Novel Aspects of Soret and Dufour in Entropy Generation Minimization for Williamson Uid Ow

Scientia Iranica B (2020) 27(5), 2451{2464

Sharif University of Technology Scientia Iranica Transactions B: Mechanical Engineering http://scientiairanica.sharif.edu

Novel aspects of Soret and Dufour in entropy generation minimization for Williamson uid ow

T. Hayata,b, F. Masooda, S. Qayyuma,*, M. Ijaz Khana, and A. Alsaedib a. Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan. b. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia.

Received 27 December 2018; received in revised form 4 February 2019; accepted 27 April 2019

KEYWORDS Abstract. Soret and Dufour e ects on MHD ow of Williamson uid between two rotating disks are examined in this study. Impacts of strati cation, viscous dissipation, Williamson uid; and activation energy are also considered. Bejan number and entropy generation for Rotating disks; the strati ed ow are discussed. The governing Partial Di erential Equations (PDEs) Activation energy; are converted into ODE using von Karmantransformations. The convergent solution Viscous dissipation; of complicated ODE is found using a homotopic procedure. The results of physical Entropy generation; quantities are discussed through plots and numerical values. It is noted that axial and Soret and Dufour radial velocities are higher in the case of greater Weissenberg number. Temperature and e ects; concentration pro les are the decreasing functions of thermal and solutal strati cation Strati cation. parameters, respectively. Entropy and Bejan number show the opposite trends for higher Weissenberg number and Brinkman number.

1. Introduction was numerically analyzed by Zehra et al. [3]. Khan et al. [4] and Qayyum et al. [5] studied entropy gen- Recent researchers have particularly focused on the eration during heat transfer of Williamson nano uid ows of non-Newtonian uids because these liquids due to stretching sheet and rotating disks, respectively. have penetrated industrial and technological processes. Further studies on non-Newtonian uids can be seen In this regard, they have suggested a number of models in [6{10]. for such liquids due to their diverse properties. In Non-Newtonian uid ow with rotating disks has general, subclasses of di erential-type liquids, known many applications in engineering, e.g., air cleaning as second, third, and fourth grades, are extensively machine, medical equipment and aerodynamical engi- analyzed. However, Williamson uid model has re- neering processes, etc. Karman [11] rstly examined ceived insigni cant attention. Williamson [1] studied the steady ow using an in nite rotating disk. The the pseudoplastic material experimentally. Nadeem et ow generated by rotating disks with radiative heat al. [2] discussed Williamson uid ow that results ux and variable thickness was examined by Hayat from the stretching velocity of a surface. This ow, et al. [12]. Khan et al. [13] presented MHD Eyring- which is characterized by pressure-dependent viscosity, Powell uid ow with a rotating disk. Doh and Muthtamilselvan discussed Micropolar uid ow by a *. Corresponding author. rotating disk with MHD and thermophoretic particle E-mail address: [email protected] (S. Qayyum) deposition e ects [14]. Khan et al. examined magnetic eld and double di usion in the couple stress uid ow doi: 10.24200/sci.2019.52553.2772 by a rotating disk [15]. Griths et al. addressed the 2452 T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464 stability analysis of ow using a rotating disk [16]. 2. Formulation Qayyum et al. [17] discussed disorder in the system based on Williamson uid motion. Three-dimensional steady ow of Williamson uid be- It is a well-known fact that mass transfer exists in tween two rotating stretchable disks was considered. view of species concentration di erence in a mixture. This study aims to scrutinize FGM in ow with e ects The species characterized by di erent concentrations of Dufour/Soret, strati cation, and viscous dissipation. transport themselves in a mixture from a region with Chemical reaction with activation energy was also higher concentration to a region with lower concen- investigated. Flow takes place due to the stretching tration. Moreover, activation energy is the minimum of disks. The lower (z = 0) and upper (z = h) quantity of energy that must be possessed by reactants disks are characterized by respective angular velocities before any speci ed chemical reaction occurs. This of 1 and 2. Flow is caused by the stretching of lower disk. Lower and upper disks correspond to process, which is followed by a chemical reaction with ^ ^ ^ ^ activation energy in mass transfer, is usually used temperatures (T1; T2) and concentration (C1; C2). A in food processing, geothermal reservoirs, chemical uniform magnetic eld of strength (B0) is exerted in engineering, etc. Bestman investigated convection of the Z-direction (see Figure 1). a binary mixture owing in a porous space with acti- An extra stress tensor () of Williamson uid is given below: vation energy [18]. Makinde et al. discussed unsteady radiative ow in the presence of chemical reaction [19]. (~ ~ ) = ~ + 0 1 A; (1) Awad et al. described unsteady rotating ow with 1 1 _ activation energy and chemical reaction [20]. Sha que et al. studied the rotating ow containing chemically where A denotes the rst Rilvin-Erickson tensor, ~0 is reactive species and activation energy [21]. the zero shear rate viscosity, ~1 is the in nite shear In recent years, to nd optimal designs for en- rate viscosity, and > 0 is a time constant. Herein, _ is de ned below: gineering system, the second law of thermodynamics r has been applied to the analysis of Entropy Generation 1 _ = (trA2): (2) Minimization (EGM). EGM helps determine the grow- 2 ing rate of irreversibility during a process. Reversibility Through Eq. (1), one obtains the following: of heat and mass transfer and irreversibility of viscous dissipation can be determined by entropy generation. = (~0 + (~0 ~1) _ ) A; (3) During the convection process, irreversibility takes place inside the cavity. It is quite necessary to diminish which further yields: the irreversibility process to ensure energy conserva- @u^ = 2((~ ~ ) _ + ~ ) ; tion. To study the e ects of entropy generation inside rr 0 1 0 @r a thermal system, the second law of thermodynamics is considered. Firstly, Bejan introduced entropy generation [22] and explained that entropy was generated due to viscous e ects and thermal conductivity. Amani and Nobari investigated entropy generation in a curved pipe at a constant wall temperature [23]. Hayat et al. studied peristaltic rotating ow of nanoparticles with entropy generation [24]. Shit et al. found that nano uid ow resulted from exponential stretching sheet and entropy generation [25]. Some of the recent works about entropy generation can be seen in [26{30]. Here, this study analyzes the e ects of entropy generation on MHD Williamson uid ow between two rotating disks. Further strati cation, viscous dissipation, and activation energy e ects are considered. To the best of the authors' knowledge, such an attempt at studying Williamson uid has not been considered yet. Homotopy technique [31{42] is used to de- velop convergent solutions. Behaviors of temperature, velocity, entropy generation, Nusselt number, Bejan number, and skin friction are discussed via graphs and tabulated values. At the end, concluding remarks are given. Figure 1. Schematic diagram of the problem. T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464 2453

!n 1 @v^ u^ T^ E = 2((~0 ~1) _ + ~0) + ; k2(C^ C^ ) exp ; (11) r @ r r 2 ^ ^ T2 T @w^ = 2 ((~ ~ ) _ + ~ ) ; zz 0 1 0 @z :L =[(~0 ~1) _ + ~0] 1 @u^ @v^ v^ @u^ 2 @w^ 2 v^ @v^ @v^ 2 = =((~ ~ ) _ + ~ ) + ; 2 + 2 2 + r r 0 1 0 r @ @r r @r @z r @r @r @w^ @u^ v^2 u^2 @u^ 2 @v^ 2 rz = zr = ((~0 ~1) _ + ~0) + ; + + 2 + + ; (12) @r @z r2 r2 @z @z 1 @w^ @v^ = = ((~ ~ ) _ + ~ ) + : (4) u^ = a1r; v^ = 1r; w^ = 0; z z 0 1 0 r @ @z ^ ^ ^ ^ ^ ^ Eq. (5) is shown in Box I. Mathematical statements T = T1 = m1r + T0; C = C1 = m3r + C0; of the problem under consideration satisfy the following [5]: at z = 0; u^ = a2r; v^ = 2r; p = 0; @u^ u^ @w^ + + = 0; (6) C^ = C^ = m r + C^ ; T^ = T^ = m r + T^ @r r @z 2 4 0 2 2 0 @u^ @u^ v^2 at z = h; (13) w +u ^ @z @r r where, in (x; y; z) directions, the velocity components @p @ @ are (u; v; w^), respectively, p the hydrostatic pressure, = + rr + zr + rr B2u;^ (7) @r @r @z r 0 Cp speci c heat, density of uid, k thermal conductivity, Dm the e ective di usivity rate of mass, Cs the @v^ @v^ v^u^ u^ + w + susceptibility of concentration, KT thermal-di usion @r @z r ratio, kr the reaction rate, electrical conductivity, E activation energy where = 8:61 105eV/K the @r @z r 2 Boltzmann constant, n the tted rate constant, and = + + 2 B0 v;^ (8) @r @z r Tm mean temperature of uid. Suitable transformations for this analysis are @w @w @p @ @ u^ + w = + rz + zz + rz ; (9) given as follows: @r @z @z @r @z r ^ ^ @T^ @T^ k @2T^ @2T^ 1 @T^ u^ = r 1 f(); v^ = r 1 g^(); w^ = h 1 h(); u^ +w ^ = + + @r @z C @z2 @r2 r @r p C^ C^ p^ '^() = 2 ; P^() = ; ^ ^ 2 2 D K @2C^ @2C^ 1 @C^ C1 C0 h 1 +:L + m T + + ; (10) 2 2 CpCs @r @z r @r T^ T^ z ^() = 2 ; = : (14) T^ T^ h @C^ @C^ @2C^ @2C^ 1 @C^ 1 0 u^ +w ^ = D + + @r @z @r2 @z2 r @r Eq. (1) is satis ed and Eqs. (7){(13) take the following forms: D K 1 @T^ @2T^ @2T^ + m T + + ^ 2 2 h^0 + 2f^ = 0; (15) Tm r @r @r @z

v u " # u @u^ 2 @w^ 2 u^ 1 @v^ 2 1 @u^ @v^ v^ 2 @v^ 1 @w^ 2 @u^ 2 _ = t2 + + + + + + + + @w^ + : (5) @r @z r r @ r @ @r r @z r @ @r @z

Box I 2454 T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464

(1 + We_ )f^00 + Re(^g2 f^2 Mf^ h^f^0) c C^ C^ D K Pr = 0 p ;S = 2 0 ;D = m T 2 ^ ^ f k C1 C0 kCs 2 WeA ^00 ^02 00 0 ^0 + (f f +g ^ g^ f ) ^ ^ _ C1 C0 a1 a2 ;A1 = ;A2 = ; T^ T^ 1 1 2We 1 0 + (h^00h^0f^0 + 3f^02f^+g ^02f^) = 0; (16) _ ( ) T^ T^ We = 0 1 ; Sc = ;S = 2 0 ; D 1 ^ ^ WeRe 0 T1 T0 (1 + We_ )^g00 + (^g00g^02 +g ^00g^0f^0) _ 2 r2 r2 Ec = ;A2 = : (23) ^ ^ 2 2We cp(T1 T0) h (h^g^0 + 2^gf^+ Mg^) + (^g0h^0h^00 + 2^g0f^f^0) _ Herein, Re represents the local Reynold number, Ec Eckert number, Pr Prandtl number, We Weissenberg = 0; (17) number, M magnetic eld parameter, Sc Schmidt number, D Dufour number, Sr Soret number, S ther- WeA2 f 1 Re(P 0 + h^0h^) f^03 +f^0g^02 +f^00f^0h^0 +^g00g^0h^0 mal strati cation parameter, S2 solutal strati cation _ parameter, E dimensionless activation energy, dimensionless reaction rate, and temperature di erence ^0 ^00 2(1 + We _ )(f + h ) parameter. Radial and tangential shear stresses and heat 4We (2f^f^0h^0 + h^00h^02) = 0; (18) transfer rate at the lower disk are given below: _ rzjz=0 zjz=0 C ^ = ;C = ; ^00 ^ ^ ^0 ^ ^ ^ fr1 f^ Re + ( + S1) (Re)(Pr)(h + S1f + f) 0 1 1 0 1

00 rqwjz=0 + Df (Re' ^ +' ^ + S2)(Ec)(Pr)(1 + W e _ ) Nur1 = ; (24) k(T^ T^ ) 1 0 ^2 ^02 02 ^02 4f + Ref + Re^g + 2h = 0; (19) where qw is: @T^ Re' ^00 ReScf^(S +' ^) Re(Sc)' ^0h^ q = k : (25) 2 w @z + S (Re^00 + S + ^) + (' ^ + S ) The de nitions of Nusselt number and skin friction r 1 2 coecients at the lower disk are: p n E Nu = Re^0(0); (Re)Sc(1 + ^) exp '^ = 0; (20) r1 1 + ^ Cf^ with boundary conditions: r1 = 1 + (We)f^0(0) A ^ ^ f(0) = A1; h(0) = 0; g^(0) = 1; r 4f^2(0)+A2 g^02(0)+f^02(0) +2h^02(0) ; ^ '^(0) = 1 S2; (0) = 1 S1; at = 0; C ^ ^ ^ f1 0 f(1) = A2; g^(1) = ; (1) = 0;P (1) = 0; p = 1 + (We)^g (0) Re '^(1) = 0; at = 1: (21) r ^2 2 02 ^02 ^02 r 4f (0)+A g^ (0)+f (0) +2h (0) : _ = 4f^2 + A2 f^02 +g ^02 + 2h^02; (22) (26) For the upper disk (radial, tangential), shear stresses where: and Nusselt number are given below: 2 2 2 B 1r k j j M = 0 ; Re = ; = r ; rz z=h z z=h Cf^ = ;Cf = ; 1 1 r2 0 1 ^2 0 1 E D K T^ T^ T^ T^ rq j h E = ;S = m T 1 0 ; = 1 0 ; Nu = w z= : (27) ^ r ^ ^ ^ r2 ^ ^ T2 DTm C1 C0 T2 k(T1 T0) T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464 2455

Thus, the Nusselt number and skin friction for the where the constants are Zi (i = 1 5). upper disk are obtained as: p ^0 Nur2 = Re (1); 4. Convergence C ^ Auxiliary variables (~g^, ~'^, ~^, ~^ , ~ ^), play a promi- fr2 ^0 h f p = 1 + (We)f (1) nent role in the convergence analysis. Figure 2 shows ~- Re curves for the mth-order approximation. The solutions r are found to be convergent for the regions 1:7 4f^2(1)+2h^02(1)+ Re g^02(1)+f^02(1) ; ~h^ 0:3, 1:2 ~f^ 0:3, 1:0 ~g^ 0:4, 0:9 ~ 0:6, and 1:0 ~ 0:7. Table 1 ^ '^ C ^ comprises numerical values of velocity, temperature, f2 0 p = 1 + (We)^g (1) and concentration. Table 1 shows the numerical values Re of velocity concentration and temperature distribution. r Clearly, a meaningful solution of h^0(0), f^0(0),g ^0(0), 4f^2(1)+Re g^02(1)+f^02(1) +2h^02(1) :(28) ^0(0), and' ^0(0) is derived from the 20th-order approximations. Table 2 shows good agreement between the For lower and upper disks, the Sherwood number can obtained results and those in the previous literature. be written as follows:

rJ rJ Sh= w ; Sh= w ; (29) 5. Entropy ^ ^ ^ ^ D(C1 C0) z=0 D(C1 C0) z=h By considering the roles of thermal irreversibility, Joule where Jw is: heating irreversibility, viscous dissipation irreversibil- @C ity, and mass transfer irreversibility, the formulation of Jw = D : (30) @z entropy generation is presented. Its dimensional form Thus, after applying transformations, the Sherwood is represented by Eq. (35) as shown in Box II, where ^ ^ numbers take the following form: Cm and Tm are mean concentration and temperature, ^ ^ Sh respectively, and (rT ), (rC) and () are de ned p = '^0(0); below: Re ! ! ! @T^ 1 @T^ @T^ Sh 0 ^ p = '^ (1): (31) rT = e^r + e^ + e^z; (36) Re @r r @ @z ! ! ! 3. Solution technique ^ ^ ^ ^ @C 1 @C @C rC = e^r + e^ + e^z; (37) Auxiliary linear operators and initial approximations @r r @ @z are: ^0 ^00 00 2 3 Lh^ = h ; Lf^ = f ; Lg^ =g ^ ; @u @u = 0 @y + @y : (38) ^00 00 L = ; L'^ =' ^ ; (32) After applying transformation, one obtains the ^ ^ h0 = 0; f0 = A1 A1 exp() + A2(); ^ g^0 = 1 + ( 1); 0 = (1 ) (1 S1);

'^0 = (1 )(1 S2); (33) with,

Lh^ [Z1] = 0;

Lf[Z2 + Z3] = 0;

Lg^[Z4 + Z5] = 0;

L^[Z6 + Z7] = 0; Figure 2. ~-curves for axial, radial, tangential velocity, L'^[Z8 + Z9] = 0; (34) temperature, and concentration pro les. 2456 T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464

Table 1. Series solutions convergence for = 0:2, n1 = 1, A1 = 0:1, A2 = 0:4, n = 0:1, E1 = 0:5, = 0:5, = 0:5, S1 = S2 = 0:02, Df = 0:5, Ec = 0:1, M = 0:7, Sc = 1, Re = 0.7, We = 0.2, Pr = 0.6, Sr = 0.3,

~h^ = ~f^ = ~g^ = ~^ = ~'^ = 0:8: Order of h^ (0) f^0(0) g^0 (0) ^0 (0) '0 (0) approximation 1 0.20000 0.5272 1.0380 {0.11008 0.6884 5 0.20000 0.4809 1.192 {0.09493 0.6863 11 0.20000 0.4812 1.192 0.09682 0.5462 16 0.20000 0.4812 1.192 0.2655 0.4127 20 0.20000 0.4812 1.192 0.2655 0.3554 25 0.20000 0.4812 1.192 0.2655 0.3554 30 0.20000 0.4812 1.192 0.2655 0.3554 40 0.20000 0.4812 1.192 0.2655 0.3554

~ k 2 R Dh i2 DR h i S00 = + rT^ + B2(^u2 +^v2) + g rC^ + g rC:^ rT^ : G T^ T^2 T^ 0 C^ T^ | {zm } | m {z } | m {z } | m {z m }(35) Viscous dissipation irreversibility Thermal irreversibility Joule dissipation irreversibility Mass transfer irreversibility

Box II

Table 2. Comparison table for the validation of the ~ r2 2 r2T^ S00 Br = 0 1 ;N = m G ; problem with Ref. [5] by varying We. ^ ^ G ^ ^ k T1 T0 k T1 T0 C C C C We f0 g0 f0 g0 [5] [5] [Present] [Present] C^ C^ = 1 0 ; (40) 0.1 0.550420 1.12934 0.550420 1.12934 2 ^ Cm

0.2 0.590397 1.20342 0.590397 1.20342 where Br and NG are Brinkman number and local entropy generation, respectively, 1 and 2 are tem- 0.3 0.630044 1.27751 0.630044 1.27751 perature and concentration ratio parameters, and L is the di usion parameter. following: Herein, the dimensionless form of Bejan number (Be) is given by Eqs. (41) and (42) as shown in Box III. h i ^02 2 NG = 1 Re + (S1 + ) 6. Discussion + M (Re) (Br)(f^2 +g ^2) + (Br) (1 + We_ ) This section discusses a graphical interpretation of various physical parameters for velocity, temperature, entropy generation, Bejan number, skin friction, Nus- 4f^2 + Ref^02 + Re^g02 + 2h^02 selt number, and Sherwood number. h i 6.1. Velocity pro le 2 + L Re' ^02 + (' ^ + S )2 Figures 3{8 are designed to analyze the behavior of 1 1 Hartmann number (M) and Weissenberg number (We) h i on h^ (), f^(), andg ^ () in axial, radial, and transverse + L ^ + S (S +' ^) + Re' ^0^0 : (39) 1 2 directions. These quantities are discussed for both upper and lower disks. Figures 3{5 show the analysis Dimensionless parameters are given as follows: of the behavior of M for axial h^ (), radial f^(), and tangentialg ^ () velocities at both disks. It is ^ ^ T^ T^ RgD C1 C0 found that the magnitude of velocities h^ (), f^(), = 1 0 ;L = ; 1 ^ andg ^ () is reduced as the Hartman number (M) Tm k T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464 2457

Entropy generation due to heat and mass transfer Be = ; (41) Total entropy generation h i h i h i h i ^02 2 2 02 2 ^ 0 ^0 1 Re + (S1 + ) + L Re' ^ + (S1 +' ^) + L S1 + (S2 +' ^) + Re' ^ Be = h 1 i h i : (42) L S + ^ (S +' ^) + Re' ^0^0 + M (Re) (Br) f^2 +g ^2 + (S + )2 + Re^02 1h i h 2 i h 1 1 i 2 2 02 ^2 ^02 02 ^02 +L (S1 +' ^) + Re' ^ + (1 + We_ )(Br) 4f + Re f +g ^ + 2h 1

Box III

Figure 6. Axial velocity h^ () via We. Figure 3. Axial velocity h^ () via M.

^ Figure 4. Radial velocity f^() via M. Figure 7. Radial velocity f () for We.

Figure 5. Tangential velocityg ^ () via M. Figure 8. Tangential velocityg ^ () via We. 2458 T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464 increases. A resistive force is produced when the between the ambient uid temperature and the surface transverse magnetic eld is in e ect. A force acting as condition. Figure 11 shows the e ect of Weissenberg Lorentz force generates resistance and reduces velocity. number We, on ^(). Temperature is enhanced in the ^ ^ The e ect of Weissenberg number (We) on h (), f (), case of estimating higher S1. The behavior of Prandtl andg ^ () is shown in Figures 6{8. The axial and number, Pr, with respect to ^() is shown in Figure 12. radial velocities, h^ (), and f^(), rise when Weissenberg For larger Pr, thermal di usivity has a smaller value, number (We) increases and tangential velocityg ^ (), which causes a reduction in temperature ^(). decreases. At a higher We number, the rotational ^ ^ 6.3. Concentration velocity ( 1) improves and, thus, h (), and f () increase (see Figures 6 and 7). Figures 13{18 show the trend of concentration (' ^()) for variations of Schmidt number (Sc), Soret number 6.2. Temperature (Sr), dimensionless activation energy parameter (E1), Figures 9{12 illustrate the trend of temperature ^() against Dufour number (D1), thermal strati cation parameter (S1), Weissenberg number (We), and Prandtl number (Pr). Figure 9 shows the impact of D1 on ^() at lower and upper disks. Fluid temperature via D1 improves. Energy ux is enhanced due to increase in the concentration gradient for varying D1 and it leads to the enhancement of the uid temperature. Temperature close to the lower disk is higher than that to the upper disk because of the higher temperature ^ ^ of the lower disk than the upper disk, i.e., T1 T2. Figure 10 shows the impact of thermal strati cation ^ parameter S1 on (). To estimate the large number of S1, the temperature decreases due to a potential drop Figure 11. Temperature pro le via We.

Figure 9. Temperature pro le via Df . Figure 12. Temperature pro le via Pr.

Figure 10. Temperature pro le via S1. Figure 13. Temperature pro le via Sc. T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464 2459

Figure 14. Concentration pro le via Sr. Figure 17. Concentration pro le via .

Figure 15. Temperature pro le via E1. Figure 18. Concentration pro le via S2.

Generative chemical reaction is promoted due to a de- ^ T E crease in Arrhenius function ^ exp ^ . When T1 T E increases, concentration pro le' ^ () is enhanced. Figure 16 analyzes the e ect of chemical reaction parameter on' ^ (). For larger , a reduction in the concentration' ^ () is noticed. Due to concentration '^ () gradient, this behavior shows weak buoyancy e ect; thus, a reduction in' ^ () occurs. Figure 17 shows that the concentration' ^ () is reduced for the larger temperature di erence parameter . According to Figure 18, the concentration' ^ () is the decreasing function of the larger solutal strati cation parameter Figure 16. Temperature pro le via . S2. dimensionless reaction rate (), temperature di erence 6.4. Entropy generation minimization parameter (), and solutal strati cation parameter This section emphasizes the graphical interpretation (S2). At the growing values of Sc, the concentration of various physical parameters of entropy generation (' ^ ()) decreases because mass di usivity is reduced (Ng) and Bejan number (Be). Br e ects on Ng and for larger Sc (see Figure 13). The e ect of Soret Be are shown in Figures 19 and 20. The opposite number (Sr) on' ^ () is shown in Figure 14. Accord- trend for Be is observed due to a increment in the ingly, the concentration' ^ () increases for larger Sr. disorderness of system for larger Be. For a larger Temperature gradient is enhanced for larger Sr, which estimation of Br due to its dissipation, a lower rate of tends towards greater convective ow. Hence, the conduction is produced and, thus, entropy generation concentration distribution' ^ () is enhanced. Figure 15 (Ng) is enhanced. The impact of di usion L on Ng shows that the concentration' ^ () increases for the and Be is shown in Figures 21 and 22. Both Ng larger dimensionless activation energy parameter (E1). and Be experience an increase for larger L. For the 2460 T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464

Figure 19. Entropy generation (Ng) via Br. Figure 22. Bejan number (Be) via L.

Figure 20. Bejan number (Be) via Br. Figure 23. Entropy generation (Ng) via We.

Figure 21. Entropy generation (Ng) via L. Figure 24. Bejan number (Be) via We. increasing L, the di usion rate of nanoparticles is (M) is witnessed. For larger M, the uid resistance enhanced; therefore, Bejan number and total entropy grows due to rise in Lorentz force and, consequently, of the system increase. The e ect of We on Ng and Ng increases. Bejan number (Be) reduces for larger Be is discussed in Figures 23 and 24. For larger We, M. Herein, the irreversibility of uid friction prevailed the entropy generation increases, while Bejan number over the heat and mass transfer irreversibilities. decreases. Figures 25{28 show the impact of solutal and thermal strati cation parameters S2 and S1 on Ng 6.5. Skin friction and Nusselt and Sherwood and Be. Both Ng and Be are reduced for the larger numbers estimation of S1 and S2. Figures 29 and 30 clearly E ects of Reynold number (Re) on skin friction co- reveal the e ect of M on Ng, and the enhancement of ecient at lower and upper disks are analyzed in Be in Ng parallel to increase in the magnetic parameter Figures 29 and 30. There is an increase in skin friction T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464 2461

Figure 28. Bejan number (Be) via S2. Figure 25. Entropy generation (Ng) via S1.

Figure 29. Skin friction via Re.

Figure 26. Bejan number (Be) via S1.

Figure 30. Skin friction via Re.

Figure 27. Entropy generation (Ng) via S2. at both disks for the larger Re. From Figures 31 and 32, for the increasing Eckert number (Ec) the heat transfer rate near the surface of lower disk decreases, while reverse behavior is observed for the upper disk. Figures 33 and 34 are drawn to investigate the e ect of strati cation parameter (S2) on the Sherwood number. For varying S2, the Sherwood number near the surface of the lower disk is reduced. However, in proximity to the upper disk, the Sherwood number experiences a reverse behavior. Figure 31. Nusselt number via Ec at lower disk. 2462 T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464

Increasing behavior of Re for the skin friction was observed; Entropy rate through (Br), (L), and (We) was enhanced, while it decreased for thermal and solutal strati cation parameters (S1) and (S2); Bejan number showed a decreasing behavior for larger (S1), (S2), (We), and (Br), while it increased for larger (L).

Acknowledgement

Figure 32. Nusselt number via Ec at upper disk. We are grateful to Higher Education Commis- sion (HEC) of Pakistan for nancial support of this work under the project number No. 20- 3038/NRPU/R&D/HEC/13.

References

1. Williamson, R.V. \The ow of pseudoplastic materi- als", Indust. Eng. Chem., 21, pp. 1108{1111 (1929). 2. Nadeem, S., Hussain, S.T., and Lee, C. \Flow of a Williamson uid over a stretching sheet", Brazilian J. Chem. Eng., 30, pp. 619{625 (2013). 3. Zehra I., Yousaf, M.M., and Nadeem, S. \ Numerical solutions of Williamson uid with pressure dependent

Figure 33. Sherwood number via S2. viscosity", Results in Physics, 5, pp. 20{25 (2015). 4. Khan, M.I., Khan, T.A., Hayat, T., et al. \Irre- versibility analysis and heat transfer performance of Williamson nano uid over a stretched surface", Heat Transfer Research, 50(9) (2018). DOI: 10.1615/Heat- TransRes.2018026342 5. Qayyum, S., Khan, M.I., Hayat, T., et al. \Entropy generation in dissipative ow of Williamson uid between two rotating disks", Int. J. Heat Mass Transf., 127, pp. 933{942 (2018). 6. Hsiao, K. \To promote radiation electrical MHD activation energy thermal extrusion manufacturing system eciency by using Carreau-nano uid with parameters control method", Energy, 130, pp. 486{499 (2017).

Figure 34. Sherwood number via S2. 7. Hsiao, K. \Combined electrical MHD heat transfer thermal extrusion system using Maxwell uid with radiative and viscous dissipation e ects", Appl. Thermal 7. Conclusions Eng., 112, pp. 1281{1288 (2017). The following observations are worth mentioning: 8. Hsiao, K. \Micropolar nano uid ow with MHD and viscous dissipation e ects towards a stretching sheet Velocities h^ (), f^() andg ^ () were reduced for M with multimedia feature", Int. J. Heat Mass Transf., at both disks, while h^ () and f^() were increasing 112, pp. 983{990 (2017). functions of We; 9. Hayat, T., Farooq, S., and Ahmad, B. \Impact of compliant walls on magneto hydrodynamics peristalsis For larger S and Pr, the temperature ^() was 1 of Je rey material in a curved con guration", Sci. reduced, while We and D1 experienced an ascending Iranica, 25, pp. 741{750 (2018). trend; 10. Hsiao, K. \Stagnation electrical MHD nano uid mixed Opposite trends of concentration' ^ () were found convection with slip boundary on a stretching sheet", for S2 and Sc; Appl. Thermal Eng., 98, pp. 850{861 (2016). T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464 2463

11. Karman, T.V. \Uber laminare and turbulente Rei- 25. Shit, G.C., Haldar, R., and Mandal, S. \Entropy bung", ZAMM-Journal of Applied Mathematics and generation on MHD ow and convective heat transfer Mechanics/Zeitschrift fur Angewandte Mathematik in a porous medium of exponentially stretching surface und Mechanik, 1, pp. 233{252 (1921). saturated by nano uids", Adv. Powder Technol., 28, pp. 519{1530 (2017). 12. Hayat, T., Qayyum, S., Imtiaz, M., and Alsaedi, A. \Radiative ow due to stretchable rotating disk 26. Afridi, M.I. and Qasim, M. \Entropy generation and with variable thickness", Results Phys., 7, pp. 156{165 heat transfer in boundary layer ow over a thin needle (2017). moving in a parallel stream in the presence of nonlinear Rosseland radiation", Int. J. Thermal Sci., 123, pp. 13. Khan, N.A., Aziz, S., and Khan, N.A. \MHD ow of 117{128 (2018). Powell{Eyring uid over a rotating disk", J. Taiwan Inst. Chem. Eng., 45, pp. 2859{2867 (2014). 27. Govindaraju, M., Ganesh, N.V., Ganga, B., et al. \En- 14. Doh, D.H. and Muthtamilselvan, M. \Thermophoretic tropy generation analysis of magneto hydrodynamic particle deposition on magnetohydrodynamic ow of ow of a nano uid over a stretching sheet", J. Egyp. micropolar uid due to a rotating disk", Int. J. Mech. Math. Soci., 23, pp. 429{434 (2015). Sci., 130, pp. 350{359 (2017). 28. Hayat, T., Nawaz, S., and Alsaedi, A. \Entropy 15. Khan, N.A., Aziz, S., and Khan, N.A. \Numerical sim- generation in peristalsis with di erent shapes of nano- ulation for the unsteady MHD ow and heat transfer material", J. Mol. Liq., 248, pp. 447{458 (2017). of couple stress uid over a rotating disk", Plos One, 29. Hayat, T., Khan, M.I., Qayyum, S., et al. \Entropy 9, e95423 (2014). generation in ow with silver and copper nanopar- 16. Griths, P.T., Stephen, S.O., Bassom, A.P., et al. ticles", Colloids Surfaces: A Physicochemical Eng. \Stability of the boundary layer on a rotating disk Aspects, 539, pp. 335{346 (2018). for power-law uids", J. Non-Newtonian Fluid Mech., 207, pp. 1{6 (2014). 30. Rezaie, N.Z., Darasi, S.R.D., Zarandi, M.H.F., et al. \Generalized heat transfer and entropy generation of 17. Qayyum, S., Khan, M.I., Hayat, T., et al. \Entropy strati ed air-water ow in entrance of a mini-channel", generation in dissipative ow of Williamson uid be- Sci. Iranica, 24 pp. 2407{2417 (2017). tween two rotating disks", Int. J. Heat Mass Transf., 127, pp. 933{942 (2018). 31. Liao, S.J., Homotopy Analysis Method in Nonlinear Di erential Equations., Springer, Heidelberg, Ger- 18. Bestman, A.R. \Natural convection boundary layer many (2012). with suction and mass transfer in a porous medium", Int. J. Eng. Research, 14, pp. 389{396 (1990). 32. Hayat, T., Khan, M.I., Farooq, M., et al. \Stagna- tion point ow with Cattaneo-Christov heat ux and 19. Makinde, O.D., Olanrewaju, P.O., and Charles, W.M. homogeneous{heterogeneous reactions", J. Mol. Liq., \Unsteady convection with chemical reaction and ra- 220, pp. 49{55 (2016). diative heat transfer past a at porous plate moving through a binary mixture", Afrika Matematika, 22, 33. Sheikholeslami, M., Hatami, M., and Ganji, D.D. pp. 65{78 (2011). \Micropolar uid ow and heat transfer in a permeable channel using analytical method", J. Mol. Liq., 194, 20. Awad, F.G., Motsa, S., and Khumalo, M. \Heat and pp. 30{36 (2014). mass transfer ow in unsteady rotating uid ow with binary chemical reaction and activation energy", Plos 34. Hayat, T., Khan, M.I., Farooq, M., et al. \Impact of One, 9(9), e107622 (2014). Cattaneo{Christov heat ux model in ow of variable 21. Sha que, Z., Mustafa, M., and Mushtaq, A. \Bound- thermal conductivity uid over a variable thicked ary layer ow of Maxwell uid in rotating frame surface", Int. J. Heat Mass Transf., 99, pp. 702{710 with binary chemical reaction and activation energy", (2016). Results Phys., 6, pp. 627{633 (2016). 35. Sui, J., Zheng, L., Zhang, X., et al. \Mixed convec- 22. Bejan, A. \A study of entropy generation in fundamen- tion heat transfer in power law uids over a moving tal convective heat transfer", J. Heat Transf., 101 pp. conveyor along an inclined plate", Int. J. Heat Mass 718{725 (1979). Transf., 85, pp. 1023{1033 (2015). 23. Amani, E. and Nobari, M.R.H. \A numerical investi- 36. Hayat, T., Khan, M.I., Qayyum, S., et al. \Modern gation of entropy generation in the entrance region of developments about statistical declaration and prob- curved pipes at constant wall temperature", Energy, able error for skin friction and Nusselt number with 36, pp. 4909{4918 (2011). copper and silver nanoparticles", Chin. J. Phys., 55, pp. 2501{2513 (2017). 24. Hayat, T., Ra q, M., Ahmad, B., and Asghar, S. \Entropy generation analysis for peristaltic ow of 37. Turkyilmazoglu, M. \Convergence accelerating in the nanoparticles in a rotating frame", Int. J. of Heat Mass homotopy analysis method: A new approach", Adv. Transf., 108, pp. 1775{1786 (2017). Appl. Math. Mech., 10(4), pp. 925{947 (2018). 2464 T. Hayat et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2451{2464

38. Hayat, T., Qayyum, S., Imtiaz, M., et al. \Im- the leading mathematicians working in Pakistan and, pact of Cattaneo-Christov heat ux in Je rey uid currently, is a Professor of Mathematics at the Quaid- ow with homogeneous-heterogeneous reactions", Plos i-Azam University. He has 2000 plus publications. One, 11e, e0148662 (2016). 39. Hayat, T., Khan, M.I., Qayyum, S., et al. \Entropy Faria Masood is an MPhil student at Quaid-I-Azam generation in magneto hydrodynamic radiative ow University. She is doing MPhil under the kind super- due to rotating disk in presence of viscous dissipation vision of Prof. Dr. Tasawar Hayat. and Joule heating", Phys. Fluids, 30, 017101 (2018). 40. Turkyilmazoglu, M. \Parametrized adomian decom- Sumaira Qayyum is a Pakistani Mathematician who position method with optimum convergence", Trans. has made research contributions to the area of math- Modelling Comp. Simul., 27(4), pp. 1{22 (2017). DOI: ematical uid mechanics. She is currently doing PhD 10.1145/3106373 under the supervision of Prof. Dr. Tasawar Hayat. She 41. Turkyilmazoglu, M. \Determination of the correct has many research publications in the eld of uid range of physical parameters in the approximate an- mechanics. alytical solutions of nonlinear equations using the adomian decomposition method", Mediterranean J. Muhammad Ijaz Khan is a Pakistani Mathemati- Mathem., 13, pp. 4019{4037 (2016). cian who has made research contributions to the area 42. Khan, M.I., Qayyum, S., Hayat, T., et al. \Entropy of mathematical uid mechanics. He is currently doing generation minimization and statistical declaration PhD under the supervision of Prof. Dr. Tasawar Hayat. with probable error for skin friction coecient and Nusselt number", Chin. J. Phys., 56, pp. 1525{1546 He has 200 plus research publications in the eld of (2018). Fluid Mechanics.

Ahmed Alsaedi is a Saudian mathematician who Biographies has made pioneering research contributions to the area Tasawar Hayat is a Pakistani Mathematician who has of mathematical uid mechanics. He is currently made pioneering research contributions to the area of a Professor of Mathematics at the King Abdulaziz mathematical uid mechanics. He is considered one of University, Saudi Arabia.