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. © 2020 Sharif University of Technology. All rights reserved. 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 gen- eration [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 dissipa- tion, 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 follow- ing [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 conduc- tivity, 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.