# Magneto Jeffrey Nanofluid Bioconvection over a Rotating Vertical Cone due to Gyrotactic Microorganism

Authors: S. Saleem; Hunza Rafiq; A. Al-Qahtani; Mohamed Abd El-Aziz; M. Y. Malik; I. L. Animasaun
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The particular inquiry is made to envision the behavioral characteristics of gyrotactic microorganism effects on the MHD flow of Jeffrey nanofluid. Together the nanoparticles and motile microorganism are inducted into the modeled nonlinear differential equations. The optimal solutions for the governing equations are tackled by optimal homotopy analysis method. The physical characteristics of the relatable parameters are explored and deliberated in terms of graphs and numerical charts. Also, the precision of the present findings is certified by equating them with the previously published work. It is explored that rescaled density of the motile microorganisms contracts with bioconvection Peclet numberPe. It is seen that bioconvection Rayleigh number Rb shrinks the magnitude of tangential velocity. Also, bioconvection Schmidt number Sb augments the reduced density number of the motile microorganisms.


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## Body

## 1. Literature Survey

Non-Newtonian fluid mechanics is a subject which is essentially interdisciplinary in its nature and which is also wide in its area of application. Indeed, non-Newtonian behavior comes across mostly in organic and allied processing engineering. It is too useful in the excavating industry, where slurries and muds are controlled, and in applications like lubrication and biomedical flow phenomena. The imitation of nonlinear fluid flow occurrences is thus of significance to engineering. Everyday uses of specific flows are established in freezing progressions where a coolant is executed on continuously moving surface. Non-Newtonian fluids along heat and mass transfer are correspondingly crucial in food dispensation, heavy oils, and lubricants [1–5].An important area of study is mixed convectional flows, which occurs in atmospheric boundary layer flows, heat exchangers, solar collectors, and nuclear reactors. It establishes in a condition wherever the impacts of buoyantic forces due to forced convection turn to be considerable. In this existing effort, a cone is placed in a nonlinear fluid through the cone axis together with the outer flow is explored. Heat transfer analysis with combined convective flow about rotating cone-shaped tool is splendidly common in automobile and biochemical productions. Hering and Grosh [6] have conferred a variety of similarity solutions for cone shaped bodies in detail. Anilkumar and Roy [7] achieved the self-similar solutions for time dependent flow over a rotating cone. Nadeem and Saleem [8] deliberated a theoretical analysis for MHD flow above a spinning cone. The flow over conic shape objects has been studied by various investigators [9–12].In emerging world, study of nanofluids has extended too much consideration by the investigators because of its features to improve the thermal conductivity relative to base fluids. It was suggested by Choi [13]. Nanofluids have been found to possess enhanced thermos-physical properties such as viscosity, thermal conductivity, thermal diffusivity, and convective heat-transfer coefficients compared with those of base fluids like ethylene glycol, water, oil, etc. because of these enhanced properties nanofluids are used in several electronic applications (cooling of microchips, fluidic digital display devices, micro-electromechanical systems, micro-reactors, etc.), pharmaceutical processes, transportation industry, in biomedical (nano-drug delivery, cancer therapeutics, cryopreservation, nano-cryosurgery, sensing and imaging, etc.) and many others. Buongiorno [14] and Kakaç and Pramuanjaroenkij [15] have inspected an inclusive review of convective transporting nanofluids. Effects of CuO nanoparticles on heat transfer behavior of PCM in solidification process considering radiative source term were explored by Sheikholeslami et al. [16]. A number of investigational and hypothetical mechanisms connected to nanofluids are specified in [17–21].The investigational and hypothetical analysis of the magnetohydrodynamic flows is significant from the scientific point of view, since it has countless worthy applications like magnetohydrodynamic electrical power generation, geophysics, etc. Scientifically, this occurrence is stated in the configuration of renowned Navier-Stokes equations in combination with Maxwell equations. Rashidi et al. [22] deliberated the study of the second law of thermodynamics functional to incompressible nanofluid flowing above a permeable rotational disk under vertical magnetic field. Ellahi [23] observed the flow features of MHD fluid with temperature-dependent viscosity. Awais et al. [24] worked on MHD Sisko fluid flow adjacent to the axisymmetric stagnation point near a stretching cylinder.An important area of study is bioconvectional flows. Bioconvection [25] designs are generally perceived in the research workshop in narrow suspensions of arbitrarily, however on ordinary upwardly, swimming microorganisms which are a slight thicker than water and however have also been seen in situ in micro patches of zooplankton. Ahmed et al. [26] discussed the magnetohydrodynamic combined convective boundary-layer stagnation point flow of a nanofluid towards an elongating surface together with nanoparticles and gyrotactic microorganisms. Kuznetsov [27] explored the bioconvection in a horizontal layer occupied with a nanofluid. The idea was to use microorganisms to encourage or improve convection in a nanofluid. Khan and Makinde [28, 29] inspected effects of heat and mass transfer with bioconvection on a nanofluid flow. Raju and Sandeep [30] considered the thermal and mass flux activities of MHD bioconvection flow above two channels. Zaimi et al. [31] started a study of stagnation point flow to a stretching or shrinking plate in a nanofluid by gyrotactic microorganisms. Makinde and Sadia et al. [32] discussed phenomena of bioconvection in nanofluid containing gyrotactic microorganisms about a wavy cone. Makinde and Animasaun [33] investigated bioconvection nanofluid flow with nonlinear thermal radiation and quartic autocatalysis chemical reaction.To the best of authors’ information, the study associated with flow, heat, and mass transfer features of gyrotactic microorganisms containing MHD Jeffrey fluid over a vertical cone with nanoparticles has not been discovered so far. The extremely nonlinear differential equations of a Jeffrey nanofluid with prescribed heat and mass flux conditions are formerly elucidated by the optimal homotopy analysis method (OHAM) [34–40]. Likewise, the impact of appropriate parameters on the velocities, temperature, concentration fields, and gyrotactic microorganism are discovered and deliberated in detail through graphs and numerical tables. The precision of the analytical technique is also confirmed by comparing the results with the existing literature.


## 2. Mathematical Analysis

A time-dependent mixed convective boundary layer flow of Jeffrey nanofluid comprises gyrotactic microorganisms about a rotating cone. It is stated that the cone is in rotation with angular velocityΩ and creating unsteadiness in the flow field as displayed in Figure 1. The contribution of heat and mass flux is also reflected in the flow process. Further the flow is taken in MHD regime.Figure 1
Geometry of the flow field.Flow analysis [7] is as follows:(1)x∂u∂x+u+x∂w∂z=0,(2)∂u∂t+u∂u∂x+w∂u∂z-v2x=υ1+λ1∂2u∂z2+υλ21+λ1u∂3u∂x∂z2+∂2u∂z2∂w∂z+∂u∂z∂2u∂x∂z+w∂3u∂z3+∂3u∂z2∂t-σρB2u+1-C∞ρC∞βgcos⁡α∗T-T∞-ρp-ρf∞gC-C∞cos⁡α∗-n-n∞gγρm-ρf∞cos⁡α∗,(3)∂v∂t+u∂v∂x+w∂v∂z+uvx=υ1+λ1∂2v∂z2+υλ21+λ1∂3v∂z2∂t+u∂3v∂z2∂x+∂u∂z∂2v∂x∂z+w∂3v∂z3+∂w∂z∂2v∂z2-σρB2v,(4)∂T∂t+u∂T∂x+w∂T∂z=α∂2T∂z2+ρcpρcfDBCzTZ+DTT∞∂T∂z2,(5)∂C∂t+u∂C∂x+w∂C∂z=DB∂2C∂z2-DTT∞∂T∂z2,(6)∂n∂t+u∂n∂x+w∂n∂z+bwcCw-C∞∂∂zn∂C∂z=Dn∂2n∂z2.The important boundary conditions are as follows:(7)ux,0,t=wx,0,t=0,vx,0,t=Ωxsinα∗1-st∗-1,kTzx,0,t=-vw,ρDCzx,0,t=-mw,χt,x,0=χw,ux,∞,t=vx,∞,t=0,Tx,∞,t=T∞,Cx,∞,t=C∞,χx,∞,t=χ∞.T is the temperature, C is concentration and n represents the concentration of microorganism in the fluid,n∞ is the density of motile microorganisms in the fluid, γ is the average volume of microorganisms, β is the fluid volume expansion coefficient, and λ1 is the ratio of relaxation and retardation whereas λ2 is the retardation time. b is the chemotaxis constant and wcis maximum cell swimming speed.Upon application of the following transformations [7]:(8)υe=Ωxsinα∗1-st∗-1,η=Ωsinα∗υ1/21-st∗-1/2z,t∗=Ωsinα∗t,ut,x,z=-2-1Ωxsinα∗1-st∗-1f′η,vt,x,z=Ωxsinα∗1-st∗-1gη,wt,x,z=υΩsinα∗1/21-st∗-1/2fη,Tt,x,z-T∞=T0-T∞xL1-st∗-2θη,Ct,x,z-C∞=C0-C∞xL1-st∗-2ϕη,Gr1=gβcosα∗T0-T∞L3υ2,Gr2=gβcosα∗C0-C∞L3υ2,ReL=Ωsinα∗L2v,γ1=Gr1ReL2,γ2=Gr2ReL2,Nr=γ2γ1,Pr=vα,Sc=vD,A=λ2Ωsinα∗1-st∗-1.And then subject to (1) to (6) along with boundary conditions (7) developed a set of five nonlinear differential equations with single variable.(9)11+λf′′′-ff′′+12f′2-2g2-2γ1θ-Nrϕ-Rbχ-sf′+12ηf′′+A1+λ12f′f′′′+12sηfiv-12f′′2+ffiv+2sf′′′-Mf′=0,(10)11+λg′′-fg′-gf′-sg+12ηg′-A1+λ2sg′′+12sηg′′′+12g′′f′-12f′′g+fg′′′-Mg=0,(11)θ′′-Pr⁡s12ηθ′+2θ+fθ′-f′θ2+Ntθ′ϕ′+Ntθ′2=0,(12)ϕ′′-Sc⁡s12ηϕ′+2ϕ+fϕ′-f′ϕ2+NtNbθ′′=0,(13)χ′′-Sb⁡s12ηχ′+2χ+fχ′-f′χ2-Peϕ′′χ+Ω+ϕ′χ′=0.Pe, Rb, Nb, Nt, M, Nr, SbandSc are bio convection Peclet number, bioconvection Rayleigh number, Brownian motion parameter, thermophoresis parameter, Hartmann number, buoyancy ratio parameter, bioconvection, and Schmidt numbers, respectively.The dimensionless boundary conditions are(14)f=0=f′,g=1,θ′=ϕ′=-1,χ=1asη→0,f′=0=f′′,g=0=g′,θ=ϕ=0,χ=0asη→∞.The coefficients of skin-friction in both xandy –directions are correspondingly specified as(15)Cfx=2τxzz=0ρΩxsinα∗1-st∗-12,(16)Cfy=2τyzz=0ρΩxsinα∗1-st∗-12,or(17)CfxRex1/2=11+λ-f′′+A2f′f′′-3sf′′+2ff′′′+ηsf′′′η=0,CfyRex1/2=11+λ-g′-A23sg′-f′g′+2g′′f+sηg′′η=0.The Nusselt, Sherwood, and density of motile microorganisms are, respectively, given by(18)NuRex-1/2=1θ0,ShRex-1/2=1ϕ0,NnRex-1/2=-χ′0.


## 3. Analytical Solutions by OHAM

The solution of group of mixed nonlinear equations (9) to (13) in series form is computed with the optimal homotopy analysis method (OHAM). The appropriate initial guesses f0, g0, θ0, ϕ0, and χ0 are as follows:(19)f0η=0,g0η=exp⁡-η,θ0η=exp⁡-η,ϕ0η=exp⁡-η,χ0η=exp⁡-η.And the linear operators ζf, ζg, ζθ, ζϕ, and ζχ for this problem are as follows:(20)ζfη=f′′′-f′,ζgη=g′′-g,ζθη=θ′′-θ,ζϕη=ϕ′′-ϕ,ζχη=χ′′-χ.


### 3.1. Optimal Convergence-Control Parameters

The auxiliary parameters which are involved in the convergence of the homotopic solutions arec0f, c0g, c0θ, c0ϕandc0χ. We have engaged the notion of minimization by introducing the residual errors to find out the optimum values of c0f, c0g, c0θ, c0ϕandc0χ as introduced by [34].The Mathematica package BVPh2.0 is used to minimize the average residual error. Three arrays of total optimum convergence control parameters are attained at2nd, 4th, and 6th iterations.With6th iteration optimum convergence-control parameter, singular averaged squared residual errors are achieved and offered in Table 2. Averaged-squared residual errors and total averaged-squared residual errors are reduced by increasing the approximations.Table 1
Optimal convergence control parameters and total averaged squared residual errors usingBVPh2.0.
m c 0 f c 0 g c 0 θ c 0 ϕ c 0 χ E m t CPU TIME[S] 2.0 - 1.04 - 0.19 - 1.06 - 1.06 -1.39 1.34 × 10 - 3 31.65 4.0 - 1.88 - 0.42 - 0.92 - 0.97 -1.49 2.08 × 10 - 4 360.27 6.0 - 0.90 - 0.39 - 0.79 - 0.89 -1.56 4.92 × 10 - 5 12253.3Table 2
Single averaged squared residual errors using optimum values atm=6 from Table 1.
m E m f E m g E m θ E m ϕ E m χ CPU TIME[S] 6.0 1.18 × 10 - 5 5.98 × 10 - 6 4.44 × 10 - 6 9.82 × 10 - 6 1.71 × 10 - 5 32.59 12.0 6.50 × 10 - 7 1.71 × 10 - 7 3.09 × 10 - 8 2.90 × 10 - 7 7.69 × 10 - 7 128.56 18.0 3.17 × 10 - 7 4.66 × 10 - 10 1.65 × 10 - 10 5.27 × 10 - 10 5.60 × 10 - 8 394.23


## 3.1. Optimal Convergence-Control Parameters

The auxiliary parameters which are involved in the convergence of the homotopic solutions arec0f, c0g, c0θ, c0ϕandc0χ. We have engaged the notion of minimization by introducing the residual errors to find out the optimum values of c0f, c0g, c0θ, c0ϕandc0χ as introduced by [34].The Mathematica package BVPh2.0 is used to minimize the average residual error. Three arrays of total optimum convergence control parameters are attained at2nd, 4th, and 6th iterations.With6th iteration optimum convergence-control parameter, singular averaged squared residual errors are achieved and offered in Table 2. Averaged-squared residual errors and total averaged-squared residual errors are reduced by increasing the approximations.Table 1
Optimal convergence control parameters and total averaged squared residual errors usingBVPh2.0.
m c 0 f c 0 g c 0 θ c 0 ϕ c 0 χ E m t CPU TIME[S] 2.0 - 1.04 - 0.19 - 1.06 - 1.06 -1.39 1.34 × 10 - 3 31.65 4.0 - 1.88 - 0.42 - 0.92 - 0.97 -1.49 2.08 × 10 - 4 360.27 6.0 - 0.90 - 0.39 - 0.79 - 0.89 -1.56 4.92 × 10 - 5 12253.3Table 2
Single averaged squared residual errors using optimum values atm=6 from Table 1.
m E m f E m g E m θ E m ϕ E m χ CPU TIME[S] 6.0 1.18 × 10 - 5 5.98 × 10 - 6 4.44 × 10 - 6 9.82 × 10 - 6 1.71 × 10 - 5 32.59 12.0 6.50 × 10 - 7 1.71 × 10 - 7 3.09 × 10 - 8 2.90 × 10 - 7 7.69 × 10 - 7 128.56 18.0 3.17 × 10 - 7 4.66 × 10 - 10 1.65 × 10 - 10 5.27 × 10 - 10 5.60 × 10 - 8 394.23


## 4. Graphical Outcomes and Discussion

The intention of concern segment is to inspect the impacts of various parameters on flow, heat, concentration, and gyrotactic microorganisms. The influence of ratio of relaxation and retardation timesλ, Deborah number A, and bioconvection Rayleigh number Rb on the tangential velocity is sketched in Figures 2(a)–2(c). It is depicted that tangential velocity -f′η has a decreasing behavior for AandRb but increasing behavior for λ. Similarly, the features of λ, A, and Rb are examined on the azimuthal velocity gη(see Figures 3(a)–3(c)). Here positive values of A and λ result in reduction of azimuthal velocity gη; meanwhile Rb enhances the magnitude of boundary layer thickness. Also it is noted that the behavior of λ>0 is opposite for -f′η and gη.Figure 2 (a)
Variation of tangential velocity-f′η against A (b)
Variation of tangential velocity-f′η against λ (c)
Variation of tangential velocity-f′η against RbFigure 3 (a)
Variation of azimuthal velocityg(η) against A (b)
Variation of azimuthal velocityg(η) against λ (c)
Variation of azimuthal velocityg(η) against RbFurther, Figures4(a)–4(c) show that the magnitude of temperature for Nb and Nt is increased and reduced for Pr. Its notable, Brownian motion of nanoparticles offers a main role in nanofluids to augment the heat transfer. Therefore, it can be encouraged that nanofluids and magnetic field intensity create variation in the fluid’s temperature and as such, the use of nanofluids in a magnetic field will be noteworthy in the cooling and heating processes. Additional heat is produced by the contact of nanoparticles and the base fluid because of Brownian motion and thermophoresis. So, the thermal boundary layer develops thicker and the effect is so noticeable that strong temperature exceeds are eminent in the vicinity of the cone for elevating Nb and Nt. And intense particles are dragged from hot surface to the cold area for thermophoresis phenomena. Thus the temperature of the fluid rises. Actually for escalation in Pr, the thermal diffusivity shrinks which significantly decays the temperature. On the other hand, the effects of Brownian motion parameter Nb, thermophoresis parameter Nt, and Schmidt number Sc on concentration are also discussed in Figures 5(a)–5(c). It is portrayed that the concentration of fluid particles reduces in magnitude for Nb and Sc, while the magnitude goes higher for Nt. Schmidt number shows converse relation for coefficient of Brownian diffusion. Greater values of Sc are a source of smaller Brownian diffusion coefficient which is responsible to decrease the concentration field.Figure 4 (a)
Variation of temperatureθη for Nb (b)
Variation of temperatureθη for Nt (c)
Variation of temperatureθη for PrFigure 5 (a)
Variation concentration profilesϕη for Nb (b)
Variation concentration profilesϕη for Nt (c)
Variation concentration profilesϕη for ScFigures6(a) and 6(b) demonstrate the impact of gyrotactic microorganism parameters on density of motile microorganism χη. It is explored that the density of motile microorganism is a decreasing function of microorganism Peclet number Pe and bioconvection Schmidt number Sb. In fact Peclet number is proportion of extreme cell swimming speed to diffusivity of microorganisms as escalation in Pe enhances cell swimming speed so rise in Peclet number shrinks the density of motile microorganisms.Figure 6 (a)
Variation rescaled density of motile microorganismχη for Pe (b)
Variation rescaled density of motile microorganismχη for SbThe influences ofPe and Sb on -χ′0 are plotted in Figure 7. It is very clear from the figure that -χ′0 has a direct relation with both Pe and Sb for accelerating flow (i.e., s>0). On the other hand when flow is retarding the aptitude of -χ′0 is opposite for Pe and Sb. This determines that bioconvection Schmidt number creates huge resistance to the motion of the fluid. Thus, Sb is much more effective in the boundary layer region. This is predictable since the viscous diffusion rate escalates due to a rise in Sb. which results in significant reduction in -χ′0.Figure 7
Variation of reduced density number of the motile microorganisms-χ′0 with bioconvection parameters.In addition, to verify the stability of present results, comparison in Table3 is computed in the deficiency of buoyancy force, Peclet number, and bioconvection Schmidt number with the results given by Sparrow and Cess [10]. The outcomes are seen to be in decent compatible order with each other.Table 3
Comparative results for important physical quantities for special casePe=Sb=A=Rb=λ=0.
Present results Numerical results [7] M C f x R e x 1 / 2 C f y R e x 1 / 2 C f y R e x 1 / 2 C f y R e x 1 / 2 0.0 1.0205 0.6154 1.0207 0.6159 1.021∗ 0.616∗ 0.5 0.7735 0.8485 0.7730 0.8488 0.770∗ 0.849∗ 1.0 0.6197 1.0695 0.6194 1.0692 0.619∗ 1.069∗ 2.0 0.4617 1.4411 0.4613 1.4418 0.461∗ 1.442∗ 3.0 0.3817 1.7476 0.3813 1.7477 0.381∗ 1.748∗ ∗Sparrow and Cess [10].Table4 is prepared for values of skin friction coefficients against ratio of relaxation and retardation times λ, magnetic parameter M, buoyancy parameter γ1, and ratio of buoyancy forces N. It is obvious that tangential skin friction coefficient rises with larger γ1andN, but decreases with both λandM. Besides this, azimuthal skin friction coefficient is directly proportional to all these pertinent parameters expect λ. Actually, we can say that closing the walls of cone the temperature of the wall is more prominent than the temperature of the fluid which eventually rises Gr2 as compared to Gr1; therefore greater N offers the larger skin friction values.Table 4
Surface skin friction coefficients for some significant parameters.
λ M γ 1 N r C f x R e x 1 / 2 C f y R e x 1 / 2 0.5    3.06172 0.58831 1.0    2.92640 0.52921 1.5    1.92763 0.48527 0.5   4.4517 1.2593 1.0   4.3485 2.3509 1.5   4.2310 3.3476 1.0  2.7583 0.7576 1.5  3.5482 0.8572 2.0  4.2382 0.9438 0.5 3.4620 0.5072 1.0 4.3947 0.6373 1.5 5.1926 0.7493The behavior of Pr,Sc, and N on heat and mass transfer rate coefficient is deliberated in Table 5. It is found that Pr and N boost the heat transfer rate coefficient while the variation is opposite for Sc. Also the mass transfer rate coefficient shows an increasing aptitude for changed values of NandSc. On the contrary, Pr reduces the magnitude of mass transfer rate coefficient.Table 5
Variation of local Nusselt and Sherwood numbers for certain noteworthy parameters.
Pr S c N r N u R e x - 1 / 2 S h R e x - 1 / 2 3.0   0.845 0 0.8495 7.0   1.1035 0.8492 12.0   1.2431 0.8485 1.0  0.6684 0.8790 2.0  0.6676 1.1570 3.0  0.6667 1.7639 0.5 0.6184 0.8703 1.0 0.6354 0.8623 1.5 0.6512 0.8576


## 5. Conclusions

This investigation gives analytical solutions of flow, heat, and mass transfer in MHD Jeffrey nanofluid on a rotating cone with motile microorganism. Based on the calculated outcomes, the subsequent summary is reached at the following:(i)
The tangential skin friction coefficient varies as an inverse function ofλ.(ii)
Rescaled density of the motile microorganisms shrinks with bothPeandSb.(iii)
The impression ofM is to decline the major skin friction while increasing the minor skin friction.(iv)
The effect of Nb is opposite for temperature and concentration fields.(v)
Growing values ofPeandSb augment the reduced density number of the motile microorganisms for positive s.(vi)
This study is pertinent in aerospace manufacturing for design and improvement of apparatus.



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