The present work is focused on behavioral characteristics of gyrotactic microorganisms to describe their role in heat and mass transfer in the presence of magnetohydrodynamic (MHD) forces in Powell-Eyring nanofluids. Implications concerning stretching sheet with respect to velocity, temperature, nanoparticle concentration and motile microorganism density were explored to highlight influential parameters. Aim of utilizing microorganisms was primarily to stabilize the nanoparticle suspension due to bioconvection generated by the combined effects of buoyancy forces and magnetic field. Influence of Newtonian heating was also analyzed by taking into account thermophoretic mechanism and Brownian motion effects to insinuate series solutions mediated by homotopy analysis method (HAM). Mathematical model captured the boundary layer regime that explicitly involved contemporary non linear partial differential equations converted into the ordinary differential equations. To depict nanofluid flow characteristics, pertinent parameters namely bioconvection Lewis number *Lb*, traditional Lewis number *Le*, bioconvection Péclet number *Pe*, buoyancy ratio parameter *Nr*, bioconvection Rayleigh number *Rb*, thermophoresis parameter *Nt*, Hartmann number *M*, Grashof number *Gr*, and Eckert number *Ec* were computed and analyzed. Results revealed evidence of hydromagnetic bioconvection for microorganism which was represented by graphs and tables. Our findings further show a significant effect of Newtonian heating over a stretching plate by examining the coefficient values of skin friction, local Nusselt number and the local density number. Comparison was made between Newtonian fluid and Powell-Eyring fluid on velocity field and temperature field. Results are compared of with contemporary studies and our findings are found in excellent agreement with these studies.

## I. INTRODUCTION

Magnetohydrodynamics deals with the fluid’s dynamic activities that if manifests towards different surfaces during flow. Fluids either Newtonian or non-Newtonian not only show electrically conducting characteristics but also exhibit magnetic field effects during flow. At present, diverse applications of MHD flow with combined heat and mass transfer effects have received attention in geothermal field, mechanical, biological and chemical fields including power generation and aerodynamics. Several practical studies have examined MHD flows recently such as Acharya et al., (2017) assessed the effect of curvature parameter improvisation on temperature by studying boundary layer flow past a stretching cylinder which was continuously moving and involved the partial slip mechanism and the presence of a non-uniform heat source within the flow field. Nanoparticles of copper, silver, titanium oxide and aluminum oxide were suspended in water by Ali et al., (2016) who used it as a Brinkman type nanofluid flowing over a vertically located plate embedded in a porous medium to investigate the unsteady MHD flow of the nanofluid. Convective heat and mass transfers in the three dimensional MHD slip flow of water-graphene and water-magnetite nanofluids past a stretching surface of variable thickness under the conditions of thermophoresis and Brownian motion by Babu and Sandeep (2016). Their findings indicated that heat and mass transfer rates in water-graphene nanofluids were higher as compared to the water-magnetite nanofluid. Hayat et al., (2016) studied the magnetohydrodynamic flow of a Jeffrey fluid past a surface having the property of being nonlinearly stretched under the conditions of Newtonian and Joule heating, discovering that the skin friction coefficient and temperature were inversely related to the nonlinear parameter of the stretched sheet. Hayat et al., (2017a) also examined the MHD flow of an electrically conducting Powell-Eyring nanofluid past a non-linear slendering (possessing variable thickness) stretching sheet, with a magnetic field being applied transverse to the sheet surface. Their findings suggested that greater thickness of the stretching sheet walls resulted in the reduction of velocity distribution, and higher values of thermophoresis enhanced both temperature and concentration profiles.

Newtonian heating is among one of the four convection processes that operates in nature for the purpose of heat transfer. The distinct feature of this type of convection is that it does not depend on local surface temperature to act as heat sink (Chaudhary and Jain, 2006) and hence Newtonian heating conditions enhances the heat transfer application to a relatively wider range. Merkin (1994) practically demonstrated the phenomenon of free-convection boundary layer for the first time followed by overwhelming number of studies where Newtonian heating applications were investigated (Lesnic et al., 2000; Lesnic et al., 2004; Salleh et al., 2011). Initial studies pertaining to convective heat transfer revealed that Newtonian conditions were first simply analyzed for vertical flat plate immersed in a viscous fluid (Merkin (1994) later on complex conditions where vertical and horizontal surfaces embedded in a porous medium were also considered (Lesnic et al., 2004).

Newtonian heating effects have amazed the commercial outputs owing to which industrial use of stretching surfaces has become multidimensional. Ranging from plastic polymers, extrusion of metals and polymer sheets and metal spinning, use of non-Newtonian fluids holistically covers different chemical engineering and biological processes. Although the study to characteristics of flow over stretching surface was pioneered by Crane (Crane 1970), followed by several significant contributions that led to the develop variety of rheological properties (Quemada 1978; Chhabra and Richardson, 2011), a complexity is generated while examining the relationship between non-Newtonian fluids in the presence of shear stress and rate of strains.

A growing recognition was observed for fluid flows in high-tech industry, aerodynamics, and engineering industries. With the advent of Powell-Eyring fluids it is now possible to simulate various aspects heat transfer kinetics of liquids, which was previously limited to describing only empirical relations of non Newtonian fluids. It is naïve to suppose that heat transfer models used to describe industrial fluids such as oils, paper pulps, polymer solution, and other non Newtonian fluids confront boundary layer conditions. Over the period of years, there has been rational progress in the investigations of Powell-Eyring fluids. Since the inception of this concept in 1944 (Powell and Eyring 1944), the intensity of drawing advantages of Powell-Eyring fluids is observed to have an overarching effect ranging from pressure distribution (Yürüsoy, 2003) to overcome slider bearing problem (Islam et al., 2009), and from peristaltic transfer of heat and mass (Shaaban and Abou–Zeid, 2013) to proposing solutions for flow effect encountering magnetic field effects (Akbar et al., 2015) and from moving surface (Hayat et al., 2012) to stretching surface (Hayat et al., 2017b). The impact of thermo diffusion and thermal radiation on Williamson nanofluid moving on a porous sheet was investigated by Bhatti and Rashidi (2016a). These authors have also demonstrated numerical simulation for nanofluids in megnetohydrodynamics conditions using point flow over stretching surface (Bhatti and Rashidi, 2016b) as well as on porous sheet with emphasis on heat transfer (Bhatti et al. 2016a). For simulation of entropy generation, Bhatti et al., (2016b) conducted MHD based a numerical study using Carreau nanofluids. In addition, permeable stretching surface is used to examine the Powell-Eyring nanofluids flow for entropy generation (Bhatti et al., 2016c). How fluids operate in versatile surfaces in relation to low and high shear rates especially in changing viscosity condition such as in the case of stretching cylinder, is now correctly modeled with the help of Powell-Eyring fluids (Freidoonimehr et al., 2017; Mackley et al., 2017). These investigations have utilized the non-Newtonian Eyring-Powell fluid diligence in carrying out transformations leading from partial differential equations into ordinary differential equations to propose numerical solutions by taking into account necessary thermal conditions beside optimizing viscosity, boundary layer and convective flow attributes.

The microorganisms such as microalgae and bacteria have high density than water hence on the average, they swim in upward direction against the gravity. Gathering of microorganisms make the top layer of suspension too dense than the lower layer causing unstable density distribution. As a result, convective instability takes place and may generate convection patterns. These spontaneous and random movement patters of microorganisms in the suspension are described as bioconvection. Geotactic stimulus (Winet and Jahn, 1974; Kuznetsov and Jiang 2003), chemotatic (Yanaoka et al., 2009) and gyrotactic (Ghorai and Hill 2000; Sharma and Kumar, 2012). The reason of investigation of swimming pattern of microorganisms in water or in other liquids denser than water is their involvement ecological, commercial and industrial products such as ecological fuels, ethanol, fertilizers, fuel cells. Climate change and global warming scenarios associated with carbon dioxide emission in atmosphere requires an efficient mechanism to sequester carbon from environment. Microalgae are successfully used to extract CO_{2} from exhausts of industries through a flu gas combustion system (Zhao et al., 2015; Cheah et al., 2015).

Bioconvection flow of microorganisms due to gyrotactic stimuli and water based nanofluids involving heat and mass transfer (Siddiqa et al., 2016). The heat and mass transfer in man-made systems and MHD flows both in viscous and water-based porous medium has wide ranging industrial applications. To understand the behavior of nanofluids, the combined effects of pertinent parameters such as Brownian motion, thermophoresis, buoyancy driven bioconvection (Uddin et al., 2016) were tackled using nonlinear partial differential equations to ordinary differential equations by employing various techniques such as Runge-Kutta order method (Makinde and Animasaun, 2016). Although in real-world solutions to solve these problems are very important, the extensive modeling and mathematical simulations has contributed in nanofluid bioconvection as unprecedented achievement. Various aspects of convective flow was in focus such as oxytactic (Hillesdon and Pedley, 1996), non-linear (Metcalfe and Pedley, 2001), numerical modeling through porous medium (Becker and Kuznetsov 2004). Recently investigations focused on heat transfer that lead to bioconvection have extensively used gyrotactic microorganisms. Here viscosity of fluid has been examined to optimize the flow such as carried out by Bhatti and Zeeshan (2016) for peristaltic flow of two-phase flow using Jeffrey fluid model. In addition, chemical reaction effects were mathematically modeled for MHD nanofluid flow having gyrotactic microorganisms to explain the influence of thermal radiations (Bhatti et al., 2016d). Gyrotactic microorganism have also illustrated peristaltically induced motions that enabled researchers to present a better picture of simultaneous effects of coagulation and variable magnetic field on flow characteristics that involve Jeffrey nanofluids (Bhatti et al., 2017).

A comprehensive review of literature has revealed that there exist scanty information about bioconvection flow of Powell-Eyring nanofluid and Newtonian heating effects. Hence our aim is to examine the impact of magnetohydrodynamics bioconvective flow of Powell Eyring nanofluid towards a stretching plate in the presence of Newtonian heating. Using suitable variables, the governing partial differential equations are transformed to ordinary differential equations, which are solved along with the corresponding boundary conditions using versatile application of homotopy analysis method (HAM) (Abbasbandy et al., 2017; Hayat et al., 2017c; Shukla et al., 2017). As this problem has not been considered before so that the results are new and original.

### A. Model formulation

The bioconvective magnetohydrodynamic flow of Powell-Eyring nanofluid over a stretching plate in the presence of gyrotactic microorganisms is considered (see Fig. 1). The flow is characterized by the presence of a uniform transverse magnetic field $B\xb0$. Furthermore, without voltage application, the Reynolds number is also small. Hence the induced magnetic field has insignificant effect. The condition generates mixed bioconvection which was taken into account along with Newtonian heating effects. We further assume that nanoparticles do not have any influence on both the swimming velocity and swimming direction of microorganisms. It is taken for granted that stability of resulting nanofluid suspension is maintained and it is dilute (nanoparticles concentration less than 1%) to uphold the gyrotactic microorganisms bioconvection in optimum manner. Since high viscosity of base fluid pose negative influence on bioconvection, it is imperative to dilute the suspension containing nanoparticles (Kuznetsov, 2012; Mosayebidorcheh et al., 2017). Following the approach given by Mutuku and Makinde (2014) we used the above mentioned boundary layer approximations i.e. continuity, momentum, energy, nanoparticle concentration, and conservation for gyrotactic microorganism and hence the equations can be written as:

subjected boundary conditions are

Where $u$ and $v$ represent velocity components in *x* and *y* direction respectively. Stretching velocity is given by $Uw$, fluid and microorganism density are $\rho f$ and $\rho m$, the pressure fluid is represented as *p*, kinematic density is $\nu f$, whereas material parameters are given by $\beta f$ and *c*, the electrical conductivity is $\sigma ,$ concentration of fluid is *C*, temperature is *T*, and the concentration of microorganisms in the fluid is given by *n*. The nanoparticle volume fraction, temperature and density of motile microorganisms are represented by $C\u221e,$$T\u221e$, and $n\u221e$ which exist far away from the stretching plate. Here the gravity vector is shown by *g*, while the fluid volume expansion coefficient is represented by $\beta $, the average volume of microorganisms is given by $\gamma $, thermal diffusivity of base fluid is symbolized by $\alpha $, the ratio of effective heat capacitance of the nanoparticles is shown as $\tau =(\rho C)p(\rho C)f$, while to represent Brownian diffusion coefficient, thermophoretic diffusion coefficient and diffusivity of microorganisms *D*_{B}, *D*_{T}, *D*_{m} are used respectively, where as the dynamic viscosity is symbolized as $\mu $, and the chemotaxis constant and maximum cell swimming speed are given by *b* and *W*_{c} respectively.

Upon application of following transformations

where prime exhibits differentiate on with respect to $\eta $, and material fluid parameters are represented by $\mathit{\epsilon}$ and $\delta $, *Pr* is the Prandtl number, $\lambda $ is the Grashof number, *M* is Hartmann number, *Nr* is the Buoyancy ratio parameter, *Rb* is the bioconvection Rayleigh number, *Nb* is the Brownian motion parameter, *Nt* is the thermophoresis parameter, *Le* is the traditional Lewis number, *Ec* is the Eckert number, *Lb* is the bioconvection Lewis number, $\Omega $ is the microorganism concentration difference parameter, *Pe* is the bioconvection Peclet number and $\gamma $ denotes Newtonian heating parameter. Dimensionless skin friction, local Nusselt number and local density of motile microorganism become

The physical quantities employed in this study are the skin friction *C*_{f}, Nusselt numbers *Nu*, and density number of the motile microorganisms *Nn* defined as

where shear stress, surface heat flux and motile surface microorganisms flux are $\tau w$, $qw$ and *q*_{n} which are defined as

in which $Rex=U\xb0x\upsilon $ is the local Reynold number.

## II. HOMOTOPIC SOLUTIONS

It is not possible to derive or present the exact solution for equations (1) to (11), hence we take into account the homotopy analysis method (HAM) to derive solution. Since HAM provides robust way to smooth the application of differential equations and to present solutions for governing equations related to nanofluids based heat transfer (Khodaparast et al., 2017; Yang and Liao, 2017). The initial guesses and corresponding linear operators for the dimensionless momentum, energy, concentration and motile microorganisms equations can be decided in the following forms:

with

where *A*_{i}(*i* = 1 − 9) are the arbitrary constants.

### A. Zeroth-order problem

The zeroth order problems are presented as

Here $Nf,$$N\mathit{\theta},$$N\varphi $ and $N\xi $ are corresponding non-linear operators which are given below:

here $\u210ff$, $\u210f\mathit{\theta}$, $\u210f\varphi $ and $\u210f\xi $ are non-zero auxiliary parameters whereas $q\u22080,1$ denotes the embedding parameter.

### B. *m*th-order deformation equations

The *m*th-order deformation problems may be represented as under:

We can express *q* = 0 and *q* = 1 in the following form

By virtue of variations of *q* from 0 to 1, then $f^(\eta ,q)$, $\mathit{\theta}^\eta ,q$, $\varphi ^(\eta ,q)$ and $\xi ^(\eta ,q)$, reveals from the initial solutions $f0(\eta )$, $\mathit{\theta}0\eta $, $\varphi 0(\eta )$ and $\xi 0(\eta )$ to the final solutions $f(\eta ),$$\mathit{\theta}\eta $, $\varphi (\eta )$ and $\xi (\eta )$ respectively.

In order to converge series solutions, the values of auxiliary parameters are selected accordingly. The general solutions $fm*,\mathit{\theta}m*,\varphi m*,\chi m*$ are specified as under:

in which $Aii=1,2,3,\u2026,9$ are the involved constants.

## III. CONVERGENCE EXAMINATION

Here it is noteworthy that resulting series solution to be constructed may essentially be categorized as convergent. This leads to develop a condition where outcome solutions seems best fit with auxiliary parameters due to the fact that convergence analysis overwhelmingly accentuate by the involvement of auxiliary parameters. Consequently the *h*-curves are constructed in Figs. 2 and 3 which provides considerable evidence of convergence of series solutions to corresponding equations. The admissible ranges of the auxiliary parameters $\u210ff,$ $\u210f\mathit{\theta},$ $\u210f\varphi $ and $\u210f\xi $ which are $\u22121.08\u2264\u210ff\u22640.0$, $\u22121.3\u2264\u210f\mathit{\theta}\u22640.1,$ $\u22121.05\u2264\u210f\varphi \u2264\u22120.15,$ $\u22121.3\u2264\u210f\xi \u22640.25.$

## IV. RESULTS AND DISCUSSION

Illustrations are given to depict the impact of fluid parameters $\mathit{\epsilon}$ and $\lambda $ on velocity, temperature, concentration of nanofluids and motile density of microorganisms. Fig. 4 is plotted to examine influence of fluid parameters on velocity profile $f\u2032.$ Enhanced velocity field was attributed due to rise in $\mathit{\epsilon}$ and $\lambda $. Impact of $\delta $ and $M$ on velocity profile is revealed in Fig. 5. The velocity profile decreases by increasing $\delta $ and *M*. It is worth mentioning that physical role of Lorentz force must be taken into account while interpreting the velocity function of fluid due to resistive nature of this force. Its increase is directly related with increase in magnetic parameter consequently velocity decreases. Behavior of bioconvection Rayleigh number $Rb$ and buoyancy ratio parameter $Nr$ on velocity profile $f\u2032$ is presented in Fig. 6. It is evident that velocity profile decreases when bioconvection Rayleigh number $Rb$ and buoyancy ratio parameter $Nr$ are increased. Higher buoyancy ratio parameter $Nr$ seems responsible for larger buoyancy force due to the concentration and that is responsible for decrease in velocity profile. Variation of ratio of Brownian motion parameter $Nb$ and thermophoresis parameter $Nt$ on temperature profile is plotted in Fig. 7. It is observed that for increasing Brownian motion parameter $Nb$ the temperature profile decreases however, rise in thermophoresis parameter $Nt$ has resulted in increase of temperature profile. Fig. 8 portrays influence of both the parameters $Nr$ and $Rb$ on temperature profile which indicates an increase in these two parameters results in temperature enhancement. Effects of dimensionless parameter $Le$ on temperature profile is displayed in Fig. 9 which showed that temperature profile increase with increase of parameter. The impact of Eckert number $Ec$ and Prandtl number $Pr$ on temperature profile is illustrated in Fig. 10. Here increase in $Ec$ results in rise in temperature profile where with the increase in $Pr\u2009$ a decline in temperature profile is observed as shown in Fig. 10. It is important to note that nanoparticle concentration profile increase with the increase of parameter *Nt* in Fig. 11. Thermophoresis appears to facilitate thermal and nanoparticle diffusion especially in case when $Nt>0$, that corresponds to cold plate. As a result, movement of nanoparticles takes place from heated fluid regime to surface which has promoted the distribution of nanoparticles in fluid. The effect of Brownian motion parameter $Nb$ on nanoparticle concentration profile is given in Fig. 12 in which it is evident that nanoparticle concentration profile decrease when parameter Brownian motion parameter $Nb$ increase. Fig. 13 reveals that nanoparticle concentration increases with the rise in buoyancy ratio parameter $Nr$ and Rayleigh number *Rb*. We estimated a significant difference between concentration of nanoparticles present in surroundings and stretched surface that has contributed in concentration profile enhancement. Variation in parameter $Le$ for nanoparticle concentration profile is interpreted through Fig. 14 which shows a decreasing behavior of nanoparticle concentration profile. As *Le* exists in inverse proportion to Brownian diffusion coefficient, a corresponding rise in $Le$ has resulted in reducing the concentration profile. Fig. 15 displayed the effect of $Nt$ and $Nb$ on motile microorganism density profile which indicates that with the increase of thermophoresis parameter $Nt$ and Brownian motion parameter $Nb$ motile microorganism density profile decrease. Enhancement in $Nb$ bring motion of nanoparticles at higher state which increase the fluid temperature. Hence motile microorganism density profile decreased. In Fig. 16, increase of buoyancy ratio parameters $Nr$ and $Rb$ has shown to cause an increase of motile microorganism density profile. Increase in buoyancy force is attributed to large $Rb$ values, which causes bioconvection to take place as a result density of fluid increase. The effects of bioconvection Lewis number $Lb$ and Peclet number $Pe$ are manifested in Fig. 17 where increase of parameters $Lb$ and $Pe$ results in motile microorganism density profile to decrease. For higher values of Lewis number *Lb*, the diffusivity of microorganism decreases consequently, the fluid density of motile microorganisms has gradually lowered. Comparison between Newtonian fluid and Powell-Eyring fluids for both velocity and temperature distributions are presented in Figs. 18 and 19. The comparison of Newtonian fluid and Powell-Eyring fluid shows that magnitude of velocity is greater in case of Powell-Eyring fluid.

We present the series solutions for convergence in Table I. It is noticed that convergence of momentum equation takes place at 20th order of approximation, whereas the convergence of energy equation, concentration and microorganisms occur at 30th order of approximation. The physical parameters are presented in Table II where we have checked their behavior on skin friction coefficient $Re1\u22152Cfx$. We found that due to increase of $\lambda $ and $\delta $, there is a considerable enhancement of skin friction while for increase in $\mathit{\epsilon}$, *M*, *Nr*, and *Rb* parameters, the skin friction decreases. In Table III, the impact on local Nusselt number $Rex\u22121\u22152Nu$ for various parameters is given which shows that Nusselt number decreases for parameters *M*, *Nr*, *Rb*, *Nb*, *Nt*, *Ec*, *Le* and *Pe*. Whereas enhancement was observed for parameters $\lambda ,$ $Pr,$ *Lb* and $\gamma $. The effect of local motile microorganism number was presented in Table IV. Here values of density number of motile microorganisms increase when *M*, *Nr*, *Rb*, *Nt*, *Ec* and *Le* increase and decrease when $\mathit{\epsilon},$ $\delta ,$ $Pr,$ *Nb* and *Lb*. Comparison of magnetic parameter of present and previous studies is given in Table V. There is excellent agreement observed between the results of current study and the findings reported earlier.

Order of approximation . | $\u2212f\u2032\u20320$ . | $\u2212\mathit{\theta}\u20320$ . | $\varphi \u20320$ . | $\u2212\xi \u2032(0)$ . |
---|---|---|---|---|

1 | 1.124 | 0.1178 | 0.1178 | 0.7935 |

3 | 1.149 | 0.1276 | 0.1276 | 0.5916 |

5 | 1.148 | 0.1311 | 0.1311 | 0.5151 |

15 | 1.143 | 0.1415 | 0.1415 | 0.4602 |

20 | 1.140 | 0.1456 | 0.1459 | 0.4394 |

30 | 1.140 | 0.1468 | 0.1468 | 0.4372 |

35 | 1.140 | 0.1468 | 0.1468 | 0.4372 |

50 | 1.140 | 0.1468 | 0.1468 | 0.4372 |

Order of approximation . | $\u2212f\u2032\u20320$ . | $\u2212\mathit{\theta}\u20320$ . | $\varphi \u20320$ . | $\u2212\xi \u2032(0)$ . |
---|---|---|---|---|

1 | 1.124 | 0.1178 | 0.1178 | 0.7935 |

3 | 1.149 | 0.1276 | 0.1276 | 0.5916 |

5 | 1.148 | 0.1311 | 0.1311 | 0.5151 |

15 | 1.143 | 0.1415 | 0.1415 | 0.4602 |

20 | 1.140 | 0.1456 | 0.1459 | 0.4394 |

30 | 1.140 | 0.1468 | 0.1468 | 0.4372 |

35 | 1.140 | 0.1468 | 0.1468 | 0.4372 |

50 | 1.140 | 0.1468 | 0.1468 | 0.4372 |

$\mathit{\epsilon}$ . | M
. | $\delta $ . | $\lambda $ . | Nr
. | Rb
. | $\u2212Rex1\u22152Cfx$ . |
---|---|---|---|---|---|---|

0.5 | 1 | 0.5 | 0.1 | 0.1 | 0.1 | 1.589 |

1 | 1.819 | |||||

1.5 | 2.040 | |||||

0.5 | 0.5 | 1.402 | ||||

0.7 | 1.484 | |||||

1 | 1.589 | |||||

1 | 0.5 | 1.589 | ||||

1 | 1.468 | |||||

1.5 | 1.343 | |||||

0.5 | 0.1 | 1.589 | ||||

0.2 | 1.571 | |||||

0.3 | 1.558 | |||||

0.1 | 0.1 | 1.589 | ||||

0.2 | 1.590 | |||||

0.3 | 1.596 | |||||

0.1 | 0.1 | 1.589 | ||||

0.2 | 1.594 | |||||

0.3 | 1.597 |

$\mathit{\epsilon}$ . | M
. | $\delta $ . | $\lambda $ . | Nr
. | Rb
. | $\u2212Rex1\u22152Cfx$ . |
---|---|---|---|---|---|---|

0.5 | 1 | 0.5 | 0.1 | 0.1 | 0.1 | 1.589 |

1 | 1.819 | |||||

1.5 | 2.040 | |||||

0.5 | 0.5 | 1.402 | ||||

0.7 | 1.484 | |||||

1 | 1.589 | |||||

1 | 0.5 | 1.589 | ||||

1 | 1.468 | |||||

1.5 | 1.343 | |||||

0.5 | 0.1 | 1.589 | ||||

0.2 | 1.571 | |||||

0.3 | 1.558 | |||||

0.1 | 0.1 | 1.589 | ||||

0.2 | 1.590 | |||||

0.3 | 1.596 | |||||

0.1 | 0.1 | 1.589 | ||||

0.2 | 1.594 | |||||

0.3 | 1.597 |

M
. | $\lambda $ . | Nr
. | Rb
. | Pr
. | Nb
. | Nt
. | Ec
. | Le
. | Lb
. | Pe
. | $\gamma $ . | $Rex\u22121\u22152Nux$ . |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.5 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3630 |

0.7 | 0.3405 | |||||||||||

1.0 | 0.3133 | |||||||||||

1.0 | 0.2 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3158 |

0.3 | 0.3182 | |||||||||||

0.4 | 0.3204 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

0.2 | 0.3132 | |||||||||||

0.3 | 0.3129 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

0.2 | 0.3124 | |||||||||||

0.3 | 0.3116 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

2 | 0.3559 | |||||||||||

3 | 0.3667 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

0.2 | 0.3667 | |||||||||||

0.4 | 0.3699 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

0.2 | 0.3104 | |||||||||||

0.3 | 0.3076 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

0.2 | 0.2402 | |||||||||||

0.3 | 0.2047 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

3 | 0.3127 | |||||||||||

4 | 0.3126 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 0.5 | 1 | 0.1 | 0.3125 |

1.0 | 0.3133 | |||||||||||

1.5 | 0.3137 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1.0 | 1 | 0.1 | 0.3133 |

2 | 0.4531 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1.0 | 3 | 2 | 0.6314 |

3 | 0.6426 | |||||||||||

4 | 0.6481 |

M
. | $\lambda $ . | Nr
. | Rb
. | Pr
. | Nb
. | Nt
. | Ec
. | Le
. | Lb
. | Pe
. | $\gamma $ . | $Rex\u22121\u22152Nux$ . |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.5 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3630 |

0.7 | 0.3405 | |||||||||||

1.0 | 0.3133 | |||||||||||

1.0 | 0.2 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3158 |

0.3 | 0.3182 | |||||||||||

0.4 | 0.3204 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

0.2 | 0.3132 | |||||||||||

0.3 | 0.3129 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

0.2 | 0.3124 | |||||||||||

0.3 | 0.3116 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

2 | 0.3559 | |||||||||||

3 | 0.3667 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

0.2 | 0.3667 | |||||||||||

0.4 | 0.3699 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

0.2 | 0.3104 | |||||||||||

0.3 | 0.3076 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

0.2 | 0.2402 | |||||||||||

0.3 | 0.2047 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.3133 |

3 | 0.3127 | |||||||||||

4 | 0.3126 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 0.5 | 1 | 0.1 | 0.3125 |

1.0 | 0.3133 | |||||||||||

1.5 | 0.3137 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1.0 | 1 | 0.1 | 0.3133 |

2 | 0.4531 | |||||||||||

1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1.0 | 3 | 2 | 0.6314 |

3 | 0.6426 | |||||||||||

4 | 0.6481 |

$\mathit{\epsilon}$ . | M
. | $\lambda $ . | Nr
. | Rb
. | Pr
. | Nb
. | Nt
. | Ec
. | Le
. | Lb
. | Pe
. | $\gamma $ . | $\u2212\xi \u2032(0)$ . |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.5 | 1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

1 | 0.4715 | ||||||||||||

1.5 | 0.4955 | ||||||||||||

0.5 | 0.5 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4738 |

0.7 | 0.4585 | ||||||||||||

1 | 0.4372 | ||||||||||||

1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 | |

0.2 | 0.4413 | ||||||||||||

0.3 | 0.4450 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

0.2 | 0.4370 | ||||||||||||

0.3 | 0.4369 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

0.2 | 0.4360 | ||||||||||||

0.3 | 0.4348 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

2 | 0.4497 | ||||||||||||

3 | 0.4570 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

0.2 | 0.4954 | ||||||||||||

0.3 | 0.5151 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

0.2 | 0.3232 | ||||||||||||

0.3 | 0.3076 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

0.2 | 0.4324 | ||||||||||||

0.3 | 0.4285 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

3 | 0.4339 | ||||||||||||

4 | 0.4310 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

1.5 | 0.7769 | ||||||||||||

2 | 0.7770 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

2 | 0.3242 | ||||||||||||

3 | 0.2158 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 2 | 1.413 |

3 | 1.315 | ||||||||||||

4 | 1.271 |

$\mathit{\epsilon}$ . | M
. | $\lambda $ . | Nr
. | Rb
. | Pr
. | Nb
. | Nt
. | Ec
. | Le
. | Lb
. | Pe
. | $\gamma $ . | $\u2212\xi \u2032(0)$ . |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.5 | 1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

1 | 0.4715 | ||||||||||||

1.5 | 0.4955 | ||||||||||||

0.5 | 0.5 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4738 |

0.7 | 0.4585 | ||||||||||||

1 | 0.4372 | ||||||||||||

1 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 | |

0.2 | 0.4413 | ||||||||||||

0.3 | 0.4450 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

0.2 | 0.4370 | ||||||||||||

0.3 | 0.4369 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

0.2 | 0.4360 | ||||||||||||

0.3 | 0.4348 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

2 | 0.4497 | ||||||||||||

3 | 0.4570 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

0.2 | 0.4954 | ||||||||||||

0.3 | 0.5151 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

0.2 | 0.3232 | ||||||||||||

0.3 | 0.3076 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

0.2 | 0.4324 | ||||||||||||

0.3 | 0.4285 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

3 | 0.4339 | ||||||||||||

4 | 0.4310 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

1.5 | 0.7769 | ||||||||||||

2 | 0.7770 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 0.1 | 0.4372 |

2 | 0.3242 | ||||||||||||

3 | 0.2158 | ||||||||||||

0.5 | 1.0 | 0.1 | 0.1 | 0.1 | 1 | 0.1 | 0.1 | 0.1 | 2 | 1 | 1 | 2 | 1.413 |

3 | 1.315 | ||||||||||||

4 | 1.271 |

M
. | Malik et al.,, (2015) . | Alsaedi et al.,, (2017) . | Present study . |
---|---|---|---|

0.0 | 1.00000 | 1.00000 | 1.00000 |

0.5 | 1.11802 | 1.11803 | 1.11802 |

1.0 | -1.41419 | 1.41421 | 1.41420 |

M
. | Malik et al.,, (2015) . | Alsaedi et al.,, (2017) . | Present study . |
---|---|---|---|

0.0 | 1.00000 | 1.00000 | 1.00000 |

0.5 | 1.11802 | 1.11803 | 1.11802 |

1.0 | -1.41419 | 1.41421 | 1.41420 |

## V. CONCLUSIONS

The bioconvective flow of Powell-Eyring nanofluids towards a stretched surface with Newtonian heating effects was investigated. Key findings are presented as under:

Velocity profile decreases by increasing

*M*and $\delta $.Velocity profile is decreasing function by increasing

*Rb*and*Nr*.Both the temperature and concentration of nanoparticle have increase with the corresponding increase in

*Nt*values.For increase of

*Rb*, we found velocity profile decrease whereas temperature profile increase.Motile density profile decrease for

*Lb*and*Pe*increase.

## NOMENCLATURE

- $\rho f\u2003\u2003$
Fluid density (kg/m

^{3})- $\rho m\u2009\u2009\u2009\u2009\u2009\u2009\u2009$
Microorganisms density (kg/m

^{3})- $\alpha \u2003\u2003\u2009$
Thermal diffusivity of base fluid

- $DT\u2003\u2003\u2009\u2009$
Thermophoretic diffusion coefficient

- $\sigma \u2003\u2003\u2009$
Electrical conductivity of fluid ($\Omega $m)

^{−1}- $g\u2003\u2003\u2009\u2009$
Gravity vector (m/s

^{2})- $b\u2003\u2003\u2009\u2009$
Chemotaxis constant

- $\beta \u2003\u2003\u2009$
Volume expansion coefficient of the fluid

- $T\u221e\u2003\u2003\u2009\u2009$
Atmosphere temperature (°K)

- $Tw\u2003\u2003\u2009$
Surface temperature (°K)

- $nw\u2003\u2003$
Surface density of motile organisms (kg/m

^{3})- $Nb\u2003\u2003$
Brownian motion parameter

- $Cfx\u2003\u2003$
Skin friction coefficient

- $Nnx\u2003\u2003$
Density number of motile microorganisms

- $\lambda $
Grashof number

- $\varphi \u2003\u2003$
Nanoparticle concentration profile

- $\tau w\u2003\u2003$
Shear stress

- $Lb\u2003\u2003$
Bioconvection Lewis number

*Nr*Buoyancy force parameter

- $p\u2003\u2003\u2009\u2009$
Fluid pressure (N m

^{−2})- $T\u2003\u2003\u2009$
Local temperature (°K)

- $DB\u2003\u2003\u2009\u2009$
Brownian diffusion coefficient

*D*_{m}Diffusivity of microorganisms

- $\gamma \u2003\u2003\u2009\u2009$
Volume of microorganisms

- $\mu \u2003\u2003\u2009\u2009\u2009$
Viscosity (kg m

^{−1}s^{−1})- $Wc\u2003\u2003$
Maximum cell swimming speed

*c*_{p}specific heat of nanoparticle (J kg

^{−1}k^{−1})- $C\u221e\u2003\u2003\u2009\u2009\u2009$
Atmosphere concentration (kg/m

^{3})*M*Hartman number

- $Nt\u2003\u2003$
Thermophoresis parameter

- $Pe\u2003\u2003$
Peclet number

*Nu*_{x}Nusselt number

*c*_{f}Specific heat capacity of the fluid (J kg

^{−1}k^{−1})- $\xi \u2003\u2003$
Motile microorganism density profile

- $\mathit{\theta}\u2003\u2003\u2009\u2009$
Temperature profile

- $\upsilon \u2003\u2003$
Kinematic viscosity (m

^{2}s^{−1})*Le*Lewis number

## REFERENCES

_{2}and flue gas-containing CO

_{2}by microalgae