Gyrotactic microorganism and bio-convection during flow of Prandtl-Eyring nanomaterial


 Our main intension behind this work is to investigate Prandtl-Eyring nanomaterial in presence of gyrotactic microorganisms. Flow is generated via stretching sheet and is subject to melting heat effect. Radiation and dissipation are addressed. Entropy rate is also reported. Nanofluid effects are explored through Buongiorno model for nanofluid by considering Brownian motion and thermophoresis impacts. Problem related modelling is done by obtaining PDEs and these PDEs are then transmitted into ODEs by using appropriate similarity variables. Homotopic technique has been employed to obtain a convergent series solution of the considered problem. Graphical results have been presented to investigate the impact of different prominent variables over fluid velocity, temperature distribution, nanofluid concentration and on microorganism concentration. Entropy analysis has been discussed and the physical quantities such as surface drag force, Nusselt number, local Sherwood number and microorganism density number for the current problem is obtained. Velocity boost against higher melting and fluid parameters. Temperature of the fluid reduces with an increment in melting and radiation parameters while it intensifies through Prandtl and Eckert number, Brownian motion and thermophoresis parameters. Decay in concentration is noticed against higher values of melting and thermophoresis parameters while it increases for higher Schmidt number and Brownian motion parameter. Microorganism field boosts with higher values of Peclet number and microorganism concentration difference parameter. Moreover entropy generation rate intensifies against higher radiation parameter and Brickman number.


Introduction
Recently advancements in nanotechnology got much attention from scientists and researchers due to its enormous applications in industrial and technological zones. The introduction of nano uid brings a lot of its applications to the real world. In the beginning Choi [1] observed that the heat transportation capacity (thermal conductivity) of normal uids can be improved by adding small nanosize particles with the characteristics of high thermal conductivity. He presented the idea of replacing normal/traditional uids with these more advance uids with high thermal e ciency. These advance uids are known as nano uids. Nano uid plays a vital role in technological devices, heat-related equipment, nuclear reactor, space technology, processing of generic drugs, microelectronics, radiators, cooling system, bio-sensors and many more. Buongiorno [2] explained the transportation of mass and heat transfer in nano uids. He presented the seven slip mechanism and concluded that out of all these thermophoresis and Brownian di usion is of particular interest. Ahmad et al. [3] investigated the nanomaterial ow of magnetohydrodynamic non-Newtonian uid. Turkyilmazoglu [4] studied the mass and heat transport of unsteady natural convective ow of nanoliquid with radiation past a vertical surface. MHD forced convection in nano uid is reported to Sheikholeslami and bhatti [5]. Hayat et al. [6] considered melting e ect in CNTs ow with stagnation point by variable thickness surface. Some recent investigations on nano uid can be see through Refs. [7][8][9][10][11][12][13][14][15][16][17].
Bio-convection appears due to the up swimming of microorganisms in a particular direction since the density of microorganisms are higher than the base uid. Therefore, the upper surface of the uid becomes denser due to the gathering of microorganism. Thus upper surface of uid is unstable and eventually the microorganism falls down causing bioconvection. The up swimming of microorganisms back to the top carry on bioconvection process. The utilization of gyrotactic microorganism in di erent nanoliquid enhances mass transfer, microscale mixing and also enhances the stability of nano uids. Khan et al. [18] studied bioconvection nano uid ow considering gyrotactic microorganism. Mutuku and Makinde [19] explored MHD bioconvection ow of nanoliquid past a porous vertical surface with gyrotactic microorganism. Hayat et al. [20] examined bioconvection in ow of Walter-B nanomaterials. Kuznetsov [21] examined the bioconvection nano uid ow with the suspension of both gyrotactic microorganism and nanoparticles.
There is no doubt that the phenomenon of melting heat transfer has been studied extensively due to its ever rising usages in industrial and technological processes. Melting heat transfer plays an essential role in coil exchanger, magma solidi cation, semiconductor material preparation, welding process, crystal growth, permafrost melting, thermocouple and heat engines. Krishnamurthy et al. [22] considered chemical reactions and radiation during nano uid ow with melting heat. Prasannakumara et al. [23] investigated melting heat, thermal radiation, MHD and heat source e ects doing ow in region of stagnation point. Both melting heat transfer phenomenon and thermal radiation in magnetohydrodynamic ow past a moving surface is addressed by Das [24]. Further explorations on melting heat transfer phenomenon can be seen through Refs. [25][26][27][28][29].
Very little attention is paid by the researchers towards ow of non-Newtonian uids in presence of microorganisms. This consideration narrow down when melting heat is considered. Here we have considered melting e ect in Prandtl-Eyring nano uid ow containing gyrotactic microorganism past a stretching sheet. Viscous dissipation and thermal radiation e ects are also addressed. Entropy generation is accounted. Appropriate variables are used to transform governing expressions (PDEs) into ODEs. HAM technique is employed for solutions development. Entropy, velocity, concentration, microorganism and temperature under involved physical variables is evaluated graphically.

Mathematical formulation
Two dimensional Prandtl-Eyring nano uid with motile gyrotactic microorganisms is considered by a stretched sheet. Sheet is subject to melting phenomena. Thermal radiation and viscous dissipation elaborates heat transport phenomena. Surface is stretched with the velocity (u = ax), with constant (a). Flow geometry is given in Figure 1. After executing aforementioned assumptions and boundary layer approximations we obtain [30,31]: With boundary conditions Consider the transformations [6,12]: Making use of these transformations in Eqs. (1-6) we get with transformed boundary conditions

Solution procedure
Obtained non-linear ODEs with BCs are tackled analytically through homotopy analysis method (HAM). This technique is very helpful for solving highly non-linear ODEs. This method works on basis of de ning appropriate initial guesses and linear operators. Thus initial guesses and linear operators are [32][33][34]: Where p ∈ [ , ] is embedding parameter while ( f , θ , ϕ , χ) are convergence control parameters. The non-linear operators are written as The solution in series form is written as

Discussion
The investigations of di erent in uential parameters over velocity eld, temperature distribution, mass concentration, microorganism concentration and on entropy rate is the main aim behind this section

. Velocity
Variations of melting and uid parameters (Me,α) for velocity eld (f (η)) are depicted in Figures 3-4. Figure 3 displays outcomes of melting parameter for velocity. Velocity enhances against higher melting variable (Me). Physically with an increment in melting (Me) the heat transfer enhances due to high convection from hot uid to cold melting surface and as a result velocity boosts. Figure 4 elucidate impact of material parameter (α) for velocity (f (η)).
Clearly an augmentation in velocity occurs against higher material parameter (α). It is quite clear that absolute viscosity and uid parameter (α) are related inversely. Hence higher uid parameter (α) leads to intensi cation in velocity of uid (f (η)).

. Temperature eld
In uences of di erent prominent variables on temperature are displayed in Figures 5-10). Figure 5 depicts melting parameter (Me) variations for temperature distribution. Higher melting parameter decays temperature distribution. Physically for higher melting (Me) convective ow enhances which cause more heat transfer from hot uid towards melting surface and as a results the tempera- ture decays. Figure 6 illustrates variation of Brownian motion parameter against temperature. Temperature rises for higher (N b ). Outcomes of radiation (Rd) for temperature is displayed in Figure 7. A reduction in temperature occurs against larger radiation parameter. Figure 8 elucidate temperature variations for thermophoresis parameter (N t ). An augmentation in thermophoresis variable results in temperature enhancement. Figure 9 presents Prandtl number (Pr) outcomes for temperature. Temperature boosts against higher Prandtl number. Decline in thermal di usivity occurs for higher Prandtl number and temperature enhances against higher (Pr). Figure 10 depict Schmidt number (Sc) variations for temperature. Clearly temperature augmentation occurs for higher Schmidt number.  Table 1: Convergence of f ( ),θ ( ),ϕ ( ),χ ( ) when α = β = Pr = Pe = . , N t = N b = Rd = . , Me = Sc = . , Lb = . at various approximations.

. Concentration
Variations of concentration for di erent variables are presented in Figures 11-14. Figure 11 display outcome of melting parameter for concentration (ϕ(η)). An augmentation in melting parameter result in concentration decay. Physically molecular di usivity decreases with augmentation in Brownian motion parameter which results in concentration enhancement. Figure 13 display thermophore-  sis outcome on concentration (ϕ(η)). A decline in concentration is observed for larger values of thermophoresis (N t ). Figure 14 elucidate outcome of Schmidt number for concentration (ϕ(η)). With an increment in Schmidt number concentration boosts. Basically mass transfer decays form hot uid to cold surface which is responsible for concentration enhancement.

. Microorganism
In uences of bio convection lewis number (Lb), microorganism concentration di erence (Ω), Peclet number (Pe) and melting parameter (Me) on microorganism eld are displayed in Figures 15-18. Figure 15 displays impact of bio convection Lewis number on microorganism eld. A de-

. Entropy rate
Variations of radiation (Rd) and Brinkman number (Br) on entropy rate are presented in Figures 19-20. Figure 19 depicts outcomes of Brinkman number for entropy rate (S G (η)). An augmentation in entropy rate occurs against higher Brinkman number. Since viscous forces increases for higher Brinkman number which causes disturbance in the system and consequently entropy rate boosts against Brinkman number. Figure 20 shows outcome of radiation   for entropy rate (S G (η)). Clearly entropy rate boosts against higher radiation variable (Rd).

Conclusion
Presented work deals with ow of Prandtl uid in the presence of microorganisms. Surface is subjected to melting heat. Boungiorno model for nano uid describes nano uid characteristics while heat transfer character-istics are explored via thermal radiation and viscous dissipation. It is concluded that velocity increases with an increase in uid and melting parameters. Decline in temperature of the uid is noticed for both radiation and melting parameters while it intensi es for larger Brownian motion and thermophoresis parameters. Higher melting parameter causes reduction in concentration while it increases with higher Brownian motion parameter and Schmidt number. Higher Peclet number leads to intensi cation in microorganism eld while it decays against bio-concentration Lewis number. Entropy production rate is higher for Brinkman number and radiation parameter.
In future this work can be extended for cavity problems and can be handled by nite element method.

Funding information:
The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Con ict of interest:
The authors state no con ict of interest.