the momentum boundary layer thickness as γ increases. An increase in E=k*−1EaT∞−1 results in an increase in the Arrhenius activation energy, reducing the chemical reaction and increasing the nanofluid concentration profile significantly. The influences of E and γ are minimal in both the velocity and temperature profiles.

Figure 2: (a) Variation in the Gη(η) curves with γ and E (solid and dashed lines represent the active and passive control of nanoparticles, respectively). (b) Variation in the θ(η) curves with γ and E (solid and dashed lines represent the active and passive control of nanoparticles, respectively). (c) Variation in the θ(η) curves with γ and E (solid and dashed lines represent the active and passive control of nanoparticles, respectively)

Fig. 3 presents the impacts of Rd and δ on the θ(η) and ϕ(η) profiles. The increasing values of Rd and δ boost the temperature profile and reduce the concentration profile. In contrast to the δ case, Rd considerably affects both the temperature and concentration boundary layers. Figs. 2 and 3 show that the thicknesses of the velocity, thermal, and concentration boundary layers are higher for active control than for the passive control of nanoparticles. The numerical values of the concentration profile are negative in case of passive control.

Figure 3: (a) Variation in the θ(η) curves with Rd and δ (solid and dashed lines represent the active and passive control of nanoparticles, respectively). (b) Variation in the θ(η) curves with Rd and δ (solid and dashed lines represent the active and passive control of nanoparticles, respectively)

The impacts of E, γ, λ1, and λ2 with M on the local skin friction ( CfRex ), local Nusselt number ( NuxRex ), and local Sherwood number ( ShxRex−1/2 ) are presented in Figs. 4 and 5. The increase in M and γ reduces CfRex and NuxRex & increases ShxRex . The presence of Lorentz force in the flow field because of the applied magnetic field and the chemical reaction processes reduce the surface drag as well as the heat transfer rate and enhance the mass transfer rate. The parameter E is an increasing function of the activation energy, Ea. Thus, an increase in E increases the surface drag ( CfRex ) and the heat transfer rate ( NuxRex ) and reduces the mass transfer rate ( ShxRex ). Fig. 5 indicates that the convective heat transfer rate increases because of the presence of thermal and concentration buoyancy (λ1 and λ2, respectively). The same buoyancy parameters exert an opposite impact on the mass transfer rate in case of the active and passive control of nanoparticles.

Figure 4: (a) Variation in CfRex with γ , E, and M (solid and dashed lines represent the active and passive control of nanoparticles, respectively). (b) Variation in NuxRex with γ , E, and M (solid and dashed lines represent the active and passive control of nanoparticles, respectively). (c) Variation in ShxRex with γ , E, and M (solid and dashed lines represent the active and passive control of nanoparticles, respectively)

Figure 5: (a) Variation in NuxRex with λ1, λ2, and M (solid and dashed lines represent the active and passive control of nanoparticles, respectively). (b) Variation in ShxRex with λ1, λ2, and M (solid and dashed lines represent the active and passive control of nanoparticles, respectively)

By comparing the active and passive control of the nanoparticles, high heat and mass transfer rates can be observed in the passive control case and a high surface drag can be observed in the active control case.

5  Conclusions

In this study, the effects of the Arrhenius activation energy on the momentum, energy, and mass transport of a second-grade magneto nanofluid flow with elastic deformation effects were analyzed. The governing parameters of the flow include the thermal buoyancy, concentration buoyancy, magnetic field, elastic deformation, nonlinear thermal radiation, binary-order chemical reaction, and Arrhenius activation energy. Further, numerical results were obtained and discussed. The notable findings are given below:

•   The Arrhenius activation energy significantly increases the thickness of the second-grade nanoconcentration boundary layer, whereas the chemical reaction reduces it. A slight increase is observed in the second-grade nano momentum boundary layer thickness as the Arrhenius activation energy increases. The opposite trend is observed in the second-grade nanothermal boundary layer with the activation energy.

•   The elastic deformation and nonlinear thermal radiation increase the nanothermal boundary layer thickness and shrink the nanoconcentration boundary layer.

•   The numerical amount of skin friction and the local Nusselt number increase with the activation energy and decrease with the chemical reaction rate. An increase in the thermal and concentration buoyancy increases the local Nusselt number and decreases the local Sherwood number.

•   The thickness of the nanoboundary layers can be increased via active control of the nanoparticles in the boundary.

•   The Nusselt and local Sherwood numbers increase in the passive control case, and the skin friction increases in the active control case.

Acknowledgement: The authors wish to express their sincere thanks to the honorable referees for their valuable comments and suggestions to improve the quality of the paper. In addition, the authors would like to express their gratitude to the United Arab Emirates University, Al Ain, UAE for providing financial support with Grant No. 31S363-UPAR (4) 2018.

Funding Statement: This work was supported by United Arab Emirates University, Al Ain, UAE with Grant No. 31S363-UPAR (4) 2018.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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