# Evaluating the Higher-Order Slip Consequence in Bioconvection Nanofluid Flow Configured by a Variable Thick Surface of Disk

Authors: Hassan Waqas; Sumeira Yasmin; Nesreen Althobaiti; Ebenezer Bonyah; Ahmed Alshehri; Zahir Shah
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

For innovations in manufacturing and engineering scientific fields, the devices (electrical and computer systems) with large thermal effectiveness are needed. As a result, their thermal efficiency has become a very hot problem for many canvassers. With the novelty of this analysis, a mathematical study is performed to estimate the Darcy-Forchheimer flow of viscous magnetized fluid with Arrhenius activation energy and bioconvection effects through a variable thick surface of a rotating disk. The impact of thermal conductivity, heat source, and nonlinear thermal radiation is considered. The higher-order velocity slip impacts are also scrutinized. The system of partial differential equations (PDEs) and specific boundary restrictions is altered into a system of ODEs by adopting the suitable similarity transformations. The reduced ODE’s system is tackled with the aid of shooting scheme under (bvp4c) built-in tool commercial software MATLAB. Moreover, the effects of different parameters over velocity components, thermal conductivity, concentration, and microorganism’s fields are also examined. The confirmation of our findings is also explained through tables and graphical results. The results revealed that the radial velocity increases with the growing estimations of mixed convection parameter. The second-order velocity slip in radial direction causes a decrement in the estimation of axial velocity. Temperature distribution increases with a larger temperature ratio parameter. The concentration field of species and microorganism profile is reduced via a Brownian motion parameter and Peclet number, respectively.


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## Body

## 1. Introduction

### 1.1. Literature Survey

Nanofluid is a fluid constructed with nanosized (1 to 100 nm) materials or molecules, named “nanomaterials” and otherwise “nanoparticles.” Such substances are designed colloidal suspension of nanopowders in a continuous phase fluid. Nanoparticles utilized in nanofluids are constructed with oxidations, metals, and carbon nanotubes including carbides. Continuous phase fluids include ethanol glycol oil as well as water. Nanofluids have novel characteristics that build them potentially helpful in a broad range of heat transformation applications such as fuel cells, microprocessors, hybrid powered engines, pharmacological mechanisms, cooling equipment, chillers, refrigerators, and heat radiators in grinding, among others. Nanofluids are combinations of nanoparticles and the base fluids that can generate numerous heterogeneous nanofluids that are described for their thermophysical characteristics (thermal diffusivity and thermal conductivity, as well as viscosity) as a cooling system in heat transmission in comparison to base fluid, which increased with growing volumetric fraction of nanomaterials. Choi and Eastman [1] were firstly introduced to the basic idea of nanoparticles in constant phase fluids. Buongiorno [2] addressed nanoparticle study by developing a model to evaluate the thermal properties of continuous phase fluids. He predicted that the increased conductance of constant phase fluid would be caused by the tiny-sized and lower volume fraction of supporting nanoelements. Nanotechnology is extremely important in numerous fields including chemical as well as metallurgical operating systems, transportation, macroscopic objects, cancer treatment, and electricity generation. Eid and Mabood [3] reported the suspending magnetohydrodynamic (MHD) flowing of micropolar dusty nanoparticles impinging on a permeability expanding sheet. Umar et al. [4] studied numerically the 3-dimensional incompressible Eyring-Powell nanofluid flow across a stretched surface including velocity slip as well as activation energy. Muhammad et al. [5] explored numerical simulations for 3-D Eyring–Powell (EP) nanoliquid under the nonlinear thermal radiation with changed heat and solutal fluxes. Rasool et al. [6] researched the Marangoni convective Casson-type flow of nanofluid impacted by the existence of Lorentz forces introduced into the modeling by an organized arrangement of magnets in the shape of the Riga configuration. Mahanthesh et al. [7] described the effect of quadratic thermal radiation as well as convection on the boundary layer 2-phase flowing of a dusty flow of nanoliquid through a vertical surface. Aaiza et al. [8] evaluate the magneto-nanoliquid flow across a channel in the presence of different nanoparticle shape factors. Hussanan et al. [9] investigate the thermal transportation phenomenon in micropolar water-based fluid. Khalid et al. [10] scrutinized the exact solution of nanofluid flow. Ali et al. [11] discussed the Brinkman-type nanofluid flow through vertical surface. Zin et al. [12] introduced the free convection behavior in Jeffery fluid through vertical porous surface.The word activation energy plays a significant role in chemical reactions. The scientist Svante Arrhenius was the principal who initiated the description on activation energy in 1889 and that it is the minimum amount of energy required to initiate chemical reactions to a state in which they could experience material change. The application includes compound construction, food processing, transportation structures, a geothermal stored, and businesses. Heat and mass distribution aspects manage various mechanisms that were reacting synthetically which include the species composite reactions as well as codification strength having application in oil production and geothermal configuration. In actuality, it is the least amount of energy required to transform the reactants into substances. Activation energy can take the shape of kinetic and potential energy. It is essential to generate theoretical observations in addition to experimental contributions to measure the flow influence of activation energy. There have been very several theoretical efforts on this discussion. Indeed, the relationship between mass transfer and chemical reaction becomes tiresome. Such interactions can be observed both within the fluid and in mass transmission during the production process and employ of reactants at various levels. Bestman [13] clarified that viscoelastic fluid flow of thermal and mass moved in which restricts divider movements in its own plane utilizing a simplistic scientific theory of twofold material reaction with Arrhenius actuation energy. In this inspection, he obtained an analytical curriculum of action for the problem by employing an irritation method. Guo et al. [14] examined the kinetic model and thermodynamic of oxidant pyrolysis of microalgae wastes using a double distribution activation energy concept and simulating annealing. Hayat et al. [15] evaluated the effect of activation energy on entropy generation (EG) in a 3-dimensional magnetohydrodynamic (MHD) rotational flowing of nanofluids containing a binary chemical process. Araújo et al. [16] scrutinized the kinetic modeling and Arrhenius activation energy distributions in complicated systems with Hopfield Neural Network-based system. Elangovan and Natarajan [17] reported the primary treatment influences on qualitative characteristics, hydration diffusivity, and Arrhenius activation energy of solar drying gourd. The numerical analysis of unsteady Maxell nanofluid is simulated by Bilal et al. [18]. The effect of magneto nanofluid flow over disk with viscous dissipation is analyzed by Saeed et al. [19].Microorganism molecules have been widely used in the production of manufacturing and industrial products such as ethanol, waste-derived biofuel, and fertilizers. They are also utilized in water therapeutic facilities. These microorganisms generate hydrogen gas and biofuel, a favorable dispatchable energy source. As a result, we must investigate the swimming structures and mass transmission properties of microorganisms in order to make their applications more successful, profitable, and widespread for the benefit of humanity. Bioconvection is the production of various types of irregular fluid structures at the microscopic level therefore of the unexpected swimming of self-propelled microorganisms found in water or those certain denser fluids. Natural hypotheses such as searching for nutrients, oxygen for breathing, and improving light absorption for photosynthesis influence the swimming of such microorganisms. Platt in 1961 [20] introduced the term “bioconvection” to illustrate the methodology of improvement of manners in depth suspensions of motile microorganisms at constant temperatures, in comparison to those reported under convectional conditions. Kuznetsov [21] proposed the concept of biothermal convection caused by temperature gradients as well as microorganism swimming. Tlili et al. [22] scrutinized the effect of bioconvection micropolar nanoliquid flow including gyrotactic motile microorganisms on thermal and solutal stratifications at the boundary layer. Al-Mubaddel et al. [23] examined Sisko-based nanofluid under the bioconvection radiation flow with specific thermal and solutal fluxes. Abbasi et al. [24] illustrated the flow of viscoelastic nanoparticles containing gyrotactic motile microorganisms across a rotating stretched disk under convective and zero mass flux conditions. Shehzad et al. [25] scrutinized the bioconvection of a Maxwell-based nanoliquid above an isolated rotational disk under the effect of double diffusional Cattaneo–Christov (C-C) concepts. Aziz et al. [26] evaluated the effects of motile microorganisms on unstable Williamson nanoliquid caused by a bidirectional accelerating surface. Alizadeh and Ganji [27] analyzed the two-phase thermosyphon utilizing RSM. The thermal transportation features and thermal resistance are discussed by Alizadeh and Ganji [28]. Some important research work about fluid flow and heat transfer can be studied in [29–31].Taking the higher-order velocity slip consider, this communication extends the bioconvection nanofluid flow through a stretching disk. The Darcy-Forchheimer porous medium is considered. The heat transfer is incorporated in the presence of nonlinear thermal radiation, joule heating, thermophoresis, and Brownian diffusion. The significance of Arrhenius activation energy is considered. The dimensionless system is tackled by utilizing a shooting scheme with the bvp4c function of MATLAB. The effects of flow controlling parameters are analyzed.


## 1.1. Literature Survey

Nanofluid is a fluid constructed with nanosized (1 to 100 nm) materials or molecules, named “nanomaterials” and otherwise “nanoparticles.” Such substances are designed colloidal suspension of nanopowders in a continuous phase fluid. Nanoparticles utilized in nanofluids are constructed with oxidations, metals, and carbon nanotubes including carbides. Continuous phase fluids include ethanol glycol oil as well as water. Nanofluids have novel characteristics that build them potentially helpful in a broad range of heat transformation applications such as fuel cells, microprocessors, hybrid powered engines, pharmacological mechanisms, cooling equipment, chillers, refrigerators, and heat radiators in grinding, among others. Nanofluids are combinations of nanoparticles and the base fluids that can generate numerous heterogeneous nanofluids that are described for their thermophysical characteristics (thermal diffusivity and thermal conductivity, as well as viscosity) as a cooling system in heat transmission in comparison to base fluid, which increased with growing volumetric fraction of nanomaterials. Choi and Eastman [1] were firstly introduced to the basic idea of nanoparticles in constant phase fluids. Buongiorno [2] addressed nanoparticle study by developing a model to evaluate the thermal properties of continuous phase fluids. He predicted that the increased conductance of constant phase fluid would be caused by the tiny-sized and lower volume fraction of supporting nanoelements. Nanotechnology is extremely important in numerous fields including chemical as well as metallurgical operating systems, transportation, macroscopic objects, cancer treatment, and electricity generation. Eid and Mabood [3] reported the suspending magnetohydrodynamic (MHD) flowing of micropolar dusty nanoparticles impinging on a permeability expanding sheet. Umar et al. [4] studied numerically the 3-dimensional incompressible Eyring-Powell nanofluid flow across a stretched surface including velocity slip as well as activation energy. Muhammad et al. [5] explored numerical simulations for 3-D Eyring–Powell (EP) nanoliquid under the nonlinear thermal radiation with changed heat and solutal fluxes. Rasool et al. [6] researched the Marangoni convective Casson-type flow of nanofluid impacted by the existence of Lorentz forces introduced into the modeling by an organized arrangement of magnets in the shape of the Riga configuration. Mahanthesh et al. [7] described the effect of quadratic thermal radiation as well as convection on the boundary layer 2-phase flowing of a dusty flow of nanoliquid through a vertical surface. Aaiza et al. [8] evaluate the magneto-nanoliquid flow across a channel in the presence of different nanoparticle shape factors. Hussanan et al. [9] investigate the thermal transportation phenomenon in micropolar water-based fluid. Khalid et al. [10] scrutinized the exact solution of nanofluid flow. Ali et al. [11] discussed the Brinkman-type nanofluid flow through vertical surface. Zin et al. [12] introduced the free convection behavior in Jeffery fluid through vertical porous surface.The word activation energy plays a significant role in chemical reactions. The scientist Svante Arrhenius was the principal who initiated the description on activation energy in 1889 and that it is the minimum amount of energy required to initiate chemical reactions to a state in which they could experience material change. The application includes compound construction, food processing, transportation structures, a geothermal stored, and businesses. Heat and mass distribution aspects manage various mechanisms that were reacting synthetically which include the species composite reactions as well as codification strength having application in oil production and geothermal configuration. In actuality, it is the least amount of energy required to transform the reactants into substances. Activation energy can take the shape of kinetic and potential energy. It is essential to generate theoretical observations in addition to experimental contributions to measure the flow influence of activation energy. There have been very several theoretical efforts on this discussion. Indeed, the relationship between mass transfer and chemical reaction becomes tiresome. Such interactions can be observed both within the fluid and in mass transmission during the production process and employ of reactants at various levels. Bestman [13] clarified that viscoelastic fluid flow of thermal and mass moved in which restricts divider movements in its own plane utilizing a simplistic scientific theory of twofold material reaction with Arrhenius actuation energy. In this inspection, he obtained an analytical curriculum of action for the problem by employing an irritation method. Guo et al. [14] examined the kinetic model and thermodynamic of oxidant pyrolysis of microalgae wastes using a double distribution activation energy concept and simulating annealing. Hayat et al. [15] evaluated the effect of activation energy on entropy generation (EG) in a 3-dimensional magnetohydrodynamic (MHD) rotational flowing of nanofluids containing a binary chemical process. Araújo et al. [16] scrutinized the kinetic modeling and Arrhenius activation energy distributions in complicated systems with Hopfield Neural Network-based system. Elangovan and Natarajan [17] reported the primary treatment influences on qualitative characteristics, hydration diffusivity, and Arrhenius activation energy of solar drying gourd. The numerical analysis of unsteady Maxell nanofluid is simulated by Bilal et al. [18]. The effect of magneto nanofluid flow over disk with viscous dissipation is analyzed by Saeed et al. [19].Microorganism molecules have been widely used in the production of manufacturing and industrial products such as ethanol, waste-derived biofuel, and fertilizers. They are also utilized in water therapeutic facilities. These microorganisms generate hydrogen gas and biofuel, a favorable dispatchable energy source. As a result, we must investigate the swimming structures and mass transmission properties of microorganisms in order to make their applications more successful, profitable, and widespread for the benefit of humanity. Bioconvection is the production of various types of irregular fluid structures at the microscopic level therefore of the unexpected swimming of self-propelled microorganisms found in water or those certain denser fluids. Natural hypotheses such as searching for nutrients, oxygen for breathing, and improving light absorption for photosynthesis influence the swimming of such microorganisms. Platt in 1961 [20] introduced the term “bioconvection” to illustrate the methodology of improvement of manners in depth suspensions of motile microorganisms at constant temperatures, in comparison to those reported under convectional conditions. Kuznetsov [21] proposed the concept of biothermal convection caused by temperature gradients as well as microorganism swimming. Tlili et al. [22] scrutinized the effect of bioconvection micropolar nanoliquid flow including gyrotactic motile microorganisms on thermal and solutal stratifications at the boundary layer. Al-Mubaddel et al. [23] examined Sisko-based nanofluid under the bioconvection radiation flow with specific thermal and solutal fluxes. Abbasi et al. [24] illustrated the flow of viscoelastic nanoparticles containing gyrotactic motile microorganisms across a rotating stretched disk under convective and zero mass flux conditions. Shehzad et al. [25] scrutinized the bioconvection of a Maxwell-based nanoliquid above an isolated rotational disk under the effect of double diffusional Cattaneo–Christov (C-C) concepts. Aziz et al. [26] evaluated the effects of motile microorganisms on unstable Williamson nanoliquid caused by a bidirectional accelerating surface. Alizadeh and Ganji [27] analyzed the two-phase thermosyphon utilizing RSM. The thermal transportation features and thermal resistance are discussed by Alizadeh and Ganji [28]. Some important research work about fluid flow and heat transfer can be studied in [29–31].Taking the higher-order velocity slip consider, this communication extends the bioconvection nanofluid flow through a stretching disk. The Darcy-Forchheimer porous medium is considered. The heat transfer is incorporated in the presence of nonlinear thermal radiation, joule heating, thermophoresis, and Brownian diffusion. The significance of Arrhenius activation energy is considered. The dimensionless system is tackled by utilizing a shooting scheme with the bvp4c function of MATLAB. The effects of flow controlling parameters are analyzed.


## 2. Mathematical Description

### 2.1. Flow Analysis

Here, our main purpose is to scrutinize the steady three-dimensional, incompressible, and axially symmetric laminar flow of nanofluid including bioconvection with motile microorganism over a stretching disk surface. The schematic view of flow problem and the system of coordinates is depicted in Figure1; here, the surface of the disk is stretchable through the rate of α and moving along own axis under an angular frequency Ω. Due to the stretchable and angular frequency as well as electrical conduction in the presence of electromagnetic field, the flow is produced. The suitable boundary conditions are applied at the disk surface. Here, no suction/injection is present at the disk surface, that is, w=0. The velocity components denoted as u,v,andw are along the r,φ,andz axis, respectively. The temperature of fluid Tf, nanoparticle concentration of fluid Cf, bioconvection fluid Nf, ambient temperature T∞, and ambient volumetric concentration C∞ as well as ambient swimming organisms N∞ are also presented at the surface of a stretching disk.Figure 1
Schematic configuration of the flow problem.The governing partial deferential equations are expressed as [32, 33]
(1)ur+ur+wz=0,(2)u∂ru−v2r+wzu=νfuzz−σfβ02ρfu−νfk∗u−Fu2+1ρf1−Cfρfβ∗∗g∗T−T∞−ρp−ρfg∗C−C∞−N−N∞g∗γρm−ρf,(3)uvr+uvr+wzv=νfvzz−σfβ02ρfv−νfk∗v−Fv2,(4)urT+wTz=1ρcpKTzTzz+kfρcpfTzz+σfβ02ρcpfu2+v2+μfρcpfuz2+vz2+τDBTz∂zC+DTT∞Tz2+16σ∗3k∗ρcfTz3Tzz+Q∗ρcpfT−T∞,(5)uCr+wCz=DBCzz+DTT∞Tzz−kr2C−C∞TT∞mexp−EaκT,(6)uNr+wNz+NzCzzbWcCw−C∞=DmNzz.


### 2.2. With Boundary Restrictions

(7)u=ra+λ1uz+λ2uzz,v=rΩ+λ3vz+λ4vzz,w=0,−kTz=h1Tf−T,−DBCz=h2Cf−C,−DmNz=h3Nf−N,atz=a∗rR0+1−ς,(8)u⟶0,v⟶0,T⟶T∞,C⟶C∞,N⟶N∞,when z⟶∞.In the above governing equations, the velocity components areu,v,andw along r,φ,andz directions, νf represent the kinematic viscosity; σf is the electrical conductivity of momentum; ρf is the fluid density of microorganism; K∗ demonstrate the porous space permeability; β0 is the magnetic field strength; F=Cb∗/rK∗ illustrate the coefficient of nonuniform inertia; g is the gravity; volume expansion coefficient is indicated by β∗∗; Dm represents the microorganism diffusion parameter; m exemplify the power law index; chemotaxis constant is expressed as b; the cell swimming speed is identified as We; Cb∗ display the drag coefficient; T,C,andN are the basic temperature, volumetric concentration, and swimming bioconvection of fluid; kf signify the thermal conductivity; cp represents specific heat capacity; the heat capacitance ratio is symbolized by τ=ρcpp/ρcpf; DT represents thermophoresis diffusion coefficient; T∞ stand for ambient temperature; m is the fitted rate constant; DB represent the Brownian motion coefficient; the dynamic viscosity is μf, C∞andN∞ are ambient concentration and ambient density microorganism, respectively; the rate of chemical reaction is kr2; β1,β2,β3,andβ4 are linear/nonlinear thermal-based and nanoparticle concentration expansions; the exponential function is exp; the angular frequency is Ω; the activation energy is symbolized by Ea; the stretching constant is a; κ=8.61×10−5eV/K expressed the Stefan Boltzmann constant; λ1λ2<0 and λ3λ4<0 are the first- and second-order slip coefficients, respectively; and h1 and h2 are the heat mass transfer coefficient and the mass transfer coefficient, respectively.


### 2.3. Similarities

The similarity variables are(9)u=r∗R0ΩFη,v=r∗R0ΩGη,w=R0Ω1+r∗−ςΩR02ρfμf−1/n+1Hη,Θ=T−T∞Tf−T∞,Φ=C−C∞Cf−C∞,Χ=N−N∞Nf−N∞,η=zR01+r∗ςΩR02ρfμf1/n+1.In above transformations, radius isR0; the power law index is n; the thickness power law index is ς; the components of velocity (radial, tangential, and axial) are symbolized by F, G, and H; and r∗ is the constant of nondimensional radius. The temperature variable conductivity is addressed as
(10)KT=k∞1+∈1T−T∞ΔT.


### 2.4. Reduced Equations

The governing dimensionless equations after applying the suitable similarity transformation are expressed as(11)2F+Hη+ηεςFη=0,(12)Re1−n/1+n1+r∗2ςFηη−2Fη2+G2−HF−FFηςηε−MF−K1∗F−FrF2+λΘ−NrΦ−NcΧ=0,(13)Re1−n/1+n1+r∗2ςGηη−2FG−HGη−FGηςηε−MG−K1∗G−FrG2=0,(14)1prRe1−n/1+n1+r∗2ς1+∈1Θηη+∈1Θη2+43RdΘηη+θw−133Θ2Θη2+Θ3Θηη+3Θw−122ΘΘη2+Θ2Θηη+3Θw−1Θη2+ΘΘηη−HΘη−FΘηςηε+NtRe1−n/1+n1+r∗2ςΘη2+NbRe1−n/1+nΘηΦη+MEcF2+G2+EcRe1−n/1+n1+r∗ςFη2+Gη2+QΘ=0,(15)NtNbRe1−n/1+n1+r∗2ςΘηη+Re1−n/1+n1+r∗2ςΦηη−ScςηεFΘη−ScHΦη−K1Φ1+α1Θmexp−E11+α1Θ=0,(16)Re1−n/1+n1+r∗2ςΧζζ−LbHΧζ+LbςηεFΧη−PeΦηηΧ+Ω0+ΧηΦη=0,with
(17)Fα=A+L1Fηα+L2Fηηα,Gα=1+L3Gηα+L4Gηηα,Hα=0,Θηα=−Ω11−Θα,Φηα=−Ω21−Φα,Χηα=−Ω31−Χα,(18)F∞⟶0,G∞⟶0,Θ∞⟶0,Φ∞⟶0,Χ∞⟶0.The following are currently considered for introducing the innovative similarity transformation to modify the origin fromα to 0, that is,
(19)Hη=hη−α=hζ,Fη=fη−α=fζ,Gη=gη−α=gζ,Θη=θη−α=θζ,Φη=ϕη−α=ϕζ,Χη=χη−α=χζ.We get(20)2f+hζ+ζ+αςεFζ=0,(21)Re1−n/1+n1+r∗2ςfζζ−2fζ2+g2−hf−ffζςη+αε−Mf−K1∗f−Frf2+λθ−Nrϕ−Ncχ=0,(22)Re1−n/1+n1+r∗2ςgζζ−2fg−hgζ−fgζςη+αε−Mg−K1∗g−Frg2=0,(23)1prRe1−n/1+n1+r∗2ς1+∈1θζζ+∈1θζ2+43Rdθζζ+θw−133θ2θζ2+θ3θζζ+3θw−122θθζ2+θ2θζζ+3θw−1θζ2+θθζζ−hθζ−fθζςζ+αε+NtRe1−n/1+n1+r∗2ςθζ2+NbRe1−n/1+nθζϕζ+MEcf2+g2+EcRe1−n/1+n1+r∗ςfζ2+gζ2+Qθ=0,(24)NtNbRe1−n/1+n1+r∗2ςθζζ+Re1−n/1+n1+r∗2ςϕζζ−Scςζ+αεfθζ−Schϕζ−K1ϕ1+α1θmexp−E11+α1θ=0,(25)Re1−n/1+n1+r∗2ςχζζ−Lbhχζ+Lbςζ+αεfχζ−Peϕζζχ+Ω+χζϕζ=0,through boundary conditions
(26)f0=A1+L1fζ0+L2fζζ0,g0=1+L3gζ0+L4gζζ0,h0=0,θζ0=−Ω11−θ0,ϕζ0=−Ω21−ϕ0,χζ0=−Ω31−χ0,(27)f∞⟶0,g∞⟶0,θ∞⟶0,ϕ∞⟶0,χ∞⟶0.


### 2.5. Dimensionless Prominent Parameters

Now, the dimensionless prominent parameters are given in Table1.Table 1
Dimensionless prominent parameters.
Reynolds numberRe=ΩR02ρf/μfMixed convection variableλ=β∗∗g∗1−C∞Tf−T∞/rΩ2Buoyancy ratio parameterNr=ρp−ρfCf−C∞/ρf1−C∞Tf−T∞β∗∗Bioconvection Rayleigh numberNc=γ∗ρm−ρfNf−N∞/ρf1−C∞Tf−T∞β∗∗Dimensionless constantsr∗=r/R02,ε=r∗/1+r∗Magnetic parameterM=σfβ02/ρfΩLocal porosity parameterK1∗=νf/k∗ΩForchheimer numberFr=Fr,Fr=Cb∗/k∗Eckert numberEc=r2Ω2/cpTf−T∞Brownian motion parameterNb=τDBCf−C∞/νfPrandtl numberPr=ρcpνf/kfThermal source parameterQ=Q∗/ρcpfΩNonlinear thermal radiation parameterRd=4σ∗T∞3/k∗Thermophoretic parameterNt=τDTTf−T∞/νfT∞Activation energy parameterE1=Ea/κT∞Schmidt numberSc=νf/DBStretching rate to angular frequencyA1=a/ΩFirst-order slip parameters in radial directionL1=λ1/R01+r∗ςΩR02ρf/μf1/n+1>0,L3=λ3/R01+r∗ςΩR02ρf/μf1/n+1>0Tangential directional second-order slip parametersL2=λ2/R021+r∗2ςΩR02ρf/μf2/n+1<0,L4=λ4/R021+r∗2ςΩR02ρf/μf2/n+1<0Thermal Biot numberΩ1=h1/kfR0/1+r∗ςμf/ΩR02ρf1/n+1Solutal Biot numberΩ2=h2/DmR0/1+r∗ςμf/ΩR02ρf1/n+1Microorganism Biot numberΩ3=h3/DmR0/1+r∗ςμf/ΩR02ρf1/n+1Chemical reaction parameterK1=kr2/ΩTemperature ratio parameterα1=Tf−T∞/T∞Disk thickness coefficientα=a/R02ΩR02ρf/μf−1/n+1Bioconvective Lewis numberLb=ν/DmPeclet numberPe=bWc/DmMicroorganism difference numberΩ0=N∞/Nf−N∞,


### 2.6. Physical Quantities

Here, the local skin friction coefficient (LSF)Cf, local Nusselt number (LNN) Nux, local Sherwood number (LSW) Shx, and density motile microorganism (LMN) Snx are represented as follows:Local skin friction coefficient(28)Cf=τzr2+τzθ2ρfΩr2.Here,τzrandτzθ are expressed as
(29)τzr=μfuzz=0=μfr∗Ω11+r∗ςΩR02ρfμf1/n+1fζ0,(30)τzθ=μfvzz=0=μfr∗Ω11+r∗ςΩR02ρfμf1/n+1gζ0.Here,τw is identified as
(31)τw=τzr2+τzθ2.Finally, local skin friction coefficient(32)CfxRen−1/n+1=τwz=0ρfrΩ2=1r∗1+r∗ςfζ02+gζ021/2.Here,τzr denoted the shear stress along the radial direction, τzθ represent the shear stress along the direction of tangential, and τw denoted the shear stress.Local Nusselt number(33)Nux=R0qwkfTf−T∞z=0.Here,qw is symbolized as
(34)qwz=0=−kfTzz=0=−kfTf−T∞1+r∗ςΩR02ρfμf1/n+1θζ0.Finally, the local Nusselt number(35)Re−1/n+1Nux=−1+r∗ςθζ0.Local Sherwood number(36)Shx=R0qmDBCf−C∞z=0.Here,qm is denoted as
(37)qm=−DBCzz=0.Finally, the local Sherwood number(38)ShxRe−1/n+1=−1+r∗ςϕζ0.Density motile microorganism number(39)Snx=R0qnDmNf−N∞z=0.Here,qn is identified as
(40)qn=−DmNzz=0.Finally, the density motile microorganism number(41)SnxRe−1/n+1=−1+r∗ςχζ0.


## 2.1. Flow Analysis

Here, our main purpose is to scrutinize the steady three-dimensional, incompressible, and axially symmetric laminar flow of nanofluid including bioconvection with motile microorganism over a stretching disk surface. The schematic view of flow problem and the system of coordinates is depicted in Figure1; here, the surface of the disk is stretchable through the rate of α and moving along own axis under an angular frequency Ω. Due to the stretchable and angular frequency as well as electrical conduction in the presence of electromagnetic field, the flow is produced. The suitable boundary conditions are applied at the disk surface. Here, no suction/injection is present at the disk surface, that is, w=0. The velocity components denoted as u,v,andw are along the r,φ,andz axis, respectively. The temperature of fluid Tf, nanoparticle concentration of fluid Cf, bioconvection fluid Nf, ambient temperature T∞, and ambient volumetric concentration C∞ as well as ambient swimming organisms N∞ are also presented at the surface of a stretching disk.Figure 1
Schematic configuration of the flow problem.The governing partial deferential equations are expressed as [32, 33]
(1)ur+ur+wz=0,(2)u∂ru−v2r+wzu=νfuzz−σfβ02ρfu−νfk∗u−Fu2+1ρf1−Cfρfβ∗∗g∗T−T∞−ρp−ρfg∗C−C∞−N−N∞g∗γρm−ρf,(3)uvr+uvr+wzv=νfvzz−σfβ02ρfv−νfk∗v−Fv2,(4)urT+wTz=1ρcpKTzTzz+kfρcpfTzz+σfβ02ρcpfu2+v2+μfρcpfuz2+vz2+τDBTz∂zC+DTT∞Tz2+16σ∗3k∗ρcfTz3Tzz+Q∗ρcpfT−T∞,(5)uCr+wCz=DBCzz+DTT∞Tzz−kr2C−C∞TT∞mexp−EaκT,(6)uNr+wNz+NzCzzbWcCw−C∞=DmNzz.


## 2.2. With Boundary Restrictions

(7)u=ra+λ1uz+λ2uzz,v=rΩ+λ3vz+λ4vzz,w=0,−kTz=h1Tf−T,−DBCz=h2Cf−C,−DmNz=h3Nf−N,atz=a∗rR0+1−ς,(8)u⟶0,v⟶0,T⟶T∞,C⟶C∞,N⟶N∞,when z⟶∞.In the above governing equations, the velocity components areu,v,andw along r,φ,andz directions, νf represent the kinematic viscosity; σf is the electrical conductivity of momentum; ρf is the fluid density of microorganism; K∗ demonstrate the porous space permeability; β0 is the magnetic field strength; F=Cb∗/rK∗ illustrate the coefficient of nonuniform inertia; g is the gravity; volume expansion coefficient is indicated by β∗∗; Dm represents the microorganism diffusion parameter; m exemplify the power law index; chemotaxis constant is expressed as b; the cell swimming speed is identified as We; Cb∗ display the drag coefficient; T,C,andN are the basic temperature, volumetric concentration, and swimming bioconvection of fluid; kf signify the thermal conductivity; cp represents specific heat capacity; the heat capacitance ratio is symbolized by τ=ρcpp/ρcpf; DT represents thermophoresis diffusion coefficient; T∞ stand for ambient temperature; m is the fitted rate constant; DB represent the Brownian motion coefficient; the dynamic viscosity is μf, C∞andN∞ are ambient concentration and ambient density microorganism, respectively; the rate of chemical reaction is kr2; β1,β2,β3,andβ4 are linear/nonlinear thermal-based and nanoparticle concentration expansions; the exponential function is exp; the angular frequency is Ω; the activation energy is symbolized by Ea; the stretching constant is a; κ=8.61×10−5eV/K expressed the Stefan Boltzmann constant; λ1λ2<0 and λ3λ4<0 are the first- and second-order slip coefficients, respectively; and h1 and h2 are the heat mass transfer coefficient and the mass transfer coefficient, respectively.


## 2.3. Similarities

The similarity variables are(9)u=r∗R0ΩFη,v=r∗R0ΩGη,w=R0Ω1+r∗−ςΩR02ρfμf−1/n+1Hη,Θ=T−T∞Tf−T∞,Φ=C−C∞Cf−C∞,Χ=N−N∞Nf−N∞,η=zR01+r∗ςΩR02ρfμf1/n+1.In above transformations, radius isR0; the power law index is n; the thickness power law index is ς; the components of velocity (radial, tangential, and axial) are symbolized by F, G, and H; and r∗ is the constant of nondimensional radius. The temperature variable conductivity is addressed as
(10)KT=k∞1+∈1T−T∞ΔT.


## 2.4. Reduced Equations

The governing dimensionless equations after applying the suitable similarity transformation are expressed as(11)2F+Hη+ηεςFη=0,(12)Re1−n/1+n1+r∗2ςFηη−2Fη2+G2−HF−FFηςηε−MF−K1∗F−FrF2+λΘ−NrΦ−NcΧ=0,(13)Re1−n/1+n1+r∗2ςGηη−2FG−HGη−FGηςηε−MG−K1∗G−FrG2=0,(14)1prRe1−n/1+n1+r∗2ς1+∈1Θηη+∈1Θη2+43RdΘηη+θw−133Θ2Θη2+Θ3Θηη+3Θw−122ΘΘη2+Θ2Θηη+3Θw−1Θη2+ΘΘηη−HΘη−FΘηςηε+NtRe1−n/1+n1+r∗2ςΘη2+NbRe1−n/1+nΘηΦη+MEcF2+G2+EcRe1−n/1+n1+r∗ςFη2+Gη2+QΘ=0,(15)NtNbRe1−n/1+n1+r∗2ςΘηη+Re1−n/1+n1+r∗2ςΦηη−ScςηεFΘη−ScHΦη−K1Φ1+α1Θmexp−E11+α1Θ=0,(16)Re1−n/1+n1+r∗2ςΧζζ−LbHΧζ+LbςηεFΧη−PeΦηηΧ+Ω0+ΧηΦη=0,with
(17)Fα=A+L1Fηα+L2Fηηα,Gα=1+L3Gηα+L4Gηηα,Hα=0,Θηα=−Ω11−Θα,Φηα=−Ω21−Φα,Χηα=−Ω31−Χα,(18)F∞⟶0,G∞⟶0,Θ∞⟶0,Φ∞⟶0,Χ∞⟶0.The following are currently considered for introducing the innovative similarity transformation to modify the origin fromα to 0, that is,
(19)Hη=hη−α=hζ,Fη=fη−α=fζ,Gη=gη−α=gζ,Θη=θη−α=θζ,Φη=ϕη−α=ϕζ,Χη=χη−α=χζ.We get(20)2f+hζ+ζ+αςεFζ=0,(21)Re1−n/1+n1+r∗2ςfζζ−2fζ2+g2−hf−ffζςη+αε−Mf−K1∗f−Frf2+λθ−Nrϕ−Ncχ=0,(22)Re1−n/1+n1+r∗2ςgζζ−2fg−hgζ−fgζςη+αε−Mg−K1∗g−Frg2=0,(23)1prRe1−n/1+n1+r∗2ς1+∈1θζζ+∈1θζ2+43Rdθζζ+θw−133θ2θζ2+θ3θζζ+3θw−122θθζ2+θ2θζζ+3θw−1θζ2+θθζζ−hθζ−fθζςζ+αε+NtRe1−n/1+n1+r∗2ςθζ2+NbRe1−n/1+nθζϕζ+MEcf2+g2+EcRe1−n/1+n1+r∗ςfζ2+gζ2+Qθ=0,(24)NtNbRe1−n/1+n1+r∗2ςθζζ+Re1−n/1+n1+r∗2ςϕζζ−Scςζ+αεfθζ−Schϕζ−K1ϕ1+α1θmexp−E11+α1θ=0,(25)Re1−n/1+n1+r∗2ςχζζ−Lbhχζ+Lbςζ+αεfχζ−Peϕζζχ+Ω+χζϕζ=0,through boundary conditions
(26)f0=A1+L1fζ0+L2fζζ0,g0=1+L3gζ0+L4gζζ0,h0=0,θζ0=−Ω11−θ0,ϕζ0=−Ω21−ϕ0,χζ0=−Ω31−χ0,(27)f∞⟶0,g∞⟶0,θ∞⟶0,ϕ∞⟶0,χ∞⟶0.


## 2.5. Dimensionless Prominent Parameters

Now, the dimensionless prominent parameters are given in Table1.Table 1
Dimensionless prominent parameters.
Reynolds numberRe=ΩR02ρf/μfMixed convection variableλ=β∗∗g∗1−C∞Tf−T∞/rΩ2Buoyancy ratio parameterNr=ρp−ρfCf−C∞/ρf1−C∞Tf−T∞β∗∗Bioconvection Rayleigh numberNc=γ∗ρm−ρfNf−N∞/ρf1−C∞Tf−T∞β∗∗Dimensionless constantsr∗=r/R02,ε=r∗/1+r∗Magnetic parameterM=σfβ02/ρfΩLocal porosity parameterK1∗=νf/k∗ΩForchheimer numberFr=Fr,Fr=Cb∗/k∗Eckert numberEc=r2Ω2/cpTf−T∞Brownian motion parameterNb=τDBCf−C∞/νfPrandtl numberPr=ρcpνf/kfThermal source parameterQ=Q∗/ρcpfΩNonlinear thermal radiation parameterRd=4σ∗T∞3/k∗Thermophoretic parameterNt=τDTTf−T∞/νfT∞Activation energy parameterE1=Ea/κT∞Schmidt numberSc=νf/DBStretching rate to angular frequencyA1=a/ΩFirst-order slip parameters in radial directionL1=λ1/R01+r∗ςΩR02ρf/μf1/n+1>0,L3=λ3/R01+r∗ςΩR02ρf/μf1/n+1>0Tangential directional second-order slip parametersL2=λ2/R021+r∗2ςΩR02ρf/μf2/n+1<0,L4=λ4/R021+r∗2ςΩR02ρf/μf2/n+1<0Thermal Biot numberΩ1=h1/kfR0/1+r∗ςμf/ΩR02ρf1/n+1Solutal Biot numberΩ2=h2/DmR0/1+r∗ςμf/ΩR02ρf1/n+1Microorganism Biot numberΩ3=h3/DmR0/1+r∗ςμf/ΩR02ρf1/n+1Chemical reaction parameterK1=kr2/ΩTemperature ratio parameterα1=Tf−T∞/T∞Disk thickness coefficientα=a/R02ΩR02ρf/μf−1/n+1Bioconvective Lewis numberLb=ν/DmPeclet numberPe=bWc/DmMicroorganism difference numberΩ0=N∞/Nf−N∞,


## 2.6. Physical Quantities

Here, the local skin friction coefficient (LSF)Cf, local Nusselt number (LNN) Nux, local Sherwood number (LSW) Shx, and density motile microorganism (LMN) Snx are represented as follows:Local skin friction coefficient(28)Cf=τzr2+τzθ2ρfΩr2.Here,τzrandτzθ are expressed as
(29)τzr=μfuzz=0=μfr∗Ω11+r∗ςΩR02ρfμf1/n+1fζ0,(30)τzθ=μfvzz=0=μfr∗Ω11+r∗ςΩR02ρfμf1/n+1gζ0.Here,τw is identified as
(31)τw=τzr2+τzθ2.Finally, local skin friction coefficient(32)CfxRen−1/n+1=τwz=0ρfrΩ2=1r∗1+r∗ςfζ02+gζ021/2.Here,τzr denoted the shear stress along the radial direction, τzθ represent the shear stress along the direction of tangential, and τw denoted the shear stress.Local Nusselt number(33)Nux=R0qwkfTf−T∞z=0.Here,qw is symbolized as
(34)qwz=0=−kfTzz=0=−kfTf−T∞1+r∗ςΩR02ρfμf1/n+1θζ0.Finally, the local Nusselt number(35)Re−1/n+1Nux=−1+r∗ςθζ0.Local Sherwood number(36)Shx=R0qmDBCf−C∞z=0.Here,qm is denoted as
(37)qm=−DBCzz=0.Finally, the local Sherwood number(38)ShxRe−1/n+1=−1+r∗ςϕζ0.Density motile microorganism number(39)Snx=R0qnDmNf−N∞z=0.Here,qn is identified as
(40)qn=−DmNzz=0.Finally, the density motile microorganism number(41)SnxRe−1/n+1=−1+r∗ςχζ0.


## 3. Numerical Scheme

Significantly, the method of finding the exact solution for momentum, temperature, nanoparticle concentration, and bioconvection equations through corresponding initial conditions is very complicated and doubtful about the significance of results. Researchers have attempted to mathematically research the nanoliquid flowing past the stretching disk. Chemical processes, including activation energy, are also used to research the characteristics of mass transformation. Choose the initial guesses, and the dimensionless highly linear governing equations ((20))–((25)) with the related boundary conditions ((26))-((27)) are numerically integrated by utilizing the computational software MATLAB through built-in rule bvp4c (shooting method). The bvp4c method is a powerful way of resolving an initial value problem and a well-known methodology to find more than one solution. In order to solve these equations, first, we converted the higher-order differential equation into a first-order system by using the following technique.We identify new variables as(42)h=r1,hζ=r1′,f=r2,fζ=r3,fζζ=r3′,g=r4,gζ=r5,gζζ=r5′,θ=r6,θζ=r7,θζζ=r7′,ϕ=r8,ϕζ=r9,ϕζζ=r9′,χ=r10,χζ=r11,χζζ=r11′,(43)r1′=−2r2−ζ+αςεr3,(44)r3′=2r32−r42+r1r2+r2r3ςζ+αε+Mr2+K1∗r2+Frr22−λr6−Nrr8−Ncr10Re1−n/1+n1+r∗2ς,(45)r5′=2r2r4+r1r5+r2r5ςζ+αε+Mr4+K1∗r4+Frr42Re1−n/1+n1+r∗2ς,(46)r7′=r1r7−∈1r72+r2r7ςζ+αε−NtRe1−n/1+n1+r∗2ςr72−NbRe1−n/1+nr7r9−MEcr22+r42−EcRe1−n/1+n1+r∗ςr32+r52−4/3Rdθw−133r62r72+3θw−122r6r7ζ2+3θw−1r72−Qr61/prRe1−n/1+n1+r∗2ς1+∈1+4/3Rd1+θw−13r63+3θw−12r62+3θw−1r6,(47)r9′=−Nt/NbRe1−n/1+n1+r∗2ςr7′+Scςζ+αεr2r7+Scr1r9+K1r81+α1r6mexp−E1/1+α1r6Re1−n/1+n1+r∗2ς,(48)r11′=Lbr1r11−Lbςζ+αεr2r11+Per9′r10+Ω+r11r9Re1−n/1+n1+r∗2ς,(49)r20=A1+L1r30+L2r3′0,r40=1+L3r50+L4r5′0,r10=0,r70=−Ω11−r60,r90=−Ω21−r80,r110=−Ω31−r100,(50)r2∞⟶0,r4∞⟶0,r6∞⟶0,r8∞⟶0,r10∞⟶0.


## 4. Result and Discussion

The aim of this portion is to envisage variations in velocity components, thermal field, concentration of nanoparticles, and microorganism profile due to interesting involved parameters introduced during the flow of bioconvective nanofluid that are demonstrated in Figures2–10. Figure 2 explains the effects of second-order slip parameter and stretching rate to angular frequency. The escalating estimations of stretching rate to frequency exaggerate the axial velocity component. From this scenario, it can be detected that axial velocity of nanofluid decays via larger variations in second-order velocity slip parameter. Figure 3 communicates the impacts of second-order velocity slip as well as magnetic parameter versus tangential velocity. It can be observed that tangential velocity boosts up via larger second-order velocity slip parameter. Here, we also observe that the larger estimations in magnetic parameter diminishes the tangential velocity of nanofluid flow. In terms of physics, the magnetic parameter is associated to the Lorentz force, which is a resistive force to the fluid flow. As the magnetic parameter increases, the resistance forces increase, and the velocity decreases. Figure 4 demonstrates the features of mixed convection parameter as well as bioconvection Rayleigh number over a radial velocity component. It is noticed that velocity is improved by growing the estimations of mixed convection parameter while it depresses via a greater bioconvection Rayleigh number. The features of buoyancy ratio parameter and second-order velocity slip parameter versus radial component of velocity are elaborated through Figure 5. The radial velocity is a decreasing function of second-order velocity slip parameter and buoyancy ratio parameter. Physically, for a given buoyancy impact, bioconvection inhibits the up movement of solid particles that arise in nanofluid; however, for a higher buoyancy impact, the fluid resists the fluid, resulting in fluid movement decline.Figure 2
Fluctuation in axial velocity component against two different parameters.Figure 3
Fluctuation in tangential velocity component against two different parameters.Figure 4
Fluctuation in radial velocity component against two different parameters.Figure 5
Fluctuation in radial velocity component against two different parameters.Figure 6
Fluctuation in thermal field against two different parameters.Figure 7
Fluctuation in thermal field against two different parameters.Figure 8
Fluctuation in concentration field against two different parameters.Figure 9
Fluctuation in the microorganism’s field against two different parameters.Figure 10
Fluctuation in the microorganism’s field against two different parameters.Figure6 examines the behavior of thermal conductivity and temperature ratio parameter via thermal distribution of nanomaterials. It is witnessed that temperature field upsurges due to an increment in thermal conductivity and temperature ratio parameter. Figure 7 portrays the impression of Prandtl number and thermal Biot number versus temperature field. It is mentioned that improving Prandtl number reduces the temperature distribution. From this communication, we analyzed that enhancing the thermal Biot number escalates thermal field. Physically, when the Prandtl number rises, the thermal diffusivity diminishes. Fluid temperature drops as a result of the lower thermal diffusivity. Figure 8 presents the trend of the Brownian motion coefficient and Prandtl number against solutal field. Here, concentration reduces via a greater amount of Brownian motion coefficient as well as Prandtl number. Micromixing and heat conduction in the nanofluid are aided by enhancing the Brownian motion parameter, causing the temperature to rise and the nanoparticles to scatter more widely.Figure9 shows the nature of bioconvection Lewis number and Peclet number over a microorganism’s field. It is concluded that the microorganism field diminishes by increasing the variations of bioconvection Lewis number and Peclet number. The bioconvection Lewis number has an converse relationship with microorganism diffusivity. As the bioconvection Lewis number rises, the diffusivity decreases, and the microorganism profile drops. The features of microorganism Biot number and bioconvection Rayleigh number are mentioned in Figure 10. It is noted that larger magnitudes of microorganism Biot number and bioconvection Rayleigh number increases the microorganism’s profile. Here, it was analyzed that good validation in results between published literature and current outcomes is presented in Table 2.Table 2
Validation in results between published literature and current outcomes are present.
f″0g′0θ′0Xun et al. [29]0.510230.615920.39627Anderson et al. [30]0.5100.6160.3963Ming et al. [31]0.510210.615910.39632Khan et al. [33]0.510820.615950.3958Current results0.510840.615980.3958


## 5. Conclusion

Computational analysis is conducted on bioconvective viscous nanofluid flow past a stretching disk with higher-order slips and nonlinear thermal radiation. The main outcomes are listed as follows:(i)
Axial velocity component escalates versus a larger amount of stretching rate to angular frequency(ii)
Tangential velocity is increases against a second-order slip parameter(iii)
The increment in radial velocity has been analyzed along with increments in mixed convection parameter(iv)
The larger slip parameter and buoyancy ratio parameter reduce the radial velocity of nanofluid(v)
Thermal field of species rises against temperature-dependent thermal conductivity(vi)
Temperature distribution is increased via the temperature ratio parameter(vii)
Greater values of the thermal Biot number boost the temperature field(viii)
Concentration is reduced against the Brownian motion parameter(ix)
The microorganism’s profile boosts via the microorganism’s Biot number while it diminishes against the Peclet number(x)
The microorganism’s field is depressed against the bioconvection Lewis number



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*Source: 2766317-2022-02-14.xml*