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

Authors: F. Ali; K. Loganathan; E. Prabu; S. Eswaramoorthi; M. Faizan; A. Zaib; Dinesh Kumar Chaudhary
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

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.


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## Body

## 1. Introduction

The non-Newtonian fluid flow past the stretching surface has massive features in science and engineering technology. Various kinds of non-Newtonian fluid models are paid widely remarkable attention due to their outstanding features such as polymers solution, multiple types of engine oil, and paint. The power-law model has been achieved by the rheological properties of complex fluids such as oil and polymer solution. Sutterby fluid model [1, 2] validates the phenomenon of pseudoplastic and dilatants and obtains comprehensive mathematical expression. The hyperbolic tangent flow with the Dufour effect due to stretching surface was attempted by Gangadhar et al. [3]. MHD flow of Sutterby fluid in a Darcy surface has been conducted by Bilal et al. [4]. The thermal behavior of Sutterby nanomaterial with stratification is discussed by Nazeer and others [5]. Khan et al. [6] described Sutterby nanofluid over a rotating disk. The combined effect of heat and mass transfer in the MHD flow of Sutterby nanoliquid due to stretching cylinder is deliberated by Sohail and Naz [7]. Rehman and others [8] reported the thermal stratification phenomenon on the Sutterby nanofluid with zero mass flux condition. The 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. The 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. There 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. The 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. The 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. The impact of nanofluid characteristics and heat exchange inside a twisted tube over a revolving disk is pointed out by Alempour et al. [17]. The 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. The 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 third-grade nanofluid flow over a stretching plate with entropy generation. They 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. They 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. The current analysis is dedicated to inspecting the entropy production on bioconvection for the flow of Sutterby fluid because a heated convective sheet is examined. Thermal radiation, heat sink/source, Brownian motion, and thermophoresis are also considered. The main aim of this study is that the non-Newtonian Sutterby fluid is selected to examine the rheological properties of shear-thinning/thickening with the effect of thermal radiation and microorganism. The similarity conversion is used to alter the nonlinear PDE into ODE to solve modeled equations. The 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. The 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.


### 1.1. Mathematical Formulation

The incompressible, laminar, time-dependent flow of Sutterby nanomaterial past stretching sheet subject to the convective condition and entropy production is examined. The stretching sheet velocity isUwx,t=ax/1−γt. The 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). The mathematical formulation for Sutterby nanofluid in a steady flow is taken in [25], and it converted to be unsteady in the following flow equations.Figure 1
Physical model.(1)∂u^∂x+∂v^∂y=0,(2)∂u^∂t+u^∂u^∂x+v^∂u^∂y=μ0ρ∂2u^∂y2+mB22∂u^∂y2∂2u^∂y2−σρB02u^,(3)∂T^∂t+u^∂T^∂x+v∂T^∂y=kρCp∂2T^∂y2+τDB∂C^∂y∂T^∂y+DTT^∞∂T^∂y2−1ρCp∂qr∂y+Q0ρCpT^−T^∞,(4)∂C^∂t+u^∂C^∂x+v^∂C^∂y=DB∂2c^∂y2+DTT^∞∂2T^∂y2−κ0C^−C^∞,(5)∂χ^∂t+u^∂χ^∂x+v^∂χ^∂y+bWcC^w−C^∞∂∂yχ^∂C^∂y=Dm∂2χ^∂y2.With boundary condition,(6)u^=Uwx,t,v^=0,−k∂T^∂y=htT^w−T^∞,C^=C^w,χ^=χ^wy=0,u^⟶0,T^⟶T^∞,C^⟶C^∞,χ^⟶χ^∞aty⟶∞.In equation (2), m shows three conditions. If m<0, then it shows pseudoplastic or represents fluid with decreasing viscosity; if m>0, it demonstrates the dilatant or fluid with increasing viscosity; if m=0, it reflects the Newtonian fluid.The Rosseland approximation is crucial in thermal radiative heat flow. The Rosseland approximation requires an optically thick medium and radiation that travels only a short distance before being scattered or absorbed. As a result, a simplified model for the Radiative Transfer Equation (RTE) based on the Rosseland approximation is provided as follows:(7)qr=4σ13k∗∂T^4∂y=−16σ∗3k∗T^3∂T^4∂y,where T^4 can be expanded as follows:(8)T^4≅4T^∞3T^−3T^∞4.Replacing equation (7) into equation (8),(9)qr=16σ∗T∞33k∗∂T^∂y.Similarity expressions are(10)u^=ax1−γtf′η,v^=−av1−γtfη,η=av1−γty,θ=T^−T^∞T^w−T^∞,ϕ=C^−C^∞C^w−C^∞,N=χ^−χ^∞χ^w−χ^∞,,(11)f‴+ff″−f′2−Af′+η2f″+12mδRef″2f‴+Mf′=0,(12)θ″1+43Rd−Pr12ηAθ′+PrNtθ′2+PrNbθ′ϕ′+Prεθ=0,(13)ϕ″+Scfϕ′−12ηScAϕ′+NbNtθ″−Scκϕ=0,(14)N″−Peϕ″N+ϖ+ϕ′N′−Lb12ηAN′+fN′=0.The converted boundary conditions are(15)f0=0,f′0=1,θ′0=−Bi+Biθ0,ϕ0=1f′∞=0,θ∞=0,ϕ∞=0,N∞=0.The Reynolds numberRe=ax2/v, Deborah number δ=B2a2/v, magnetic parameter M=σB02/ρa, unsteady parameter A=γ/a, heat source parameter ε, radiation parameter Rd=4σ∗T∞3/kk∗, Prandtl number Pr=v/α, Brownian motion parameter Nb=τDBC^w−C^∞/v, thermophoresis parameter Nt=τDTT^w−T^∞/T⌢∞v, Schmidt number Sc=v/DB, Biot number Bi=d/kv/a, microorganisms concentration ϖ=χ^∞/χ^m−χ^∞, Bioconvection Peclet number Pe=bWc/Dm. Bioconvection Lewis number Lb=v/Dm are shown.


## 1.1. Mathematical Formulation

The incompressible, laminar, time-dependent flow of Sutterby nanomaterial past stretching sheet subject to the convective condition and entropy production is examined. The stretching sheet velocity isUwx,t=ax/1−γt. The 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). The mathematical formulation for Sutterby nanofluid in a steady flow is taken in [25], and it converted to be unsteady in the following flow equations.Figure 1
Physical model.(1)∂u^∂x+∂v^∂y=0,(2)∂u^∂t+u^∂u^∂x+v^∂u^∂y=μ0ρ∂2u^∂y2+mB22∂u^∂y2∂2u^∂y2−σρB02u^,(3)∂T^∂t+u^∂T^∂x+v∂T^∂y=kρCp∂2T^∂y2+τDB∂C^∂y∂T^∂y+DTT^∞∂T^∂y2−1ρCp∂qr∂y+Q0ρCpT^−T^∞,(4)∂C^∂t+u^∂C^∂x+v^∂C^∂y=DB∂2c^∂y2+DTT^∞∂2T^∂y2−κ0C^−C^∞,(5)∂χ^∂t+u^∂χ^∂x+v^∂χ^∂y+bWcC^w−C^∞∂∂yχ^∂C^∂y=Dm∂2χ^∂y2.With boundary condition,(6)u^=Uwx,t,v^=0,−k∂T^∂y=htT^w−T^∞,C^=C^w,χ^=χ^wy=0,u^⟶0,T^⟶T^∞,C^⟶C^∞,χ^⟶χ^∞aty⟶∞.In equation (2), m shows three conditions. If m<0, then it shows pseudoplastic or represents fluid with decreasing viscosity; if m>0, it demonstrates the dilatant or fluid with increasing viscosity; if m=0, it reflects the Newtonian fluid.The Rosseland approximation is crucial in thermal radiative heat flow. The Rosseland approximation requires an optically thick medium and radiation that travels only a short distance before being scattered or absorbed. As a result, a simplified model for the Radiative Transfer Equation (RTE) based on the Rosseland approximation is provided as follows:(7)qr=4σ13k∗∂T^4∂y=−16σ∗3k∗T^3∂T^4∂y,where T^4 can be expanded as follows:(8)T^4≅4T^∞3T^−3T^∞4.Replacing equation (7) into equation (8),(9)qr=16σ∗T∞33k∗∂T^∂y.Similarity expressions are(10)u^=ax1−γtf′η,v^=−av1−γtfη,η=av1−γty,θ=T^−T^∞T^w−T^∞,ϕ=C^−C^∞C^w−C^∞,N=χ^−χ^∞χ^w−χ^∞,,(11)f‴+ff″−f′2−Af′+η2f″+12mδRef″2f‴+Mf′=0,(12)θ″1+43Rd−Pr12ηAθ′+PrNtθ′2+PrNbθ′ϕ′+Prεθ=0,(13)ϕ″+Scfϕ′−12ηScAϕ′+NbNtθ″−Scκϕ=0,(14)N″−Peϕ″N+ϖ+ϕ′N′−Lb12ηAN′+fN′=0.The converted boundary conditions are(15)f0=0,f′0=1,θ′0=−Bi+Biθ0,ϕ0=1f′∞=0,θ∞=0,ϕ∞=0,N∞=0.The Reynolds numberRe=ax2/v, Deborah number δ=B2a2/v, magnetic parameter M=σB02/ρa, unsteady parameter A=γ/a, heat source parameter ε, radiation parameter Rd=4σ∗T∞3/kk∗, Prandtl number Pr=v/α, Brownian motion parameter Nb=τDBC^w−C^∞/v, thermophoresis parameter Nt=τDTT^w−T^∞/T⌢∞v, Schmidt number Sc=v/DB, Biot number Bi=d/kv/a, microorganisms concentration ϖ=χ^∞/χ^m−χ^∞, Bioconvection Peclet number Pe=bWc/Dm. Bioconvection Lewis number Lb=v/Dm are shown.


## 2. Engineering Interest Quantities

(16)Cfx=τwρuw2,Nux=xqwkT^w−T^∞,Shx=xqmDbC^w−C^∞,Nn=xqnDmχ^w−χ^∞.Shear wallτw, heat qw, mass qm, and motile density qn have been displayed as(17)τw=−μ0∂u^∂y+m6Reδ∂u^∂y3y=0,qw=−k∂T^∂y|,qm=−k∂C^∂y|y=0,qn=−Dm∂χ^∂y|y=0.In the use of (10) and (17),(18)CfRex0.5=f″0+m6δRef″03,NuxRex1/2=−θ′01+43Rd,ShxRex1/2=−ϕ′0,NnRex1/2=−N′0.


## 3. Entropy Generation Analysis

The entropy production rate (SG) for the Sutterby fluid [25] under magnetic field, thermal radiation, and diffusion effect can be written as(19)SG=kT^0∂T^∂x2+∂T^∂y2++16σ∗T^∞33kk∗∂T^∂y2+μT^∞∂u^∂y21+mReδ6∂u^∂y2+RDmC^∞∂C^∂y2+RDmT^∞∂T∂y∂C^∂y+RDmχ^∞∂χ^∂y2+RDmT^∞∂χ^∂y∂T^∂y+σB02T^∞u2.The dimensionless entropy generation number is expressed as(20)Ns=1+Rdθ′2+ReBrΠ1+mReδ3f″2f″2+ReΓΠ2ϕ′2+ReΓΠϕ′θ′+ReξΠ2N′2+ReξΠN′θ′+BrΠMf′2.The Bejan number (Be) is described as(21)Be=1+Rdθ′2+ReΓ/Π2ϕ′2+ReΓ/Πϕ′θ′+Reξ/Π2N′2+Reξ/ΠN′θ′1+Rdθ′2+ReBr/Π1+mReδ/3f″2f″2+ReΓ/Π2ϕ′2+ReΓ/Πϕ′θ′+Reξ/Π2N′2+Reξ/ΠN′θ′+Br/ΠMf′2.


### 3.1. 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.(22)f=ϒ1,f′=ϒ2,f″=ϒ3,f′″=ϒ3′θ=ϒ4,θ′=ϒ5,θ″=ϒ5′ϕ=ϒ6,ϕ′=ϒ7,ϕ″=ϒ7′N=ϒ8,N′=ϒ9,N″=ϒ9′,(23)ϒ3′=−ϒ1ϒ3+ϒ22−Mϒ21+1/2mδReϒ32,(24)ϒ5′=−Prϒ1ϒ5+αϒ1ϒ2ϒ5−PrNbϒ5ϒ7+Ntϒ521+Rd−αϒ2,(25)ϒ7′=−ScSrϒ5′−Scϒ1ϒ5+ϒ5′NtNb−Scκϒ6,(26)ϒ9′=Peϒ7′ϒ+ϖ−ϒ7ϒ9+Lbϒ9.The converted boundary condition is(27)ϒ10=0,ϒ20=1,ϒ50=−γ1−ϒ40,ϒ60=1,ϒ80=1ϒ2∞=0,ϒ4∞=0,ϒ6∞=0,ϒ8∞=0.By using (27) with boundary conditions, the results are using a finite value for ηmax .(28)f′ηmax⟶0,θηmax⟶0,ϕηmax⟶0,Nηmax⟶0.The step size is measuredΔη=0.001 and 10−6 is a convergent criterion to repeat and attain the numeric solution.


## 3.1. 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.(22)f=ϒ1,f′=ϒ2,f″=ϒ3,f′″=ϒ3′θ=ϒ4,θ′=ϒ5,θ″=ϒ5′ϕ=ϒ6,ϕ′=ϒ7,ϕ″=ϒ7′N=ϒ8,N′=ϒ9,N″=ϒ9′,(23)ϒ3′=−ϒ1ϒ3+ϒ22−Mϒ21+1/2mδReϒ32,(24)ϒ5′=−Prϒ1ϒ5+αϒ1ϒ2ϒ5−PrNbϒ5ϒ7+Ntϒ521+Rd−αϒ2,(25)ϒ7′=−ScSrϒ5′−Scϒ1ϒ5+ϒ5′NtNb−Scκϒ6,(26)ϒ9′=Peϒ7′ϒ+ϖ−ϒ7ϒ9+Lbϒ9.The converted boundary condition is(27)ϒ10=0,ϒ20=1,ϒ50=−γ1−ϒ40,ϒ60=1,ϒ80=1ϒ2∞=0,ϒ4∞=0,ϒ6∞=0,ϒ8∞=0.By using (27) with boundary conditions, the results are using a finite value for ηmax .(28)f′ηmax⟶0,θηmax⟶0,ϕηmax⟶0,Nηmax⟶0.The step size is measuredΔη=0.001 and 10−6 is a convergent criterion to repeat and attain the numeric solution.


## 4. Result and Discussion

The current study involves the governing parameters on velocityf′η, temperature θη, concentration θη , density of motile microorganisms Nη , entropy generation Nsη, and Bejan number Beη. Table 1 presents the code validation of −θ0 for different Pr values. This validation provides that the result of our computation is an optimum one.Table 1
Comparing of−θ0 for the diverse value of the Prandtl number Pr.
PrGoyal and Bhargava [33] (FEM)G. B. Chandra Mouli et al. [34]Current solution0.70.45390.453916160.45605115144420.91130.911357680.91132118233571.89541.895403261.895381877516203.35393.353900353.353886926545Figures2(a)–2(d) delineate the different values of the Deborah number δ on velocity distribution f′η, temperature field θη, entropy generation Nsη, and Bejan number Beη. Figure 2(a) presents the larger value of the Deborah number δ results, the fluid velocity reduces for pseudoplastic flow m<0, and fluid velocity enhances as the Deborah number δ upsurges for the dilatant flow m<0. The Deborah number fluidity for scenarios when m>0 and m<0 is represented. In both circumstances, increasing the value of the Deborah number results in the opposite behavior of the velocity fields. The flow of fluid accelerates as the Deborah number increases but decreases as the Deborah number increases in the form of the pseudoplastic fluid. Increases in the values of the Deborah number play an important part in refining elastic impacts, and as a result, dilatant fluids increase fluid flow; nevertheless, the pseudoplastic fluid exhibits quite different behavior. An increment in the value of the Deborah number as thermal layer thickness depreciates for pseudoplastic flow m<0 in Figure 2(b). Also, the temperature field θη reduces with the increasing value of the Deborah number δ in the shear thickening fluid m<0. Figures 2(c) and 2(d) interestingly show that entropy production rate Nsη enhances for dilatant m<0 and reduces for pseudoplastic m<0 through the escalation in the Deborah parameter δ while the Bejan number Beη is an escalating function of the Deborah number δ for both pseudoplastic flow m<0 but reduces dilatant flow m<0. The plot of various values of power-law index parameter (m) is shown in Figures 3(a) and 3(b). In Figure 3(a), the velocity profile f′η escalates as (m) rises. Moreover, the velocity profile f′η is interpreted through (m=0, m<0, m>0). The Newtonian fluid m=0, pseudoplastics m<0, and dilatants fluid m<0 are analyzed. The Newtonian, shear-thinning, and shear-thickening fluid is described in Figure 3(b). The temperature profile θη decays with a larger value of m. Figures 4(a)–4(d) display the behavior of unsteady variable A on f′η , θη, Nsη, and Beη. For both cases m>0andm<0, Figures 4(a) and 4(b) represent that both velocity and temperature profiles decay due to greater value of A. Actually, the unsteady variable reduces the flow strength of velocity profile f′η and temperature field θη. The reason behind this is that pseudoplastic and dilatant fluid was not affected on both profiles. Similar performance for Nsη and Beη was noted in Figures 4(c) and 4(d).Figure 2
Impact ofδ on (a) f′η, (b) θη, (c) Nsη, and (d) Beη.
(a)(b)(c)(d)Figure 3
Impact of (m) on (a)f′η and (b) θη.
(a)(b)Figure 4
Impact of A on (a)f′η, (b) θη, (c) Nsη, and (d) Beη.
(a)(b)(c)(d)Figures5(a)–5(d) depict the feature of Re for the velocity distribution f′η, temperature field θη, entropy generation Nsη, and Bejan number Beη. From Figure 5(a), the arising value of the Reynolds number Re diminishes the velocity profile f′η for viscous force and velocity field augment due to the larger value of the Reynolds number for dilatant fluid. 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. The plots of the magnetic variable on the velocity and temperature profiles are exposed in Figure 6(a)–6(d). The 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 of magnetic number for both cases pseudoplastic and dilatants whereas its opposite behavior is seen in Figure 6(d).Figure 5
Impact of (Re) on (a)f′η, (b) θη, (c) Nsη, and (d) Beη.
(a)(b)(c)(d)Figure 6
Impact of M on (a)f′η, (b) θη, (c) Nsη, and (d) Beη.
(a)(b)(c)(d)The view ofNb on temperature field θη and concentration profile ϕη is disseminated in Figures 7(a) and 7(b). The temperature profile θη 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 fluid temperature. However, deceleration in the concentration field is observed in Figure 7(b). Figures 8(a) and 8(b) display thermophoretic estimation Nt on temperature θη and concentration profiles θη. Larger values of Nt for both θη and ϕη have been intensified. Nanoparticles have dragged from hotter to cold surfaces. That is the reason of the increase in thermal and concentration profiles. The plots for the Prandtl Pr and Schmidt number Sc on temperature ϕη and concentration profiles ϕη are carried out in Figures 9(a) and 9(b). It can be shown from these figures that enlargement in the Pr causes a dazzling depreciation in the fluid temperature. The concentration profile reduces with a larger value of Sc. Physically, the Schmidt number Sc is dependent on the proportion of momentum to mass diffusivity. By increasing the value of Sc, momentum diffusivity 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 profiles. The Biot number means the ratio of convection over conduction inside the boundary at the surface. The 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 profile. The features of the Bioconvection Lewis number Lb and Peclet number Pe on motile density microorganism Nη were seen in Figures 11(a) and 11(b). For the mounting value of Lb as the diffusivity microorganism upsurges, so, the density of motile microorganism has increased. Figure 11(b) shows the influence of the Peclet number on microorganism density that has been assumed. Enlarging the Peclet number climbs the movement of fluid particles. Due to this, microorganism density enhances as their concentration diminishes.Figure 7
Impact of Nb on (a)θη and (b) ϕη.
(a)(b)Figure 8
Impact ofNt on (a) θη and (b) ϕη.
(a)(b)Figure 9
Impact ofPr on (a) θη and (b) Sc onϕη.
(a)(b)Figure 10
Impact of (a) Bi onθη and (b) κ on ϕη.
(a)(b)Figure 11
Impact of (a) Lb and (b) Pe onNη.
(a)(b)The impact of Re and δ on skin friction coefficient was presented in Figure12 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 values and the opposite trend was obtained in pseudo-plastic case. Figure 13 illustrates the variations of Nu for different 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. The 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 figure that the motile density escalates when escalating the vales of Pe and its downturns for enriching the Lb quantity.Figure 12
Impact of Re andδ versus skin friction.Figure 13
Impact of Rd and Nb versus the Nusselt number.Figure 14
Impact ofκ and Sc versus the Sherwood number.Figure 15
Impact of Lb and Pe versus the Motile density.Numerical estimation of drag friction for Re, δ andM 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 improving the Pr, κ, Nb and Nt values and its downfalls when larger quantity of Sc values. The 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. The 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), 16(b) and 17(a), 17(b). Figures 18 explains the skin friction coefficient for different values of δ for dilatant and pseudo‐plastic cases. It is found that pseudo‐plastic has a more pronounced than the dilatant case. The 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 pseudo‐plastic case than the dilatant case.Table 2
Numerical simulation values ofRex1/2Cfx with numerous variables Re, M,δ.
ReδMRex1/2Cfxm = −0.5m = 0.50.10.20.20.88960.88920.30.87680.91490.50.86380.92740.40.83060.95800.60.79650.98780.70.77901.00240.40.70350.85120.50.65780.84280.60.69430.8372Table 3
Numerical values ofRex−1/2Nux with numerous variables of Rd,Nb,Nt,Pr,ε.
RdNbNtPrRex−1/2Nuxm = −0.5m = 0.50.10.30.70.70.05940.05980.20.06120.06090.30.06680.06710.40.05380.05340.50.03930.03970.60.02440.02480.50.05810.05840.40.06990.07020.30.07930.07960.70. 08240. 08251.10.09840.0987Table 4
Numerical values ofRex−1/2Shx with numerous variables of Nb,Nt,Pr,Sc.
PrScκNbNtRex−1/2Shx0.52.30.40.10.84290.60.32180.71.06240.81.90740.91.16711.01.07412.41.10922.51.14252.61.17700.31.17790.41.17850.51.18100.21.41610.31.48140.41.4918Table 5
Numerical values ofRex−1/2Nhx with numerous variables of Nb,Pe,Lb,ϖ.
LbPeϖNbRex−1/2Nhx0.90.90.90.72.06581.2.04281.12.02061.02.30021.2.43511.12.59811.12.97031.23.16411.33.56370.33.10720.43.86280.53.9162Table 6
List of symbols.
ExpressionNameUnitReLocal Reynolds numberDimensionlesstTimeSecδDeborah numberDimensionlessMMagnetic parameterDimensionlessmPower index numberDimensionlessAUnsteady parameterDimensionlessNbBrownian motion parameterDimensionlessNtThermophoresis parameterDimensionlessPrPrandtl numberDimensionlessRdRadiation parameterDimensionlessScSchmidt numberDimensionlessLbBioconvection Lewis numberDimensionlessPeBioconvection Peclet numberDimensionlessεHeat source/sink parameterDimensionlessκChemical reaction parameterDimensionlessCfSkin friction coefficientDimensionlessNuxNusselt numberDimensionlessShxSherwood numberDimensionlessNhxMicroorganism density numberDimensionlessfDimensionless velocity profileDimensionlessθDimensionless temperature profileDimensionlessϕDimensionless concentration profileDimensionlessNDimensionless microorganism profileDimensionlessϖMicroorganism concentration difference parameterDimensionlessηSimilarity variableDimensionlessu^,v^Components of velocity alongx,y directionms−1τRatio of the effective heat capacityDimensionlessυKinematic viscositym2s−1aStretching rates−1ρDensitykgm−1CpSpecific heatJkg−1K−1DBBrownian diffusion coefficientm2s−1DTThermophoretic diffusion coefficientm2s−1DmMicroorganism’s diffusion coefficientm2s−1BoConstant magnetic fieldkgs=2A−1C^Concentrationkgm−3C^∞Ambient concentrationkgm−3C^wSurface concentration of nanoparticleskgm−3kThermal conductivitymkgs−3K−1T^TemperatureKT^∞Ambient temperatureKT^wWall temperatureKhtCoefficient of heat transferWm−2K−1χ^Microorganism’s profileDimensionlessχ^wSurface concentration of microorganismskgm−3χ^∞Ambient concentration of microorganismskgm−3WcMaximum cell swimming speedms−1x,yCartesian coordinatesmσElectrical conductivityS3m2kg−1uwVelocity of the sheetms−1SGLocal volumetric entropy generation rateWm−3K−1NsEntropy generationDimensionlessBeBejan numberDimensionlessFigure 16
Streamlines for (a) Unsteady flow. (b) Steady flow.
(a)(b)Figure 17
Streamline for (a)m = -0.3 and (b) m = 0.3.
(a)(b)Figure 18
Bar graph for skin friction.Figure 19
Bar graph for Nusselt number.


## 5. 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. The 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. This 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.(i)
Velocity profile is reducing while temperature, entropy generation, and Bejan number accelerated due to the larger value of the Deborah number form<0.(ii)
The velocity of fluid mounted whereas temperature, entropy generation, and Bejan number declined due to the larger value of the Deborah number form>0.(iii)
A higher value inPr causes the reduction in the temperature profile.(iv)
Increases in the Deborah number for heat flow result in a decrease in the temperature profile while increasing the heat source/sink parameter.(v)
Skin friction coefficient augments form<0,m>0 but reduced the Nusselt number.



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