# On the Role of Bioconvection and Activation Energy for MHD-Stretched Flow of Williamson and Casson Nanofluid Transportation across a Porous Medium Past a Permeable Sheet

Authors: M. Eswara Rao; M. Siva Sankari; Ch. Nagalakshmi; S. Rajkumar
Journal: Journal of Nanomaterials
 (2023)

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## Abstract

The goal of the current study is to theoretically evaluate the heat and mass transfer behavior of magnetohydrodynamic Casson and Williamson fluids as they pass through a stretched sheet with Brownian and thermophoresis effects. The evolution of high-density heat devices necessitates efficient thermal transportation. For these requirements, the concept of nanofluid plays an active role. This article describes the effects of Cattaneo–Christov. This study introduces new concepts such as motile microorganism bioconvection, non-Fourier heat flux, and activation energy. A Runge–Kutta-based shooting procedure is used to solve the problem. The effects of the relevant parameters on velocity, thermal, concentration, and motile microorganisms are depicted graphically. The computed reduced Nusselt, Sherwood, and motile density numbers are displayed in tables in this study. When compared to the Williamson fluid, the thermal and concentration fields of the Casson fluid are heavily influenced by the parameters.


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## Body

## 1. Introduction

Most of the increase in the viscosity of non-Newtonian fluids in modern times is due to trade encroachment. Non-Newtonian fluids transpire not barely in environment, for illustration, mountainside and debris flows, but level in designer mechanization as well as an ample diversity of trade meadows remaining to its vital relevance efficacy in organic material, substances, victuals belongings, and delicate heed yield, among others.Fluids oscillate in their tackiness, which might differ depending on the speed of twist and assure liquids contain a resilient constituent in temperament, which is alleged as nonfluids [1]. Casson fluid is well-classified when such linear equipment is owed to conventional trim lessening and condensing uniqueness once circumspectly reading the non-Newtonian fluid subclass was studied by Ogunseye et al. [2] and Raju et al. [3]. For its microspore’s uniqueness, Casson fluid is non-Newtonian owing to its trim pressure and twist connection. This product is an excellent option for shear-thinning submissions due to its high level of adhesion and yielding resistance to shear. Casson was the originator of the replica, which was ingeniously conceived for ink generation and silicon dispersion.A mathematical investigation was conducted, and it was concluded that the Casson term drew the drift motion due to increased tackiness. The Casson liquid is one of the most often used viscoelastic non-Newtonian equipment. The energy stint was uncovered to have elevated the Casson rejoinder temperature gradient. Though, the trade worth of Casson liquid can be enlarged once it is collected with Williamson liquid [4], while the Casson and Williamson liquids are intermingled amid manufacturing colloidal nanoparticles in a tacky liquid, the convectional temperature conveys, and the heat conductivity potency of the solvents is augmented for the finest efficiency of nanotechnological and trade products. Bejawada et al. [5] resolute the Williamson and Casson liquids flow past a stretching sheet with heat and mass transfer in the occurrence of heat-reliant heat resource. Magnetohydrodynamics (MHD) is studied by looking at the magnetic properties of electrically conducting liquids.Explicitly, the magnetic fields acquire an indispensable part in establishing the leads. Owing to this foremost reimbursement, the investigators are incessantly investigating the MHD drifts. MHD drifts are indispensable in a widespread assortment of mechanized and procedural conventions, such as MHD generators, the intend of the nuclear reactor, and flow meters. MHD is a discipline that scrutinizes the drift of electrically accomplishing elucidations in the occurrence of a magnetic field. Abundant inventive and estimated inquiries on usual electrically directing drifts explain that the implication of a magnetic field radically amends their haulage and heat transfer channels. Nuclear influence shrubbery preservation using liquid sodium, the direction force estimate based on liquid probable inequality vertical to motion and the magnetic field, and other tools essential to the study of MHD were analyzed by Rao and Sreenadh [6] and Li et al. [7]. The behavior of the MHD Casson nanoliquid drifted over a permeated sheet was observed by Yahya et al. [8]. The investigators had conferred the brunt of Cattaneo–Christov heat flux on the fluid flow.Nanofluids are tincture deferments of nanometer dimension elements in a transporter liquid. The nanostructures are deemed in common organization of metals, pewter oxidations, carbon nanospouts and carbides, while the haulers solvent are ethylene and paraffin oil. Lately, nanoliquids change to swift perceive from concocts and researchers since their pervasive procedural, traded, and manufacturing applications like geothermal procedures, power-driven freezing, drug liberation, cancer therapy, heat relocate mechanisms, converter cooling, etc. were discussed by Yousef et al. [9].Bioconvection refers to the phenomenon in which swimming microorganisms experience an increase in their mobility due to the macroscopic movement of the fluid caused by the spatial fluctuation of density over a region. Gyrotactic microbes in a nanofluid flowing through a porous media causing bioconvection was studied by Ahmad et al. [10]. Williamson Sutterby nanofluid transports due to a stretching surface with a convective boundary, the role of bioconvection and Cattaneo–Christov heat flux was carried out by Yahya et al. [8]. Abdal et al. [11] studied the investigation of the MHD stretched flow of Williamson Maxwell nanofluid through a porous matrix over a penetrated sheet while taking into account bioconvection and activation energy.The microorganisms that move on their own tend to move the base fluid in a certain direction, creating a bioconvective stream. Different forms of motile microbes, such as those that move via gyrotactic motion, negative gravitaxis, or chemotactic chemicals, have been identified. Awan et al. [12] studied the effects of bioconvection on the flow of Williamson nanofluids caused by an exponentially increasing heat source and a motile microbe over a stretching sheet. Nanobioconvective Williamson fluid that allows for bacteria to swim in the bloodstream was studied by Rana et al. [13]. Williamson nanofluid bioconvection radiative flow across a vertical-stretched cylinder with activation energy and swimming microorganisms in a numerical simulation was explored by Zhang et al. [14].When compared to motile microbes, the nanoparticles are not self-driven; rather, their motion is influenced by Brownian motion and the thermophoresis effect. Thermophoresis and Brownian motion’s effects on thermally and chemically reactive substances flow of Casson nanofluid over a sheet that is being linearly stretched were studied by Tawade et al. [15]. Kumaran and Sandeep [16] studied the parabolic motion of multicomponent, multihybrid (MHD) Casson–Williamson fluids with a Brownian moment and thermophoresis. Thermophoresis and Brownian motion’s effects on the heat transfer and entropy formation in nanofluids were discussed by Mahmoodi and Kandelousi [17]. In a suspension of gyrotactic microorganisms over an incline-stretched sheet, the thermo solutal Marangoni effect on bioconvection has been seen by Kairi et al. [18]. If the nanoparticle concentration is low, bioconvection in the nanofluids should be possible because the nanoparticles will not be able to cause a significant increase in the viscosity of the base fluid.The influence of bioconvection of microorganism for two different drifts of Casson nanofluid and Williamson nanofluid is rarely deliberated in the present article. It aids to deal with the probable subsiding of nanoentities, and thus enrichment of thermal conductivity of the common liquids can be recognized. The novelty of this work affects Cattaneo–Christov heat flux, bioconvection, and activation energy. The buoyancy effects and magnetic field strength impart differentiated stimulus on drift temperature and concentration features. The character of these valuable and workable physical features for two substantial non-Newtonian liquids is counted and their comparative upshots benefit widen our understanding and utilization of these drifts. Utilization of similarity transmutes enables to yield numerical solution of nonlinear coupled system with the execution of Runge–Kutta process and shooting procedure. By exploiting the parametric progress of bioconvection, nanofluid slips, and Cattaneo–Christov factor, the findings are estimated for Casson liquid and Williamson liquid. These outcomes are valuable for thermal management of heat exchangers of the evolving machineries.


## 2. Outline of the Problem’s Mathematical Analysis

A steady two-dimensional surge of Casson and Williamson nanofluids among activation energy and Cattaneo–Christov dissemination entrenched in a porous medium is deliberated in the continuation of biotransformation microfollicles. The liquids drift remaining to a stretching sheet, the wall velocity isUw=Dx, D is stretching persistent. Uw is the stretching velocity of sheet along x-axis. It is reputed that the actions of the drifts rectilinear remain to the impervious stretching sheet. Here, ua and vb are the velocities of liquids drift along x-axis and y-axis, respectively. The T, C, and N values in Figure 1 represent the fluid temperature, microbial concentration, and density, respectively, at close proximity to the sheet. We did not stipulate an interior temperature foundation other than the fact that the reaction between substances is first order. We also restrained fractious propagation possessions by Brownian effort and thermophoresis. We disregarded tacky intemperance and persuaded magnetic field possessions. By the expectations detailed above, the governing equivalence in positions of similarity inconstant can be articulated in the following equations [11].Figure 1
Geometry of the problem figure.(1)∂ua∂x+∂vb∂x=0.(2)ua∂ua∂x+vb∂x∂vb∂x=υ1+1β∂2ua∂y2+2Γ∂ua∂y∂2ua∂y2−σB02uaρ−υk′ua−1ρ1−C∞·ρβT−T∞−ρp−ρgC−C∞−N−N∞gγρm−ρ.(3)ua∂T∂x+vb∂T∂y=α∂2T∂y2+ρpCpρCDB∂C∂y∂T∂y+vbDTT∞∂T∂y2τua∂ua∂x∂T∂x+vb∂vb∂x∂T∂x+ua∂vb∂x∂T∂x+vb∂ua∂y∂T∂x+2uavb∂2T∂x∂y+ua2∂∂x∂T∂x+vb2∂∂y∂T∂y.(4)ua∂C∂x+vb∂C∂y=DB∂∂y∂C∂y+DTT∞∂∂y∂T∂y−Kr2Cw−C∞·TT∞mexp−Eak2T.(5)ua∂N∂x+vb∂N∂y+bWc∂∂ynΔC∂C∂y=Dm∂∂y∂N∂y.(6)ua=uw=Dx,vb=vw,T−Twx=0,C−Cwx=0,N−Nwx=0,aty=0ua⟶0,T⟶T∞,C⟶C∞,N⟶N∞aty⟶∞.Using the similar variables,(7)η=Dυy,ua=Dxf′η,vb=−Dυfη,θη=T−T∞Tw−T∞,ϕη=C−C∞Cw−C∞,ψη=N−N∞Nw−N∞.Equation (1) is gratified and Equations (2)–(6) become,(8)1+1βf‴−f′2+ff″+λf″f‴−M+KPf′+ωθ−Nrφ−Rbψ=0.(9)θ″+prfθ′+NcLeθ′ϕ′+NcLe∗Nbtθ′2+bf2θ″+ff′θ′=0.(10)ϕ″+Scfϕ′+1Nbtθ′′−Sc∗Aϕ1+δθmexp−E1+δθ=0.(11)ψ″+Lbfψ′−peϕ″ψ+Ω+ψ′ϕ′=0.(12)f0=S,f′0=1,θ0=1,ϕ0=1,atη=0,f′∞⟶0,θ∞⟶0,ϕ∞⟶0,ψ⟶∞,asη⟶∞.The nondimensional constraints are in their particular order,where,(13)β=λ1D,M=σB2oDρ,KP=υk1D,ω=β1g1−C∞Tw−T∞D2x.(14)Nr=ρp−ρCw−C∞β1ρ1−C∞Tw−T∞,Rb=ρm−ργN∞1−C∞ρβT∞.(15)b=τ1D,Sc=υDB,A=k2rD,δ=Tw−T∞T∞,E=Eak2T∞.(16)Lb=υDm,Pe=b1WcDm,Ω=N∞Nw−N∞,S=−υwDυ.(17)Pr=υα,Le=αDB,Nbt=DBT∞Cw−C∞DTTw−T∞.The objective quantities are declared as:Cfx, viscous drag coefficient; Nux, local Nusselt number; Shx, local Sherwood number; and Nnx, local density of microfollicles are known in Equation (18).(18)Cfx=τωρU2w,Nux=xqwkTw−T∞,Shx=xqmDBCw−C∞,Nnx=xqnDmNw−N∞,where τw,qw,qm, and qn denote shear stress, surface heat flux, surface mass flux, and motile microfollicles flux are given by (at y = 0),(19)τw=μ1+β1+Γ2∂ua∂y∂ua∂y,qw=−k∂T∂y,qm=−DB∂C∂y,qn=−Dm∂N∂y.On solving these quantities with the help of known similarity transformation, we acquire,(20)CfxRex−1/2=1+βf′′0+λ2f′′02,NuxRex−1/2=−θ′0,ShxRex−1/2=−ϕ′0,NnxRex−1/2=−ψ′0where Rex=xUw/ν is the local Reynolds number.


## 3. Solution Procedure

For numerical results, the Runge–Kutta method of order four and the shooting technique are used to solve the nonlinear ordinary differential Equations (8)–(11) with the given boundary conditions (12). To use this method, the new variable is written as follows [11]:(21)h1′=h2h2′=h3h3′=1/1+1/βh22+λh3h3′+M+KPh2−wθ−Nrϕ−Rbχh4′=h5h5′=−prh1h5−NcLeh5h7−NcLe∗Nbth52−bh12h5′+h1h1′h4′h6′=h7h7′=−Sch1h6′−1Nbth7′−Sc∗Ah61+δh4mexp−E1+δh4h8′=h9h9′=Peh7′χ+Ω+h8′h6′−Lbh1h8′.


## 4. Results and Discussion

This exertion obtainable is a representation of non-Newtonian Casson–Williamson nanoliquid surge wizened by the momentum, energy, and concentration equation and persuaded by the magnetic field, biotransformation, porous medium, microfollicle, heat creation, and chemical reaction. We exploit the fourth-order Runge–Kutta amalgamation design and a shooting approach to unravel the structure of extremely erratic ordinary differential Equations (11), (12), and (18) in concert with the boundary conditions (19) and (20), which articulate that problem. The tabular methods are used to show the numerical investigations. The influence of nanoliquid physical strictures on heat and mass transference tariffs, as well as the local drag friction factor, local Nusselt number, local Sherwood number, and local motile density number are also calculated and obtainable in Tables 1 and 2. The upshots of liquid strictures on the Casson and Williamson nanoliquid velocity f′η, temperature θ(η), concentration φ(η), and motile density contour χ(η) are obtainable explicitly and deliberated in this sector.Table 1
Values ofCfRex, Nux/Rex, Shx/Rex, and Nnx/Rex for values of M, KP, w, Nr, Rb, Pr, Nc, Nbt, b, E, A, Le, Ω, Lb, and Pe (for Williamson fluid (λ = 0.2 and β = ∞)).
MKPwNrRbbLePrNcEADLbPeCfRexNux/RexShx/RexNnx/Rex0.01.2635330.1222900.7947550.2196250.31.3845840.1227450.7818620.2192580.01.1776200.1217450.8047160.2199230.51.3845840.1227450.7818620.2192580.11.3845840.1227450.7818620.2192580.31.3845840.1227450.7818620.21925811.3845840.1227450.7818620.21925821.4334620.1229350.755780.219072.11.3845840.1227450.7818620.21925821.4560390.1233620.7696581.45603911.3845840.1227450.7818620.21925821.3838600.1131540.7921640.7921640.11.3845840.1227450.7818620.2192580.21.3913300.1957850.7031570.2263100.51.3845840.1227450.7818620.2192581.01.3939630.1860410.7035280.2256220.11.3845840.1227450.7818620.2192580.51.3583370.0016370.9746570.20530511.3845840.1227450.7818620.21925821.3850670.1234380.7592290.2214790.11.3845840.1227450.7818620.2192580.51.3816950.1186690.9292600.2047040.01.3844750.1227450.7818920.2219150.051.3845840.1227450.7818620.2192580.11.3845840.1227450.7818620.2192580.21.3826000.1226820.7824900.2522410.01.3822440.1225670.7873780.2819910.11.3844950.1226170.7867780.218770Table 2
Values ofCfRex, Nux/Rex, Shx/Rex, and Nnx/Rex for values of M, KP, w, Nr, Rb, Pr, Nc, Nbt, b, E, A, Le, Ω, Lb, and Pe (for Casson fluid (β = 2.0 and λ = 0)).
MKPWNrRbbLePrNcEADLbPeCfRexNux/RexShx/RexNnx/Rex0.01.0679040.1211720.8152050.2202060.31.1614740.1220410.8029010.2198140.01.0007180.1203610.8245750.2205210.51.1614740.1220410.8029010.2198140.11.1614740.1220410.8029010.2198140.31.1614740.1220410.8029010.21981411.1614740.1220410.8029010.21981421.1955970.1223550.7981000.219658.11.1614740.1220410.8029010.21981421.2129870.1229440.7926570.21942511.1614740.1220410.8029010.21981421.1606260.1101940.8160000.2186470.11.1614740.1220410.8029010.2198140.21.1662170.1926330.7289830.2265160.51.1614740.1220410.8029010.2198141.01.1686360.1857510.7260160.2261180.11.1614740.1220410.8029010.2198140.51.1421000.0018870.9868200.20618311.1614740.1220410.8029010.21981421.1618120.1226990.7810780.2219610.11.1614740.1220410.8029010.2198140.51.1594390.1181340.9459550.2056600.01.1613890.1220380.8029280.2225730.051.1614740.1220410.8029010.2198140.11.1614740.1220410.8029010.2198140.21.1598190.1219620.8034900.2555690.01.1597380.1219800.8034270.2845350.11.1614740.1220410.8029010.219814


## 5. Velocity Profile

For different data of magnetic parameter,M, the effect of velocity profile has been exposed in Figure 2(a). It is observed that as M goes up, f′η goes down. This is because a strong magnetic field helps produce a force called the Lorentz force, which slows down the drift of liquid. It is clear that Lorentz force had more of an upshot on Williamson fluid than on Casson liquid.Figure 2
(a) Influence ofM on f′η. (b) Influence of KP on f′η. (c) Influence of w on f′η. (d) Influence of Nr on f′η. (e) Influence of Rb on f′η.
(a)(b)(c)(d)(e)The enhanced porosity parameterKP, as shown in Figure 2(b), retards the flow velocity because KP has the ability to produce fluid flow resistance.Figure2(c) presents the result of the mixed transformation stricture applied to f′η for Williamson and Casson nanofluids. The higher values of the mixed convection stricture cause an upsurge as a result of the stronger buoyancy force.Figures2(d) and 2(e) explain the characteristics of the bioconvection Rayleigh number (Rb) and the buoyancy ratio parameter (Nr) as they are related to f′η.Changing both parameters causesf′η to decline. Physically, both values are related to buoyancy ratio forces that resist the motion of the two fluids.


## 6. Temperature

Figure3(a) illustrates the temperature field for a range of values of M. When the values of M go up, the temperature spreads out more. Physically, an increase in M generated the flow-resisting force (Lorentz force). This force has a propensity to reveal θη. Additionally, it is shown that the thermal boundary layer of Casson fluid is significantly influenced by the Lorentz force compared to that of Williamson fluid.Figure 3
(a) Influence ofM on θη. (b) Influence of b on θη. (c) Influence of Le on θη. (d) Influence of Pr on θη. (e) Influence of Le on θη.
(a)(b)(c)(d)(e)Figure3(b) presents an illustration of the effects that b has on θη. By increasing the Cattaneo–Christov parameter b, θη is decreased. Consequently, Casson fluid diffuses more heat than Williamson fluid, and the temperature curve for Casson fluid is steeper than that for Williamson fluid. λ is obviously a ratio between buoyant forces and viscous forces. Consequently, a small change induces a decrease in the θη.Figure3(c) demonstrates the effect of the Lewis number Le on θη. θη depreciates as a result of its direct association with the diffusivity of mass, which leads to a decrease in the temperature distribution.The influence of Pr is shown asθη in Figure 3(d). It is obvious that raising Pr values results in a decrease in θη. Physically, it results from Pr’s opposite relationship to thermal diffusivity.The relationship between Nc andθη is illustrated in Figure 3(e); a rise in Nc results in an increase in θη.


## 7. Concentration

The concentration fields are shown in Figure4(a) for various magnetic field parameter values. The concentration distribution φη is enhanced by raising the values of the magnetic field parameter. Physically, increasing the magnetic field parameter caused the flow to experience a resistive force (Lorentz force). This force has a propensity to make the concentration field visible. Additionally, it is shown that, in contrast to Williamson fluid, the concentration boundary layers of Casson fluid are strongly influenced by the Lorentz force.Figure 4
(a) Influence ofA on φ(η). (b) Influence of E on φ(η). (c) Influence of A on φ(η). (d) Influence of Le on φ(η).
(a)(b)(c)(d)As shown in Figure4(b), an upsurge in the activation energy E persuades an upsurge in φη.Figure4(c) depicts a decline in φη as the chemical reaction rate parameter A is exceeded. Larger inputs of A result in a more rapid chemical reaction that decreases φη while the increase in A generates greater resistance to the flow of Casson fluids compared to Williamson fluids.With the exceeding of the Lewis number Le,φη decreases as shown in Figure 4(d). In addition, a decrease in mass diffusivity makes Casson fluid less resistant to flow than Williamson fluid.


### 7.1. Motile Density

With an increase inM, the motile density profile χη becomes more pronounced in Figure 5(a). It is a result of the flow slowing. The motile density profile χη is increased as a result of the curves produced by M.Figure 5
(a) Influence ofM on χ(η). (b) Influence of D on χ(η). (c) Influence on Pe on χ(η). (d) Influence of Lb on χ(η).
(a)(b)(c)(d)By raising the value of the microorganism difference parameter, which is shown in Figure5(b), it also exhibits an addition.Figure5(c) illustrates the outcomes of Peclet number Pe. A greater Peclet number decreases microorganism diffusivity. The increase in Peclet number Pe reduces the microorganism boundary layer thickness and motile density profile χη.Figure5(d) depicts the results of the Lewis number Lb. Furthermore, it is evident that a lower value of χη for the spread of microorganisms results from a bigger value of Lb.


## 7.1. Motile Density

With an increase inM, the motile density profile χη becomes more pronounced in Figure 5(a). It is a result of the flow slowing. The motile density profile χη is increased as a result of the curves produced by M.Figure 5
(a) Influence ofM on χ(η). (b) Influence of D on χ(η). (c) Influence on Pe on χ(η). (d) Influence of Lb on χ(η).
(a)(b)(c)(d)By raising the value of the microorganism difference parameter, which is shown in Figure5(b), it also exhibits an addition.Figure5(c) illustrates the outcomes of Peclet number Pe. A greater Peclet number decreases microorganism diffusivity. The increase in Peclet number Pe reduces the microorganism boundary layer thickness and motile density profile χη.Figure5(d) depicts the results of the Lewis number Lb. Furthermore, it is evident that a lower value of χη for the spread of microorganisms results from a bigger value of Lb.


## 8. Conclusions

This paper investigates the significance of magnetic field, biotransformation, and activation energy for Williamson and Casson nanofluid on a porous prolonged pane with Cattaneo–Christov dissemination. The key findings of this study are as follows:(1)
It is scrutinized that for the superior efforts of magnetic strictureM and porous its stricture, the velocity diminishes(2)
Among the intensifying significances of Rayleigh number Rb, buoyancy ratio stricture Nr, velocity moderates, while conflicting conduct be revealed forw(3)
Temperature diminishes amid the heckling significances of Cattaneo–Christov stricturesb, Lewis number Le, and Prandtl number Pr(4)
Casson fluid is further exaggerated with these austere than Williamson fluid. On contrary,M and Nc show opposite behavior for it(5)
Le sources reduction into concentration, while these strictures seize hefty values. On further dispense, the concentration demonstrates conflicting conduct forE(6)
For motile density contour disparate strictures, i.e.,Ω(D) and Peclet number. Pe articulates extra upshot for Casson fluids as evaluated to Williamson fluids(7)
Skin friction coefficient enhanced forβ, M, KP, Nr, and Rb while decreased down for λ and ω(8)
Nusselt number reduces when uplifting the parameter Nc, while it increases for Pr, Nbt, andb(9)
Sherwood number increases for Sc,A, δ, and n while it reduces when uplifting the parameter E(10)
Motile density number increases for Lb,Ω(D), and Pe



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