Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation

Physica A xxx (xxxx) xxx

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Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation ∗

A. Kumar, R. Tripathi , R. Singh, V.K. Chaurasiya Department of Mathematics, National Institute of Technology, Jamshedpur 831014, Jharkhand, India

article

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Article history: Received 14 May 2019 Received in revised form 7 November 2019 Available online xxxx Keywords: MHD Williamson fluid Entropy generation Stretching sheet Bejan number

a b s t r a c t The present communication addresses the heat and mass transfer mechanism in MHD nanofluid flow of Williamson fluid over a stretching sheet taking the combined effects of Joule heating, nonlinear thermal radiation and viscous dissipation into consideration. For physical relevance we also analyzed the influence of chemical reactions on the flow field. The appropriate transformations are implemented to metamorphose the governing PDEs into a set of coupled ODEs. The shooting technique along with fourth order Runge– Kutta method has been implemented to get the solutions of obtained highly non-linear ODEs. The second law of thermodynamics is implemented to model the equation of entropy generation for the current analysis. Impact of different dominant parameters on velocity, temperature, concentration, entropy generation as well as Bejan number are described through graphs whereas the variation in the skin friction coefficient, heat transfer rate and mass transfer rate are studied using numerical data in the tabular form. It is observed from the obtained numerical data that the rate of heat transfer gets reduced with increase in Eckert number while the thermal radiation parameter tends to enhance it. Increase in Brinkman parameter leads to a rise in entropy generation while it (Brinkman parameter) has an adverse effect on Bejan number. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Heat and mass transfer analysis is an inspiring topic in non-Newtonian fluid flows because of its dominant role in industrial and engineering process. Fluids such as pulps, human blood, honey, concentrated juices, shampoos, jelly, sugar solutions, etc. which do not obey the Newtonian constitutive property are normally put in the category of non-Newtonian fluids. The Williamson fluid is an essential class of pseudo-plastic fluid model. The investigators have shown a great interest in studying the pseudo-plastic fluids due to their notable industrial and engineering applications such as adhesives and emulsions, blood cells, in inkjet printing, their occurrence in food processing, their use in coated photographic films etc. In the previous couple of decades, a substantial number of research works concerning Williamson fluid model have been published. In those research works, numerous researchers have put their efforts in describing the rheological behavior of Williamson fluid and to understand the effect of Weissenberg number on flow and pumping characteristics. A model was developed by Williamson in 1929 which assessed the flow of pseudoplastic liquids and the obtained results were validated experimentally. Kumaran and Sandeep [1] made a theoretical analysis to highlight the diffusion-thermo ∗ Corresponding author. E-mail address: [email protected] (R. Tripathi). https://doi.org/10.1016/j.physa.2019.123972 0378-4371/© 2020 Elsevier B.V. All rights reserved.

Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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Nomenclature a T∞ Tw

τ µ0 I

γ˙ v σ ρ ( ) ρ Cp f DB T k1

σ∗ M Nb Ec

θw Kr Nux

τw

jw NG

α1

L P C∞ Cw

µ∞ Γ A1 u

ν k C (p

ρ Cp

C DT k∗ We Pr R Nt Sc Cfx Shx qw Rex Br

α2 Be

) s

Stretching sheet parameter Free stream temperature Wall temperature Stress tensor Shear rate viscosity at zero Identity tensor Shear rate Velocity component in y-direction Electrical conductivity Density Heat capacity of base fluid Brownian diffusion coefficient Fluid temperature Reaction rate Stefan Boltzmann constant Magnetic parameter Brownian motion parameter Eckert number Temperature ratio parameter Chemical reaction parameter Nusselt number Wall shear stress Wall mass flux Entropy generation rate Temperature difference parameter Diffusion parameter Pressure Free stream concentration Wall concentration Shear rate viscosity at infinity Williamson time constant First Rivlin–Erickson tensor Velocity component in x-direction Kinematic viscosity Thermal conductivity Specific heat Heat capacity of nanofluid Concentration of nanoparticle Thermophoresis diffusion coefficient Mean absorption coefficient Weissenberg number Prandtl number Thermal radiation parameter Thermophoresis parameter Schmidt number Skin friction Sherwood number Wall heat flux Local Reynold number Brinkman number Concentration difference parameter Bejan number

Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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and thermo-diffusion effects on the parabolic flow of Williamson as well as Casson fluids under the magnetic environment using shooting technique. It is observed in their research study that the rate of heat and mass transfer for Casson fluid flow is relatively large as compared to Williamson fluid flow. Recently, Zehra et al. [2], Ramzan et al. [3], Rehman et al. [4], Hashim et al. [5], Eldabe et al. [6], Nadeem and Akram [7] etc., have carried out the theoretical analysis to study the flow of Williamson fluid using the numerical as well as analytical technique. The theoretical/experimental studies of magnetohydrodynamic (MHD) flows are of significant importance due to their varied and wide applications in various industrial and engineering processes, viz. MHD generators, designing of cooling systems with liquid metals, flow meters, nuclear reactors utilizing liquid metals and geothermal energy exploration, etc. Looking at all these application and relevance of MHD flows, several authors conducted theoretical studies on magnetohydrodynamic flows. Seth et al. [8] made a theoretical investigation to understand the effect of magnetic field on the flow of electrically conducting fluid past a shrinking/stretching sheet using shooting technique. Sheikholeslami et al. [9] presented a numerical study on the laminar flow of Al2 O3 nanofluid, where the entire flow domain is permeated by a uniform magnetic field. Other significant analysis belonging to this context are mentioned in Refs. [10–14]. During the last few decades, a lot of research studies on energy generation in various fluid flow situations have been carried out by the investigators, focusing on entropy generation. These research studies were motivated by observing the importance of entropy generation in fluid flows and their relevance in several industrial applications which arise in most of the thermal processes viz. heat pumps, refrigerators, combustion engines, air conditioners and many more. Some of the common irreversible processes in fluid dynamics include Joule heating, heat generation due to thermal resistance, fluid viscosity within a system, fluid flow through a resistive medium such as in the Joule-Thomson effect and diffusion, etc. Owing to such wide occurrence, a good amount of research is done by the researchers on entropy generation. Bejan [15] was the first one who investigated the entropy generation in an irreversible process and introduced the rate of entropy production for heat transfer flows. Freidoonimehr and Rahimi [16] investigated the entropy generation and heat transfer analysis on hydromagnetic nanofluid flow due to a stretching/shrinking sheet. Their outcomes reveal that rate of entropy generation rises with increase in the Brinkman number while this physical quantity decreases for larger values of stretching or shrinking parameter. Khan et al. [17] studied the entropy generation analysis in the chemically reactive flow of Sisko nanofluid and concluded that increment in the material parameter reduces the Bejan number. Kefayati and Tang [18] discussed the three-dimensional non-Newtonian fluid in a cubic cavity to analyze the entropy generation using the Lattice Boltzmann technique. The results show that increment in Rayleigh number results in a rise in entropy generation and diminishes the average Bejan number. Hayat et al. [19] implemented an analytical technique to investigate entropy analysis on two dimensional laminar hydrodynamic flow of nanofluid adjacent to an accelerating stretching sheet. Dalir et al. [20] presented an entropy generation analysis on two dimensional laminar flow of Jaffery nanomaterial within the magnetic environment, using implicit Keller box technique. In this article, authors concluded that thermal energy irreversibility gets reduced with the rise in ratio of ‘‘relaxation to retardation time’’ parameters. Some of the other important research investigations regarding entropy generation are done by Seth et al. [21,22], Sheremet et al. [23,24], Bondareva et al. [25] While discussing the hydromagnetic heat transfer flows, thermal radiation plays an important role in the rate of heat transfer and it may influence several manufacturing phenomena such as glass production, preparation of gas turbines, furnace design, rocket propulsion system and spacecraft re-entry vehicles’ internal combustor, etc. This inspired researchers to investigate the thermal radiation effect in hydromagnetic flows. Hashim et al. [26] carried out a study concerning the hydromagnetic flow of Williamson nanofluid due to a stretching sheet under the convective boundary conditions at the surface of the sheet, taking thermal radiation effect into consideration. Qayyum et al. [27] in their article, presented an analysis on the MHD laminar flow of Williamson nanofluid between two rotating disks, with the consideration of thermal radiation effect. They observed that rate of heat transfer is getting enhanced as the thermal radiation parameter is enhanced. Kumar et al. [28] scrutinized the effect of thermal radiation on transient natural convection flow of nanofluids, driven solely by the linear motion of the vertical plate. In their research work, authors concluded that temperature profile of nanofluid increases with the increase in the thermal radiation parameter. Some other noteworthy research investigations in this context are due to Kumar et al. [29], Hayat et al. [30], Khan et al. [31], Mohammadein et al. [32], Kumar et al. [33], etc. The reviews of above mentioned literature reveal that the flows of chemically reactive non-Newtonian Williamson nanofluid, induced by a stretching sheet with the consideration of nonlinear thermal radiation effect within the magnetic field environment has not been discussed yet and current investigation is an attempt to fill this gap. The formulation of the problem, physical quantities of interest, analysis of entropy generation and solution methodology are placed in Sections 2 to 5 respectively. Moreover, validation of results and important findings are discussed in Sections 6 and 7 while the noteworthy conclusions are reported in Section 8. 2. Formulation We have considered the two dimensional, steady, electrically conducting, hydromagnetic nanofluid flow of Williamson fluid over a stretching sheet under the influence of nonlinear thermal radiation. Viscous dissipation effect is also taken into account. A uniform magnetic field of magnitude B0 is permeating through the flow region and acting in the transversal direction to the sheet. Considering the flow situation for small magnetic Reynolds number, the induced magnetic field Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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Fig. 1. Flow geometry of the problem.

has been neglected. We have not considered the presence of an external electric field, so that the electric field produced by polarization of charges is insignificant and therefore it has been neglected. The Cartesian system is chosen to represent the fluid flow domain, as shown in Fig. 1. It is assumed that the flow is induced because of stretching of the elastic sheet by imposing a force applied at one of the edges of the sheet in a way that the sheet velocity varies linearly. The fluid flow situation is limited in the region y ≥ 0 where y-axis is taken in the direction perpendicular to the stretching sheet. Also, in the free stream, concentration and temperature of the fluid are denoted by C∞ and T∞ respectively while Tw and Cw are representing the wall temperature and concentration respectively. Under the above-considered assumptions the flow equations in the compact form are expressed as following [34,35]:

∇.V = 0,

(1)

(J × B) , ρ { } ( ) ( ) DT J2 2 ρ Cp f (V .∇) T = k∇ T + ρ Cp s DB ∇ C .∇ T + ∇ T .∇ T − ∇.qr + + τ .L′ , T∞ σ ρ (V .∇) V = v∇ 2 V +

(V .∇) C = DB ∇.(∇ C ) +

DT T∞

∇.(∇ T ) − k1 (C − C∞ ) ˆi,

(2) (3) (4)

where L′ = gradV , B ≡ [0; B0 ; 0] represents the strength of the magnetic field, qr is radiative heat flux and J ≡ σ (V × B) is the current density. With the assistance of rheological expression of Williamson fluid, we construct the governing equations. The Cauchy stress tensor for the considered fluid model (Williamson fluid) is defined as [36,37]

[ T = −PI + τ , ] ∞) τ = µ∞ + (µ10−−µ A1 , Γ γ˙

} (5)

where P denotes the pressure, τ represents the Cauchy stress tensor, (µ0 , µ∞ ) denote the shear rate viscosity at zero and infinity respectively, Γ is the material Williamson time constant, I the identity tensor, A1 the first Rivlin–Erickson tensor with A1 = gradV + (gradV)T and γ˙ represents the shear rate and defined as follows

} √ γ˙ = Π2 , ( ) Π = 12 trace A21 .

(6)

For the present study, we have considered the infinity shear rate viscosity to be zero i.e. µ∞ = 0 and Γ γ˙ < 1. Therefore, we have

τ=

[

µ0 1 − Γ γ˙

] A1

(7)

Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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or making use of binomial expansion, Eq. (7) is converted in the following form

τ = µ0 [1 + Γ γ˙ ] A1 .

(8)

The fluid flow equations for hydromagnetic Williamson nanofluid with viscous dissipation effect under the boundary layer approximations, are given as:

∂ u ∂v + =0 ∂x ∂y √ ∂ u ∂ 2u ∂u ∂ 2u σ ∂u +v = v 2 − B20 u + 2Γ , u ∂x ∂y ∂y ρ ∂{y ∂ y2 ( ) ( ) ( ) 2 ρ Cp f u ∂∂Tx + v ∂∂Ty = k ∂∂ yT2 + ρ Cp s DB ∂∂Cy ∂∂Ty + +σ u

B20 u2 2

−

∂ q′r ∂y

+ µ0 Γ

( )3 ∂u ∂y

+ µ0

(9) (10) DT T∞

( )2 ∂u ∂y

(

,

)2 }⎫ ⎪ ∂T ⎪ ⎬ ∂y

(11)

⎪ ⎪ ⎭

∂C ∂C ∂ C DT ∂ T +v = DB 2 + − k1 (C − C∞ ) ∂x ∂y ∂y T∞ ∂ y2 2

(12)

The boundary conditions at the stretching sheet and in the free stream are as follows u = uw (x) = ax, v = 0, C = Cw , T = Tw at y = 0, u → 0, C → C∞ , T → T∞ when y → ∞.

} (13)

Incorporating the Rosseland approximation, the radiative heat flux q′r is expressed as q′r = −

4

(

3

σ∗

)

k∗

∂T 4 16 =− ∂y 3

(

σ∗

)

k∗

T3

∂T . ∂y

(14)

Where σ indicates electrical conductivity, ρ is density, Cp is specific heat, (ρ Cp )f represents heat capacity of base fluid, k denotes thermal conductivity, DB indicates Brownian diffusion coefficient, (ρ Cp )s denotes heat capacity of nanofluid, DT represents thermophoresis diffusion coefficient, Γ is relaxation time, k1 denotes reaction rate, k∗ is the mean absorption coefficient, ν is kinematic viscosity, and σ ∗ denotes Stefan Boltzmann constant. In order to recast the PDEs (9) to (12) into ODEs, following similarity transforms [38] are introduced:

√

u = axf ′ (η) , ν = − aν f (η) , η =

θ (η) =

T −T∞ Tw −T∞

, φ (η) =

C −C∞ Cw −C∞

√a } y, ν

(15)

.

Implementing (15) into (10)–(12), we obtain the following ODEs in the dimensionless form f ′′′ + Wef ′′ f ′′′ − Mf ′ − f ′2 + ff ′′ = 0,

(16)

) ⎫ θ ′′ + R(1 + (θw − 1) θ)2 3 (θw − 1) θ ′ 2 + θ ′′ (1 + (θw − 1) θ) ⎬ ( ) + Pr f θ ′ + Nbθ ′ φ ′ + Nt θ ′ 2 + EcMf ′ 2 + √12 WeEcf ′′ 3 + Ecf ′′ 2 = 0,⎭

(17)

(

1 Nt ′′ 1 ′′ φ + f φ′ + θ − Kr φ = 0. Sc Sc Nb Associated conditions at the boundary in non-dimensional form are given as: f (0) = 0, f ′ (0) = 1, φ (0) = 1, θ (0) = 1,

(18)

} (19)

f ′ (∞) → 0, φ (∞) → 0, θ (∞) → 0.

(ρ Cp )f v σ B2 Γ x represents the Weissenberg number, M = ρ ao denotes magnetic parameter, Pr = is the k ( ) 3 (ρ Cp )s DB (Cw −C∞ ) 16σ ∗ T∞ Prandtl number, Nb = signifies the Brownian motion parameter, R = 3k∗ k symbolizes the thermal (ρ Cp )f v ( ) 2 2 (ρ Cp )s DT (Tw −T∞ ) radiation parameter, Ec = C (Ta x−T ) indicates the Eckert number, Nt = denotes the thermophoresis ∞ P w (ρ Cp )f vT∞ k parameter, θw = TTw represents the temperature ratio parameter, Sc = Dv symbolizes the Schmidt number and Kr = a1 ∞ where We =

√

2a3

v

represents the chemical reaction parameter.

B

3. Physical quantities The quantities of physical interests such as coefficient of skin friction Cfx , Nusselt number Nux and Sherwood number Shx are defined as Cfx =

−2τw xqw xjw and Shx = − . , Nux = − ρ u2w k (Tw − T∞ ) DB (Cw − C∞ )

(20)

Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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In the above expressions, τw , qw and jw are defined as

⎫ ( )2 ] ( )⏐⏐ ∂u ⏐ ⎪ τw = µo + 2 ∂∂ uy , ⎪ ⎪ ∂y ⏐ ⎪ y=0 ⎪ ⎬ ) ( )⏐ ( ∗ 3 ∂T ⏐ σ T +k , qw = − 163k ⏐ ∗ ∂y ⎪ y=0 ⎪ ⎪ ( )⏐ ⎪ ⎪ ∂C ⏐ ⎭ jw = −DB ⏐ . [

∂u ∂y

Γ √

∂y

(21)

y=0

Making use of (15) and (21) in (20), the dimensionless form of (20) is given as 0.5

Cfx (Rex )

[

= −2 f (0) + ′′

We 2

f

′′ 2

] (0) ,

(22)

Nux (Rex )−0.5 = − 1 + R(θ (0) (θw − 1) + 1)3 θ ′ (0) ,

(23)

Shx (Rex )−0.5 = −φ ′ (0) .

(24)

(

)

ax2

where τw is the wall shear stress, Rex = v denotes the local Reynold number, Cfx represents the skin friction coefficient, qw the wall heat flux, Nux denotes the Nusselt number, Shx is the Sherwood number and jw is the wall mass flux. 4. Modeling of entropy generation According to Bejan [39,40], Entropy is the portion of a system’s thermal energy/temperature which is unavailable for conversion into mechanical work. It is influenced by fluid friction, mass transfer, heat transfer etc. The mathematical expression for the volumetric rate of entropy generation is defined as SG =

(

k T2

1+

∞

16σ ∗ T 3

)(

3kk∗

∂T ∂y

)2



RD

+

C∞

∂C ∂y



+

)2 +

RD T∞

Joule dissipation irreversibility

(

∂T ∂y

)(

∂C ∂y



Ω

⎫ ⎪ ⎪ ⎪ ⎪ T∞ ⎪  ⎪ ⎪ ⎪ ⎬

+

T

∞   



Thermal irreversibility

(

σ B2o u2

Fluid friction irreversibility

) 

Diffusive irreversibility

(25)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

where Ω is defined by

Ω = µo

[(

∂u ∂y

)2

+Γ

(

∂u ∂y

)3 ] (26)

In the right-hand side of (25), the first term represents the thermal irreversibility or the entropy generation due to heat transfer, the second term refers to the entropy generation due to the Joule dissipation irreversibility, second last term denotes the viscous dissipation while the last term represents the diffusive or concentration irreversibility. Implementing (15) into (25), the entropy generation rate takes the following dimensionless form

α2 ′2 φ + Lθ ′ φ ′ , (27) α1 ( ) represents the Brinkman number, α1 = TwT−T∞ indicates temperature ∞

We ′′3 NG = 1 + R(θ (θw − 1) + 1)3 α1 θ ′2 + Brf ′′2 + Br √ f + MBrf ′2 + L

(

)

2

in the above expression (27), Br = difference parameter, α2 =

(

µo a2 x2 k(Tw −T∞ )

)

Cw −C∞ denotes the concentration difference C∞ RD(Cw −C∞ ) is the diffusion parameter. k

parameter, NG =

SG T∞ v k(Tw −T∞ )a

specifies the

entropy generation rate and L = The evaluation of Bejan number Be (which is basically a ratio of heat transfer irreversibility to total irreversibility) help us in understanding the irreversibility of heat transport and this number ranges between 0 to 1. Bejan number (in the dimensionless form) is defined as Be = (

α

1 + R(θ (θw − 1) + 1)3 α1 θ ′ + L α2 φ ′ + Lθ ′ φ ′

(

)

2

2

1 ) α We ′′ 3 1 + R(θ (θw − 1) + 1)3 α1 θ ′ 2 + Brf ′′ 2 + Br √ f + MBrf ′ 2 + L α2 φ ′ 2 + Lθ ′ φ ′ 2 1

(28)

From the above expression (28), it can be said that when the range of Bejan number is between 0 to 0.5, then fluid friction, magnetic field and diffusive irreversibilities dominate over heat transfer irreversibility. Whereas, if it varies from 0.5 to 1 then irreversibility due to heat transfer is the dominant one. Be = 0.5 represents the scenario where the irreversibility due to heat transfer and irreversibilities effect due to fluid friction and magnetic field have equal proportion in the entropy generation. Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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Table 1 Comparison of skin friction coefficient when M = 0. We

Skin friction Present investigation

Nadeem et al. [41]

0.1 0.2 0.3 0.4

1.000010 0.9765272 0.939853 0.882718

1.000000 0.976588 0.939817 0.882720

Fig. 2. Comparison of results (velocity) with those of Nadeem et al. [41].

5. Solution methodology In order to obtain the solution of a coupled system of non-linear ODEs (16)–(18) along with the boundary conditions (19), shooting technique along with fourth order Runge–Kutta method is employed. In order to make the numerical computations feasible, a range of values of the similarity variable is considered. However, for η = 10, the velocity, thermal as well as concentration distribution are converging to their respective values in the free stream region. Thus the maximum value of η is taken as 10 for all numerical computations. We have considered the uniform step size h = 0.01 through the numerical computations along η direction while the residual of boundary conditions at infinity is assumed to be 10−5 . 6. Validation of results In order to verify the accuracy of the numerical method used in this paper, a comparison between our results and a previously published research paper is established. This comparison is shown in Table 1 as well as in Fig. 2. Table 1 presents the values of skin friction coefficient Cfx (Rex )0.5 of our problem and that of Nadeem et al. [41] for different values of We (in the absence of magnetic field i.e. M = 0). The values of the other parameters involved in the governing equations are zero. As evident from Table 1 and Fig. 2, A very good agreement between these two results is obtained. Therefore, we can use this numerical technique for further computations. 7. Result and discussion In this section, the effect of various parameters like: Weissenberg number We, radiation parameter R, Brownian diffusion parameter Nb, magnetic parameter M, thermophoresis parameter Nt, temperature ratio parameter θw , Eckert number Ec, Schmidt number Sc, chemical reaction parameter Kr, Brinkman number Br, diffusion parameter L, temperature difference parameter α1 and concentration difference parameter α2 on velocity f ′ (η), temperature θ (η), concentration φ (η), entropy generation Ng and Bejan number Be are discussed and displayed in the graphical form while the numerical value of skin friction coefficient Cfx (Rex )0.5 , rate of heat transfer Nux (Rex )−0.5 and rate of mass transfer Shx (Rex )−0.5 are calculated and presented in tabular form. The default values are chosen as We = 0.2, M = 0.5, Nt = 0.2, Nb = 0.3, Ec = 0.2, Pr = 6.2, Kr = 0.3, Sc = 1.5, R = 0.5, θw = 1.1, α1 = 1.1, α2 = 1.4, Br = 0.4, L = 0.6, and L = 0.6, until otherwise specified particularly. 7.1. Velocity profiles The effects of Weissenberg number We and magnetic parameter M on f ′ (η) are plotted in Figs. 3 and 4. Fig. 3 is drawn to analyze the behavior of velocity distribution f ′ (η) for various values of We(= 0.0, 0.2, 0.4, 0.6). It is noticed that velocity of the fluid is getting decelerated on increasing We. Since, for viscoelastic fluids, Weissenberg number We presents the Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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Fig. 3. Graph of f ′ (η) against We.

Fig. 4. Graph of f ′ (η) against M.

ratio of elastic forces to viscous forces, an increase in We means the relaxation time of fluid particles increases and due to this resistance, a rise in We causes a reduction in fluid velocity. It is witnessed from Fig. 4 that M has the similar effect on the velocity profile as We, i.e. rise in the value of (M = 0.0, 0.3, 0.6, 0.9) tends to decelerate the fluid velocity f ′ (η). It is happening because the presence of magnetic field produces a force which is resistive in nature and it works in the opposite direction to the fluid motion, consequently instigating the fluid velocity to decelerate. 7.2. Temperature profiles To study the behavior of temperature profile θ (η) against parameters like Ec , M , R, Nt , Nb and θw , Figs. 5–10 are sketched. Impact of parameter Ec on θ (η) is illustrated in Fig. 5. As per this figure, θ (η) is getting increased on increasing (Ec = 0.0, 0.5, 1.0, 1.5) because Eckert number Ec suggests the relationship between kinetic energy and enthalpy, an improvement in Ec means the dissipative heat or extra kinetic energy is stored in the fluid particles via frictional heating, which increases the overall temperature of fluid. The impact of magnetic parameter M on θ (η) is shown in Fig. 6. It is examined that θ (η) is an increasing function of magnetic parameter M (M = 0.0, 0.3, 0.6, 0.9). Physically speaking, externally applied magnetic field tends to enhance the fluid temperature owing to dissipation of energy generated by Lorentz force. Further, Fig. 7 demonstrates the impact of parameter R on θ (η) and as per the observation of this figure, it is noticed that θ (η) is getting increased on increasing parameter R (R = 0.0, 0.5, 1.0, 1.5). This is due to the reason that the fluid absorbs more heat when the value of R is increased, due to which θ (η) is increased. Figs. 8 and 9 reveal the variation of θ (η) under the influence of Nt and Nb respectively. Fig. 8 displays that θ (η) gets enhanced with the increment in the values of (Nt = 0.0, 0.2, 0.4, 0.6). Moreover, it is also, witnessed from Fig. 9 that an increase in θ (η) is observed on increasing (Nb = 0.1, 0.5, 0.9, 1.3). Temperature is behaving this way because of the fact that Brownian motion is the chaotic motion of nanoscale particles in the regular fluid. Consequently, for larger value of Nb, the intensity of this chaotic motion leads to increment in the kinetic energy of the nanoscale particles and therefore, temperature is increasing. Fig. 10 is sketched to analyze the nature of temperature distribution θ (η) for various values of (θw = 1.1, 1.3, 1.5, 1.8). It is found that fluid temperature θ (η) and the associated thickness, both are getting enhanced on increasing θw . 7.3. Concentration profiles Figs. 11 and 12 are portrayed to display the behavior of concentration distribution φ (η) against the variation in Kr and Sc respectively. Fig. 11 is drawn to see the nature of φ (η) for various value of Kr. It is witnessed from Fig. 11 that Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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Fig. 5. Graph of θ (η) against Ec.

Fig. 6. Graph of θ (η) against M.

Fig. 7. Graph of θ (η) against R.

Fig. 8. Graph of θ (η) against Nt.

Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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A. Kumar, R. Tripathi, R. Singh et al. / Physica A xxx (xxxx) xxx

Fig. 9. Graph of θ (η) against Nb.

Fig. 10. Graph of θ (η) against θw .

Fig. 11. Graph of φ (η) against Kr.

φ (η) decreases for higher values of Kr. It is happening since the positive values of Kr corresponds to destructive kind of chemical reaction, therefore larger values of Kr cause reduction of species with a better rate. Hence, a diminishing effect of φ (η) is observed with a rise in Kr. It is witnessed from Fig. 12 that concentration distribution φ (η) decreases on increasing the values of Sc. This phenomenon is in excellent agreement with the fact that Sc suggests the ratio between momentum diffusivity and mass diffusivity. A rise in Sc means a reduction in the molecular diffusion. Therefore, imposition of high chemical molecular diffusion causes φ (η) to increase comprehensively. 7.4. Entropy generation and Bejan number The effects of Br , α1 , α2 and L on the entropy generation Ng and Bejan number Be are shown in Figs. 13 to 20. Figs. 13 and 14 are sketched to display the behavior of Ng and Be against Br. From these two Figures Higher value of Ng is observed for larger Br while opposite scenario is noticed for Be i.e. Bejan number decreases on increasing Br. Physically, Br has direct relationship with heat generated by fluid friction and heat transfer via molecular conduction. Hence, more heat is produced in the system with an increment in Br which rises the disorderliness of system. That is why Ng improves. Fig. 14 shows that Be is decreasing on increasing Br. Figs. 15 and 16 are portrayed to display the behavior of Ng and Be against the variation in temperature difference parameter α1 on Ng and Be, respectively. The higher the difference in temperature, Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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Fig. 12. Graph of φ (η) against Sc.

Fig. 13. Graph of Ng against Br.

Fig. 14. Graph of Be against Br.

the more will be disorderliness in the system. It consequences in a rise in entropy generation. For larger values of α1 ; the thermal irreversibilities dominate over the fluid friction, magnetic field and viscous diffusion irreversibilities. Therefore, Bejan number Be is increased. Influence of α2 on Ng and Be are shown in Figs. 17 and 18, respectively. It is revealed from these two figures that both Ng and Be get enhanced for larger values of concentration difference parameter α2 . Figs. 19 and 20 display the effect of diffusion parameter L on Ng and Be. Both the figures show that Ng and Be, both are getting improved on increasing the value of L. Since for larger values of L, diffusivity in the fluid particle is enhanced which rises disorderliness. Therefore, with a rise in the value of diffusive variable, there is an increment in the entropy generation. 7.5. Quantities of physical interest The change in the behavior of skin friction coefficient Cfx (Rex )0.5 , Nusselt number Nux (Rex )−0.5 and Sherwood number Shx (Rex )−0.5 are computed for various flow parameter and presented in Tables 2–4 respectively. Table 2 is prepared to highlight the impact of M and We on the skin friction coefficient (surface drag force). From this table it is found that magnitude of Cfx (Rex )0.5 increases on increasing the different estimations of M(= 0.0, 0.3, 0.6, 0.9, 1.2) while a completely opposite behavior is observed for Weissenberg number (We = 0.0, 0.2, 0.4, 0.6, 0.8) i.e. comparatively smaller Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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Fig. 15. Graph of Ng against α1 .

Fig. 16. Graph of Be against α1 .

Fig. 17. Graph of Ng against α2 .

Fig. 18. Graph of Be against α2 .

Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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Fig. 19. Graph of Ng against L.

Fig. 20. Graph of Be against L.

Table 2 Numerical values of skin friction when Nt = 0.2, Nb = 0.1, Ec = 0.5, Kr = 0.3, R = 0.5, and θw = 1.1. M

We

Cfx

0.0 0.3 0.6 0.9 1.2 0.3 0.3 0.3 0.3 0.3

0.2 0.2 0.2 0.2 0.2 0.0 0.2 0.4 0.6 0.8

1.92111 2.17878 2.40523 2.60874 2.79448 2.28039 2.17878 2.05049 1.66667 1.25000

Table 3 Numerical values of Nusselt number when We = 0.3, Nt = 0.2, Nb = 0.1, Kr = 0.3 and θw = 1.1. R

Ec

M

−θ ′ (0)

0.0 0.5 1.0 1.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.5 0.5 0.5 0.5 0.0 0.3 0.5 0.8 0.5 0.5 0.5 0.5

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.0 0.3 0.5 0.6

0.254994 0.333572 0.403955 0.475624 0.662497 0.465231 0.333572 0.135859 0.319611 0.333572 0.349701 0.357474

Please cite this article as: A. Kumar, R. Tripathi, R. Singh et al., Simultaneous effects of nonlinear thermal radiation and Joule heating on the flow of Williamson nanofluid with entropy generation, Physica A (2020) 123972, https://doi.org/10.1016/j.physa.2019.123972.

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A. Kumar, R. Tripathi, R. Singh et al. / Physica A xxx (xxxx) xxx Table 4 Numerical values of Sherwood number when M = 0.3, Ec = 0.5, We = 0.3, R = 0.5, and θw = 1.1. Kr

Nt

Nb

−φ ′ (0)

0.0 0.3 0.5 0.9 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

0.2 0.2 0.2 0.2 0.0 0.2 0.4 0.6 0.2 0.2 0.2 0.2

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.5 0.9 1.3

0.419015 0.844075 1.016420 1.267990 0.787501 0.844075 0.946077 1.088510 0.844075 0.813770 0.808880 0.806075

skin fraction coefficient is observed for larger Weissenberg number. The nature of Nux (Rex )−0.5 under the variation of radiation parameter R, Eckert number Ec and M is elucidated in Table 3. One can conclude from this table the rate of heat transfer Nux (Rex )−0.5 gets reduced on increasing Eckert number (Ec = 0.0, 0.3, 0.5, 0.8) while an improvement in rate of heat transfer is observed on increasing R and M. Table 4 is prepared to demonstrate the influences of chemical reaction parameter Kr, Brownian diffusion parameter Nb and thermophoresis parameter Nt on the rate of mass transfer Shx (Rex )−0.5 (Sherwood number). Tabulated values depict that mass transfer rate is reduced on increasing Brownian diffusion parameter while it is getting enhanced on increasing chemical reaction and thermophoresis parameters (Kr = 0.0, 0.3, 0.5, 0.9 and Nt = 0.0, 0.2, 0.4, 0.6). 8. Concluding remarks Our aim in carrying out this research work was to investigate the features of entropy generation on two-dimensional hydromagnetic chemically reactive flow of non-Newtonian Williamson nanofluid with nonlinear thermal radiation. Noteworthy outcomes of the current study are summarized as follows:

• The fluid velocity gets lowered with the increment in the values of governing flow parameters such as magnetic parameter and Weissenberg number.

• Temperature of the fluid is perceived to rise on increasing the value of Eckert number, magnetic parameter, Brownian diffusion parameter, radiation parameter, temperature difference parameter and thermophoresis parameter.

• Concentration of nanoparticles is reduced on increasing either of chemical reaction parameter or Schmidt number. • Strengthening of Brinkman parameter, diffusion parameter, temperature and concentration difference parameters lead to a rise in entropy generation. However, diffusion parameter, temperature and concentration difference parameters tend to enhance the Bejan number significantly. • An increase in the magnetic parameter and Weissenberg number lead to a reduction in the skin friction coefficient. The rate of heat transfer gets reduced with increase in Eckert number while this physical quantity is enhanced on increasing radiation parameter. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] G. Kumaran, N. Sandeep, Thermophoresis and Brownian moment effects on parabolic flow of MHD Casson and Williamson fluids with cross diffusion, J. Molecular Liquids 233 (2017) 262–269. [2] I. Zehra, M.M. Yousaf, S. Nadeem, Numerical solutions of Williamson fluid with pressure dependent viscosity, Results Phys. 5 (2015) 20–25. [3] M. Ramzan, M. Bilal, J.D. Chung, MHD stagnation point Cattaneo–Christov heat flux in Williamson fluid flow with homogeneous–heterogeneous reactions and convective boundary condition—A numerical approach, J. Molecular Liquids 225 (2017) 856–862. [4] K.U. Rehman, A.A. Khan, M. Malik, U. Ali, M. Naseer, Numerical analysis subject to double stratification and chemically reactive species on Williamson dual convection fluid flow yield by an inclined stretching cylindrical surface, Chin. J. Phys. 55 (4) (2017) 1637–1652. [5] Hashim, M. Khan, A. Hamid, Numerical investigation on time-dependent flow of Williamson nanofluid along with heat and mass transfer characteristics past a wedge geometry, Int. J. Heat Mass Transfer 118 (2018) 480–491. [6] N. Eldabe, M.A. Elogail, S. Elshaboury, A.A. Hasan, Hall effects on the peristaltic transport of Williamson fluid through a porous medium with heat and mass transfer, Appl. Math. Model. 40 (1) (2016) 315–328. [7] S. Nadeem, S. Akram, Peristaltic flow of a Williamson fluid in an asymmetric channel, Commun. Nonlinear Sci. Numer. Simul. 15 (7) (2010) 1705–1716.

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