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.

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 (Crane1970), followed by several significant contributions that led to the develop variety of rheological properties (Quemada1978; 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 Eyring1944), 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 Jiang2003), chemotatic (Yanaoka et al., 2009) and gyrotactic (Ghorai and Hill2000; 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 CO2 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 Kuznetsov2004). 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.

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°. 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:

ux+vy=0,
(1)
uux+vuy=υf+1ρfβfc2uy212ρfβfc3uy22uy2σB°2ρfu+1ρf  1CρfβgTT(ρpρf)gCCnngγ(ρmρf)  ,
(2)
uTx+vTy=αT2x2+T2y2+ασB°2ku2+τDBCyTy+DTTTx2+Ty2+1(ρc)p(υf+1ρfβfc)uy216ρfβf3c3uy4,
(3)
uCx+vCy=DB2Cx2+2Cy2+DTT2Tx2+2Ty2,
(4)
unx+vny+bWcCwCynCy+xnCx=Dm2nx2+2ny2+22nxy,
(5)

subjected boundary conditions are

u=Uw(x),  v=0,  Ty=h1T,  DBdCdy+DTTdTdy=0,  n=nw  at  y=0,
(6)
u=0,  CC,  TT,  nn  aty.

Where u and v represent velocity components in x and y direction respectively. Stretching velocity is given by Uw, fluid and microorganism density are ρf and ρm, the pressure fluid is represented as p, kinematic density is νf, whereas material parameters are given by βf and c, the electrical conductivity is σ, 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,T, and n 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 β, the average volume of microorganisms is given by γ, thermal diffusivity of base fluid is symbolized by α, the ratio of effective heat capacitance of the nanoparticles is shown as τ=(ρC)p(ρC)f, while to represent Brownian diffusion coefficient, thermophoretic diffusion coefficient and diffusivity of microorganisms DB, DT, Dm are used respectively, where as the dynamic viscosity is symbolized as μ, and the chemotaxis constant and maximum cell swimming speed are given by b and Wc respectively.

FIG. 1.

Physical model.

Upon application of following transformations

η=aν12y,ψ=(aν)12xf(η),𝜃(η)=TTT,ϕ(η)=CCCwC,ξ(η)=nnnwn.
(7)

Using Eq (7) into Eqs. (16), we get

(1+𝜖)f+fff2𝜖δf2fMf+λ(𝜃NrϕRbξ)=0,
(8)
𝜃+𝜃(Prf+Nbϕ)+Nt𝜃2+PrEc(1+𝜖)f2𝜖δ3f4+Mf2=0,
(9)
ϕ+Lefϕ+NtNb𝜃=0,
(10)
ξ+LbfξPeϕ(ξ+Ω)+ξϕ=0,
(11)
f(0)=1,f(0)=0,𝜃(0)=γ1  1+𝜃  0,Nbϕ(0)+Nt𝜃(0)=0,ξ(0)=1,
(12)
f()=0,𝜃()=0,ϕ()=0,ξ()=0,
(13)

where prime exhibits differentiate on with respect to η, and material fluid parameters are represented by 𝜖 and δ, Pr is the Prandtl number, λ 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, Ω is the microorganism concentration difference parameter, Pe is the bioconvection Peclet number and γ denotes Newtonian heating parameter. Dimensionless skin friction, local Nusselt number and local density of motile microorganism become

𝜖=1ρfβCν,  δ=a3x22νc2,  M=σB°2aρf,  λ=βg(1C)(T)aUNr=ρpρfCwCβρf1CT,  Rb=γρmρfnwnβρf1CTPr=να,   Nb=τDBCwCα,   Nt=τDTα,  Ec=αU°2cpT,  cp=kραLe=υDB,  Lb=υDnn,  Pe=bWcDmn,  Ω=nnwn.
(14)

The physical quantities employed in this study are the skin friction Cf, Nusselt numbers Nu, and density number of the motile microorganisms Nn defined as

Cf=τwρfU°2,  Nu=ϰqwk(TwT),  Nn=ϰqnDn(nwn).

where shear stress, surface heat flux and motile surface microorganisms flux are τw, qw and qn which are defined as

τw=μuy+1βc(uy)16βcq(uy)3y=0,  qw=kTyy=0,qn=Dnuy.y=0
(15)

using (15) into (14) we have

Re12Cfx=1+𝜖f(0)𝜖δ3f03,
(16)
Nux=Rex12Nu=γ11+1𝜃(0),
(17)
Nnx=Rex12Nn=ξ(0).
(18)

in which Rex=U°xυ is the local Reynold number.

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:

f0(η)=1exp[η],  𝜃0η=γ1exp[η]1γ1,  ϕ0(η)=γ1Ntexp[η]1γ1Nb,  ξ0(η)=exp[η],
(19)
Lf=d3fdη3dfdη,  L𝜃=d2𝜃dη2𝜃,  Lϕ=d2ϕdη2ϕ,  Lξ=d2ξdη2ξ,
(20)

with

LfA1+A2expη+A3expη=0,
(21)
L𝜃A4expη+A5expη=0,
(22)
LϕA6expη+A7expη=0,
(23)
LξA8expη+A9expη=0,
(24)

where Ai(i = 1 − 9) are the arbitrary constants.

The zeroth order problems are presented as

(1q)Lf[f^(η,q)f0(η)]=qfNff^(η,q),𝜃^(η,q),ϕ^(η,q),ξ^η,q,
(25)
f^(η,q)η=0=0,f^(η,q)ηη=0=1,f^(η,q)ηη=0,
(26)
(1q)L𝜃[𝜃^(η,q)𝜃0(η)]=q𝜃N𝜃f^(η,q),𝜃^η,q,ϕ^(η,q),
(27)
𝜃^(η,q)ηη=0+γ11+𝜃^(η,q)η=0=0,𝜃^(η,q)η=0,
(28)
(1q)Lϕ[ϕ^(η,q)ϕ0(η)]=qϕNϕf^(η,q),𝜃^η,q,ϕ^(η,q),
(29)
Nbϕ^(η,q)η=0  +Nt𝜃^η,q=0,ϕ^(η,q)η=0,
(30)
(1q)Lξ[ξ^(η,q)ξ0(η)]=qξNξf^(η,q),ϕ^(η,q),ξ^η,q,
(31)
ξ^(η,q)|η=0=1,ξ^(η,q)η=0.
(32)

Here Nf,N𝜃,Nϕ and Nξ are corresponding non-linear operators which are given below:

Nff^(η,q),𝜃^η,q,ϕ^(η,q),ξ^η,q=(1+𝜖)3f^(η,q)η3+f^(η,q)2f^(η,q)η2f^(η,q)η2𝜖δ2f^(η,q)η223f^(η,q)η3Mf^(η,q)η+λ[𝜃^η,qNrϕ^(η,q)Rbξ^(η,q)],
(33)
N𝜃f^η,q,𝜃^(η,q),ϕ^(η,q)=2𝜃^(η,q)η2+𝜃^(η,q)η(Prf^(η,q)+Nbϕ^η,q)+PrEc1+𝜖2f^(η,q)η2_𝜖δ32f^(η,q)η24+PrMEcf^(η,q)η2+Nt𝜃^(η,q)η2,
(34)
Nϕf^η,q,𝜃^η,q,ϕ^η,q=2ϕ^η,qη2+Lef^η,qϕ^η,qη+NtNb2𝜃^η,qη2,
(35)
Nξf^η,q,𝜃^η,q,ϕ^η,q,ξ^η,q=2ξ^η,qη2+Lbf^η,qξ^η,qηPeξη,q+Ω2𝜃^η,qη2+ξ^η,qηϕ^η,qη,
(36)

here f, 𝜃, ϕ and ξ are non-zero auxiliary parameters whereas q0,1 denotes the embedding parameter.

The mth-order deformation problems may be represented as under:

Lffmηχmfm1η=fRmfη,
(37)
f^m(η,q)η=0=0,f^m(η,q)ηη=0=0,f^m(η,q)ηη=0,
(38)
L𝜃𝜃mηχm𝜃m1η=𝜃Rm𝜃η,
(39)
𝜃^m(η,q)ηη=0+γ1(1+𝜃^m(η,q)η=0)=0,𝜃^m(η,q)η=0,
(40)
Lϕϕmηχmϕm1η=ϕRmϕη,
(41)
Nbϕ^m(η,q)ηη=0+Ntϕ^m(η,q)ηη=0=0,ϕ^m(η,q)η=0,
(42)
Lξξmηχmξm1η=ξRmξη,
(43)
ξ^m(η,q)η=0  =0,ξ^m(η,q)η=0,
(44)
Rmfη=(1+𝜖)fm1+k=0m1fm1kfkfm1kf𝜖δk=0m1fm1kl=0kfk1kflM2fm1+λ𝜃m1Nrϕm1Rbξm1,
(45)
Rm𝜃η=𝜃m1η+Prk=0m1fm1k𝜃k+Nbk=0m1ϕm1k𝜃k+Ntk=0m1𝜃m1k𝜃k
+PrEck=0m11+𝜖fm1kfkPrEc𝜖δ3k=0m1fm1kfkll=0sflss=01fs
+PrMEck=0m1fm1kfk,
(46)
Rmϕη=ϕm1+Lek=0m1fm1kϕk+NtNb𝜃m1,
(47)
Rmξη=ξm1+Lbk=0m1fm1kξkPek=0m1ξm1kϕk+ξm1kϕk+PeΩ𝜃m1,
(48)
χm=0,  m11,m>1.
(49)

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

f^(η,0)=f0(η),  f^(η,1)=f(η),  𝜃^(η,0)=𝜃0(η),  𝜃^(η,1)=𝜃(η),
(50)
ϕ^(η,0)=ϕ0(η),  ϕ^(η,1)=ϕ(η),  ξ^(η,0)=ξ0(η),  ξ^(η,1)=ξ(η).
(51)

By virtue of variations of q from 0 to 1, then f^(η,q), 𝜃^η,q, ϕ^(η,q) and ξ^(η,q), reveals from the initial solutions f0(η), 𝜃0η, ϕ0(η) and ξ0(η) to the final solutions f(η),𝜃η, ϕ(η) and ξ(η) respectively.

In order to converge series solutions, the values of auxiliary parameters are selected accordingly. The general solutions fm*,𝜃m*,ϕm*,χm* are specified as under:

fm(η)=fm*(η)+A1+A2expη+A3expη,
(52)
𝜃m(η)=𝜃m*(η)+A4expη+A5expη,
(53)
ϕm(η)=ϕm*(η)+A6expη+A7expη,
(54)
ξm(η)=ξm*(η)+A8expη+A9expη,
(55)

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

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 f,𝜃,ϕ and ξ which are 1.08f0.0, 1.3𝜃0.1,1.05ϕ0.15,1.3ξ0.25.

FIG. 2.

h-curves for f(η) and 𝜃(η).

FIG. 2.

h-curves for f(η) and 𝜃(η).

Close modal
FIG. 3.

h-curves for 𝜃(η) and ξ(η).

FIG. 3.

h-curves for 𝜃(η) and ξ(η).

Close modal

Illustrations are given to depict the impact of fluid parameters 𝜖 and λ 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. Enhanced velocity field was attributed due to rise in 𝜖 and λ. Impact of δ and M on velocity profile is revealed in Fig. 5. The velocity profile decreases by increasing δ 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 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 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.

FIG. 4.

Effects of 𝜖 and λ on f(η).

FIG. 4.

Effects of 𝜖 and λ on f(η).

Close modal
FIG. 5.

Effects of M and δ on f(η).

FIG. 5.

Effects of M and δ on f(η).

Close modal
FIG. 6.

Effects of Rb and Nr on f(η).

FIG. 6.

Effects of Rb and Nr on f(η).

Close modal
FIG. 7.

Effects of Nt and Nb on 𝜃(η).

FIG. 7.

Effects of Nt and Nb on 𝜃(η).

Close modal
FIG. 8.

Effect of Nr and Rb on 𝜃(η).

FIG. 8.

Effect of Nr and Rb on 𝜃(η).

Close modal
FIG. 9.

Effects of Le and γ on 𝜃(η).

FIG. 9.

Effects of Le and γ on 𝜃(η).

Close modal
FIG. 10.

Effects of Pr and Ec on 𝜃(η).

FIG. 10.

Effects of Pr and Ec on 𝜃(η).

Close modal
FIG. 11.

Effects of Nt on ϕη.

FIG. 11.

Effects of Nt on ϕη.

Close modal
FIG. 12.

Effects of Nb on ϕ(η).

FIG. 12.

Effects of Nb on ϕ(η).

Close modal
FIG. 13.

Effects of Nr and Rb on ϕ(η).

FIG. 13.

Effects of Nr and Rb on ϕ(η).

Close modal
FIG. 14.

Effects of Le on ϕ(η).

FIG. 14.

Effects of Le on ϕ(η).

Close modal
FIG. 15.

Effects of Nt and Nb on ξ(η).

FIG. 15.

Effects of Nt and Nb on ξ(η).

Close modal
FIG. 16.

Effects of Nr and Rb on ξ(η).

FIG. 16.

Effects of Nr and Rb on ξ(η).

Close modal
FIG. 17.

Effects of Lb and Pe on ξ(η).

FIG. 17.

Effects of Lb and Pe on ξ(η).

Close modal
FIG. 18.

Effect of Powell-Eyring fluid and Newtonian fluid on f(η).

FIG. 18.

Effect of Powell-Eyring fluid and Newtonian fluid on f(η).

Close modal
FIG. 19.

Effect of Powell-Eyring fluid and Newtonian fluid on 𝜃(η).

FIG. 19.

Effect of Powell-Eyring fluid and Newtonian fluid on 𝜃(η).

Close modal

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 Re12Cfx. We found that due to increase of λ and δ, there is a considerable enhancement of skin friction while for increase in 𝜖, M, Nr, and Rb parameters, the skin friction decreases. In Table III, the impact on local Nusselt number Rex12Nu 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 λ,Pr,Lb and γ. 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 𝜖,δ,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.

TABLE I.

Convergence of series solutions for various order of approximations when 𝜖=δ=0.5,  λ=Ω=γ=Nr=Rb=Nb=Nt=0.1,M=Lb=Pe=Pr=1 and Le = 2.

Order of approximationf0𝜃0ϕ0ξ(0)
1.124 0.1178 0.1178 0.7935 
1.149 0.1276 0.1276 0.5916 
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 approximationf0𝜃0ϕ0ξ(0)
1.124 0.1178 0.1178 0.7935 
1.149 0.1276 0.1276 0.5916 
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 
TABLE II.

Numerical values of skin friction Rex12Cfx.

𝜖MδλNrRbRex12Cfx
0.5 0.5 0.1 0.1 0.1 1.589 
     1.819 
1.5      2.040 
0.5 0.5     1.402 
 0.7     1.484 
     1.589 
 0.5    1.589 
     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 
𝜖MδλNrRbRex12Cfx
0.5 0.5 0.1 0.1 0.1 1.589 
     1.819 
1.5      2.040 
0.5 0.5     1.402 
 0.7     1.484 
     1.589 
 0.5    1.589 
     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 
TABLE III.

Numerical values of Nusselt number Rex12Nux.

MλNrRbPrNbNtEcLeLbPeγRex12Nux
0.5 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3630 
0.7            0.3405 
1.0            0.3133 
1.0 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.3158 
 0.3           0.3182 
 0.4           0.3204 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
  0.2          0.3132 
  0.3          0.3129 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
   0.2         0.3124 
   0.3         0.3116 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
           0.3559 
           0.3667 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
     0.2       0.3667 
     0.4       0.3699 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
      0.2      0.3104 
      0.3      0.3076 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
       0.2     0.2402 
       0.3     0.2047 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
           0.3127 
           0.3126 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.5 0.1 0.3125 
         1.0   0.3133 
         1.5   0.3137 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 1.0 0.1 0.3133 
           0.4531 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 1.0 0.6314 
           0.6426 
           0.6481 
MλNrRbPrNbNtEcLeLbPeγRex12Nux
0.5 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3630 
0.7            0.3405 
1.0            0.3133 
1.0 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.3158 
 0.3           0.3182 
 0.4           0.3204 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
  0.2          0.3132 
  0.3          0.3129 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
   0.2         0.3124 
   0.3         0.3116 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
           0.3559 
           0.3667 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
     0.2       0.3667 
     0.4       0.3699 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
      0.2      0.3104 
      0.3      0.3076 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
       0.2     0.2402 
       0.3     0.2047 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.3133 
           0.3127 
           0.3126 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.5 0.1 0.3125 
         1.0   0.3133 
         1.5   0.3137 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 1.0 0.1 0.3133 
           0.4531 
1.0 0.1 0.1 0.1 0.1 0.1 0.1 1.0 0.6314 
           0.6426 
           0.6481 
TABLE IV.

Numerical values of motile microorganism density number.

𝜖MλNrRbPrNbNtEcLeLbPeγξ(0)
0.5 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
            0.4715 
1.5             0.4955 
0.5 0.5 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4738 
 0.7            0.4585 
            0.4372 
 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
  0.2           0.4413 
  0.3           0.4450 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
   0.2          0.4370 
   0.3          0.4369 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
    0.2         0.4360 
    0.3         0.4348 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
            0.4497 
            0.4570 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
      0.2       0.4954 
      0.3       0.5151 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
       0.2      0.3232 
       0.3      0.3076 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
        0.2     0.4324 
        0.3     0.4285 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
            0.4339 
            0.4310 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
          1.5   0.7769 
            0.7770 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
            0.3242 
            0.2158 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 1.413 
            1.315 
            1.271 
𝜖MλNrRbPrNbNtEcLeLbPeγξ(0)
0.5 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
            0.4715 
1.5             0.4955 
0.5 0.5 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4738 
 0.7            0.4585 
            0.4372 
 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
  0.2           0.4413 
  0.3           0.4450 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
   0.2          0.4370 
   0.3          0.4369 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
    0.2         0.4360 
    0.3         0.4348 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
            0.4497 
            0.4570 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
      0.2       0.4954 
      0.3       0.5151 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
       0.2      0.3232 
       0.3      0.3076 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
        0.2     0.4324 
        0.3     0.4285 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
            0.4339 
            0.4310 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
          1.5   0.7769 
            0.7770 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.4372 
            0.3242 
            0.2158 
0.5 1.0 0.1 0.1 0.1 0.1 0.1 0.1 1.413 
            1.315 
            1.271 
TABLE V.

A comparison of findings of present study with Malik et al.,, (2015), Alsaedi et al.,, (2017) of skin friction for various of M when 𝜖=δ=λ=Nr=Rb=0.

MMalik 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 
MMalik 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 

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 δ.

  • 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.

ρf  

Fluid density (kg/m3)

ρm       

Microorganisms density (kg/m3)

α   

Thermal diffusivity of base fluid

DT    

Thermophoretic diffusion coefficient

σ   

Electrical conductivity of fluid (Ωm)−1

g    

Gravity vector (m/s2)

b    

Chemotaxis constant

β   

Volume expansion coefficient of the fluid

T    

Atmosphere temperature (°K)

Tw   

Surface temperature (°K)

nw  

Surface density of motile organisms (kg/m3)

Nb  

Brownian motion parameter

Cfx  

Skin friction coefficient

Nnx  

Density number of motile microorganisms

λ

Grashof number

ϕ  

Nanoparticle concentration profile

τw  

Shear stress

Lb  

Bioconvection Lewis number

Nr

Buoyancy force parameter

p    

Fluid pressure (N m−2)

T   

Local temperature (°K)

DB    

Brownian diffusion coefficient

Dm

Diffusivity of microorganisms

γ    

Volume of microorganisms

μ     

Viscosity (kg m−1s−1)

Wc  

Maximum cell swimming speed

cp

specific heat of nanoparticle (J kg−1k−1)

C     

Atmosphere concentration (kg/m3)

M

Hartman number

Nt  

Thermophoresis parameter

Pe  

Peclet number

Nux

Nusselt number

cf

Specific heat capacity of the fluid (J kg−1k−1)

ξ  

Motile microorganism density profile

𝜃    

Temperature profile

υ  

Kinematic viscosity (m2s−1)

Le

Lewis number

1.
Abbasbandy
,
S.
,
Shivanian
,
E.
,
Vajravelu
,
K.
, and
Kumar
,
S.
, “
A new approximate analytical technique for dual solutions of nonlinear differential equations arising in mixed convection heat transfer in a porous medium
,”
International Journal of Numerical Methods for Heat & Fluid Flow
27
(
2
) (
2017
).
2.
Acharya
,
N.
,
Das
,
K.
, and
Kundu
,
P. K.
, “
Framing the features of MHD boundary layer flow past an unsteady stretching cylinder in presence of non-uniform heat source
,”
Journal of Molecular Liquids
225
,
418
425
(
2017
).
3.
Akbar
,
N. S.
,
Ebaid
,
A.
, and
Khan
,
Z. H.
, “
Numerical analysis of magnetic field effects on Eyring-Powell fluid flow towards a stretching sheet
,”
Journal of Magnetism and Magnetic Materials
382
,
355
358
(
2015
).
4.
Ali
,
F.
,
Gohar
,
M.
, and
Khan
,
I.
, “
MHD flow of water-based Brinkman type nanofluid over a vertical plate embedded in a porous medium with variable surface velocity, temperature and concentration
,”
Journal of Molecular Liquids
223
,
412
419
(
2016
).
5.
Alsaedi
,
A.
,
Khan
,
M. I.
,
Farooq
,
M.
,
Gull
,
N.
, and
Hayat
,
T.
, “
Magnetohydrodynamic (MHD) stratified bioconvective flow of nanofluid due to gyrotactic microorganisms
,”
Advanced Powder Technology
28
(
1
),
288
298
(
2017
).
6.
Babu
,
M. J.
and
Sandeep
,
N.
, “
3D MHD slip flow of a nanofluid over a slendering stretching sheet with thermophoresis and Brownian motion effects
,”
Journal of Molecular Liquids
222
,
1003
1009
(
2016
).
7.
Becker
,
S. M.
,
Kuznetsov
,
A.
, and
Avramenko
,
A.
, “
Numerical modeling of a falling bioconvection plume in a porous medium
,”
Fluid Dynamics Research
35
(
5
),
323
339
(
2004
).
8.
Bhatti
,
M. M.
and
Rashidi
,
M. M.
, “
Effects of thermo-diffusion and thermal radiation on Williamson nanofluid over a porous shrinking/stretching sheet
,”
Journal of Molecular Liquids
221
,
567
573
(
2016a
).
9.
Bhatti
,
M. M.
and
Rashidi
,
M. M.
, “
Numerical simulation of entropy generation on MHD nanofluid towards a stagnation point flow over a stretching surface
,”
International Journal of Applied and Computational Mathematics
,
1
15
(
2016b
).
10.
Bhatti
,
M. M.
and
Zeeshan
,
A.
, “
Analytic study of heat transfer with variable viscosity on solid particle motion in dusty Jeffery fluid
,”
Modern Physics Letters B
30
(
16
),
1650196
(
2016
).
11.
Bhatti
,
M. M.
,
Abbas
,
T.
, and
Rashidi
,
M. M.
, “
A new numerical simulation of MHD stagnation-point flow over a permeable stretching/shrinking sheet in porous media with heat transfer
,”
Iranian Journal of Science and Technology, Transactions A: Science
,
1
7
(
2016a
).
12.
Bhatti
,
M. M.
,
Abbas
,
T.
,
Rashidi
,
M. M.
, and
Ali
,
M. E. S.
, “
Numerical simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet
,”
Entropy
18
(
6
),
200
(
2016b
).
13.
Bhatti
,
M. M.
,
Abbas
,
T.
,
Rashidi
,
M. M.
,
Ali
,
M. E. S.
, and
Yang
,
Z.
, “
Entropy generation on MHD Eyring–Powell nanofluid through a permeable stretching surface
,”
Entropy
18
(
6
),
224
(
2016c
).
14.
Bhatti
,
M. M.
,
Mishra
,
S. R.
,
Abbas
,
T.
, and
Rashidi
,
M. M.
, “
A mathematical model of MHD nanofluid flow having gyrotactic microorganisms with thermal radiation and chemical reaction effects
,”
Neural Computing and Applications
,
1
13
(
2016d
).
15.
Bhatti
,
M. M.
,
Zeeshan
,
A.
, and
Ellahi
,
R.
, “
Simultaneous effects of coagulation and variable magnetic field on peristaltically induced motion of Jeffrey nanofluid containing gyrotactic microorganism
,”
Microvascular Research
110
,
32
42
(
2017
).
16.
Chaudhary
,
R. C.
and
Jain
,
P.
, “
Unsteady free convection boundary-layer flow past an impulsively started vertical surface with newtonian heating
,”
Romanian Journal of Physics
51
(
9/10
),
911
925
(
2006
).
17.
Cheah
,
W. Y.
,
Show
,
P. L.
,
Chang
,
J. S.
,
Ling
,
T. C.
, and
Juan
,
J. C.
, “
Biosequestration of atmospheric CO2 and flue gas-containing CO2 by microalgae
,”
Bioresource technology
184
,
190
201
(
2015
).
18.
Chhabra
,
R. P.
and
Richardson
,
J. F.
(
2011
).
Non-Newtonian flow and applied rheology: engineering applications
.
Butterworth-Heinemann
.
19.
Crane
,
L. J.
, “
Flow past a stretching plate
,”
Z. Angew. Math. Phy.
21
,
645
647
(
1970
).
20.
Freidoonimehr
,
N.
,
Rashidi
,
M. M.
,
Momenpour
,
M. H.
, and
Rashidi
,
S.
, “
Analytical approximation of heat and mass transfer in MHD non-Newtonian nanofluid flow over a stretching sheet with convective surface boundary conditions
,”
International Journal of Biomathematics
10
(
01
),
1750008
(
2017
).
21.
Ghorai
,
S.
and
Hill
,
N. A.
, “
Wavelengths of gyrotactic plumes in bioconvection
,”
Bulletin of Mathematical Biology
62
(
3
),
429
450
(
2000
).
22.
Hayat
,
T.
,
Bashir
,
G.
,
Waqas
,
M.
, and
Alsaedi
,
A.
, “
MHD flow of Jeffrey liquid due to a nonlinear radially stretched sheet in presence of Newtonian heating
,”
Results in Physics
6
,
817
823
(
2016
).
23.
Hayat
,
T.
,
Iqbal
,
Z.
,
Qasim
,
M.
, and
Obaidat
,
S.
, “
Steady flow of an Eyring Powell fluid over a moving surface with convective boundary conditions
,”
International Journal of Heat and Mass Transfer
55
(
7
),
1817
1822
(
2012
).
24.
Hayat
,
T.
,
Khan
,
M. I.
,
Waqas
,
M.
, and
Alsaedi
,
A.
, “
Effectiveness of magnetic nanoparticles in radiative flow of Eyring-Powell fluid
,”
Journal of Molecular Liquids
231
,
126
133
(
2017b
).
25.
Hayat
,
T.
,
Khan
,
M. I.
,
Waqas
,
M.
, and
Alsaedi
,
A.
, “
Mathematical modeling of non-Newtonian fluid with chemical aspects: A new formulation and results by numerical technique
,”
Colloids and Surfaces A: Physicochemical and Engineering Aspects
518
,
263
272
(
2017c
).
26.
Hayat
,
T.
,
Ullah
,
I.
,
Alsaedi
,
A.
, and
Farooq
,
M.
, “
MHD flow of Powell-Eyring nanofluid over a non-linear stretching sheet with variable thickness
,”
Results in Physics
7
,
189
196
(
2017a
).
27.
Hillesdon
,
A. J.
and
Pedley
,
T. J.
, “
Bioconvection in suspensions of oxytactic bacteria: Linear theory
,”
Journal of Fluid Mechanics
324
,
223
259
(
1996
).
28.
Islam
,
S.
,
Shah
,
A.
,
Zhou
,
C. Y.
, and
Ali
,
I.
, “
Homotopy perturbation analysis of slider bearing with Powell–Eyring fluid
,”
Zeitschrift für Angewandte Mathematik und Physik
60
(
6
),
1178
(
2009
).
29.
Khodaparast
,
H. H.
,
Madinei
,
H.
,
Friswell
,
M. I.
,
Adhikari
,
S.
,
Coggon
,
S.
, and
Cooper
,
J. E.
, “
An extended harmonic balance method based on incremental nonlinear control parameters
,”
Mechanical Systems and Signal Processing
85
,
716
729
(
2017
).
30.
Kuznetsov
,
A. V.
, “
Nanofluid bioconvection: interaction of microorganisms oxytactic upswimming, nanoparticle distribution, and heating/cooling from below
,”
Theoretical and Computational Fluid Dynamics
26
(
1
),
291
310
(
2012
).
31.
Kuznetsov
,
A. V.
and
Jiang
,
N.
, “
Bioconvection of negatively geotactic microorganisms in a porous medium: The effect of cell deposition and declogging
,”
International Journal of Numerical Methods for Heat & Fluid Flow
13
(
3
),
341
364
(
2003
).
32.
Lesnic
,
D.
,
Ingham
,
D. B.
, and
Pop
,
I.
, “
Free convection from a horizontal surface in a porous medium with Newtonian heating
,”
Journal of Porous Media
3
(
3
) (
2000
).
33.
Lesnic
,
D.
,
Ingham
,
D. B.
,
Pop
,
I.
, and
Storr
,
C.
, “
Free convection boundary-layer flow above a nearly horizontal surface in a porous medium with Newtonian heating
,”
Heat and Mass Transfer
40
(
9
),
665
672
(
2004
).
34.
Mackley
,
M. R.
,
Butler
,
S. A.
,
Huxley
,
S.
,
Reis
,
N. M.
,
Barbosa
,
A. I.
, and
Tembely
,
M.
, “
The observation and evaluation of extensional filament deformation and breakup profiles for non Newtonian fluids using a high strain rate double piston apparatus
,”
Journal of Non-Newtonian Fluid Mechanics
239
,
13
27
(
2017
).
35.
Makinde
,
O. D.
and
Animasaun
,
I. L.
, “
Bioconvection in MHD nanofluid flow with nonlinear thermal radiation and quartic autocatalysis chemical reaction past an upper surface of a paraboloid of revolution
,”
International Journal of Thermal Sciences
109
,
159
171
(
2016
).
36.
Malik
,
M. Y.
,
Khan
,
I.
,
Hussain
,
A.
, and
Salahuddin
,
T.
, “
Mixed convection flow of MHD Eyring-Powell nanofluid over a stretching sheet: A numerical study
,”
AIP Advances
5
(
11
),
117118
(
2015
).
37.
Merkin
,
J. H.
, “
Natural-convection boundary-layer flow on a vertical surface with Newtonian heating
,”
International Journal of Heat and Fluid Flow
15
(
5
),
392
398
(
1994
).
38.
Metcalfe
,
A. M.
and
Pedley
,
T.
, “
Falling plumes in bacterial bioconvection
,”
Journal of Fluid Mechanics
445
,
121
149
(
2001
).
39.
Mosayebidorcheh
,
S.
,
Tahavori
,
M. A.
,
Mosayebidorcheh
,
T.
, and
Ganji
,
D. D.
, “
Analysis of nano-bioconvection flow containing both nanoparticles and gyrotactic microorganisms in a horizontal channel using modified least square method (MLSM)
,”
Journal of Molecular Liquids
227
,
356
365
(
2017
).
40.
Mutuku
,
W. N.
and
Makinde
,
O. D.
, “
Hydromagnetic bioconvection of nanofluid over a permeable vertical plate due to gyrotactic microorganisms
,”
Computers & Fluids
95
,
88
97
(
2014
).
41.
Powell
,
R. E.
and
Eyring
,
H.
, “
Mechanisms for the relaxation theory of viscosity
,”
Nature
154
,
427
428
(
1944
).
42.
Quemada
,
D.
, “
Rheology of concentrated disperse systems II. A model for non-Newtonian shear viscosity in steady flows
,”
Rheologica Acta
17
(
6
),
632
642
(
1978
).
43.
Salleh
,
M. Z.
,
Nazar
,
R.
,
Arifin
,
N. M.
,
Pop
,
I.
, and
Merkin
,
J. H.
, “
Forced-convection heat transfer over a circular cylinder with Newtonian heating
,”
Journal of Engineering Mathematics
69
(
1
),
101
110
(
2011
).
44.
Shaaban
,
A. A.
and
Abou-Zeid
,
M. Y.
, “
Effects of heat and mass transfer on MHD peristaltic flow of a non-Newtonian fluid through a porous medium between two coaxial cylinders
,”
Mathematical Problems in Engineering
(
2013
).
45.
Sharma
,
Y.
and
Kumar
,
V.
, “
The effect of high-frequency vertical vibration in a suspension of gyrotactic micro-organisms
,”
Mechanics Research Communications
44
,
40
46
(
2012
).
46.
Shukla
,
N.
,
Rana
,
P.
,
Bég
,
O. A.
, and
Singh
,
B.
(
2017
), “
Effect of chemical reaction and viscous dissipation on MHD nanofluid flow over a horizontal cylinder: Analytical solution
,” In
Chamola
,
B. P.
and
Kumari
,
P.
(Eds.),
AIP Conference Proceedings
(Vol. 1802, No. 1, p.
020015
).
AIP Publishing
.
47.
Siddiqa
,
S.
,
Gul e H
,
Begum
,
N.
,
Saleem
,
S.
,
Hossain
,
M. A.
, and
Reddy Gorla
,
R. S.
, “
Numerical solutions of nanofluid bioconvection due to gyrotactic microorganisms along a vertical wavy cone
,”
International Journal of Heat and Mass Transfer
101
,
608
613
(
2016
).
48.
Uddin
,
M. J.
,
Kabir
,
M. N.
, and
Bég
,
O. A.
, “
Computational investigation of Stefan blowing and multiple-slip effects on buoyancy-driven bioconvection nanofluid flow with microorganisms
,”
International Journal of Heat and Mass Transfer
95
,
116
130
(
2016
).
49.
Winet
,
H.
and
Jahn
,
T. L.
, “
Geotaxis in protozoa I. A propulsiongravity model for tetrahymena (Ciliata)
,”
Journal of Theoretical Biology
46
(
2
),
449
465
(
1974
).
50.
Yanaoka
,
H.
,
Inamura
,
T.
, and
Suzuki
,
K.
, “
Numerical analysis of bioconvection generated by chemotactic bacteria
,”
Journal of Fluid Science and Technology
4
(
3
),
536
545
(
2009
).
51.
Yang
,
Z.
and
Liao
,
S.
, “
A HAM-based wavelet approach for nonlinear ordinary differential equations
,”
Communications in Nonlinear Science and Numerical Simulation
48
,
439
453
(
2017
).
52.
Yürüsoy
,
M.
, “
A Study of pressure distribution of a slider bearing lubricated with Powell-Eyring fluid
,”
Turkish Journal of Engineering and Environmental Sciences
27
(
5
),
299
304
(
2003
).
53.
Zhao
,
B.
,
Su
,
Y.
,
Zhang
,
Y.
, and
Cui
,
G.
, “
Carbon dioxide fixation and biomass production from combustion flue gas using energy microalgae
,”
Energy
89
,
347
357
(
2015
).