Entropy generation and activation energy mechanism in nonlinear radiative flow of Sisko nanofluid: rotating disk.

The theme of the present communication is to explore the novel analysis of entropy generation optimization, binary chemical reaction and activation energy for nonlinear convective flow of Sisko model on a radially stretchable rotating disk in the presence of a uniform vertical magnetic field. Nonlinear mixed convection, nonlinear thermal radiation, MHD, viscous dissipation, Joule heating and non-uniform heat generation/absorption are also considered. Nanofluid model includes significant slip mechanism of Brownian motion and thermophoresis. Apposite transformations are endorsed to get the nonlinear coupled ODEs system. The resultant system of ordinary differential equations is endeavoured for series solutions through homotopic technique. Total entropy generation is inspected through numerous emerging flow variables. Comparative study is made for temperature, velocity, heat transfer rate, Bejan number, entropy generation and mass transfer Nusselt number by considering shear thickening and thinning fluids. Finally, a comparison is specified with the previous existing results.

Introduction

Fluid flow over rotating surfaces has been analysed extensively in view of its engineering and industrial applications. The fluid flow due to rotating surfaces is used in electric-power generating system, air cleaning machine, medical equipment, gas turbines, turbine system, food processing technology and aero-dynamical engineering [1, 2, 3]. Therefore experimental and theoretical work related to this type of flow appears to be very fascinating. Firstly, Karman [4] has introduced significant similarity transformations for the fluid flow problem over a disk. Heat and mass transfer analysis of viscous fluid through a porous rotating disk is numerically investigated by Turkyilmazoglu [5]. Mair et al. [6] discussed the diffusion and slip mechanism for Carreau–Yasuda fluid over a rotating disk. Heat transfer of hydro-magnetic Couette flow in a rotating system with variable viscosity and Hall effects is considered by Makinde et al. [7]. Radiative flow problem due to a stretchable rotating disk in presence of variable thickness is measured by Hayat et al. [8] through homotopic technique. Babu and Sandeep [9] investigated thermophoresis and Brownian motion effects for 3-dimensional flow of nanofluids with MHD and slip effects. Hayat et al. [10] explored the slip mechanism by a variable thickness rotating disk subject to MHD. The study of entropy generation on hydro-magnetic nanoliquid flow over a porous rotating disk was thought out by Rashidi et al. [11]. Hayat et al. [12] exposed the impact of MHD flow of Cu–H₂O based nanoliquid due to a rotating disk with slip mechanism. Hayat et al. [13] deliberated the combined influences of thermal radiation and heat generation/absorption on Maxwell nanofluid over a stretched surface. A numerical analysis has been executed by Raju et al. [14] for Cu-kerosene nanoliquid with MHD over a cone surface.

The consequence of thermal radiation on MHD flow and heat transfer problem is gradually becoming important in many industries. Heat transfer by thermal radiation has enormous applications in different technological processes such as nuclear power plants, satellites and space vehicles, gas turbines and the numerous propulsion devices for aircraft. The linear radiation based on linearized Rosseland approximation is not critically effective/valid for high temperature differences. Since linearized Rosseland approximation depends upon unique Prandtl number [15], whereas in the nonlinear Rosseland approximation, the problem is governed by three main factors, known as radiation parameter, Prandtl number and the temperature ratio parameter. Primarily, Pantokratoras [16] considered the impact of Rosseland radiation for linear and nonlinear phenomena on natural convection along a vertical isothermal plate using another radiation factor stated as film radiation parameter. Khan et al. [17] discussed thermal diffusion stagnation point Maxwell nanofluid flow with solar radiation and thermal conductivity. Heat transfer analysis through nonlinear Rosseland approximation in presence of MHD over a stretched sheet is observed by Mushtaq et al. [18]. Ijaz and Ayub [19] presented radiative flow of ferromagnetic Maxwell fluid in presence of dual stratification and magnetic dipole. Parida et al. [20] reflected the radiative flow of heat transfer over a flat surface with thermophoresis and slip mechanism. Cortell [21] performed numerical investigated for the flow problem of quiescent fluid in view of nonlinear Rosseland thermal radiation over a stretched sheet. Some significant investigations related to the topic can be viewed via Refs. [22, 23, 24, 25].

The multi-complex rheology of biological fluids has stimulated investigations including numerous non-Newtonian liquids. In recent years, studies of non-Newtonian liquids have gained extra significance due to their connection in mechanical and industrial applications. Mainly such fluids are experienced in the synthetic and automatic processes, material handling, businesses, oil store building foods stuffs and many others. Stress tensor in such material is associated to the shear rate by non-linear relation. The Sisko [26] fluid model predicts the rheology of both dilatant and pseudoplastic fluids based on the significant range of shear rates. This model can be acknowledged as more generalized version of power law liquids. It includes both power law and viscous models. Hayat et al. [27] elaborated the effects of mass suction/injection for unsteady, incompressible Sisko liquid over a stretched surface. Malik et al. [28] has investigated the magneto-hydrodynamic flow of Sisko fluid and heat transmission with convective boundary condition over a nonlinearly stretching sheet. Convective flow of Sisko model over a bidirectional stretched sheet is presented by Munir et al. [29]. Convective heat transfer and variable thermal conductivity of Sisko nanofluid towards a stretched cylinder is discussed by Khan et al. [30].

The main theme of the present communication is to examine irreversibility of nonlinear convective flow of Sisko nanofluid over a stretched rotating disk in the presence of binary chemical reaction and activation energy. The originality of problem is to discover the entropy generation optimization for Sisko model with significant features of non-uniform heat source/sink, binary chemical reaction, nonlinear thermal radiation, nonlinear mixed convection, and activation energy. Sisko nanomaterial model is used which defines the important slip mechanism namely Brownian and thermophoresis diffusions. Total entropy generation rate of system is estimated through several flow variables. Convergent solutions of highly nonlinear problem are obtained by using homotopy analysis method [31, 32, 33, 34, 35, 36, 37, 38, 39]. Homotopy analysis method (HAM) is one of the best efficient methods in solving different type of nonlinear differential equations such as coupled, decoupled, homogeneous and non-homogeneous. Entropy generation rate, Bejan number, Nusselt, Sherwood numbers and fluid flow features are discussed in detail through graphs and tables.

Model

Consider the incompressible boundary layer nonlinear convective flow of Sisko nanofluid with detail analysis of entropy generation and Bejan numbers due to rotating stretchable disk having angular velocity around z-axis. Effects of nonlinear thermal radiation, binary chemical reaction, activation energy, Joule heating, nonlinear mixed convection and non-uniform heat generation/absorption are accounted in the present flow problem. The velocity components are adopted in the direction of increasing . The flow configuration of the system is presented in Fig. 1. It is also assumed that a uniform magnetic field of intensity acts in the z-direction. Magnetic Reynolds number is considered to be small so that the induced magnetic field is negligible in comparison with the applied magnetic field. Flow is generated due to linear stretching of disk in radial direction with stretching rate . The governing laws of mass, momentum, energy and concentrations [40, 41] for the present flow situation are presented in Eqs. (1), (2), (3), (4), and (5).

Fig. 1

Physical model.

Apposite boundary conditions (Eq. (6)) are

The non-uniform heat source/sink [42] is formulated by Eq. (7)Where and delineates the coefficients of space and temperature dependent heat generation/absorption, respectively. The absolute values of and signify internal heat generation, while non-positive values characterize the internal heat absorption. Here and portray for coefficient of thermal and solutal expansion parameters for linear and nonlinear form respectively, for fluid density, T for fluid temperature, for specific heat, for thermal diffusivity, defines for electrical conductivity, for Brownian and thermophoretic diffusion coefficients, stands for high shear rate viscosity, for consistency index respectively. The expression is the modified Arrhenius function with as Boltzmann constant, for chemical reaction rate constant and for fitted rate constant, respectively.

Suitable transformations (Eq. (8)) for present flow problem are [43].

The governing flow expressions under above transformations take the following form as mentioned in Eqs. (9), (10), (11), (12), and (13)

The transformed end conditions (Eq. (14)) are

In the above expressions, identifies for material parameter, symbolize for magnetic field parameter, for mixed convective parameter, for stretching parameter, for ratio of concentration to thermal buoyancy forces, for nonlinear thermal convection parameter, for Prandtl number, for nonlinear solutal mixed convection parameter, for Schmidt number, for Lewis number, for Brownian and thermophoresis parameters, for radiation parameter, for reaction rate constant, for activation energy, for temperature difference and for chemical reaction parameter, respectively.

Entropy generation modeling

Entropy generation [44, 45, 46] for Sisko nano-model over a stretchable rotating disk is mentioned in Eq. (16)Where expression for is presented in (Eq. (16))

We get Eq. (17) by substituting Eq. (16) into Eq. (15).

Eq. (18) presents four factors: thermal irreversibility, Joule heating irreversibility, viscous dissipation on entropy generation in Sisko liquid flow and concentration irreversibility.

Entropy generation rate (Eq. (18)) in dimensionless form is

Non-dimensional parameters are presented in Eq. (19)Where and defines Brinkman number, temperature difference parameter, concentration difference variable and entropy generation rate respectively.

Bejan number (Eq. (20)) in non-dimensional form is

Skin friction coefficient, local Nusselt and Sherwood numbers are presented in Eq. (21).

Methodology

The homotopy analysis method (HAM) is an analytic approximation method for highly nonlinear problems. Thus, the resulting nonlinear system of ordinary differential equations has been investigated through homotopic technique [47, 48] for numerous values of the flow parameters. In this method auxiliary parameters are involved which offered us freedom to adjust the convergence region for velocity , temperature and concentration profiles. The appropriate initial guesses and linear operators for the momentum, energy and concentration laws are expressed in Eqs. (22) and (23).

With properties (Eq. (24))

Here, arbitrary constants are signified by .

Analysis

One can see that permissible values for interval of convergence include and respectively (see Fig. 2). It is noted that 20, 25, 25, 36 and 30 order of approximations, are sufficient for the convergence of momentum, temperature and concentration fields respectively (see Table 1). Tables 2 and 3 are studied to check the numerical estimation of Nusselt and Sherwood numbers for different emerging parameters. A comparison of present results is performed with that of Mitschka and Andersson (see Table 4). It is an obvious observation from this table that our results agree very well with their results, which confirms that code used in the present work is valid .

Fig. 2

h

Table 1

Convergence of HAM solutions when.

Order of approx.	Hˆ^′(0)	Fˆ^′(0)	Gˆ^′(0)	Θ^′(0)	ϕ^′(0)
01	−1.0138	−0.4375	−0.3165	−1.0819	−0.7398
10	−1.2344	−0.4262	−0.2964	−1.0778	−0.7374
20	−1.4362	−0.3963	−0.2801	−1.0645	−0.7360
25	−1.5360	−0.3963	−0.2855	−1.0582	−0.7357
30	−1.5360	−0.3963	−0.2885	−1.0457	−0.7352
36	−1.5360	−0.3963	−0.2885	−1.0430	−0.7352
50	−1.5360	−0.3963	−0.2885	−1.0430	−0.7352

Table 2

Numerical values of when

Bs		Pr	Nˆ_R	ξ_2	Nˆ_t	Nu_z(Re_β_1)^−1(n+1)
						n = 0.6	n = 1.8
0.5	0.2	1.2	0.6	0.3	0.3	1.9245	2.0124
1.0						1.9819	2.0959
1.5						2.0485	2.1905
	0.2					1.9245	2.0124
	0.4					1.8931	1.9728
	0.6					1.8362	1.9423
		7.2				1.8733	1.9478
		1.2				1.9245	2.0124
		2.5				2.0318	2.1247
			0.2			1.7991	1.8645
			0.4			1.8847	1.9638
			0.6			1.9245	2.0124
				0.1		1.9247	2.0127
				0.3		1.9245	2.0124
				0.5		1.9241	2.0120
					0.3	1.9245	2.0124
					0.5	1.8905	1.9767
					0.7	1.8147	1.9054

Table 3

Numerical values of when

Bs		Pr	Nˆ_R	ξ_2	Nˆ_t	Sh_z(Re_β_1)^−1(n+1)
						n = 0.6	n = 1.8
0.5	0.2	1.2	0.6	0.3	0.3	1.9179	2.0480
1.0						1.9448	2.0811
1.5						1.9760	2.1217
	0.2					1.9179	2.0480
	0.4					1.9565	2.0926
	0.6					1.9957	2.1743
		7.2				1.9578	2.0915
		1.2				1.9179	2.0480
		2.5				1.8653	2.0075
			0.2			1.8466	2.0098
			0.4			1.8834	2.0161
			0.6			1.9179	2.0480
				0.1		1.8904	2.0217
				0.3		1.9179	2.0480
				0.5		1.9450	2.0740
					0.3	1.9179	2.0480
					0.5	1.8834	2.0276
					0.7	1.8802	2.0061

Table 4

Comparison of and with [49] and [50] when

n	Fˆ^′(0)			Gˆ^′(0)
	Present (HAM)	Ref. [49]	Ref. [50]	Present (HAM)	Ref. [49]	Ref. [50]
0.6	0.500	0.500	0.501	0.677	0.677	0.676
0.8	0.504	0.504	0.504	0.635	0.636	0.636
1.0	0.511	0.510	0.510	0.616	0.616	0.616
1.8	0.529	0.529	0.529	0.601	0.601	0.601

Results and discussion

The influence of sundry parameters like material parameter, Hartmann number and the buoyancy ratio parameter radiation parameter, nonlinear convective parameters, heat generation/absorption parameters, temperature ratio factor , Brownian motion and thermophoresis parameters , activation energy parameter , Schmidt number Lewis number (Le), on velocity, temperature and concentration graphs are elucidated. Figs. 3, 4, and 5 examine the influences of material parameter, stretching parameter , Hartmann number and the buoyancy ratio parameter on axial , radial and tangential velocity distributions are observed for shear-thinning and shear-thickening fluids. Influence of material parameter on radial and tangential velocity distributions are shown in Fig. 3 (a, b). Here radial and tangential velocity curves rises for greater estimation of for both shear-thinning and shear-thickening fluids. Higher values of correspond to reduce fluid viscosity and fluid particles feel free to move randomly. As a result, velocity enhances. It is notify that velocity in shear thinning case is more prominent than compared with shear thickening liquids due to weaker viscosity. Salient features of Hartmann number on radial and tangential velocity distributions is displayed in Fig. 4 (a, b) when and . Here radial is found to be diminishing for superior values of when and . Physically with the rise in Lorentz force develops extra resistance between fluid particles and thus velocity profile slows down for both shear thickening and thinning fluids. While opposite impact of tangential velocity is noticed against . Fig. 5 (a, b) is plotted to study the impact of velocity profiles for higher values of . Increasing trend of axial and radial velocity is noticed for higher estimation of for both shear thickening and thinning fluids. This performance of velocity is because of an expansion in extending rate of plate. Figs. 6, 7, and 8 are sketched to analyze the effects of radiation parameter, nonlinear convective parameter due to temperature, heat generation/absorption parameters, temperature ratio factor and Brownian motion parameter on distribution respectively. Fig. 6 (a, b) defines the effect of and on. Increasing tendency of is observed for greater values of. Since larger corresponds to decay in mean absorption factor that eventually upsurges the temperature (see Fig. 6 (a)). Temperature is found to be increasing function of nonlinear convection parameter (see Fig. 6 (b)). Since motion of fluid particles increases for higher approximation of for shear thinning and shear thickening fluids. In fact, for higher, temperature difference enhances which is accountable for velocity augmentation and ultimately temperature rises. Fig. 7 (a, b) is schemed to study the characteristics of for greater non-uniform heat source/sink parameters respectively. Technically, increasing values of and performs as an agent to produce more heat. Since positive values of and acts like heat generators and negative values is for heat absorbers. Normally, increase in heat source becomes a source of enrichment in the thermal boundary thickness. Fig. 8 (a, b) shows the behavior of and on temperature curve. Here is found to be upsurge in view of. In fact, more heat is generated through the random collision of fluid particles within the frame of larger. Therefore increases for both cases. Equivalent enhancing trend of for shear-thinning and shear-thickening fluids against can be viewed in Fig. 8 (b). For higher pattern of liquid temperature is considerably more than the ambient temperature which upsurges thermal condition of the liquid. Fig. 9 (a, b) study the influence of activation energy and Schmidt number on concentration profile. In fact, greater diminishes the modified Arrhenius function that endorses the generative chemical reaction in the end. Hence enriches. The concentration and solutal boundary layer thickness declines for larger Sc (see Fig. 9 (b)). According to the definition, Schmidt number is ratio of momentum to mass diffusivities. Thus for larger the mass diffusivity decreases which is responsible in reduction of concentration. Fig. 10 exhibits that both concentration and boundary layer thickness are heightened for greater values of . The thermal conductivity of the fluid enriches in presence of nanoparticles. Therefore, higher exaggerate heat conductivity of the fluid. Such greater thermal conductivity becomes a source of rise in. Figs.11, 12, 13, 14, 15, and 16 are constructed to analyze the entropy number and Bejan number for greater values of involved parameters like material parameter, Hartmann number , Brinkman number , reaction rate factor , diffusion parameter and radiation parameter respectively. Consequence of on and is sketched in Fig 11 (a, b). Greater estimation of lessen entropy generation and reverse effect is detected for Bejan number. Since higher values of leads to an increase in shear rate of viscosity which decelerates the fluid motion. As a result diminishes for both and cases. Declining role of is inspected for greater values of because heat transfers dominant over the viscous effects (see Fig. 11 (b)). Variation of and for enhancing Brinkman number is displayed in Fig. 12 (a, b). It is remarked that increases for growing values of (Br) while reverse nature is identified for Bejan number. There exists massive quantity of heat in the system with an increase in. In fact, is heat transfer by molecular conduction in comparison of heat production by viscous heating, which is responsible for rise in disorderedness of the system. Additionally, is diminished for higher due to dominant role of viscous effects over heat transfer effects. Feature of on entropy generation and Bejan number is exhibited in Fig. 13 (a, b). Enhancing character of is pragmatic while opposite influence of is remarked. For greater, more resistance is developed that causes to be enhanced (see Fig. 13 (a)). In the present situation, viscous effects dominant over heat and mass transfer effects for greater estimation of. Hence diminishes. Fig. 14 (a, b) is sketched to explore the nature of radiation parameter on and. Here, entropy generation (irreversibility) and Bejan number are intensifying for greater values of. Since, there exists a direct relationship between internal energy and entropy generation with. Fig. 15 (a, b) reveals that reverse behavior of and is identified for larger values of. Since chemical reaction rate gradually enriches for larger estimation of, that is accountable for an upsurge in. Fig. 16 (a, b) indicates the impact of and for fixed values of diffusion parameter. Similar enhancing behavior of both and is noticed for higher values of. Since diffusivity in the fluid particle increases that directly corresponds to produce more dis-orderness and ultimately, entropy generation upsurges (see Fig. 16 (a)). It is found from Fig. 16 (b) that an increase in leads to rise in the Bejan number. It is due to the fact that, for higher heat and mass transfer preeminent over viscous effects. From Figs. 11, 12, 13, 14, 15, and 16, it is perceived that entropy generation is always prominent for shear thinning fluid while has larger impact for shear thickening liquids Local Nusselt number , skin friction coefficient and Sherwood number are presented in Figs. 17, 18, and 19 for emerging parameters , , and . Variation of heat transfer rate with and are discussed through Fig. 17. It is ascertained that Nusselt number is heightened for both and. Fig. 18 revealed that skin friction coefficent is an increasing function of and. For larger and, enhancing character of Sherwood number is observed for both shear thinning and shear thickening fluids (see Fig. 19).

Fig. 3

and against

Fig. 4

and against

Fig. 5

and against

Fig. 6

against and

Fig. 7

against and

Fig. 8

against and

Fig. 9

against and

Fig. 10

against

Fig. 12

and against

Fig. 13

and against

Fig. 14

and against

Fig. 15

and against

Fig. 11

and against

Fig. 16

and against

Fig. 17

with

Fig. 18

with

Fig. 19

with

Conclusion

Present study carried out speculative investigation of entropy generation optimization and activation energy for convective flow of Sisko nanofluid over a stretchable rotating disk. In addition, combined effects of Joule heating, nonlinear mixed convection, non-uniform heat generation/absorption and viscous dissipation are also factored into the analysis. The series solution of the governing set of differential equations is accomplished by employing Homotopy method. The presented analysis has the following observations:

Temperature rises for greater estimation of Brownian motion parameter nonlinear thermal mixed convection parameter radiation parameter temperature ratio factor

Remarkable behavior of temperature is noticed for pseudoplastic fluids.

Concentration curve reduces for higher values of while reverse impact is observed for both and

Magnitude of axial, radial and tangential velocity profiles enhances for greater estimation of material parameter

Entropy generation enhances for higher estimation of parameters while it decays for

Entropy generation is more for shear thinning fluids whereas Bejan number is prominent for shear thickening fluids

Bejan number is greater for while it declines for

Declarations

Author contribution statement

Misbah Ijaz: Performed the experiments; Analyzed and interpreted the data; Wrote the paper.

Muhammad Ayub, Hammad Khan: Conceived and designed the experiments; Performed the experiments.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Competing interest statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.