Entropy optimization in cubic autocatalysis chemical reactive flow of Williamson fluid subjected to viscous dissipation and uniform magnetic field
来源期刊：中南大学学报(英文版)2019年第5期
论文作者：M. IJAZ KHAN Sania JAVED Tasawar HAYAT M. WAQAS Ahmed ALSAEDI
文章页码：1218 - 1232
Key words：entropy generation; viscous dissipation; Williamson fluid; uniform magnetic felid; quartic autocatalysis chemical reaction
Abstract: This research elaborates magnetohydrodynamics (MHD) impact on non-Newtonian (Williamson) fluid flow by stretchable rotating disks. Both disks are rotating with different angular velocities and different stretching rates. Viscous dissipation aspect is considered for energy expression formulation. Entropy generation analysis is described via implementation of thermodynamic second law. Chemical processes (heterogeneous and homogeneous) subjected to entropy generation are introduced first time in literature. Boundary-layer approach is employed for modeling. Apposite variables are introduced for non-dimensionalization of governing systems. Homotopy procedure yields convergence of solutions subjected to computations of highly nonlinear expressions. The significant characteristics of sundry factors against thermal, velocity and solutal fields are described graphically. Besides, tabular results are addressed for engineering quantities (skin-friction coefficient, Nusselt number). The outcomes certify an increment in temperature distribution for Weissenberg (We) and Eckert (Ec) numbers.
Cite this article as: M. IJAZ KHAN, Sania JAVED, Tasawar HAYAT, M. WAQAS, Ahmed ALSAEDI. Entropy optimization in cubic autocatalysis chemical reactive flow of Williamson fluid subjected to viscous dissipation and uniform magnetic field [J]. Journal of Central South University, 2019, 26(5): 1218–1232. DOI: https://doi.org/ 10.1007/s11771-019-4082-y.
ARTICLE
J. Cent. South Univ. (2019) 26: 1218-1232
DOI: https://doi.org/10.1007/s11771-019-4082-y
M. IJAZ KHAN1, Sania JAVED1, Tasawar HAYAT2, M. WAQAS1, Ahmed ALSAEDI2
1. Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan;
2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract: This research elaborates magnetohydrodynamics (MHD) impact on non-Newtonian (Williamson) fluid flow by stretchable rotating disks. Both disks are rotating with different angular velocities and different stretching rates. Viscous dissipation aspect is considered for energy expression formulation. Entropy generation analysis is described via implementation of thermodynamic second law. Chemical processes (heterogeneous and homogeneous) subjected to entropy generation are introduced first time in literature. Boundary-layer approach is employed for modeling. Apposite variables are introduced for non-dimensionalization of governing systems. Homotopy procedure yields convergence of solutions subjected to computations of highly nonlinear expressions. The significant characteristics of sundry factors against thermal, velocity and solutal fields are described graphically. Besides, tabular results are addressed for engineering quantities (skin-friction coefficient, Nusselt number). The outcomes certify an increment in temperature distribution for Weissenberg (We) and Eckert (Ec) numbers.
Key words: entropy generation; viscous dissipation; Williamson fluid; uniform magnetic felid; quartic autocatalysis chemical reaction
Cite this article as: M. IJAZ KHAN, Sania JAVED, Tasawar HAYAT, M. WAQAS, Ahmed ALSAEDI. Entropy optimization in cubic autocatalysis chemical reactive flow of Williamson fluid subjected to viscous dissipation and uniform magnetic field [J]. Journal of Central South University, 2019, 26(5): 1218–1232. DOI: https://doi.org/ 10.1007/s11771-019-4082-y.
1 Introduction
Indeed entropy generation minimization (EGM) is an effective way to elaborate improvements in thermal attributes since it generates an instinct which is impossible to acquire via energy analysis. Thermodynamic irreversibility can be computed correctly through entropy rate. Consequently, a decline of entropy output results in additional effective energy transportation mechanisms. EGM concept subjected to convection transportation of heat problem was firstly elaborated by BEJAN [1]. Afterwards, numerous investigations on EGM were presented. SAHIN [2] formulated viscous fluid laminar flow considering EGM. Mathematical analysis for Couette flow subjected to EGM and variable viscosity was evaluated by EEGUNJOBI et al [3]. RASHIDI et al [4] scrutinized variable properties effectiveness in magneto slip flow by permeably rotated disk considering EGM. Thermally radiating convective peristalsis flow subjected to EGM and CNTs was examined by FAROOQ et al [5].
The theory covering mechanics of non- Newtonian materials delivers a freedom regarding utilizations of some mathematical mechanisms which are developed for analysis of nonlinear problems. Computations of nonlinear problems through such mechanisms yield their complete solutions. Besides, implementation of such mathematical mechanisms effectively elaborates the nature of obtained solutions. The materials for illustration clay coatings, drilling muds, ketchup, polymer melts and blood are non-Newtonian. Such materials cannot be elaborated as Newtonian since a solo non-Newtonian model cannot interpret their characteristics. Consequently, distinct non- Newtonian models for fluids classification are introduced (see Refs. [6–10]). The considered model (Williamson fluid) [11] addresses pseudoplastic characteristics much efficiently in comparison to other generalized Newtonian models. Recent investigations subjected to Williamson polymeric fluid model are given in Refs. [12–15].
Investigations subjected to chemical processes (heterogeneous, homogeneous) have achieved continuous attention from modern researchers. Such processes are essential in numerous systems for illustration catalysis, formation of fog, fibrous insulation, air and water pollution [16]. Homogeneous type chemical process transpires on a solitary phase while a heterogeneous type chemical process arises in phases. Homogeneous type chemical processes are simpler in comparison to heterogeneous type chemical processes because the homogeneous processes product is elaborated by characteristics of reactive materials. Few attempts subjected to such processes are revealed in Refs. [17–20].
In this investigation, we formulated non- Newtonian (Williamson) fluid flow by permeable disks rotation. Magnetic field, viscous dissipation and chemical processes (heterogeneous and homogeneous) are taken into account for analysis and formulation. Homotopy scheme [21–50] yields convergent solutions subjected to highly nonlinear systems. The significant characteristics of sundry factors against thermal, velocity and solutal fields are described graphically. Besides, tabular results are addressed for engineering quantities, such as skin-friction coefficient and Nusselt number.
2 Statement of problem
2.1 Flow expression
Here we are considering steady incompressible flow of Williamson fluid between two rotating disks. Both disks are stretchable with stretching rates a1 and a2 having angular velocity Ω1 and Ω2, respectively. Flow is considered in axisymmetric channel. Williamson fluid is addressed in the presence of entropy generation. The flow region is exposed by taking magnetic field in z–direction. The schematic flow portrayal is featured in Figure 1.
Figure 1 Schematic flow diagram
The governing flow equations for steady, incompressible fluid are expressed as [24]:
(1)
(2)
(3)
(4)
with
(5)
In the above expressions, the velocity components are denoted by
whereas the cylindrical coordinates are denoted by (r, θ, z). Besides, pressure 
density (ρ), kinematic viscosity (v), electrical conductivity (σ), magnetic strength (B0) and the distance between two disks (h) are presented.
Mathematical forms of 
and 
are given below:
(6)
(7)
After substitution of Eqs. (6) and (7) in Eqs. (2)–(4), the resulting equations are
(8)
(9)
(10)
Implementing [24]:
(11)
we obtain
(12)
(13)
(14)
(15)
(16)
(17)
where
highlights the Reynolds number; 
is the magnetic parameter; 
and 
are the ratio parameters; 
is the Weissenberg number and 
is constant.
2.2 Energy equation
The energy equation subjected to viscous dissipation irreversibility is expressed as
(18)
where
(19)
After substituting Eq.(19) into Eq.(18), one obtains
(20)
with
(21)
in which 
denotes the temperature at lower disk; k is the thermal conductivity; cp is the specific heat capacity; 
is the temperature at upper disk; Γ is the time constant and μ0 and μ∞ are the viscosities at zero and infinite deformation rate, respectively.
Considering the transformation,
(22)
we have from the energy equation,
(23)
(24)
where 
represents the Prandtl number; 
is the Eckert number and 
is constant.
2.3 Mass concentration via cubic chemical reaction
Implementing [19], the cubic isothermal autocatalytic reaction is defined as
(25)
where species 
has higher concentration at the disk surface. The first order isothermal heterogeneous reaction has the form:
(26)
in which
and
denote the chemical species; 
and 
denote the concentrations; kc and ks are the reaction rates. The concentration equation in terms of homogeneous-heterogeneous reactions is defined as follows:
(27)
(28)
with appropriate boundary conditions:
(29)
Setting
(30)
one obtains
(31)
(32)
with
(33)
in which 
indicates the Schmidt number; 
is the homogeneous reaction parameter; 
is the heterogeneous reaction parameter and 
is the diffusion ratio parameter. For comparable chemical species and 
we put
i.e., δ=1. Therefore, we have the following relation:
(34)
From Eqs.(31) and (32) we have
(35)
(36)
3 Physical quantities
3.1 Skin friction coefficients
Surface drag forces for radial and tangential velocities at lower disk are
(37)
In dimensionless form,
(38)
(39)
3.2 Nusselt number
At upper and lower disks, local Nusselt number is mathematically defined as
(40)
where qw for lower and upper disks are
(41)
Finally,
(42)
4 Entropy equation
Entropy generation in dimensional form is
(43)
where Φ is defined as
(44)
We obtain the following after putting Eq. (44) into Eq. (43),
(45)
The dimensionless form is
(46)
where
represents the Brinkman number; 
is the temperature difference parameter; 
is the dimensionless parameter; 
is the entropy generation rate; 
is the diffusion variable with respect to homogeneous reaction and 
is the diffusion variable with respect to heterogeneous reaction.
Bejan number is expressed as
(47)
(48)
5 Methodology
We employed homotopy scheme [21–25] for nonlinear analysis. Initial approximations and auxiliary linear operators are
(49)
(50)
with
where 
(i=1–9) are constants.
6 Convergence analysis
The non-dimensional equations have been tackled by homotopy analysis method (HAM). The homotopy analysis method is an analytical technique to tackle nonlinear ordinary/partial differential equations. The homotopy analysis method utilizes the idea of the homotopy from topology to create a convergent series solution for nonlinear frameworks. The series solution unequivocally relies on the auxiliary parameters. These parameters are proficient in controlling the convergence. To acquire the significant range of these parameters, the 
curves are plotted at the 14th order of approximations (see Figure 2). The exact ranges for momentum, energy and concentration equations are 
and 
Table 1 is plotted for the convergence series solutions when Pr=0.7, Re=0.01, M=0.4, A1=0.4, A2=0.7, Ω=1.0, Ec=0.1, Sc=1.0, A2=1, k1=0.4 and k2=0.7.It is noticed that the 20th, 13th, 20th, 20th and 13th orders of approximations are sufficient for convergence of
and
Figure 2 
for 
,
Table 1 Series solutions when Pr=0.7, We=0.01, Re=0.01, M=0.4, A1=0.4, A2=0.7, Ω=1.0, Ec=0.1, Sc=1.0, A2=1, k1=0.4, and k2=0.7
7 Discussion
In this section we analyze the behaviors of numerical results of skin friction and Nusselt number, graphical results of velocity, temperature and concentration profiles and also discuss the entropy generation and Bejan number which signifies its leading role in the formation of the present network.
7.1 Skin friction coefficients and Nusselt number
The opposition delivered because of the association between the fluid and the surface of solid is known as skin friction coefficient. In heat transfer at a boundary inside a fluid, the Nusselt number is the proportion of convective to conductive heat transfer over the boundary. Numerical outcomes are featured in Tables 2 and 3 for skin friction coefficient and Nusselt number at both lower and upper disks separately for larger estimations of Prandtl number (Pr), Hartmann number (M) and Weissenberg number (We). In Table 2, it is easily observed that in radial and transverse direction, the magnitude of drag force for lower disk shows increasing behavior for various values of We. Similarly, for large values of M opposite trend is noticed. In radial direction, drag force shows decreasing behavior and in transverse direction it shows accelerating behavior.
Table 2 Numerical outcomes for Cf0 and Cg0 when We and M are fixed
In Table 3, heat transfer rate decays at lower disk for both vital parameters Pr and We while at upper disk Pr increases heat transfer rate and for (We) it decreases.
Table 3 Numerical outcomes for 
and 
when Pr and We are fixed
7.2 Velocity components: axial 
radial 
tangential 
velocities
In this subsection, Figures 3–14 are designed to elaborate the features of M, A1, A2 and We versus 
and 
This is exactly why velocities development is one of the main focuses of the present system. In order to fully utilize the role of involved parameters, all velocities should be addressed on their own comparative studies to strengthen the system and play their respective role in the rotation, stretching etc. In order to achieve sustainable growth in this regard, Figures 3–5 are plotted for the impact of M on axial 
radial and tangentialvelocity components due to increasing value of magnetic parameter (M). The effect of Hartmann number is to diminish the velocity magnitude. Magnetic field creates Lorentz force, which acts against in the direction of fluid flow. This kind of opposing force diminishes fluid velocity. So the magnetic field is working towards weakening the velocities in the region along the disks. There is also a strong focus on stretching in the field of circulatory motion. It is worth mentioning the effect of stretching parameters A1 and A2 at lower and upper disk in Figures 6–11. It can be clearly seen that for increasing values of A1 and A2, stretching rate becomes greater on their respective disks, which enhances axial and radial motion on a large scale, while for tangential velocity , it declines slowly as it is in an inverse relation with rotational velocity. Figures
12–14 illustrate that axial 
and tangential velocities enhance for large values of We while it decays for radial velocity
due to increment in viscosity.
Figure 3 
against M
Figure 4 
against M
Figure 5 
against M
7.3 Temperature
Figures 15–18 are plotted to discuss the impact of various variables like Hartmann number (M), Eckert number (Ec), Prandtl number (Pr) and Weissenberg number (We) on 
Figure 15 shows the higher temperature subjected to Hartmann number. As it is associated with Lorentz forces which act as a resistive force causes high kinetic energy inside the fluid flow. The curves in Figure 16 shows increment in temperature due to Eckert number. It has additionally been exposed that an increase in estimations of Eckert number results in an increment of temperature distribution between two rotating disks as thermal boundary layer thickness is increased due to frictional heating. From Figure 17, it is very clear to observe the fall in temperature distribution through the influence of Prandtl number. Physically, Pr is the ratio of momentum to thermal diffusivity. Temperature decays because it slows down the rate of thermal diffusion as Pr increases. Therefore, it controls the heat transfer process within the fluid flow. In Fig. 18, it shows that temperature evolves with the Weissenberg number because of increment in resistance for greater viscosity. A non-newtonian parameter like Weissenberg number (We) claims a high percentage of shares in performance so there is significant room to increase the performance.
Figure 6 against A1
Figure 7 against A1
Figure 8 against A1
Figure 9 against A2
Figure 10 against A2
Figure 11 against A2
Figure 12 against We
Figure 13 against We
Figure 14 against We
7.4 Concentration
Figures 19–21 are achieved to observe the influence of homogeneous-heterogeneous reactions (k1, k2) and Schmidt number (Sc) against concentration 
It predicts that as the strength of homogeneous and heterogeneous reaction opens up through k1 and k2, the concentration 
decays. It is certain that the concentration of fluid particles on both disks decays as k1 and k2 evolve. Figure 21 reveals the illustrative representation between Schmidt number Sc and the concentration . Larger values of Schmidt number Sc are well responsible for the low concentration of fluid particles because it is a ratio of momentum (viscosity) and mass diffusivity.
Figure 15 
against M
Figure 16 against Ec
Figure 17 against Pr
Figure 18 against We
Figure 19 
against k1
Figure 20 against k2
7.5 Entropy generation and Bejan number
Figures 22–29 exhibit a diversity of different physical parameters such as Hartmann number M, Brinkman number Br, Weissenberg number We and scaled stretching parameter A1 for entropy generation 
and Bejan number Be.
Figure 21 against Sc
Figure 22 is plotted for the impact of rising estimations of M. The magnetic field parameter M has a serious role in capitalizing on entropy generation and Bejan number Be. As M increases, it is directly associated with Lorentz forces, which provides resistance in the system and causes more disturbance for the fluid. This makes a pathway for entropy generation to stand out for M. Figure 23 demonstrates that M acts as a reducing agent for Bejan number Be as viscous dissipation irreversibility becomes more dominant in a fluid flow as compared to heat transfer irreversibility as M enhances. In Figures 24 and 25, Br plays a significant role in entropy generation and Bejan number Be. Elevation occurs in entropy generationas dissipation phenomena involved in Br, as much as Br increase less conduction rate produces. From Figure 25, for Br=0, Be attains its higher value 1 because the irreversibility of viscous dissipation vanishes and only irreversibility due to heat transfer holds. For further values of Br, Be decays gradually.Figures 26 and 27 show the pathways associated with the parameter We. Understanding the level of entropy generationand Bejan number Be, it is very important to consider the influence of We. Resistance increases for larger values of We as it increases the viscosity difference which causes a disturbance in the system which is responsible for an increment in 
Also, We causes the downfall for Bejan number in Figure 27. In Figures 28 and 29, conflicting trend is observed for entropy generation and Bejan number Be for A1. For different values of scaled stretching parameter A1, the entropy generation increases and Bejan number Be decreases.
Figure 22 
against M
Figure 23 Be against M
Figure 24 against Br
8 Conclusions
1) A very diverse behavior is observed for We. Axial and tangential velocities strengthen for increasing values of We. On the other hand, radial velocity decreases.
2) An increment is found in temperature distribution because of the effect of We and Ec.
3) More disturbance is created in the fluid flow for larger values of Ec and We so that is why entropy generation rises for Ec and We.
4) For Br=0, Bejan number is equal to 1 and then acts as a reducing agent for the values Br>0.
5) The increasing values of Weissenberg number We revealed the increment of surface drag force for radial and transverse direction.
6) Conflicting impact of Weissenberg number We is observed for heat transfer rate.
Figure 25 Be against Br
Figure 26 against We
Figure 27 Be against We
Figure 28 against A1
Figure 29 Be against A1
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(Edited by YANG Hua)
中文导读
Williamson流体在黏性耗散和均匀磁场作用下四次自催化化学反应的熵优化
摘要：阐述了可伸缩的旋转盘对非牛顿磁流体动力学(MHD)的影响。两个圆盘以不同的角速度和不同的伸缩率旋转。在能量表达式中考虑黏性耗散，利用热力学第二定律分析熵的生成，文中首次介绍了熵产生的化学过程(多相和均相)。采用边界层方法建立模型, 引入变量对控制系统进行无量纲化。通过高度非线性表达式的计算，得到同伦过程解的收敛性。用图解法描述了各种因素对热场、速度场和溶质场的影响特征。此外，以表格形式给出了工程质量包括表面摩擦系数、Nusselt数的计算结果。结果证实了因Weissenberg(We)(韦斯森伯格)和Eckert(Ec)(埃克特)数而导致的温度梯度分布的增加。
关键词：熵产生；黏性耗散；Williamson流体；均匀磁场；四次自催化化学反应
Received date: 2018-10-30; Accepted date: 2018-12-18
Corresponding author: M. IJAZ KHAN, Assistant Professor; E-mail: mikhan@math.qau.edu.pk; ORCID: 0000-0002-9041-3292
