Physical aspects of irreversibility in radiative flow of viscous material with cubic autocatalysis chemical reaction

Analysis of irreversibility in flow by a stretchable surface has gained much consideration in recent years. Entropy optimization properly computes the second law thermodynamic irreversibilities. Therefore, deterioration of entropy proficiency results in a more useful energy transport process. In this article, a physical aspect of irreversibility in radiative flow of viscous material with quartic autocatalysis chemical reaction is addressed. The flow is discussed between two stretchable rotating disks. Heat transfer occurring in this physical problem is modelled through thermal radiation, Joule heating and viscous dissipation. This is the first time the concept of homogeneous-heterogeneous reactions has been studied with entropy generation. The nonlinear flow expressions are made dimensionless. The obtained equations are then tackled through the homotopy concept. The analysis discloses that the radiation parameter and Eckert number play a vital role in the enhancement of temperature field. The tangential velocity decreases versus the magnetic parameter. The radial component of velocity boosts close to lower disks and it decreases near the upper disks versus the Reynolds number. The variations in the Nusselt number and skin friction are presented graphically with various emerging variables. It is noticed that entropy rate can be controlled by minimizing the impact of Brinkman and Reynolds numbers.


Introduction
The flow by a stretchable rotating disk has gained much consideration from investigators. It is because of its considerable applications in mechanical and industrial engineering processes like medical equipment, manufacturing, spin coating, centrifugal pumps, food processing technology, pumping of liquid metals versus high melting point, air cleaning machine, turbo-machinery, gas turbines, electric generating systems, etc. Keeping such motivation in mind, Karman [1] initially investigated the flow behavior by a rotating disk. Hayat et al. [2] explored irreversibility aspects in MHD radiative flow by a rotating disk via Joule heating and dissipations. Dissipative flow of second grade fluid is scrutinized by Hayat et al. [3]. Mustafa [4] analyzed MHD partial slip flow of nanomaterials by a rotating disk. Wu et al. [5] analyzed the two-phase air liquid flow through a rotating disk system with flow pattern maps. Hassan et al. [6] examined the mixed convective flow of ferroliquid with iron nanomaterials due to a rotating stretchable disk. Mehmood et al. [7] worked for MHD flow subject to a rotating disk. A comparative study of five different shape nanomaterials in rotating disk flow with velocity slip and Joule heating was discussed by Sumaira et al. [8]. Xun et al. [9] highlighted the heat transport in flow of Ostwald-de Waele liquid with index decreasing over a rotating disk. Lok et al. [10] considered axisymmetric stagnation point rotational flow by a permeable rotating disk. Some latest findings regarding this direction are listed in refs. [11][12][13][14][15].
The entropy optimization in recent years has been taken into consideration by many engineers and scientists to achieve the optimal design of thermal devices. In numerous heat transport mechanisms, a heat exchanger from higher to lower is the conventional heat transport tool. Thermal properties are improved through active and passive techniques. Numerous nanomaterials are utilized to enhance the heat transfer [16,17]. Ijaz et al. [18] have discussed irreversibility associated with flow and heat transport in Sisko nanoliquid by a rotating disk. Manay et al. [19] analyzed the nanomaterial flow in microchannel heat generation with entropy concept. Entropy generation associated with heat conduction in a fixed (adiabatic) cylinder was discussed by Tian and Wang [20]. Nanomaterial entropy optimization with helical twisted tapes was explored by Li et al. [21]. Kefayati et al. [22] studied diffusive double forced convective flow of Carreau liquid with entropy generation by a cold cylinder. Khan et al. [23] highlighted the entropy generation in tangent hyperbolic nanomaterial via nonlinear mixed convection and activation energy. Sadaf et al. [24] scrutinized the entropy generation in peristalsis flow with various nanomaterials shapes. Xie and Jian [25] investigated two-layer MHD electroosmotic flow with entropy generation via microparallel channels. Related investigations regarding entropy concept are listed in refs. [26][27][28][29][30].
Here entropy generation in radiative flow of viscous material between two rotating disks is addressed. The impacts of homogeneous-heterogeneous reactions are considered. The energy equation is modelled subject to dissipation, Joule heating and radiation. Through the implementation of the second law of thermodynamics the total entropy rate is calculated. The present flow expressions are made dimensionless by suitable transformations. The homotopy analysis method [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] is used to obtain the convergent series solutions. Nusselt numbers and skin friction coefficients at both upper and lower disks are discussed.

Flow expression
Flow of viscous material in the presence of entropy generation and radiation is addressed. The flow is studied between two rotating disks. The lower disk (at z = 0) is rotating with Ω 1 in axial direction while the upper disk (at z = h) rotates with Ω 2 . Both disks are stretched respectively with stretching rates a 1 and a 2 . The magnetic field is implemented in the z-direction. The schematic flow description is highlighted in fig. 1. The basic equations of the governing problem in vector form are The flow equations are presented as [14] ∂ũ ∂r +ũ r w ∂w ∂r with the boundary conditions [39] In the above expressions, the velocity components are denoted byũ,ṽ,w and r, θ, z are the Cartesian coordinates, ρ denotes the density,p represents the pressure, ν denotes the kinematic viscosity, σ represents the electrical conductivity, B 0 and h indicate the distance between two disks. Considering the following transformations, the flow expressions take the following form: and the boundary conditionŝ where Re(= Ω1h 2 ν ) highlights the Reynolds number, M (= B 2 0 σ ρΩ1 ) denotes the magnetic parameter, A 1 (= a1 Ω1 ) and A 2 (= a2 Ω1 ) represents the ratio parameters and ε highlights the constant pressure. Simplifying eq. (7) and omitting ε, one haŝ From eq. (7), the pressure term is Integrating eq. (9) w.r.t ξ, one obtainsP

Energy equation
Mathematically, the energy equation subject to dissipation, thermal radiation and Joule heating is expressed as From eqs. (14) and (15), we get the following expression: withT in whichT 1 denotes the temperature at the lower disk, k • represents the mean absorption coefficient, c p indicates the specific heat capacity,T 2 denotes the temperature atthe upper disk, k represents the thermal conductivity, μ presents the dynamic viscosity and σ • indicates the Stefan Boltzmann constant. Considering the following transformations for energy equation, we have from energy equation where Pr(= cp(T1−T2) ) represents the Eckert number and A(= h 2 r 2 ).

Mass concentration via quartic chemical reaction
The mathematical form of the isothermal chemical reaction is defined as where species B has a higher concentration at the disk surface. The first-order isothermal heterogeneous reaction is of the form in which A, B denotes the chemical species, k c , k s the reaction rates andĈ 1 andĈ 2 the concentrations. The concentration equation in terms of homogeneous-heterogeneous reactions is defined as with appropriate boundary conditions Implementing the following transformations,φ we obtain the following expressions: ) represents the heterogeneous reaction parameter and δ(= ) denotes the diffusion ratio parameter. For comparable chemical species A and B, we put D C1 = D C2 , i.e., δ = 1. Therefore we have the following relation: From eqs. (27) and (28) we have 3 Physical quantities

Skin friction coefficients
The shear stresses, i.e., (τ zr ) and (τ zθ ) at the lower disk are expressed as The total shear stress τ w is Finally,

Nusselt number
At upper and lower disks, it is mathematically defined as where q w for lower and upper disks are Finally, In the above expressions C f 1 denotes the skin friction at the lower disk, C f 2 represents the skin friction at the upper disk, τ zr highlights the shear stress in the radial direction, τ zθ denotes the shear stress in the tangential direction, τ w represents the total shear stress, Nu x1 indicates the Nusselt at the lower disk, Nu x2 highlights the Nusselt number at the upper disk and R r (= rΩ1h ν ) denotes the local Reynolds number.

Entropy equation
We have The dimensionless form is where Br(= The Bejan number is expressed as Be = Entropy generation due to heat and mass transf er T otal entropy generation .

m-th-order problems
We have The

Convergence analysis
In the homotopy analysis technique, the convergence control parameters have an essential role on convergence and approximation rate for series solutions. For suitable values of these parameters, theh-curves are sketched at 13-th-order of approximations in fig. 2. The exact ranges for momentum, energy and concentration equations are −2.2 ≤ hf ≤ 0.8,   Table 1

Discussion
A mathematical model is presented for the flow of viscous fluid subject to thermal radiation and dissipation between two rotating stretchable disks. The homotopy analysis method is implemented for the development of series solutions. In this portion, the effects of the different pertinent variables are discussed through plots. Table 1 is plotted for the convergent series solutions. Table 2 highlights the numerical results for skin friction coefficients in radial and tangential directions. From table 2 it is noted that higher values of magnetic parameter reduces the magnitude of (C f 1 ) at the lower disk while the magnitude of (C f 2 ) increases at the upper disk. Furthermore, the magnitude of (C f 1 ) and (C f 2 ) monotonically decreases versus higher values of the Reynolds number and (A 1 ). It is also noted that these two quantities have a reverse response at the upper disk. The impacts of the radiation parameter, Eckert number and Prandtl number on Nusselt numbers (Nu x1 , Nu x2 ) at lower and upper disks are highlighted in table 3. Clearly, the transfer rate decreases at the lower disk when (Pr) and (Ec) are increased and increases versus larger (R). Physically for a larger Eckert number, the kinetic energy of the system increases, due to which temperature increases and thus the heat transfer rate decreases. Furthermore, the heat transfer rate increases at the upper disk for (Pr), (R) and (Ec).

Velocity components: Radial (f (ξ)), tangential (ĝ(ξ)), axial (f(ξ)) velocities
Figures 3-5 are plotted for the impact of (Re) on radial (f (ξ)), tangential (ĝ(ξ)), axial (f (ξ)) velocity components. In fig. 3, it is noted that the magnitude of (f (ξ)) decreases when (Re) is enhanced, i.e. (Re = 0, 2, 4, 6). Physically the Reynolds number has a direct relation with inertial forces, therefore axial velocity is decreased. Dual behavior is noticed for (f (ξ)) versus larger (Re). Initially radial velocity increases closed to the lower disk and then it boosts slowly when the Reynolds number takes the maximum value. Physically inertial forces enhance which directly affect the velocity field. That is why velocity at the upper disk is higher than at the lower disk ( fig. 4). As expected (ĝ(ξ)) decreases versus larger (Re) (fig. 5). Figure 6 describes the salient attributes of the magnetic parameter on tangential velocity. Physically magnetic parameter is the ratio of electromagnetic to viscous forces. Therefore, enhancement in the magnetic parameter creates the Lorentz force which causes flow to run in the opposite direction. Figures 7  and 8 report (f (ξ)) and (f (ξ)) versus A 1 . As expected, the axial component of velocity is enhanced for larger A 1 . Physically stretching rate increases for higher A 1 (fig. 7). Dual effect of radial velocity is observed versus A 1 .     Initially velocity increases closed to the lower disk and then it diminishes when A 1 reaches to the maximum value, i.e. A 1 = 0.9. The outcome of A 2 on axial velocity is displayed in fig. 9. It is shown that the magnitude of axial velocity decreases at both upper and lower disk for A 2 . Figure 10 depicts the behavior of A 2 on (f (ξ)). The magnitude of (f (ξ)) decreases at the lower disk and it boosts at the upper disk through higher A 2 . Note that the stretching rate is more at the upper disk than at the lower disk surface. Therefore, radial velocity enhances at the upper disk. Tangential velocity in fig. 11 rises versus rotation Ω.

Temperature field
In this subsection, figs. 12-15 show the graphical results for different variables like Pr, R, Ec and M on (θ(ξ)). Figure 12 delineates the impact of Pr on (θ(ξ)). It is examined that temperature decreases through higher Pr. The thermal layer thickness also diminishes slowly for higher Pr, i.e. Pr = 0.50, 0.55, 0.60, 1.0. Physically larger Pr is responsible for thermal diffusivity which declines the temperature of liquid. Figure 13 detects the characteristics of radiation on (θ(ξ)). It is analyzed that temperature and layer thickness increase through higher R. A larger radiation parameter produces more heat in the working liquid through the radiation process which consequently boosts the thermal field. The curves of (θ(ξ)) versus (Ec) are highlighted in fig. 14. Physically, the ratio of kinetic energy to enthalpy or dynamic temperature to the temperature is called the Eckert number. Larger Ec boosts the kinetic energy of working fluid particles which results in the augmentation of the thermal field. Variation in the magnetic parameter on (θ(ξ)) is shown in fig. 15. Physically the magnetic parameter depends on the Lorentz force which is the resistive force to the liquid flow. It rises the kinetic energy on inside molecules or atoms. That is why temperature boosts.

Concentration
In this subsection, we established the graphical results for flow variables, i.e., homogeneous reaction variable (k 1 ), Schmidt number (Sc) and heterogeneous variable (k 2 ) on (ϕ(ξ)). Figure 16 depicts the effect of (k 1 ) on (ϕ(ξ)). As expected, concentration decreases versus (k 1 ). Physically the reactants are consumed through larger (k 1 ) during the homogeneous reaction. Figure 17 has been plotted to detect the physical behavior of the heterogeneous reaction variable on (ϕ(ξ)). It is examined that the concentration of liquid particles on the disk surface decreases which directly affects   the concentration field. Figure 18 explores the physical explanation regarding the Schmidt number on (ϕ(ξ)). Since the ratio of momentum to mass diffusivities is known as the Schmidt number. Therefore momentum diffusivity intensifies through higher (Sc), which in turn increases (ϕ(ξ)).         Here the opposite trend is examined for (Be) and (N G (ξ)) versus higher (A 1 , A 2 ). Physically through larger (A 1 , A 2 ), the stretching rates enhance which creates more disturbance in liquid particles and consequently boosts the disorderedness in the system. That is why     (k 2 = 0.0, 0.3, 0.6, 0.9) on (Be) and (N G (ξ)). Here both (Be) and (N G (ξ)) monotonically boost versus homogeneous reaction and heterogeneous reaction parameters. Figures 33 and 34 are displayed to discuss how (Be) and (N G (ξ)) vary via larger (Br = 0.0, 0.4, 0.8, 1.2). Here (N G (ξ)) is increased for larger (Br). Physically for a larger Brinkman number the dissipation phenomenon generates less conduction rate which consequently boosts (N G (ξ)). In fig. 34 it is noticed that when Br = 0 then Be = 1.0. Physically it means that for (Br = 0), the irreversibility disappears for   viscous dissipation and only irreversibility associated with heat transfer is retained. There Be is maximum (Br = 0.0). When we enhance the estimation of the Brinkman number then Be gradually decreases. Figures 35-38 are plotted to analyze the salient aspects of (L 1 ) and (L 2 ) on (Be) and (N G (ξ)). Here both (Be) and (N G (ξ)) versus (L 1 ) decrease (see figs. 35 and 36). An opposite trend is noticed in the presence of (L 2 ) on (Be) and (N G (ξ)) (see figs. 37 and 38).

Concluding remarks
Here irreversibility in radiative flow of viscous material between two rotating disks via quartic autocatalysis chemical reaction is addressed. The main outcomes of the present work are listed below: -Magnitude of axial velocity decreases versus larger (Re).
-Tangential velocity is decreased by the magnetic parameter.
-(Be) and (N G (ξ)) show opposite trends for the magnetic variable.
-A dual trend is noticed for both (Be) and (N G (ξ)) via Eckert number. Open Access This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.