Computational Approach to Dynamic Systems through Similarity Measure and Homotopy Analysis Method for Renewable Energy

Abstract

To achieve considerably high thermal conductivity, hybrid nanofluids are some of the best alternatives that can be considered as renewable energy resources and as replacements for the traditional ways of heat transfer through fluids. The subject of the present work is to probe the heat and mass transfer flow of an ethylene glycol based hybrid nanofluid (Au-ZnO/C2H6O2) in three dimensions with homogeneous-heterogeneous chemical reactions and the nanoparticle shape factor. The applications of appropriate similarity transformations are done to make the corresponding non-dimensional equations, which are used in the analytic computation through the homotopy analysis method (HAM). Graphical representations are shown for the behaviors of the parameters and profiles. The hybrid nanofluid (Au-ZnO/C2H6O2) has a great influence on the flow, temperature, and cubic autocatalysis chemical reactions. The axial velocity and the heat transfer increase and the concentration of the cubic autocatalytic chemical reactions decreases with increasing stretching parameters. The tangential velocity and the concentration of cubic autocatalytic chemical reactions decrease and the heat transfer increases with increasing Reynolds number. A close agreement of the present work with the published study is achieved.

1. Introduction

Energy has a crucial role in the prosperity and development of any country. The daily consumed energy resources like natural gas, oil, and coal are certain to vanish with the passage of time because these are huge sources of energy and are being depleted due to their limited availability. To cope with such a situation, the replenishment of the world’s energy is of utmost concern, making it is a basic requirement to search for some reliable and affordable energy alternatives. Such problems apply to renewable energy systems. Nanoparticles have been shown to solve such constraints because of their remarkable heat transfer capabilities. The application of nanoparticles in the industrial, biomedical, and energy sectors is due to their thermophysical properties. Nanoparticles have seen applications in energy conversion (e.g., fuel cells, solar cells, and thermoelectric devices), energy storage (e.g., rechargeable batteries and super capacitors), and energy saving (e.g., insulation such as aerogels and smart glazes, efficient lightning like light emitting diodes and organic light-emitting diodes). To combat climate change, clean and sustainable energy sources need to be rapidly developed. Solar energy technology converts solar energy directly into electricity, for which high performance cooling, heating, and electricity generation are among the inevitable requirements. In solar collectors, the absorbed incident solar radiation is converted to heat. The working fluid conveys the generated heat for different uses [1]. Ettefaghi et al. [2] worked on a bio-nanoemulsion fuel based on biodegradable nanoparticles to improve diesel engines’ performance and reduce exhaust emissions. Gunjo et al. [3] investigated the melting enhancement of a latent heat storage with dispersed Cu, CuO, and Al₂O₃ nanoparticles for a solar thermal application. Khanafer and Vafai [4] presented a review on the applications of nanofluids in the solar energy field.

Nanofluids reduce the process time, enhance the heating rates, and improve the lifespan of machinery [5]. Nanofluids have seen applications in power saving, manufacturing, transportation, healthcare, microfluidics, nano-technology, microelectronics, etc. Recently, nano-technology has attracted great attraction from scientists [6]. Nanoparticles are the most interesting technology to introduce novel, environmentally friendly chemical and mechanical polishing slurries to fabricate effective materials [7]. Thermal conductivity is of great importance and is enhanced by the incorporation of nanoparticles in the base fluid [8]. Hamilton and Crosser [9] studied the thermal conductivity of a heterogeneous two component system. Nanofluids were obtained by the addition of nanoparticles to the base fluids, and they have gained popularity since the work of Choi and Eastman [10]. Vallejo et al. [11] analyzed the internal aspects of the fluid for six carbon-based nanomaterials in a rotating rheometer with a double conic shape containing a typical sheet. Alihosseini and Jafari [12] investigated a three-dimensional computational fluid dynamics model for an aluminum foam and nanoparticles with heat transfer using a number of cylinders having different configurations through a permeable medium. Sheikholeslami et al. [13], working with a ethylene glycol nanofluid, discussed the electric field, thermal radiation, and nanoparticle shape factors of a ferrofluid by showing that the platelet shape led to enhanced convective flow. Al-Kouz et al. [14] applied computational fluid dynamics to analyze entropy generation in a rarefied time dependent, laminar two-dimensional flow of an air-aluminum oxide nanofluid in a cavity with a square shape having more than one solid fin at the heated wall where the optimization procedure was adopted to show the conditions by which the overall entropy generation was reduced. Atta et al. [15] modified the asphaltenes isolated from crude oil to work as capping agents for the synthesis of hydrophobic silica to investigate the surface charge of hydrophobic silica nanoparticles, the chemical structure, the particle size, and the surface morphology. Rout et al. [16] presented the three and higher order nonlinear thin film study and optics fabricated with gold nanoparticles. They obtained the solution via spin-coating techniques to achieve the highest values of nonlinear absorption coefficient, nonlinear refractive index and saturation intensity. Alvarez-Regueiro et al. [17] experimentally determined the heat transfer coefficients and pressure drops of four functionalized graphene nanoplatelet nanofluids for heat transfer enhancement to discuss the nanoadditive loading, temperature and Reynolds number. Alsagri et al. [18] elaborated the heat and mass transfer flow of single walled and multi walled carbon nanotubes past a stretchable cylinder by investigating that the heat transfer enhances with the high values of nanoparticles concentration of single walled carbon nanotubes compared to that of multi walled carbon nanotubes. Working on transverse vibration, Mishra et al. [19] comparatively investigated a computational fluid dynamic model for water based nanofluid through a pipe subject to superimposed vibration, applied to the wall to increase the heat transfer in axial direction while vibration effect is decreased for pure liquid and is increased for nanofluid. Abbas et al. [20] achieved the results that in the heat and mass transfer flow of Cross nanofluid, the Bejan number was intensified for the high values of thermal radiation parameter. Some discussion on nanofluids and other relevant studies can be found in the references [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55].

Mono-nanofluids represent enhanced thermal conductivity and good rheological characteristics, but still they have some weak characteristics necessary for a particular purpose. By the hybridization process, different nanoparticles are added in a base fluid to make the hybrid nanofluid which has enhanced thermophysical properties and thermal conductivity as well as rheological properties. Ahmad et al. [56] investigated the hybrid nanofluid with activation energy and binary chemical reaction through a moving wedge taken into account the Darcy law of porous medium, heat generation, thermal slip, radiation, and variable viscosity. Dinarvand and Rostami [57] presented the ZnO-Au hybrid nanofluid when 15 gm of nanoparticles are added into the 100 gm base fluid, the heat transfer enhances more than 40% compared to that of the regular fluid.

Homogeneous-heterogeneous chemical reactions have important applications in chemical industries. Ahmad and Xu [58] worked on homogeneous-heterogeneous chemical reactions in which the reactive species were of regular size reacting with other species in a nanofluid to show more realistic mathematical model physically. Hayat et al. [59] elaborated the Xue nanofluid model to study the carbon nanotubes nanofluids in rotating systems incorporating Darcy–Forchheimer law, homogeneous-heterogeneous chemical reactions and optimal series solutions. Suleman et al. [60] addressed the homogeneous-heterogeneous chemical reactions in Ag-H₂O nanofluid flow past a stretching sheet with Newtonian heating to prove that concentration field was decreased for the increasing strength of homogeneous-heterogeneous chemical reactions.

In the literature, interesting studies exists like [5] which investigates the electrical conductivity, structural and optical properties of ZnO. In study [6], the theoretical and experimental results of electric current and thermal conductivity of H₂O-ethylene glycol based TiO₂ have been obtained. The study [7] relates to the oxide-ethylene glycol nanofluid with different sizes of nanoparticles. Due to the applications of the above studies, it is desire to investigate the ethylene glycol based Au-ZnO hybrid nanofluid flow with heat transfer and homogeneous-heterogeneous chemical reactions in rotating system. The present study has the applications in renewable energy technology, thermal power generating system, spin coating, turbo machinery etc. The solution of the problem is obtained through an effective technique known as homotopy analysis method [61]. Investigations are shown through graphs and discussed in detail.

2. Methods

A rotating flow of hydromagnetic, time independent and incompressible hybrid nanofluid between two parallel disks in three dimensions is analyzed. Homogeneous-heterogeneous chemical reactions are also considered. The lower disk is supposed to locate at z = 0 while the upper disk is at a constant distance H apart. The velocities and stretching on these disks are (_Ω1,_Ω2) and (a₁, a₂), respectively while the temperatures on these disks are T₁ and T₂, respectively. A magnetic field of strength B₀ is applied in the direction of z-axis (please see Figure 1). Ethylene glycol is chosen for the base fluid in which zinc oxide and gold nanoparticles are added.

For cubic auto-catalysis, the homogeneous reaction is

2C+B→3C,rate=^c2b_kc.

The first order isothermal reaction on the surface of catalyst is

B→C,rate=b_ks,

where B and C denote the chemical species with concentrations b and c, respectively. k_c and k_s are the rate constants.

Cylindrical coordinates (r,ϑ , z), are applied to provide the thermodynamics of hybrid nanofluid as [57,58,59]

∂w∂z+∂u∂r+ur=0,

_ρhnf−^v2r+∂u∂ru+∂u∂zw=_μhnf^∂2u∂^z2+^∂2u∂^r2−u^r2+∂u∂r1r−_{σhnf B02}u−_μhnfS^u2−_S1 ^u2−∂P∂r,

_ρhnfuvr+w∂v∂z+u∂v∂r=_μhnf^∂2v∂^z2+^∂2v∂^r2−v^r2+1r∂v∂r−_{σhnf B02}v−_μhnfS^v2−_S1 ^v2,

_ρhnfw∂w∂z+u∂w∂r=−∂P∂z+_μhnf^∂2w∂^z2+^∂2w∂^r2+1r∂w∂r−_μhnfS^w2−_S1 ^w2,

_(ρcp)hnfw∂T∂z+u∂T∂r=_khnf+16_{T13 σ1}3_k0^∂2T∂^z2+^∂2T∂^r2+1r∂T∂r+_{σhnf B02}(^v2+^u2),

w∂b∂z+u∂b∂r=−^c2b_kc+_DB^∂2b∂^z2+^∂2b∂^r2+1r∂b∂r,

w∂c∂z+u∂c∂r=^c2b_kc+_DC^∂2c∂^z2+^∂2c∂^r2+1r∂c∂r.

The boundary conditions are

atz=0,_DC∂c∂z=_ksc,_DB∂b∂z=_ksb,T=_T1,w=0,v=r_Ω1,u=r_a1,

atz=H,c→0,b→_b0,T=_T2,P→∞,w=0,v=r_Ω2,u=r_a2,

where u(r, ϑ, z), v(r, ϑ, z) and w(r, ϑ, z) are the components of velocity, P is the pressure. S is the permeability of porous medium, S₁ =_Cbr^S12is the non-uniform inertia coefficient of porous medium with C_b as the drag coefficient. Temperature of hybrid nanofluid is T and B = (0, 0, B₀) is the magnetic field._σ1is the Stefan Boltzmann constant and k₀ is the absorption coefficient. For the hybrid nanofluid, the important quantities are_ρhnf(density),_μhnf(dynamic viscosity),_σhnf(electrical conductivity), (c_p)_hnf(heat capacity) and k_hnf (thermal conductivity). The subscript “hnf” shows the hybrid nanofluid. For the thermal conductivity, the mathematical formulation is obtained via Hamilton–Crosser model [9] as

_{knf kf}=_k1+(_n1−1)_kf−(_n1−1)(_kf−_k1)_ϕ1k1+(_n1−1)_kf+(_kf−_k1)_ϕ1,

where n is the empirical shape factor for the nanoparticle whose value is given in Table 1.

The subscript “f” denotes the base fluid namely ethylene glycol and the subscript “nf” is used for nanofluid._ρsand (c_P)_s are the density and heat capacity at specified pressure of nanoparticles, respectively._ϕ1is the first nanoparticle volume fraction while_ϕ2 is the second nanoparticle volume fraction which can be formulated as [57].

_ρs=(_ρ1×_m1)+(_ρ2×_m2)_m1+_m2,

_(cP)s=(_(cP)1×_m1)+(_(cP)2×_m2)_m1+_m2,

_ϕ1=_{m1 ρ1m1 ρ1}+_{m2 ρ2}+_{mf ρf},

_ϕ2=_{m2 ρ2m1 ρ1}+_{m2 ρ2}+_{mf ρf},

ϕ=_ϕ1+_ϕ2,

where m₁, m₂ and m_f are, respectively the mass of first nanoparticle, mass of the second nanoparticle and mass of the base fluid.ϕis the total volume fraction of zinc oxide and gold nanoparticles.

The thermophysical properties of C₂H₆O₂ as well as nanoparticles are given in Table 2.

The mathematical formulations for_ρhnf(density),_μhnf(dynamic viscosity),_σhnf(electrical conductivity), (c_p)_hnf (heat capacity) are given in Table 3 where_ϕsshows the particle concentration.

Following transformations are used

^f′(ζ)_Ω1r=u,v=r_Ω1g(ζ),−2f(ζ)H_Ω1=w,−_T2+T−_T2+_T1=θ(ζ),_{Ω1 ρf νf}ϵ^r22^H2+P(ζ)=P,φ_b0=b,c=_{b0 φ1},zH=ζ,

where_νf=_{μf ρf}is the kinematic viscosity andϵis the pressure parameter.

Using the values from Equation (18) in Equations (4)–(11), the following eight Equations (19)–(26) are obtained

_B1 ^f‴+Re2^ff″− ^f′2+^g2−M_B2 ^f′−ϵ−_k2Re_B1 ^f′−_k3Re1_ρhnf^(f′)2=0,

_B1 ^g″+Re2f^g′−M_B2 ^g′−_k2Re_B1g−_k3Re1_ρhnf^(g)2=0,

^P′=2_k2−4Ref^f′−^f″,

_{B3khnf kf}^θ″+1RdPrRe2f^θ′+MEc_B4^g2+^(f′)2=0,

ScRe2^φ′f−_k4φ_φ12+^φ″=0,

_φ1″+ScRe2_φ1′f+_k4φ_φ121_k5=0,

f=0,^f′=_k6,g=1,θ=1,^φ′=_k7φ,_{k4 φ1′}=−_k7φ,P=0atζ=0,

f=0,^f′=_k8,g=Ω,θ=0,φ=1,_φ1=0atζ=1,

where (^′) represents the derivative with respect toζ. B₁ =^{1−_{m1 ρ1m1 ρ1}+_{m2 ρ2}+_{mf ρf}−2.5}×^{1−_{m1 ρ1m1 ρ1}+_{m2 ρ2}+_{mf ρf}+_{m1 ρ1m1 ρ1}+_{m2 ρ2}+_{mf ρfρs ρf}−1}, B₂ = 1 +3_{σ1 ϕ1}+_{σ2 ϕ2σf}−(_ϕ1+_ϕ2)2+_{σ1 ϕ1}+_{σ2 ϕ2}(_ϕ1+_ϕ2)_σf−_{σ1 ϕ1}+_{σ2 ϕ2σf}−(_ϕ1+_ϕ2), B₃ =_(ρcP)f1−_{m1 ρ1m1 ρ1}+_{m2 ρ2}+_{mf ρfρf}+1−_{m1 ρ1m1 ρ1}+_{m2 ρ2}+_{mf ρfρs}×1−_{m1 ρ1m1 ρ1}+_{m2 ρ2}+_{mf ρf(cP)f}+1−_{m1 ρ1m1 ρ1}+_{m2 ρ2}+_{mf ρf(cP)s}, B₄ =_{σhnf ρhnf}. k₂ =_νfS_Ω1is the porosity parameter, k₃ =_Cb ^S12is the inertial parameter due to Darcy Forchheimer effect. The other non-dimensional parameters areΩ=_{Ω2 Ω1}, Re =_Ω1 ^H2_νf, M =_{σf B02ρf Ω1}, Rd =16_{σ1 T13}3_{kf k0}, Pr =_{(ρcP )hnf νfkf}, Ec =^r2 _Ω12cP(_T1−_T2), Sc =_{νf DB}, k₄ =_{kc b02Ω1}, k₅ =_{DC DB}, k₆ =_{a1 Ω1}, k₇ =_ksH_DBand k₈ =_{a2 Ω1}which are known as rotation parameter, Reynolds number, magnetic field parameter, thermal radiation parameter, Prandtl number, Eckert number, Schmidt number, homogeneous chemical reaction parameter, diffusion coefficient ratio, stretching parameter for lower disks, heterogeneous chemical reaction parameter and stretching parameter at upper disk, respectively.

Regarding the homogeneous-heterogeneous chemical reaction, the quantities B and C may be considered in a special case, i.e., if D_B is equal to D_C, then in such a case k₅ equals unity, which leads to

1=_φ1(ζ)+φ(ζ).

Using Equation (27), Equations (23) and (24) generate

0=ScRe^1−φ2 _k4φ+2^φ′f+^φ″,

whose corresponding boundary conditions become

^φ′=_k7φforζ=0whileφ=1forζ=1.

By taking derivative of Equation (19) with respect toζ, it becomes

_B1 ^f‴′+Re2f^f‴+2g^g′−M_B2 ^f″−_k2Re_B1 ^f″−2_k3Re1_ρhnf^{f′ f″}=0.

Considering Equation (21), Equations (25) and (26), the quantityϵis computed as

ϵ=^f‴(0)−Re−^g(0)2+^f′(0)2+M_B2 ^f′(0)+1_{B1 k2} ^f′(0).

Integrating Equation (21) with respect toζby using the limit 0 toζfor evaluating P as

P=−2Re^(f)2+1_k2∫0ζf^f′−^f′(0).

Skin Frictions and Nusselt Numbers

The important physical quantities are defined as

_Cf1 (Localskinfrictionatlowerdisk)=_τ|z=0ρhnf ^(r_Ω1)2,_Cf2 (Localskinfrictionatatupperdisk)=_τ|z=Hρhnf ^(r_Ω1)2,

where

τ=^(_τzr)2+^(_τzθ)2,

denotes the sum of shear stress of tangential forces_τzrand_τzθalong radial and tangential directions which are defined as

_τzr(Shearstressfrictionatlowerdisk)=_μhnf∂u∂z_|z=0=_μhnfr_Ω1 ^f″(0)Hand_τzθ=_μhnf∂v∂z_|z=0=_μhnfr_Ω1 ^g′(0)H.

Using the information of Equations (34) and (35), Equation (33) proceeds to

_Cf1 =1R_er^{1−_{m1 ρ1m1 ρ1}+_{m2 ρ2}+_{mf ρf}−2.5 f″(0)2+g′(0)212},

_Cf2 =1R_er^{1−_{m1 ρ1m1 ρ1}+_{m2 ρ2}+_{mf ρf}−2.5 f″(1)2+g′(1)212},

whereR_er=r_Ω1H_νhnfis the Reynolds number.

Another important physical quantity is

N_ur1 (LocalNusseltnumberatlowerdisk)=H_qwkf(_T1−_T2)_|z=0,N_ur2 (LocalNusseltnumberatupperdisk)=H_qwkf(_T1−_T2)_|z=H,

where q_wis the surface temperature defined as

_qw(Atlowerdisk)=−_khnf∂T∂z_|z=0=−_khnfT1−_T2H^θ′(0).

Taking information from Equation (39), Equation (38) becomes

N_ur1 =−_{khnf kf}^θ′(0),N_ur2 =−_{khnf kf}^θ′(1).

3. Computational Methodology

Following the HAM, choosing the initial guesses and linear operators for the velocities, temperature and homogeneous-heterogeneous chemical concentration profiles as

_f0(ζ)=^ζ3(_k6+_k8)−^ζ2(2_k6+_k8)+ζ_k6,_g0(ζ)=ζΩ+1−ζ,_θ0(ζ)=−ζ+1,_φ0(ζ)=ζ_k7+1_k7+1,

^φ″=_Lφ,^f‴′=_Lf,^g″=_Lg,^θ″=_Lθ,

characterizing

_LfE1+_E2ζ+_E3 ^ζ2+_E4 ^ζ3 = 0,_LgE5+_E6ζ = 0,_LθE7+_E8ζ = 0,_LφE9+_E10ζ = 0,

where E_i(i = 1–10) are the arbitrary constants.

3.1. Zeroth Order Deformation Problems

Introducing the nonlinear operator ℵ as

_ℵf[f(ζ,j),g(ζ,j)]=_B1^∂4f(ζ,j)∂^ζ4+Re2f(ζ,j)^∂3f(ζ,j)∂^ζ3+2g(ζ,j)∂g(ζ,j)∂ζ−M_B2^∂2f(ζ,j)∂^ζ2−_k2Re_B1^∂2f(ζ,j)∂^ζ2−2_k3Re1_ρhnf∂f(ζ,j)∂ζ^∂2f(ζ,j)∂^ζ2,

_ℵg[f(ζ,j),g(ζ,j)]=_B1^∂2g(ζ,j)∂^ζ2+Re2f(ζ,j)∂g(ζ,j)∂ζ−M_B2∂g(ζ,j)∂ζ−_{k2 B1}g(ζ,j)−_k31_ρhnf^g(ζ,j)2,

_ℵθ[f(ζ,j),g(ζ,j),θ(ζ,j)]=_{B3khnf kf}^∂2θ(ζ,j)∂^ζ2+1RdPrRe2f(ζ,j)∂θ(ζ,j)∂ζ+M_B4Ec^{∂f(ζ,j)∂ζ2}+^g(ζ,j)2,

_ℵφ[f(ζ,j),φ(ζ,j)]=^∂2φ(ζ,j)∂^ζ2+ReSc2f(ζ,j)∂φ(ζ,j)∂ζ+_k4φ(ζ,j)^{(1−φ(ζ,j))2},

where j is the homotopy parameter such that j ∈ [0, 1].

Moreover

(1−j)_Lf[f(ζ,j)−_f0(ζ)]=j_{ℏf ℵf}[f(ζ,j),g(ζ,j)],

(1−j)_Lg[g(ζ,j)−_g0(ζ)]=j_{ℏg ℵg}[f(ζ,j),g(ζ,j)],

(1−j)_Lθ[θ(ζ,j)−_θ0(ζ)]=j_{ℏθ ℵθ}[f(ζ,j),g(ζ,j),θ(ζ,j)],

(1−j)_Lφ[φ(ζ,j)−_φ0(ζ)]=j_{ℏφ ℵφ}[f(ζ,j),φ(ζ,j)],

where_ℏφ,_ℏf,_ℏθand_ℏgare the convergence control parameters.

Boundary conditions of Equation (48) are

f(0,j)=0,^f′(0,j)=_k6,f(1,j)=0,^f′(1,j)=_k8.

Boundary conditions of Equation (49) are

g(0,j)=1,g(1,j)=Ω.

Boundary conditions of Equation (50) are

θ(0,j)=1,θ(1,j)=0.

Boundary conditions of Equation (51) are

^φ′(0,j)=_k7φ(0,j),φ(1,j)=1.

Characterizing j = 0 and j = 1, the calculations obtained as

j=0⇒f(ζ,0)=_f0(ζ)andj=1⇒f(ζ,1)=f(ζ),

j=0⇒g(ζ,0)=_g0(ζ)andj=1⇒g(ζ,1)=g(ζ),

j=0⇒θ(ζ,0)=_θ0(ζ)andj=1⇒θ(ζ,1)=θ(ζ),

j=0⇒φ(ζ,0)=_φ0(ζ)andj=1⇒φ(ζ,1)=φ(ζ).

f(ζ,j) becomes f₀(ζ) and f(ζ) as j assumes the values zero and one. g(ζ,j) becomes g₀(ζ) and g(ζ) as j assumes the values zero and one. θ(ζ,j) becomes θ₀(ζ) and θ(ζ) as j assumes the values zero and one. Finally, φ(ζ,j) becomes φ₀(ζ) and φ(ζ) as j assumes the values zero and one.

Applying Taylor series expansion on the Equations (56)–(59), the results are obtained as

f(ζ,j)=_f0(ζ)+∑m=1∞_fm(ζ)^jm,_fm(ζ)=1m!^∂mf(ζ,j)∂^jm_∣j=0,

g(ζ,j)=_g0(ζ)+∑m=1∞_gm(ζ)^jm,_gm(ζ)=1m!^∂mg(ζ,j)∂^jm_∣j=0,

θ(ζ,j)=_θ0(ζ)+∑m=1∞_θm(ζ)^jm,_θm(ζ)=1m!^∂mθ(ζ,j)∂^jm_∣j=0,

φ(ζ,j)=_φ0(ζ)+∑m=1∞_φm(ζ)^jm,_φm(ζ)=1m!^∂mφ(ζ,j)∂^jm_∣j=0.

_ℏφ,_ℏf,_ℏθand_ℏgare adjusted to obtain the convergence for the series in Equations (60)–(63) at j = 1, so Equations (60)–(63) transform to

f(ζ)=_f0(ζ)+∑m=1∞_fm(ζ),

g(ζ)=_g0(ζ)+∑m=1∞_gm(ζ),

θ(ζ)=_θ0(ζ)+∑m=1∞_θm(ζ),

φ(ζ)=_φ0(ζ)+∑m=1∞_φm(ζ).

3.2. mth Order Deformation Problems

Considering Equations (48) and (52) for homotopy at mth order as

_Lf[_fm(ζ)−_{χm fm−1}(ζ)]=_{ℏf Rmf}(ζ),

_fm(0)=0,_fm(1)=0,_fm′(0)=0,_fm′(1)=0,

_Rmf(ζ)=_{B1 fm−1‴′}+Re∑k=om−1_{fm−1−k fk‴}+2_{gm−1−k gk′}−M_{B2 fm−1″}−_k2Re_{B1 fm−1″}−2_k3Re1_ρhnf∑k=om−1_{fm−1−k′ fk‴}.

Considering Equations (49) and (53) for homotopy at mth order as

_Lg[_gm(ζ)−_{χm gm−1}(ζ)]=_{ℏg Rmg}(ζ),

_gm(0)=0,_gm(1)=0,

_Rmg(ζ)=_{B1 gm−1″}+Re∑k=om−12_{fm−1−k gk′}−M_{B2 gm−1′}−_{k2 B1 gm−1}−_k31_ρhnf∑k=om−1_{gm−1−k gk}.

Considering Equations (50) and (54) for homotopy at mth order as

_Lθ[_θm(ζ)−_{χm θm−1}(ζ)]=_{ℏθ Rmθ}(ζ),

_θm(0)=0,_θm(1)=0,

_Rmθ(ζ)=_{B3khnf kfθm−1″}+1RdPrRe2∑k=om−1_{fm−1−k θk′}+M_B4Ec∑k=om−1_{fm−1−k′ fk′}+∑k=om−1_{gm−1−k gk}.

Considering Equations (51) and (55) for homotopy at mth order as

_Lφ[_φm(ζ)−_{χm φm−1}(ζ)]=_{ℏφ Rmφ}(ζ),

_φm′(0)=0,_φm(1)=0,

_Rmφ(ζ)=_φm−1″+ReSc2∑k=om−1_{fm−1−k φk′}+_k4φm−1+_φm−1−k∑l=ok_{φk−l φl}−2∑k=om−1_{φm−1−k φk},

_χm=0,m≤11,m>1.

Adding the particular solutions f_m*(ζ), g_m*(ζ), θ_m*(ζ) and φ_m*(ζ), Equations (68), (71), (74) and (77) yield the general solutions as

_fm(ζ)=_fm*(ζ)+_E1+_E2ζ+_E3 ^ζ2+_E4 ^ζ3,

_gm(ζ)=_gm*(ζ)+_E5+_E6ζ,

_θm(ζ)=_θm*(ζ)+_E7+_E8ζ,

_φm(ζ)=_φm*(ζ)+_E9+_E10ζ.

4. Results and Discussion

Results and discussion provide the analysis of the problem through the impacts of all the relevant parameters. The non-dimensional Equations (20), (22), (28) and (30) with boundary conditions in Equations (25), (26) and (29) are analytically computed. The performances of different parameters on the velocity profiles with heat and concentration of homogeneous-heterogeneous chemical reactions are shown in the relevant graphs. The streamlines show the internal behaviors of flow. The physical representation of the problem is shown in Figure 1. Liao [61] introduced ℏ-curves for the convergence of the series solution to get the precise and convergent solutions of the problems. ℏ-curves are also called the convergence controlling parameters for solution in the homotopy analysis method (used for solution in the present case). These ℏ-curves specify the range of numerical values. These numerical values (optimum values) are selected from the valid region in straight line. These optimum values of ℏ-curves are selected from the straight lines parallel to the horizontal axis (please see carefully Figure 2, Figure 3, Figure 4 and Figure 5) to control the convergence of problem solution. In the present case, the valid region of each profile ℏ-curve is specified. Therefore, the admissible ℏ-curves for f(ζ), g(ζ), θ(ζ) and φ(ζ) are drawn in the ranges −10.00 ≤_ℏf≤ −4.00, −10.00 ≤_ℏg≤ −5.00, −3.5 ≤_ℏθ≤ −2.50 and −1.50 ≤_ℏφ ≤ −0.50 in Figure 2, Figure 3, Figure 4 and Figure 5, respectively.

4.1. Axial Velocity Profile

In the present study, two nanofluids namely ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂ are investigated whose behaviors are shown through the graphs under the effects of different parameters. In Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25, the green and magenta colors are used for ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂ while in Figure 24 and Figure 25, the additional colors are also used. There are solid and dashed curves in Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22 and Figure 23. The mechanism is that three positive increasing numerical values are given to one parameter in the HAM solution while all the remaining parameters are fixed to show the effect of that one parameter simultaneously on the two nanofluids namely ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂. When the solid lines locate below the dashed lines, then it shows the increasing effect and when the solid lines locate above the dashed lines, then it shows the decreasing effect. When the arrow head is from top to bottom, it shows the decreasing effect and when the arrow head is from bottom to top, it shows the increasing effect.

Figure 6 shows that for the different values of Reynolds number Re, the axial velocity f(ζ) is increased. In fact, the velocity of ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂ increase with increasing values of Reynolds number therefore overall motion is accelerated. Figure 7 shows the prominent role of stretching parameter k₆ due to lower disk in which the axial velocity f(ζ) increases. The present motion is due to stretching so if the stretching parameter is increased, the flow of fluids is also increased. In the mean time, porosity is responsible to decrease the axial flow. It shows that motion due to different nanofluids is reduced because the permeability at the edge of the accelerating surface increases. Surely, it is noted that excess of nanoparticles concentration is involved in decelerating the motion. It is worthy of notice that the axial velocity f(ζ) decreases against the inertia. Physically it means that the absorbency of the porous medium shows an increment in the thickness of the fluid. Figure 8 shows that magnetic field parameter resists the flow since due to magnetic field, the Lorentz forces are generated which resist the motion. The curves are shrink in response to the parameter effect. Figure 9 exhibits all the assigned values of Ω and axial velocity f(ζ) which offers opportunities to know about the rotating systems and shows that the flow of ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂increase.

Some interesting results have been found in case of tangential velocity g(ζ). Figure 10 shows that as the Reynolds number Re increases, the opposite tendency has been observed in the motion of ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂. The flow of mono nanofluid ZnO-C₂H₆O₂decreases while the flow of hybrid nanofluid Au-ZnO/C₂H₆O₂ shows no prominent change for increasing the Reynolds number Re. In Figure 11, the tangential velocity f(ζ) tends to decreasing. Tangential velocity assumes a likely downfall so the flow is not supported by stretching due to k₆ . Figure 12 witnesses that the tangential velocity g(ζ) shifts to the effective decreasing for hybrid nanofluid Au-ZnO/C₂H₆O₂and increases for ZnO-C₂H₆O₂ on behalf of the magnetic field parameter M. Figure 13 exhibits that rotation parameter Ω parameter resists the tangential flow of Au-ZnO/C₂H₆O₂and enhances the tangential flow of ZnO-C₂H₆O₂.

4.2. Temperature Profile

Figure 14 shows the effect of Reynolds number Re on heat transfer. The larger values of Re increase the temperature of ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂ . It has been observed in Figure 15 that as the stretching parameter k₆increases, the temperature of ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂increase. These observations indicate that the fluid temperature and its related layer are incremented for higher estimations of k₆ . The rotation parameter Ω cannot generate an extra heating to the system as shown in Figure 16. Temperature θ(ζ) is decreased on increasing the parameter Ω. The physical reason is that enhancement in Ω causes to improve the internal source of energy, that is why the fluid temperature is reduced. The system gets the parameter Pr for the designated values 1.00, 3.50, and 6.00 during the process and increases the temperature shown through Figure 17. The direct relation of Pr and thermal conductivity increases the thickness of thermal boundary layer. Larger values of Pr generate the high diffusion of heat transfer. The temperature θ(ζ) is changed to lowest level after the exchange of high values of magnetic field parameter M as shown in Figure 18. The reason is that strong Lorentz forces resist the flow of nanoparticles, so causing no high collision among the nanoparticles, consequently, the temperature is decreased. Figure 19 depicts that with the increasing values of thermal radiation parameter Rd, the temperature θ(ζ) of ZnO-C₂H₆O₂increases while the temperature of hybrid nanofluid Au-ZnO-C₂H₆O₂decreases. The reason is that radiation enhances more heat in the working fluids.

4.3. Concentration of Homogeneous-Heterogeneous Chemical Reactions

Looking at the non-dimensional Equation (28), the suitable values of Re, Sc and k₄ are the basic quantities for generating a cubic autocatalysis chemical reaction. The concentration of chemical reaction φ(ζ) is low with the Reynolds number Re as shown in Figure 20. Figure 21 shows that for the homogeneous chemical reaction parameter k₄ , the concentration of chemical reaction is decreased. From Equation (28), it is witnessed that the homogeneous chemical reaction parameter k₄is a part of performance with the multiple solutions. Enhancement in k₄ makes dominant the concentration. In Figure 22, the stretching parameter k₆upgrades concentration of chemical reaction with low level by performing active role in the rotating motion. The stretching parameter k₆ makes compact the homogeneous reaction and hence the concentration profile φ(ζ). Figure 23 stands for the outcomes of Schmidt number Sc and concentration φ(ζ). Momentum diffusivity to mass diffusivity is known as Schmidt number. The parameter Sc causes to make low the homogeneous chemical reaction.

4.4. Streamlines

Figure 24 shows the streamlines at upper disks. The size of the streamlines increases at upper disk compared to that of lower disk. Both mono nanofluid and hybrid nanofluid proceed towards the edges of disks. Figure 25 shows the streamlines for the Reynolds number Re at lower disks. The compression of streamlines are clear from Figure 25. The plumes power is strong for lower disks.

4.5. Authentication of the Present Work

The important physical quantities introduced in Section 2 are evaluated to compare the validity of the solution with the published work [8]. Table 4 shows the tabulations to the several values for the parameter Re. There exists a nice agreement with the published work [8]. Similarly in Table 5, the values of heat transfer rate are computed for the volume fraction ϕ = 0.10, 0.20, 0.30, and 0.40. These values also have the close agreement with the published work [8].

5. Conclusions

A significant modification in the mathematical model for hybrid nanofluid has been made for the analysis of flow, heat and mass transfer. Chemical species reactions are shown in hybrid nanofluid. The problem is modeled in rotating systems for the nanoparticles ZnO and Au with base fluid ethylene glycol and solved through HAM. In ethylene glycol-based fluid (C₂H₆O₂), two types of nanoparticles, namely ZnO (zinc oxide) and Au (gold), with volume fractions_ϕ1= 0.03 and_ϕ2= 0.04 are investigated, respectively. It is noted that for_ϕ1= 0.00 and_ϕ2= 0.00, the problem becomes about viscous fluid with the absence of nanoparticles volume fractions. If_ϕ1= 0.00, Ag/C₂H₆O₂is obtained and if_ϕ2= 0.00, ZnO₂/C₂H₆O₂ is constructed. Achieving better comprehension, the competencies of active parameters on flow, heat transfer and concentration of heterogeneous-homogeneous chemical reactions are noted. There exists a nice agreement between the present and published work in Table 4 and Table 5. The problem has potential for renewable energy system and researchers to investigate the thermal conductivity of nanoparticles like silver, aluminum, copper etc. with different base fluids like water, benzene, engine oil etc. The results for flow, heat transfer and concentration of homogeneous-heterogeneous chemical reactions are summarized as following.

(1)

Axial velocity f(ζ) increases for ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂with the increasing values of Reynolds number Re, stretching parameter k₆and rotation parameter Ω while axial velocity f(ζ) decreases for ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂with the increasing values of magnetic field parameter M.

(2)

Tangential velocity g(ζ) increases for ZnO-C₂H₆O₂with the increasing values of magnetic field parameter M and rotation parameter Ω while the same velocity decreases for Au-ZnO/C₂H₆O₂with the increasing values of magnetic field parameter M and rotation parameter Ω. Moreover, tangential velocity g(ζ) decreases for ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂with the increasing values of Reynolds number Re and stretching parameter k₆.

(3)

Heat transferθ(ζ)increases for ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂with the increasing values of Reynolds number Re, stretching parameter k₆. Similarly, heat transferθ(ζ)increases for ZnO-C₂H₆O₂with increasing values of thermal radiation parameter Rd while it is decreased for ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂with the increasing values of rotation parameter Ω, magnetic field parameter M. In case of Au-ZnO/C₂H₆O₂, heat transferθ(ζ)also decreases with increasing values of thermal radiation parameter Rd.

(4)

The concentration of homogeneous-heterogeneous chemical reactions φ(ζ) decreases for ZnO-C₂H₆O₂and Au-ZnO/C₂H₆O₂with the increasing values of Reynolds number Re, stretching parameter k₆and Schmidt number Sc.

(5) Streamlines are compressed at the upper portion of upper disk while these are compressed at the lower portion of lower disk when the Reynolds number Re assumes the value 0.30.

(6)

Table 4 and Table 5 show an excellent agreement of the present work with published work.

Figure 1. Geometry of the problem.

Figure 2. Illustration of theℏf -curve of f(ζ).

Figure 3. Illustration of theℏg -curve of g(ζ).

Figure 4. Illustration of theℏθ -curve of θ(ζ).

Figure 5. Illustration of theℏφ -curve of φ(ζ).

Figure 6. Illustration for the velocity f(ζ) and parameter Re = 1.00, 1.50, 2.00.

Figure 7. Illustration for the velocity f(ζ) and parameter k6= 1.00, 1.50, 2.00.

Figure 8. Illustration for the velocity f(ζ) and parameter M = 1.00, 1.50, 2.00.

Figure 9. Illustration for the velocity f(ζ) and parameter Ω = 1.00, 1.50, 2.00.

Figure 10. Illustration for the velocity g(ζ) and parameter Re = 1.00, 10.50, 20.00.

Figure 11. Illustration for the velocity g(ζ) and parameter k6= 1.00, 10.50, 20.00.

Figure 12. Illustration for the velocity g(ζ) and parameter M = 1.00, 10.50, 20.00.

Figure 13. Illustration for the velocity g(ζ) and parameter Ω = 1.00, 1.50, 2.00.

Figure 14. Illustration for the heat transfer θ(ζ) and parameter Re = 1.00, 1.50, 2.00.

Figure 15. Illustration for the heat transfer θ(ζ) and parameter k6= 1.00, 1.50, 2.00.

Figure 16. Illustration for the heat transfer θ(ζ) and parameter Ω = 1.00, 5.50, 10.00.

Figure 17. Illustration for the heat transfer θ(ζ) and parameter Pr = 1.00, 3.50, 6.00.

Figure 18. Illustration for the heat transfer θ(ζ) and parameter M = 1.00, 1.50, 2.00.

Figure 19. Illustration for the heat transfer θ(ζ) and parameter Rd = 1.00, 1.50, 2.00.

Figure 20. Illustration for the concentration φ(ζ) and parameter Re = 1.00, 1.50, 2.00.

Figure 21. Illustration for the concentration φ(ζ) and parameter k4= 1.00, 1.50, 2.00.

Figure 22. Illustration for the concentration φ(ζ) and parameter k6= 1.00, 1.50, 2.00.

Figure 23. Illustration for the concentration φ(ζ) and parameter Sc = 1.00, 1.50, 2.00.

Figure 24. Illustration for the streamlines at upper disk and parameter Re = 0.30.

Figure 25. Illustration for the streamlines for lower disks and parameter Re = 0.30.

Shapes of Nanoparticle	n	Aspect Ratio
Spherical	3	-
Brick	3.7	1:1:1
Cylinder	4.8	1:8
Platelet	5.7	1:1/8

Properties	Ethylene Glycol (C2H6O2)	Zinc Oxide (ZnO)	Gold (Au)
ρ(kg/m3)	ρf= 116.6	ρs1 = 5600	ρs2 = 19,282
cP(J/kg K)	(cP)f= 2382	(cP)s1 = 495.2	(cP)s2 = 192
k(W/m K)	kf = 0.249	ks1 = 13	ks2 = 310
σ(Um)−1	σf= 3.14	σs1 = 7.261 × 10−9	σs2 = 4.11 × 107
Nanoparticle measurement/nm	-	29 and 77	3–40

Properties	ZnO/C2H6O2
Density (ρ)	ρnf= (1 −ϕ1)ρf+ϕ1ρs
Heat capacity (ρcP)	(ρcP)nf= (1 −ϕ1)(ρcP)f+ϕ1(ρcP)s
Dynamic viscosity (μ)	μnf μf=1(1−ϕ1)2.5
Thermal conductivity (k)	knf kf=k1+(n1−1)kf−(n1−1)(kf−k1)ϕ1k1+(n1−1)kf+(kf−ks)ϕ1
Electrical conductivity (σ)	σnf σf= 1 +3(σ−1)ϕ1(σ+2)−(σ−1)ϕ1, whereσ=σs σf
Properties	Hybrid nanofluid (Au-ZnO/C2H6O2)
Density (ρ)	ρhnf= (1 − (ϕ1+ϕ2))ρf+ϕ1ρs1 +ϕ2ρs2
Heat capacity (ρcP)	(ρcP)nf= (1 − (ϕ1+ϕ2))(ρcP)f+ϕ1(ρcP)s1 +ϕ2(ρcP)s2
Dynamic viscosity (μ)	μhnf μf=11−(ϕ1+ϕ2)2.5
Thermal conductivity (k)	khnf kf=k2+(n2−1)knf−(n2−1)(knf−k2)ϕ2k2+(n2−1)knf+(knf−k2)ϕ2×k1+(n1−1)kf−(n1−1)(kf−k1)ϕ1k1+(n1−1)kf+(kf−k1)ϕ1× kf
Electrical conductivity (σhnf)	σhnf σf= 1 +3σ1 ϕ1+σ2 ϕ2σf−(ϕ1+ϕ2)2+σ1 ϕ1+σ2 ϕ2(ϕ1+ϕ2)σf−σ1 ϕ1+σ2 ϕ2σf−(ϕ1+ϕ2)

Reynolds Number (Re)	f″ (1) [8]	f″(1) (Present)	g′ (1) [8]	g′(1) (Present)
0.10	0.292991	0.292993	0.284684	0.284683
0.20	0.237792	0.237791	0.224995	0.224994
0.30	0.208284	0.208283	0.197046	0.197044
0.40	0.206995	0.206994	0.203117	0.203115

Volume Fraction (ϕ )	- θ′ (0) [8]	- θ′ (0) (Present)	- θ′ (1) [8]	- θ′(1) (Present)
0.10	3.677172	3.677170	2.26814	2.26813
0.20	4.53192	4.53190	3.128083	3.128081
0.30	2.983936	2.983935	1.648859	1.648857
0.40	3.00208	3.00207	1.624995	1.624993

Author Contributions

Conceptualization, N.S.K.; methodology, N.S.K.; software, N.S.K.; validation, N.S.K.; formal analysis, P.K.; investigation, N.S.K.; resources, P.T.; data curation, P.T.; writing-original draft preparation, N.S.K.; writing-review and editing, N.S.K.; visualization, P.K.; supervision, N.S.K.; project administration, P.K.; funding acquisition, P.K. All authors have read and agreed to the revised version of the manuscript.

Funding

This research is funded by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT.

Acknowledgments

This work was partially supported by the International Research Partnerships: Electrical Engineering Thai-French Research Center (EE-TFRC) between King Mongkut's University of Technology North Bangkok and Universite' de Lorraine under Grant KMUTNB-BasicR-64-17. The authors are cordially thankful to the honorable reviewers for their constructive comments to improve the quality of the paper. This research is supported by the Postdoctoral Fellowship from King Mongkut's University of Technology Thonburi (KMUTT), Thailand. This project is supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT.

Conflicts of Interest

The authors declare no conflict of interest.

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