Entropy Minimization on Sutterby Nanofluid past a Stretching Surface with Swimming of Gyrotactic Microorganisms and Nanoparticles

This analysis examines the flow of Sutterby nanofluid having bioconvection due to a stretching sheet and heated convective boundary condition. The heat sink/source was imposed in terms of the energy equation. The entropy minimization is also considered. Through the appropriate transformation of the system of nonlinear PDE’s setting, the existing problem has altered into nonlinear ODEs and then numerically utilized via the bvp4c method. The numerical values of skin friction, Nusselt number, Sherwood number, and motile density profiles are revealed in a tabular form. In comparison, other governing variables on velocity, temperature, concentration, and bioconvection are seen through various plots and discussed. For example, increment in the Deborah number of heat flux depreciated in temperature distribution while opposite observation was carried out for heat source and sink parameter. Moreover, it has also been investigated that the shear-thinning fluid is entirely reverse to that of the shear thickening fluid. Further, the increase in the magnetic number accelerates in the Bejan number and irreversibility ratio. Finally, the comparison has been made with previous literature and found an excellent agreement.


Introduction
e non-Newtonian uid ow past the stretching surface has massive features in science and engineering technology. Various kinds of non-Newtonian uid models are paid widely remarkable attention due to their outstanding features such as polymers solution, multiple types of engine oil, and paint. e power-law model has been achieved by the rheological properties of complex uids such as oil and polymer solution. Sutterby uid model [1,2] validates the phenomenon of pseudoplastic and dilatants and obtains comprehensive mathematical expression. e hyperbolic tangent ow with the Dufour e ect due to stretching surface was attempted by Gangadhar et al. [3]. MHD ow of Sutterby uid in a Darcy surface has been conducted by Bilal et al. [4]. e thermal behavior of Sutterby nanomaterial with strati cation is discussed by Nazeer and others [5]. Khan et al. [6] described Sutterby nano uid over a rotating disk. e combined e ect of heat and mass transfer in the MHD ow of Sutterby nanoliquid due to stretching cylinder is deliberated by Sohail and Naz [7]. Rehman and others [8] reported the thermal strati cation phenomenon on the Sutterby nano uid with zero mass ux condition. e significance of heat and mass transfer on Sutterby nanofluid past a wedge surface was discussed by Usman and others. [9]. Bhatti [10] et al. reported the chemically reactive mass transport mechanism on the Jeffrey fluid model in the presence of extrinsic magnetic impact discussed in this paper. A permeable material is used to move the fluid. e mathematical modeling of momentum and concentration equations is done via Lie group transformations. Loganathan et al. [11] conducted a steady flow of thermal analysis of Oldroyd-B flow under second-order slip and cross-diffusion effects.
In the last few decades, nanofluid established the contemplation of numerous researchers because of their various industrial and mechanical engineering applications. It is also observed that heat transport fluids efficiently depend on their physical distinctiveness, such as thermal conductivity. ere are a few base fluids containing water, oil, ethylene, and many more that have less conductivity. As a result, such types of liquids have deprived heat transfer phenomena. So, enhancing their thermal conductivity needs to be an important problem to receive the views of recent scholars. e word nanofluid was employed by [12], which showed thermal performance. Later, to augment the thermal properties of nanofluid, Buongiorno [13] added two main features of nanoliquid, namely, Brownian movement and thermophoretic. e MHD flow of micropolar nanomaterial between two rotating surfaces has been designed by Islam et al. [14]. Nonlinear radiation, swimming microorganisms, and nanoparticles with a 3D bioconvective viscoelastic nanofluid flow across a heated Riga surface were studied by Karthik et al. [15]. Ijaz et al. [16] studied the radiative flow of Sisko nanofluid over a stretching rotating circle with entropy generation. e impact of nanofluid characteristics and heat exchange inside a twisted tube over a revolving disk is pointed out by Alempour et al. [17]. e research by Abo-Elkhair et al. [18] looks at the effects of magnetic force and nonlinear thermal radiation on hybrid bio-nanofluid flow in a peristaltic channel under the influence of a high and low Reynolds number applied magnetic field. Sabir et al. [19] introduced the influence of thermal radiative on Sutterby nanofluid by using Cattaneo-Christov heat near the stagnation point. Ali and Zaib [20] studied time-dependent Eyring-Powell comprising nanoliquid due to convective conditions. Khan et al. [21] looked at the effect of heat and mass transport on third-grade nanofluid due to heat convectively stretching sheet. Ibrahim and Makinde [22] considered the power-law nanofluid over convective boundary conditions near the stagnation point. Bhatti et al. [23] address the nanofluid flow and microbe swimming across parallel rotating circular plates separated by a limited distance and containing a porous media.
In the past few years, numerous analyses into energy creation in various fluid stream conditions have been taken out by investigators' attention on the entropy generation. Many investigators are stimulated by recognizing the features of entropy production in liquid flow and some applications such as heat pumps, fire engines, air conditioners, and many more. e entropy production minimization had been coined by Bejan [24]. Later, the impact of entropy production on the Sutterby nanofluid past a stretching surface near the stagnation point was discovered by Azhar et al. [25]. Hayat et al. [26] discussed the MHD peristaltic flow of Sutterby nanofluid and entropy generation. Loganathan et al. [27] examined the thirdgrade nanofluid flow over a stretching plate with entropy generation. ey found that, for the higher thermal relaxation time parameter, the entropy production and Bejan number profiles show the opposite impact. Furthermore, the radiation constant, Biot number, suction/injection constant, Hartmann number, and Brinkman number all improve the system's entropy. Yousaf and others [28] explained the influence of entropy optimization and motile density of Williamson nanofluid past an inclined sheet. Makinde and Tshehla [29] deliberated the mixed convective flow of nanoliquid in an entropy minimization and mass suction/injection. Loganathan and Rajan [30] presented the Williamson nanofluid flow with zero mass flux and Joule heating impacts. ey also found the entropy generation of the problem. Afridi et al. [31] considered Newtonian flow for entropy minimization induced by the thermal radiative flow. Inside the cavity, the entropy production of fluid flow with the permeable surface is scrutinized by Alsabery [32].
Sutterby fluid [1,2] is one of the very important non-Newtonian fluids which shows the modeled equations of high polymer aqueous solution. e current analysis is dedicated to inspecting the entropy production on bioconvection for the flow of Sutterby fluid because a heated convective sheet is examined.
ermal radiation, heat sink/source, Brownian motion, and thermophoresis are also considered. e main aim of this study is that the non-Newtonian Sutterby fluid is selected to examine the rheological properties of shearthinning/thickening with the effect of thermal radiation and microorganism. e similarity conversion is used to alter the nonlinear PDE into ODE to solve modeled equations. e numerical solution of these modeled equations has been produced via BVP4c in Matlab. To investigate the behavior of velocity, temperature, concentration, and motile density, they are plotted against numerous variables. e numerical computation is displayed in the form of figures and tables. Manufacturing of rubber and plastic sheets, melt-spinning, glass-fiber manufacturing, and metallic plate cooling systems are some of the beneficial uses of this sort of inquiry. Also, the Bejan number is used to estimate entropy generation in power engineering and aeronautical propulsion to forecast the smartness of the overall system.

Mathematical Formulation.
e incompressible, laminar, time-dependent flow of Sutterby nanomaterial past stretching sheet subject to the convective condition and entropy production is examined. e stretching sheet velocity is U w (x, t) � ax/1 − ct. e mathematical modeling of the physical problem is designed with coordinate (x, y) with velocity component (u, v) taken in x-and y-axis (see Figure 1). e mathematical formulation for Sutterby nanofluid in a steady flow is taken in [25], and it converted to be unsteady in the following flow equations.
With boundary condition, In equation (2), m shows three conditions. If m < 0, then it shows pseudoplastic or represents uid with decreasing viscosity; if m > 0, it demonstrates the dilatant or uid with increasing viscosity; if m 0, it re ects the Newtonian uid.
e Rosseland approximation is crucial in thermal radiative heat ow. e Rosseland approximation requires an optically thick medium and radiation that travels only a short distance before being scattered or absorbed. As a result, a simpli ed model for the Radiative Transfer Equation (RTE) based on the Rosseland approximation is provided as follows: where T 4 can be expanded as follows: Replacing equation (7) into equation (8), Similarity expressions are e converted boundary conditions are e Reynolds number (Re ax 2 /v), Deborah number

Engineering Interest Quantities
Shear wall τ w , heat q w , mass q m , and motile density q n have been displayed as In the use of (10) and (17),

Entropy Generation Analysis
e entropy production rate (S G ) for the Sutterby fluid [25] under magnetic field, thermal radiation, and diffusion effect can be written as e dimensionless entropy generation number is expressed as (20) e Bejan number (Be) is described as

Solution Methodology.
To compute the numeric outcomes of formulated systems of ODEs (10)-(13) subject to boundary conditions equation (14) by bvp4c technique, the ODEs are renovated into first-order ODEs by assigning a new variable.
e converted boundary condition is By using (27) with boundary conditions, the results are using a finite value for η max .
e step size is measured Δη � 0.001 and 10 −6 is a convergent criterion to repeat and attain the numeric solution.

Result and Discussion
e current study involves the governing parameters on velocity f ′ (η), temperature θ(η), concentration θ(η) , density of motile microorganisms N(η) , entropy generation Ns(η), and Bejan number Be(η). Table 1   e Newtonian, shear-thinning, and shear-thickening fluid is described in Figure 3 Figure 5(b) shows that a larger value of the Reynolds number (Re) causes a reduction in temperature profile for shear-thinning and temperature profile decreases as there is an increment in (Re). Figures 5(c) and 5(d) display that both entropy minimization and the Bejan number reduce and upsurge with the rise of the value of Re. e plots of the magnetic variable on the velocity and temperature profiles are exposed in Figure 6(a)-6(d). e velocity of the fluid grows with a mounting value of M while an opposite observation is noted for the temperature field for both (m < 0) and (m < 0). Physically, there is an increment in the strength of the magnetic field whereas resistive force turned down the velocity. From Figure 6(c), the entropy minimization increases with a larger magnitude Mathematical Problems in Engineering of magnetic number for both cases pseudoplastic and dilatants whereas its opposite behavior is seen in Figure 6(d).
e view of (Nb) on temperature eld θ(η) and concentration pro le ϕ(η) is disseminated in Figures 7(a) and  7(b). e temperature pro le θ(η) is enhanced with enhancing the value of the Brownian motion (Nb). An augment in the (Nb) causes an enhancement in the nanoparticle collision that results in an increment in uid temperature. However, deceleration in the concentration eld is observed in Figure 7(b). Figures 8(a) and 8(b) display thermophoretic estimation (Nt) on temperature θ(η) and concentration pro les θ(η). Larger values of (Nt) for both θ(η) and ϕ(η) have been intensi ed. Nanoparticles have dragged from hotter to cold surfaces. at is the reason of the increase in thermal and concentration pro les. e plots for the Prandtl (Pr) and Schmidt number (Sc) on temperature ϕ(η) and concentration pro les ϕ(η) are carried out in Figures 9(a) and 9(b). It can be shown from these gures that enlargement in the (Pr) causes a dazzling depreciation in the uid temperature. e concentration pro le reduces with a larger value of (Sc). Physically, the Schmidt number (Sc) is dependent on the proportion of momentum to mass di usivity. By increasing the value of (Sc), momentum di usivity increases as nanoparticle concentration declines. Figures 10(a) and 10(b) show the Biot number (Bi) and chemical reaction parameter (κ) on temperature and concentration pro les. e Biot number means the ratio of convection over conduction inside the boundary at the surface. e gradually growing value of values of the Biot number climbs the temperature distribution while similar behavior is observed for increasing value of κ on the concentration pro le. e features of the Bioconvection Lewis number (Lb) and Peclet number (Pe) on motile  density microorganism N(η) were seen in Figures 11(a) and 11(b). For the mounting value of (Lb) as the di usivity microorganism upsurges, so, the density of motile microorganism has increased. Figure 11(b) shows the in uence of the Peclet number on microorganism density that has been assumed. Enlarging the Peclet number climbs the movement of uid particles. Due to this, microorganism density enhances as their concentration diminishes. e impact of Re and δ on skin friction coe cient was presented in Figure 12 for dilatant and pseudo-plastic cases. In dilatant case, the surface drag force suppresses when enhancing the δ values, and it aggravates for increasing Re    Mathematical Problems in Engineering values and the opposite trend was obtained in pseudo-plastic case. Figure 13 illustrates the variations of Nu for di erent combinations of Nb and Rd. In small quantity of Nb, the heat transfer gradient upsurges when rising the Rd variations and the reverse trend was attained for higher quantity of Nb. e consequences of κ and Sc on Sherwood number is plotted in Figure 14. In Sc 0.8, the Sherwood number enlarges for strengthening the κ values, and the Sherwood number is almost same for changing κ values at Sc 0.9. In addition, the Sherwood number decays for raising the κ values at Sc 1.0. Figure 15 gives the impact of Pe and Lb on motile density. It is detected from this gure that the motile density          escalates when escalating the vales of Pe and its downturns for enriching the Lb quantity. Numerical estimation of drag friction for Re, δ and M has been shown in Table 2 for dilatant and pseudo-plastic cases. It is noticed that the drag friction diminishes when rising the Re & δ values in the pseudo-plastic case and it is elevated in the dilatant case. In Addition, the drag friction slumps for a higher quantity of M in both cases. Table 3 presents the numerical values of Rd, Nb, Nt, Pr on Nusselt number for m −0.5 and m 0.5 . From this table we obtain Rd, Nt, Pr have an increasing tendency and the opposite behaviour is found for Nb in both cases. Table 4 provides the consequences of Pr, Sc, κ, Nb and Nt on the Sherwood number and found that the Sherwood number mounts when      8. 12 9 .2 7 9 9 1 0 .4 3 9 9 1 1 .5 9 9 9 1 2 .7 5 9 9 1 3 .9 1 9 9 1 7 . 3 9 9 9 1 6 . 2 3 9 9 1 5 . 0 7 9 9 1 3 . 9 1 9 9 1 2 . 7 5 9 9 1 1 . 5 9 9 9 1 0 . 4 3 9 9 9 . 2 7 9 9 8 . 1 2 1 8 . 5 5 9 9 1 9 . 7 1 9 9 2 0 . 8 7 9 9 2 2 . 0 3 9 9 2 3 . 1 9 9 9 2 4 . 3 5 9 9 2 5 . 5 1 9 9 2 6 . 6 7 9 9 2 7 . 8  improving the Pr, κ, Nb and Nt values and its downfalls when larger quantity of Sc values. e impact of Le, Pe, ω and Nb on motile density was displayed in Table 5. It is seen that the motile density progress when elevating the Pe, ω and Nb values and it depresses when higher values of Le. Table 6 provides the list of symbols in our analysis. e flow line for steady flow 16(a), unsteady flow 16(b), pseudo-plastic 17(a) and dilatant 17(b) was presented in Figures 16(a) It is found that pseudo-plastic has a more pronounced than the dilatant case. e Nusselt number variations for different values of κ for dilatant and pseudo-plastic cases. It is seen that the higher heat transfer gradient attains in pseudoplastic case than the dilatant case.

Conclusion
In this article, the influence of numerical simulation of MHD flow of Sutterby nanofluid with thermal radiation and heat source/sink has been investigated. Entropy production is also considered. e transmuted ODEs are obtained numerically through bvp4c. By this simulation, we have revealed the influence of emerging parameters on the flow of entropy generation and chemical reaction effects with thermal radiative for time-dependent MHD of Sutterby fluid past stretching surface. is investigation can be enlarged for Ree-Eyring fluid, Williamson fluid, and generalized fluid model. Some precise conclusions which have been produced from this research are given as follows. Data Availability e raw data supporting the conclusions of this article will be made available by the corresponding author without undue reservation.