ARTICLE

J. Cent. South Univ. (2019) 26: 1250-1259

DOI: https://doi.org/10.1007/s11771-019-4084-9

Numerical simulation for Darcy-Forchheimer 3D rotating flow subject to binary chemical reaction and Arrhenius activation energy

Tasawar HAYAT^{1, 2}, Arsalan AZIZ¹, Taseer MUHAMMAD³, Ahmed ALSAEDI²

1. Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan;

2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia;

3. Department of Mathematics, Government College Women University, Sialkot 51310, Pakistan

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: Three-dimensional Darcy-Forchheimer nanoliquid flow in the presence of rotating frame and activation energy is inspected. Flow is developed through linearly stretching of the surface. Convection of heat and mass exchange is given due consideration. The novel characteristics in regards to Brownian dispersion and thermophoresis are retained. The variation in partial differential framework (PDEs) to nonlinear ordinary differential framework (ODEs) is done through reasonable transformations. Governing differential frameworks have been computed in edge of NDSolve. Discussion regarding thermal field and concentration distribution for several involved parameters is pivotal part. Physical amounts like surface drag coefficients, transfer of heat and mass rates are portrayed by numeric esteems. It is noticed that impacts of porosity parameter and Forchheimer number on the thermal and concentration fields are quite similar. Both temperature and associated thermal layer thickness are enhanced for larger porosity parameter and Forchheimer number. Temperature and concentration fields exhibit similar trend for the higher values of rotational parameter. Effects of thermal and concentration Biot numbers on the temperature and concentration fields are qualitatively similar. Higher Prandtl and Schmidt numbers correspond to stronger temperature and concentration fields. Larger nondimensional activation energy, temperature difference parameter and fitted rate constant yield weaker concentration field. Brownian motion parameter for temperature and concentration has reverse effects while similar trend is observed via thermophoresis parameter.

Key words: nanoparticles; rotating frame; Arrhenius activation energy; Darcy-Forchheimer porous space; numerical solution

Cite this article as: Tasawar HAYAT, Arsalan AZIZ, Taseer MUHAMMAD, Ahmed ALSAEDI. Numerical simulation for Darcy-Forchheimer 3D rotating flow subject to binary chemical reaction and Arrhenius activation energy [J]. Journal of Central South University, 2019, 26(5): 1250–1259. DOI: https://doi.org/10.1007/s11771-019-4084-9.

1 Introduction

Nanofluid is a colloidal blend acquired by nanoparticles of size under 100 nm out of a working fluid. Fundamentally working fluids accommodate lower thermal exchange highlights on account of their weak warm proficiency. In this way, the expansion of nanoparticles in such sort of working liquids known as nanofluids is very appealing strategy to upgrade heat productivity of such fluids. Advance nanofluid has an essential job for cooling rate prerequisites with high warm proficiency. Directly nanofluid elements have increased astounding consideration of researchers because of their bewitching warm transport in a few down to earth zones. Today the nanofluids applications are effectively experienced in the enterprises. The utilization of nanofluids incorporates however no restricted to warm exchangers, impingement planes, car, sustainable power sources, warming and treating process, atomic reactors, electronic chips cooling, grease, ignition and drug and so forth. CHOI et al [1] discussed the structure of nanomaterials. They deduced that addition of nanoparticles in the basic working fluids is exceedingly helpful to increase cooling capacity of normal fluids. BUONGIORNO [2] created two-part connection for heat vitality transport in nanoliquids. Connection depends on two-critical slips to be specific Brownian development and thermophoresis. Some ongoing looks into on nanofluid stream can be seen through attempts [3–15].

Concentration contrast of species exists in blend with mass exchange wonder changing convergence of species in blend move from higher concentration locale to lower concentration district. The minimum mandatory vitality that is needed by reactants previously occurring synthetic response is characterized as enactment vitality. Mass exchange process alongside synthetic response with initiation vitality by and large discovers applications in mechanics of oil and water emulsions, compound designing, nourishment handling and so on. Right off the bat the normal convection stream of parallel blend in a permeable space with activation energy is studied by BESTMAN [16]. Temperamental regular convection stream with nth-arrange reaction and activation energy was explored by MAKINDE et al [17]. Further recent attempts on activation energy can be seen through the studies [18–23].

Enthusiasm of recent specialists in warmth exchange through permeable space is helpful for applications in innovative, compound and assembling forms. Unequivocal procedures toward this path may incorporate unrefined petroleum creation, the plan of atomic reactors, atomic based archives, use of geothermal essentialness, development of dampness in sinewy protection, substance synergist reactors, sun based power, units of the vitality stockpiling and a few others. Darcy’s law expresses that there exists an immediate connection between volumetric motion of liquid through medium and weight slope. Essentially, articulation of DARCY [24] is utilized generally in the writing for issues related to demonstrating and investigation of stream soaking permeable space. Customary Darcy's law is appropriate just for low velocity and littler porosity. Such law dismisses idleness and limit impacts. Changed type of traditional Darcy’s hypothesis results in non- Darcian permeable medium which joins boundary and inertial impacts. Uniqueness in porosity and high stream rates occur in numerous pragmatic conditions. Idea of such state can't be imagined because of insuitability of Darcy's law. For high velocity, flow enters a nonlinear flow locale and therefore inactivity can’t be overlooked. Subsequently FORCHHEIMER [25] employed latency through consideration of square speed term in energy articulation. MUSKAT [26] named this commitment as Forchheimer factor. The examinations on Darcy-Forchheimer stream can be cited through the investigations [27–33] and different endeavors in that.

Heat move in rotating streams is very alluring. It is a result of their tremendous applications in assembling of precious stone development, PC stockpiling gadget, warm power age and gas turbine rotors and so on. WANG [34] figured out explanatory answers for gooey liquid stream by an extended sheet in turning outline. TAKHAR et al [35] tended to be magnetohydrodynamic in pivoting extended stream. JAVED et al [36] built nearby comparative answers for pivoting stream by exponential twisting surface. ZAIMI et al [37] examined visco-elastic liquid stream in turning outline. ROSALI et al [38] investigated turning stream by exponential porous surface. MUSTAFA et al [39] talked about Maxwell liquid stream subject to variable warm conductivity and pivoting outline. Three-dimensional turning Maxwell nanoliquid stream was broke down by HAYAT et al [40]. MUSTAFA et al [41] registered diagnostic answers for pivoting stream of Oldroyd-B fluid.

HAYAT et al [42] discussed Darcy- Forchheimer flow of nanofluid subject to rotating frame. Very recently HAYAT et al [43] computed numerical solutions of Darcy-Forchheimer three- dimensional rotating flow of nanofluid with prescribed heat and mass flux conditions.

Fundamental point of current article is to address Darcy-Forchheimer 3D pivoting stream of nanoliquid initiated by straightly deformable sheet with concoction response and actuation vitality. Brownian development and thermophoresis instruments are investigated. Convective states of warmth and mass are examined. NDsolve system is actualized for arrangements. Charts portraying critical parts of present investigation are shown and dissected. In addition, surface drag coefficients and warmth and mass exchange rates are registered and examined.

2 Statement

Three-dimensional nanoliquid stretched flow in a rotating frame is analyzed. Chemical reaction subject to concept of activation energy is examined. Darcy-Forchheimer relation for porous medium is studied. Nanoliquid has qualities of Brownian development and thermophoresis. Sheet is warmed by hot fluid with temperature T_f and concentration C_f. Heat and mass transport coefficients are taken h₁ and h₂. Surface in Cartesian facilitate framework is agreed with xy-plane and fluid involves z≥0. Liquid rotates reliably about z-axis subject to constant angular velocity ω (see Figure 1). Governing boundary layer expressions are as follows:

                            (1)

     (2)

     (3)

         (4)

             (5)

with the following conditions:

             (6)

  (7)

where (u, v, w) stand for velocities in x-, y- and z-directions, v(=μ/ρ_f) for kinematic viscosity, μ for dynamic viscosity, k for thermal conductivity, ρ_f for density, k^* for permeability of porous space, F=C_b/(xk^*1/2) for variable inertial coefficient of porous space, α^*=k/(ρc)_f for thermal diffusivity, (ρc)_p for effective heat potential of nanomaterial, (ρc)_f for heat potential of the liquid, T for temperature, D_B for Brownian diffusivity, C for concentration, D_T for thermophoretic diffusivity, C_b for drag coefficient, k_r for reaction rate, n for fitted rate constant, E_a for activation energy, κ for Boltzmann constant, h₁ for heat transfer coefficient, h₂ for mass transfer coefficient, C_∞ for ambient concentration and T_∞ for ambient temperature. Setting

(8)

expression (1) is automatically verified and Eqs. (2)–(7) are reduced to

         (9)

         (10)

            (11)

           (12)

                   (13)

    (14)

Here porosity parameter, Forchheimer number, rotation parameter, Schmidt number, Prandtl number, chemical reaction parameter, Brownian movement parameter, thermal Biot number, thermophoresis parameter, nondimensional activation energy, temperature difference parameter and concentration Biot number are symbolized by λ, Fr, Ω, Sc, Pr, σ, N_b, γ₁, N_t, E, δ and γ₂ respectively. These nondimensional parameters are expressed by

                   (15)

The physical quantities are given by [44]

                         (16)

In above expressions represents the local Reynolds number.

Figure 1 Flow configuration

3 Graphical outcomes and discussion

This segment investigates effects of couple of relevant physical flow variables like thermal Biot parameter γ₁, Prandtl parameter Pr, temperature difference parameter δ, porosity parameter λ, concentration Biot parameter γ₂, Brownian movement parameter N_b, nondimensional activation energy E, Forchheimer number Fr, nondimensional reaction rate σ, Schmidt parameter Sc, thermophoresis number N_t, fitted rate constant n and rotational parameter Ω on nondimensional temperature θ(ζ) and concentration φ(ζ). Figure 2 presents curves of temperature field θ(ζ) for higher λ. It has been seen that by expanding λ, entrance profundity of θ(ζ) lifts. Physically, the presence of permeable space creates opposition in smooth movement and extreme, indicating diminishing in liquid velocity. Consequently an addition is seen for temperature and relating layer thickness. Influence of Forchheimer number Fr on temperature θ has been shown in Figure 3. Temperature and layer thickness are expanding elements of Fr. Figure 4 shows temperature θ(ζ) for particular data of rotational parameter. Biggerm Ω prompts an improvement in temperature. Figure 5 shows effect of heat Biot parameter γ₁ on temperature. Here an expansion in γ₁ causes stronger temperature.Figure 6 presents estimations θ(ζ) as for Pr. Temperature is diminished through Pr. Prandtl number has turn around relationship with thermal diffusivity. Bigger Prandtl parameter incites poor thermal diffusivity. Varieties in temperature θ(ζ) for thermophoresis variable N_t are shown in Figure 7. Truth be told, bigger thermophoresis number prompts more temperature and related layer thickness. An extension in N_t causes update in thermophoresis diffusion. This keeps in expanding temperature. Figure 8 is depicted to consider variety in temperature θ(ζ) affected by Brownian development parameter N_b. An augmentation in N_b causes an improvement in θ(ζ). Physically, the unsystematic development of nanoparticles ascends by upgrading N_b because crash of particles happens. Subsequently active vitality is changed over into thermal vitality which prompts an augmentation in temperature and related layer. Figure 9 portrays that bigger porosity parameter λ displays an expansion in concentration φ(ζ). Figure 10 shows effect of Fr on concentration φ(ζ). Both concentration and related layer are expanded by means of Fr.Figure 11 displays variety in φ(ζ) for changing Ω. It is seen that by growing turn parameter, an enhancement is seen in concentration φ and its relating layer. Figure 12 depicts effect of concentration Biot parameter γ₂ on concentration φ. Concentration and related layer are more for γ₂. Figure 13 presents impact of nondimensional activation energy E on concentration φ(ζ). Here an augmentation in E rots adjusted Arrhenius work. This in the end builds up the generative synthetic response because concentration φ(ζ) upgrades. Figure 14 displays that an improvement in nondimensional reaction rate σ shows a rot in concentration. Figure 15 illustrates impact of temperature difference parameter δ on φ(ζ). Here φ(ζ) is noted diminishing capacity of δ. Figure 16 presents concentration φ(ζ) for evolving n. It is seen that by enhancing n, entrance profundity of φ(ζ) finishes to more slender. Figure 17 shows impact of Sc on concentration φ. This figure demonstrates that bigger Sc rots concentration φ(ζ). Schmidt number includes Brownian development. An extension in Sc prompts poor Brownian development, which shows a reduction in concentration and layer thickness. Figure 18 displays concentration φ(ζ) for specific estimations of thermophoresis number N_t. Plainly higher N_t offers ascent to more concentration φ(ζ) and layer thickness. Figure 19 illustrates that concentration φ(ζ) lessens for N_b. Table 1 shows surface drag coefficients for a few estimations of λ, Fr and Ω. Drag coefficients delineate expanding pattern for higher Ω while reverse effect is watched for bigger λ and Fr. Table 2 is prepared to validate present data with previously published data by JAVED et al [36]. Here we analyzed that the present solutions have good agreement with previous solutions by JAVED et al [36]. Table 3 lists local Nusselt number by means of λ, Fr, Ω, N_t, γ₁, N_b and Pr. Heat exchange rate decays via λ, Fr, Ω, N_t and N_b. Effects of γ₁ and Pr on heat transfer rate are very comparable. Table 4 indicates numerical data of mass exchange rate by means of σ, Sc, N_t, E, δ, N_b, n and γ₂. Here we saw that mass transfer rate is higher and brings down qualities for bigger (Sc, δ, σ, N_b, n, γ₂) and (E, N_t) respectively.

Figure 2 θ(ζ) variation for λ

Figure 3 θ(ζ) variation for Fr

Figure 4 θ(ζ) variation for Ω

Figure 5 θ(ζ) variation for γ₁

Figure 6 Plots of θ(ζ) for Pr

Figure 7 θ(ζ) variation for N_t

Figure 8 θ(ζ) variation for N_b

Figure 9 φ(ζ) variation for λ

Figure 10 φ(ζ) variation for Fr

Figure 11 φ(ζ) variation for Ω

Figure 12 φ(ζ) variation for γ₂

Figure 13 φ(ζ) variation for E

Figure 14 φ(ζ) variation for σ

Figure 15 φ(ζ) variation for δ

Figure 16 φ(ζ) variation for n

Figure 17 φ(ζ) variation for Sc

Figure 18 φ(ζ) variation for N_t

Figure 19 φ(ζ) variation for N_b

Table 1 Surface drag coefficients and for varying Fr, λ and Ω

Table 2 Comparative data of –f ″(0) and –g′(0) for varying Ω when λ=Fr=0

Table 3 Heat transfer rate for varying λ, Fr, Ω, N_t, γ₁, N_b and Pr

Table 4 Mass transfer rate for varying σ, Sc, N_t, E, δ, N_b, n and γ₂

4 Conclusions

Effect of activation energy in convective Darcy-Forchheimer flow is studied. Major results of present study can be noted below:

1) Temperature and concentration have similar qualitative effects for λ and Fr.

2) Through rotation Ω, there is an enhancement of temperature θ(ζ) and concentration φ(ζ).

3) Heat and mass Biot numbers give rise to temperature and concentration.

4) Larger activation energy E shows higher concentration field φ(ζ).

5) Bigger Prandtl Pr and Schmidt Sc numbers lead to weaker temperature θ(ζ) and concentration φ(ζ).

6) Effects of δ and σ on concentration are qualitatively similar.

7) Brownian parameter N_b for temperature and concentration has reverse effects.

8) Temperature and concentration have similar trend for N_t.

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(Edited by YANG Hua)

中文导读

基于二元化学反应和Arrhenius活化能的Darcy-Forchheimer流体三维旋转数值模拟

摘要：研究了旋转框架和活化能存在下的三维Darcy-Forchheimer纳米流动。考虑了热对流和质量交换，保留了布朗色散和热泳的新特征。通过合理的变换实现了偏微分系统(PDES)向非线性常微分系统(ODES)的转换。选择NDSolve的边界条件，计算了控制差分方程。重点讨论了几个相关参数下的热场和浓度分布。用数值来描述表面阻力系数、热量传递和质量交换率等物理量。结果表明，孔隙度和Forchheimer数对热场和浓度场的影响是相似的。随着孔隙度的增大和Forchheimer数的增加，温度升高和相应的热层厚度增大。温度场和浓度场随着旋转角的增大而呈现相似的变化趋势。热和浓度Biot数对温度场和浓度场的影响在本质上是相似的。较高的Prandtl和Schmidt数对应较强的温度场和浓度场。较大的无量纲活化能、温差和拟合速率常数对应较弱的浓度场。布朗运动参数对温度和浓度的作用是相反的，而热泳参数的作用是相似的。

关键词：纳米粒子；旋转框架；Arrhenius活化能；Darcy-Forchheimer 多孔空间；数值解

Received date: 2018-09-03; Accepted date: 2018-11-29

Corresponding author: Arsalan AZIZ, PhD Candidate; Tel: +92-332-0332048; E-mail: arsalan_qau@yahoo.com; ORCID: 0000-0002- 1985-7392